Industrial Organization: A European
Perspective
Answers to Problems
Stephen Martin
Department of Economics
Krannert School of Business
Purdue University
403 State Street West
West Lafayette, Indiana 47907-2056
USA
smartin@mgmt.purdue.edu
November 2001; updated November 2002
2
Contents
1 Background
5
2 Oligopoly I
11
3 Collusion and tacit collusion
19
4 Dominance
21
5 Organization
27
6 Innovation
45
7 International Trade I
47
8 International Trade II
65
9 International Trade III
89
10 Market Integration
115
3
4
CONTENTS
Chapter 1
Background
1—1 Find monopoly output, price, deadweight welfare loss, and the Lerner
index if the market inverse demand curve is
p = 100 − Q
and marginal cost is 10.
MR = 100 − 2Q = 10 = marginal cost
Qm = 45
pm = 100 − 45 = 55
π m = (55 − 10)(45) = 2025
1
1
DW L = (90 − 45)(55 − 10) = (2025)
2
2
L=
p − 10
55 − 10
45
9
=
=
=
= 0.8181
p
55
55
11
1—2 (a) Graph the average variable, average, and marginal cost curves if the
cost function is
C(q) = 1 + 9q.
The equations of the cost curves are
AV C(q) = MC (q) = 9
AC(q) =
1
+ 9.
q
(b) Graph the average variable, average, and marginal cost curves if the cost
5
6
CHAPTER 1. BACKGROUND
p
100 •..........
..Demand curve
.........
.
.
.
.
.
... .....
..
.
....
.
• .... ......
.
.
... ......
..
... .....
....
.
.
.
.. ..
...
•
... ................
.....
...
.....
...
.....
.
.
...
.....
•
.....
...
.....
...
.....
...
.....
•
...
.....
...
•
•....
...
55
.
.
···.........
.
•
...
...
··········.........
...
··················........
...
•
...
··························.........
...
...
··································.........
.
...
•
··········································.........
...
························· .....
··Deadweight
··························· ....
...
Marginal
... ··················loss
.....
· · · · · · · · · · ...
•
... ························································..·.......
cost .....
.....
... ·································· ......
......
... ·······················································......
.....
..•.······································· .....
•
.....
...
10
.....
........................ Marginal
.....
... .
.....
revenue
curve
.
.• Q
•
•
•
• • •.
•
•
•
•
45
90 100
Figure 1.1: Monopolist’s output decision, p = 100 − Q, marginal cost = 10
7
cost
unit
...
...
...
...
...
...
...
...
...
AC(q) =
9 + 1q
...
...
...
.
.
..
...
....
...
.
.
..
...
...
....
.
.
.
....... .......
..............
..............................
....................................................................................................................................................
9
........
......
......
......
......
... AV C(q) = M C(q) = 9
7
q
Figure 1.2: Firm cost curves, C(q) = 1 + 9q
function is
C(q) = 1 + 9q − q2 + q 3 .
The equations of the cost curves are
AV C(q) = 9 − q + q 2
M C(q) = 9 − 2q + 3q 2
AC(q) =
1
+ 9 − q + q2.
q
Average variable cost and marginal cost have the same value (9) for q = 0.
They also have the same value for the output level that gives the minimum
value of average variable cost:
AV C(q) = 9 − q + q 2 = 9 − 2q + 3q 2 = M C(q)
9 − q + q 2 = 9 − 2q + 3q 2
2q 2 − q = 0 ⇒ q = 0,
1
2
8
CHAPTER 1. BACKGROUND
and the common value of average variable cost and marginal cost for q = 1/2
is
µ ¶
µ ¶
1
1
35
1 1
+3
=
= 8.75.
9− + =9−2
2 4
2
4
4
Average cost and marginal cost have same value for the output level that
gives the minimum value of average cost:
AC(q) =
1
+ 9 − q + q2 = 9 − 2q + 3q 2 = MC(q)
q
1
= 2q2 − q
q
2q 3 − q 2 − 1 = 0.
¢
¡
(q − 1) 2q 2 + q + 1 = 0
The minimum value of average cost thus occurs for q = 1:
AC(1) =
1
+ 9 − 1 + 1 = 10.
1
Find the minimum value of marginal cost:
M C(q) = 9 − 2q + 3q 2
dM C(q)
1
= −2 + 6q = 0 ⇔ q =
dq
3
and the minimum value of marginal cost is
µ ¶
µ ¶
µ ¶2
1
1
1
26
=9−2
+3
= 4.67.
MC
=
3
3
3
3
9
cost
unit
M C(q) = 9 − 2q + 3q 2
...
1
...
2
..
... AC(q) =
..
. q +9−q+q
.
.
.
...
.
..
...
....
..
.
.
.
...
..
.
...
....
..
.
.
.
... . ....
.
.........
..
.
...
.
..
...
.
.
...
.
.....
.
.
...
...
.
..
...
.
..... .
.
.
...
.
.. .
..
..
...
..... ......
..
.
....
.
.
.
.
.
..
...
....
...
.......
.....
.
..
.
.
.
.
.
.
.
.
......
....
.
...
.......
........
...
.
..
.
.
.
.
.
.
.
.
.
...........
.
.
.
..
..............•............................
10 •
.....
.
.
.
.
.
.
...
...
.....
...
.
.
.
.
.
.
.
....
..
......
.
.
....
.
.
.
.
.
.
.
.....
..
........
....
.
.
.
.
.
.
.
.
.
.
.
.
..
.....
9 ......................
............................................................................................... .................
.
.......
............................ •
8.75 •
.......
.......
.......
AV C(q) = 9 − q + q 2
•
•
7
q
0.5
1
Figure 1.3: Firm cost curves, C(q) = 1 + 9q − q 2 + q 3
10
CHAPTER 1. BACKGROUND
Chapter 2
Oligopoly Markets:
Noncooperative Behavior
2—1 For quantity-setting duopoly with inverse demand curve
p = 100 − (q1 + q2 )
(2.1)
and constant marginal cost 10 per unit, find equilibrium prices and profits if
each firm maximizes a weighted average of profit and sales,
gi = (1 − σ)π i + σpi qi .
Illustrate noncooperative equilibrium on a reaction curve diagram.
For this example
π 1 = [100 − 10 − (q1 + q2 )] q1 = [90 − (q1 + q2 )] q1
p1 q1 = [100 − (q1 + q2 )] q1
g1 = (1 − σ) [90 − (q1 + q2 )] q1 + σ [100 − (q1 + q2 )] q1
= [90(1 − σ) + 100σ − (q1 + q2 )] q1
= [90 + 10σ − (q1 + q2 )] q1 .
The equation of firm 1’s best response function is
2q1 + q2 = 90 + 10σ.
In the same way, the equation of firm 2’s best response function is
q1 + 2q2 = 90 + 10σ.
11
(2.2)
12
CHAPTER 2. OLIGOPOLY I
Given the symmetry of the model, in equilibrium firms produce the same
output:
3q = 90 + 10σ.
10
q = 30 + σ.
3
The greater the weight given to revenue in the objective function, the
greater is equilibrium output. Figure 2.1 shows best response functions for
σ = 0 and σ = 4/5.
q2
•
•...
...
...
90 •.... ....
... ...
... ...
. .
1’s best
response curve, σ = 0
• ... ...
...
... ...
.
.
.
.
.
... ...
....
... ...
4
1’s best
•
... ... ........
.. response curve, σ = 5
.
... ........
.
.
..........
....
.
.
.
.
.
.
... ...
..
•
... ...
...
.
.
.
... ...
.
... ... .......
.
.
.
.
.
.
.
•....
... ....
.........
... ...
.
.
.
.
.
.
.
•
.
.
.
.
... ...
45 .......... ..........
.
.
.
.
.
.
.
.
.
.
2’s best response
curve, σ = 0
.
.
.
.
......... ......... ..... .....
•
...
......... .......... ..
.
.
.
.
2’s best response
......... .............. E4/5,4/5
........... •........
... curve, σ = 4/5
....
•
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
•.......... .......
30 •
. .
....
. ...
....
E0 .... .............................................
.
.
.
.
... .. ......... ........ ......
......... .........
... ...
•
......... .........
... ...
......... .........
... ...
......... .........
... ...
......... .........
.
.
......... .........
•
... ...
......... .........
... ...
......... .........
... ...
......... .........
.
.
....• ....•.
.
.
•
•
• •
• • •
•
•
•
30
45
90
q1
Figure 2.1: Cournot best response curves, partial sales maximization by both
firms, σ = 45
2—2 For a price-setting duopoly with product differentiation, let the equations of the inverse demand curves be
µ
¶
1
p1 = 100 − q1 + q2
(2.3)
2
13
p2 = 100 −
µ
with corresponding demand functions
¶
1
q1 + q1 ,
2
2
q1 = (100 − 2p1 + p2 )
3
(2.4)
(2.5)
2
q2 = (100 + p1 − 2p2 ).
(2.6)
3
Let marginal cost be constant at 10 per unit.
Find equilibrium prices and profits if firm 2 maximizes profit while firm
1 maximizes a weighted average of profit and sales
g1 = (1 − σ)π 1 + σp1 q1
(2.7)
g1 = [(1 − σ)(p1 − 10) + σp1 ] q1
= [p1 − (1 − σ)10] q1
2
[p1 − (1 − σ)10] (100 − 2p1 + p2 ).
3
Set the derivative of g1 with respect to p1 equal to zero:
=
∂g1
2
= {[p1 − (1 − σ)10] (−2) + 100 − 2p1 + p2 } = 0.
∂p1
3
(2.8)
(2.9)
Rearrangement of terms gives the equation of firm 1’s price best response
function:
4p1 − p2 = 120 − 20σ.
(2.10)
Firm 2’s profit is
2
π 2 = (p2 − 10)(100 + p1 − 2p2 )
3
(2.11)
Setting the derivative of π 2 with respect to p2 equal to zero gives the
equation of firm 2’s price best response function:
−p1 + 4p2 = 120.
Note that the first-order condition implies
100 + p1 − 2p2 − 2(p2 − 10) = 0,
so that firm 2’s equilibrium profit is
4
π 2 = (p2 − 10)2
3
(2.12)
14
CHAPTER 2. OLIGOPOLY I
Write the equations of the first-order conditions as a system:
¶
µ ¶
µ ¶
µ
¶µ
1
1
4
−1
p1
= 120
− 20σ
.
p2
1
0
−1 4
(2.13)
The solution is
µ
¶
µ
¶µ ¶
µ
¶µ ¶
p1
4 1
1
4 1
1
15
= 120
− 20σ
p2
1 4
1
1 4
0
µ
¶
4
15
= 600
− 20σ
1
µ
¶
µ ¶
µ ¶
p1
1
4
3
= 120
− 4σ
p2
1
1
p1
p2
¶
µ
1
1
¶
µ
16
σ
3
4
p2 = 40 − σ.
3
p1 = 40 −
(2.14)
(2.15)
Equilibrium outputs are
q1 = 40 +
56
σ
9
(2.16)
q2 = 40 −
16
σ.
9
(2.17)
Profits are
π 1 = (p1 − 10)q1 =
µ
16
30 − σ
3
¶µ
¶
56
40 + σ ,
9
(2.18)
which falls as σ rises from 0 to 1 (Figure 2.2), and
π 2 = (p2 − 10)q2 =
µ
4
30 − σ
3
¶µ
¶
µ
¶2
16
4
4
40 − σ =
30 − σ
9
3
3
(2.19)
which also falls as σ rises from 0 to 1.
2—3 For a price-setting oligopoly with product differentiation, let the
equations of the inverse demand curves be
for i = 1, 2, ..., n and Q−i
pi = 100 − (qi + θQ−i ) ,
P
= nj6=i qj .
(2.20)
15
π
πσ
......... σ
.
.
.
.
.
.
.
.
.
.
.
1200 •................................................
............
.............................................................
.... .......................
.......................
......................
....................
1150 •
...................
•
0.1
•
0.2
•
0.3
•
0.4
•
0.5
•
0.6
•
0.7
•
0.8
•
0.9
•
1.0
σ
Figure 2.2: Sales maximization and equilibrium firm profit, price-setting
firms
The equations of the corresponding demand curves are
P
90(1 − θ) − [1 + (n − 2)θ](pi − 10) + θ nj6=i (pj − 10)
qi =
(1 − θ)[1 + (n − 1)θ]
(2.21)
If marginal cost is constant at 10 per unit, show that when firms set prices
to maximize own profit, equilibrium prices are
pB = 10 + (1 − θ)
90
,
2 + (n − 3)θ
(2.22)
so that for all θ < 1, equilibrium prices fall as the number of firms rises.
Write the system of equations of the inverse demand curves in matrix
form as




p1
q1
 p2 
 q2 




0
 = 100Jn −[(1−θ)In +θJn J ]  .  = 100Jn −[(1−θ)In +θJn J 0 ]q,
.
p=
n
n




 . 
 . 
pn
qn
(2.23)
or
p = 100Jn − [(1 − θ)In + θJn Jn0 ]q
(2.24)
where Jn is an n-element column vector of 1s and In is an n × n identify
matrix.
It proves to be convenient to express prices in terms of deviations from
marginal cost; (2.24) becomes
p − 10Jn = 90Jn − [(1 − θ)In + θJn Jn0 ]q.
(2.25)
16
CHAPTER 2. OLIGOPOLY I
Rewrite (2.25) as
[(1 − θ)In + θJn Jn0 ]q = 90Jn − (p − 10Jn ).
(2.26)
We need to find the inverse of the coefficient matrix
(1 − θ)In + θJn Jn0 .
(2.27)
Suppose the inverse takes the form
1
In + kJn Jn0 ,
1−θ
(2.28)
where the value of the parameter k is to be determined.
Then it must be that
µ
¶
1
0
0
[(1 − θ)In + θJn Jn ]
In + kJn Jn = In .
1−θ
(2.29)
Carrying out the multiplication,
θ
Jn Jn0 + kθJn Jn0 Jn Jn0 = In .
1−θ
·
¸
θ
+ nkθ Jn Jn0 = In
In + (1 − θ)k +
1−θ
In + (1 − θ)kJn Jn0 +
(2.30)
(2.31)
(using Jn0 Jn = n).
½
θ
In + [1 + (n − 1)θ]k +
1−θ
¾
Jn Jn0 = In ,
(2.32)
from which it follows that
k=−
θ
1
,
1 − θ 1 + (n − 1)θ
(2.33)
so that the inverse in question is
[(1 − θ)In +
θJn Jn0 ]−1
·
¸
1
θ
0
=
In −
Jn Jn
1−θ
1 + (n − 1)θ
(2.34)
Substituting (2.34) in (2.26) shows that the equations of the demand
curves satisfy
(1 − θ)[1 + (n − 1)θ]q =
(2.35)
(1 − θ)90Jn − {[1 + (n − 1)θ]In − θJn Jn0 } (p − 10Jn ).
17
These expressions are valid provided all quantities are nonnegative.
For example, the quantity demanded of variety 1 is
P
90(1 − θ) − [1 + (n − 1)θ](p1 − 10) + θ nj=1 (pj − 10)
q1 =
(1 − θ)[1 + (n − 1)θ]
P
90(1 − θ) − [1 + (n − 2)θ](p1 − 10) + θ nj=2 (pj − 10)
=
.
(1 − θ)[1 + (n − 1)θ]
(2.36)
Firm 1’s profit as a function of prices satisfies
(
(2.37)
(1 − θ)[1 + (n − 1)θ]π 1 =
)
n
X
(p1 − 10) 90(1 − θ) − [1 + (n − 2)θ](p1 − 10) + θ
(pj − 10) .
j=2
The first-order condition to maximize π 1 with respect to p1 is
90(1−θ)−[1+(n−2)θ](p1 −10)+θ
n
X
(pj −10)+(p1 −10) {−[1 + (n − 2)θ]} = 0
j=2
(2.38)
n
X
(pj − 10) = 90(1 − θ).
2[1 + (n − 2)θ](p1 − 10) − θ
(2.39)
j=2
Because firms in this example hold identical beliefs and have identical
cost functions, in equilibrium, all firms will charge the same price. Setting
p1 = p2 = ... = pn = pB and substituting in (2.39) gives
{2[1 + (n − 2)θ] − (n − 1)θ} (pB − 10) = 90(1 − θ)
(2.40)
[2 + (n − 3)θ](pB − 10) = 90(1 − θ)
(2.41)
pB = 10 + (1 − θ)
90
.
2 + (n − 3)θ
(2.42)
18
CHAPTER 2. OLIGOPOLY I
Chapter 3
Collusion and tacit collusion
3.1 (Measuring market share with differentiated products) Show that if
(for example, for duopoly) inverse demand curves have equations (2.41) and
(2.42),
p1 = 100 − (q1 + θq2 ) ,
(3.1)
p2 = 100 − (θq2 + q1 ) ,
(3.2)
p1 − c0 (q1 )
s1
=
,
p1
εQ1 p1
(3.3)
the expression for the Lerner index of market power that corresponds to (??)
is
where
q1
q1
≡
(3.4)
q1 + θq2
Q1
is firm 1’s market share, taking account of the imperfect substitutability of
variety 2 for variety 1, and
s1 =
εQ1 p1 ≡ −
Q1 dp1
.
p1 dQ1
(3.5)
Write the equation of firm 1’s inverse demand curve in general form as
p1 = f (Q1 ) = f (q1 + θq2 ),
where θ is a product differentiation parameter with 0 ≤ θ ≤ 1. Then firm
1’s profit is
π 1 = f(q1 + θq2 )q1 − c(q1 ).
The first-order condition to maximize π 1 with respect to q1 is
p1 + q1
dp1
= c0 (q1 )
dQ1
19
20
CHAPTER 3. COLLUSION AND TACIT COLLUSION
p1 − c0 (q1 ) = −q1
p1 − c0 (q1 )
q1
=
p1
Q1
dp1
dQ1
µ
¶
Q1 dp1
−
p1 dQ1
p1 − c0 (q1 )
s1
=
p1
εQ1 p1
Each firm’s market share is measured relative to the size of its own market,
which (because of product differentiation) differs from the markets of other
firms. The size of firm 1’s market is q1 + θq2 , the size of firm 2’s market is
θq1 + q2 .
Chapter 4
Dominance
4—1 (Price leadership with product differentiation) For a price-setting duopoly
with product differentiation, let the equations of the inverse demand curves
be
µ
¶
1
p1 = 100 − q1 + q2 ,
(4.1)
2
µ
¶
1
p2 = 100 −
q1 + q1 ,
(4.2)
2
with corresponding demand functions
2
q1 = (100 − 2p1 + p2 )
3
(4.3)
2
q2 = (100 + p1 − 2p2 ).
(4.4)
3
Let marginal cost be constant at 10 per unit.
Find equilibrium prices and profits if firm 2 sets its price p2 noncooperatively to maximize its own profit, if firm 1 knows this, and if firm 1 maximizes
its own profit, taking firm 2’s behavior into account.
Rewrite the equations of the demand curves as
q1 =
2
[90 − 2 (p1 − 10) + (p2 − 10)]
3
2
[90 + (p1 − 10) − 2(p2 − 10)] .
3
Firm 2’s payoff function is
q2 =
2
π 2 = (p2 − 10)q2 = (p2 − 10) [90 + (p1 − 10) − 2(p2 − 10)] .
3
21
(4.5)
(4.6)
(4.7)
22
CHAPTER 4. DOMINANCE
The equation of the first-order condition to maximize π 2 with respect to
p2 is
90 + (p1 − 10) − 4(p2 − 10) ≡ 0
(4.8)
1
(4.9)
[90 + (p1 − 10)]
4
Substituting the equation of firm 2’s best response function in the equation of firm 1’s demand curve and collecting terms gives the equation of firm
1’s residual demand curve:
p2 − 10 =
4
2
2
(90) − (p1 − 10) + (p2 − 10)
3
3
3
µ ¶
2
4
2 1
= (90) − (p1 − 10) +
[90 + (p1 − 10)]
3
3
3 4
µ ¶
· µ ¶
¸
2
2 1
2 1
4
= (90) +
(90) +
−
(p1 − 10)
3
3 4
3 4
3
q1 =
7
(p1 − 10) .
6
Firm 1’s payoff along its residual demand curve is
·
¸
7
π 1 = (p1 − 10) q1 = (p1 − 10) 75 − (p1 − 10)
6
= 75 −
(4.10)
(4.11)
and this is maximized for
µ ¶
7
(p1 − 10) ≡ 0
75 − 2
6
p1 = 10 +
225
1
= 10 + 32 .
7
7
The follower’s price satisfies
·
¸
1
225
855
15
p2 − 10 =
90 +
=
= 30 .
4
7
28
28
Quantities demanded are
·
µ
¶ µ
¶¸
2
225
855
75
1
q1 =
90 − 2
+
=
= 37
3
7
28
2
2
·
µ
¶
µ
¶¸
5
2
225
855
285
q2 =
90 +
−2
=
= 40 .
3
7
28
7
7
(4.12)
(4.13)
(4.14)
(4.15)
23
Payoffs are
¶µ ¶
75
16 875
5
225
=
= 1205
π 1 = (p1 − 10) q1 =
7
2
14
14
µ
¶µ
¶
855
285
243 675
47
π2 = (p2 − 10)q2 =
=
= 1243
.
28
7
196
196
µ
(4.16)
(4.17)
These compare with equilibrium payoffs of 1200 if neither firm is a leader
(See the answer to Problem 2—2 and set σ = 0).
4—2 (Limit pricing) For the price-setting market of Problem 4—1, let firm
1 be an incumbent and firm 2 a potential entrant that must pay a fixed and
sunk entry cost e if it comes into the market. If firm 1 can commit to a
post-entry price, what price must it set to make entry unprofitable? Under
what circumstances (for what values of re, where r is the interest rate used
to discount income) would firm 1 prefer to deter entry (a) if the post-entry
market would be a Bertrand (noncooperative) duopoly and (b) if firm 1 would
be a Stackelberg price leader in the post-entry market?
From (4.8), on firm 2’s best response function
q2 =
2
4
[90 + (p1 − 10) − 2(p2 − 10)] = (p2 − 10)
3
3
(4.18)
and its payoff per period is
4
1
π 2 = (p2 − 10)q2 = (p2 − 10)2 =
[90 + (p1 − 10)]2 .
3
12
(4.19)
The entrant’s present discounted value if the post-entry market is a
Bertrand duopoly is
[90 + (p1 − 10)]2
V2 =
−e
(4.20)
12r
and this is zero or negative for
[90 + (p1 − 10)]2
− e ≤ 0.
12r
(4.21)
If firm 1 is a monopolist not threatened by the possibility of entry,
monopoly price is 55. Entry is blocked if
[90 + (55 − 10)]2
−e≤0
12r
re ≥
[90 + (55 − 10)]2
(135)2
6075
=
=
= 1518.75.
12
12
4
(4.22)
24
CHAPTER 4. DOMINANCE
[90 + (p1 − 10)]2 ≤ 12re
√
90 + (p1 − 10) ≤ 2 3re
√
p1 − 10 ≤ 2 3re − 90.
If the incumbent commits to a price
√
pL = 2 3re − 80,
(4.23)
(4.24)
firm 2 stays out of the market, and the quantity demanded of firm 1 is
√
q1 = 100 − pL = 180 − 2 3re.
The incumbent’s per-period payoff if it commits to price pL is
³ √
´³
√ ´
√
2 3re − 90 180 − 2 3re = 540 3re − 12re − 16 200
(4.25)
and its value is
√ ¢
¡√
¢¡
3re − 45 90 − 3re
VL =
.
r
The incumbent’s value in a Bertrand duopoly is
4
1200
,
r
(4.26)
(4.27)
and if the alternative is Bertrand duopoly the incumbent will have at least
as great a value committing to price pL if
√ ¢
¢¡
¡√
4 3re − 45 90 − 3re
1200
≥
r
r
´³
´
³√
√
3re − 45 90 − 3re ≥ 300
√
135 3re − 3re − 4050 ≥ 300
√
135 3re − 3re − 4350 ≥ 0
√
(4.28)
45 3re − re − 1450 ≥ 0
√
45 3re−re−1450 = 0 for re = 941.24, and firm 1’s value as a Stackelberg
price leader rises with re from this value (Figure 4.1).
If the alternative to committing to an entry-deterring price is letting
the entrant into the market and acting as a Stackelberg price leader, the
incumbent’s value in the post-entry market is, from (4.16),
16 875
.
14r
(4.29)
25
∆V
70 •
.....................................
VP L −... VBert
....................
.
.
.
.
.
.
.....
.
.
.
.
.
.....
.......
.....
........
.
.
.
60 •
.
.
.
.....
.......
.....
..... . ..........
........ ......
50 •
..
....
.
.
..
40 •
....
.
.
..
...
.
.
.
..
30 •
....
.
.
.
....
.
.
.
.
20 •
..
.. .
..
10 •
.. .
.
..
.
• .•
•
•
•
•
•
•
•
re
.
.. . 1000 1100 1200 1300 1400 1500 1600
.
−10 •...
Figure 4.1: Entry cost and VP L − VBert
The incumbent’s value if it deters entry is at least as great as if it lets
the entrant into the market if
√ ¢
¡√
¢¡
4 3re − 45 90 − 3re
16 875
≥
r
14r
³√
´³
√ ´ 16 875
4
3re − 45 90 − 3re ≥
14
³√
´³
√ ´ 16 875
3re − 45 90 − 3re ≥
56
³√
´³
√ ´ 16 875
3re − 45 90 − 3re ≥
56
√
16 875
135 3re − 3re − 4050 ≥
56
√
243 675
135 3re − 3re −
≥0
56
√
81225
45 3re − re −
≥ 0.
(4.30)
56
√
45 3re−re− 81225
= 0 for re = 942.89, and firm 1’s value as a Stackelberg
56
price leader rises with re from this value.
26
CHAPTER 4. DOMINANCE
Chapter 5
Organization
14/12/01: I now believe that there is an additional avenue through which
sunk cost may affect equilibrium market structure. When some part of costs
are sunk, entry creates excess capacity and makes the shadow value of fixed
assets, at least for a time, equal to zero. This reduction in incumbents’ unit
cost reduces an entrant’s expected profit and may make entry unprofitable.
See “Sunk cost and entry,” Review of Industrial Organization 20(4), June
2002, pp. 291—304.
5—1 (fixed cost, sunk cost, market structure I) In general, let firms operate
with production function
·
¸
K −K L−L
q = min
,
aK
aL
for K ≥ K, L ≥ L, and q = 0 otherwise, where K is a minimum amount
of physical capital needed to produce at all, L is a minimum amount of
labor needed to produce at all, and aK , aL are capital and labor inputoutput coefficients, respectively. Thus if production is efficient in the sense
of minimizing cost, so that a firm employs no excess capital or labor,
q=
K−K
L−L
=
aK
aL
and input levels are
K = K + aK q
L = L + aL q.
Firms hire labor at wage rate w per period and purchase physical capital
at price pk ; for simplicity, assume both input prices are constant over time,
and assume also that physical capital does not depreciate. The rental rate
of the services of one unit of physical capital is then rpk , where r is the rate
27
28
CHAPTER 5. ORGANIZATION
of return on a safe asset (the opportunity cost of investing financial capital
in the firm). If the firm wishes to resell a unit of physical capital, it can
do so at price αpk , where the cost-sunkenness parameter α is a number that
lies between 0 and 1. If α = 0, investments in the industry are completely
sunk, in the sense that if the firm should wish to exit the industry, it would
not be able to recover any of its investment in physical capital. If α = 1,
investments in the industry are not sunk at all.
Now for specificity, let
·
¸
K − 160 L − 20
q = min
,
1
1
so that to produce at all requires hiring at least one hundred and eighty units
of capital and twenty units of labor, and that each unit of output requires
one additional unit of capital and one additional unit of labor over these
minimum amounts. Suppose also that r = 1/10, pk = 50, and w = 5.
(a) Find the cost function of a firm. Identify fixed cost, variable cost,
marginal cost, and sunk cost.
C(q) = rpk (160 + q) + w(20 + q) = 160rpk + 20w + (rpk + w)q.
Fixed cost is
160rpk + 20w = 160(5) + 20(5) = 900.
Variable cost is
(rpk + w)q = (5 + 5)q = 10q.
Marginal cost is
rpk + w = 10
per unit.
If a firm produces q units of output efficiently, its capital stock is
160 + q,
and its investment in this capital stock is
pk (160 + q) = 50(160 + q).
The portion of this investment that is sunk – the portion that could not be
recovered by sale of the assets if the firm should shut down – is
(1 − α)pk (160 + q) = (1 − α)50(160 + q).
29
(b) In a market with inverse demand curve
p = 100 − Q,
what is the long-run equilibrium number of firms in Cournot oligopoly if
firms produce efficiently? How does the level of fixed cost affect the longrun equilibrium number of firms? How does the level of sunk cost affect the
long-run equilibrium number of firms?
Write the equation of the inverse demand curve as
p = a − Q.
The cost function of a single firm is
¡
¢
C(q) = F + cq = rpk K + wL + rpk aK + waL q.
The equation of the Cournot best-response function of (say) firm 1 is
2q1 + q2 + . . . + qn = a − c.
In symmetric equilibrium all firms produce the same output,
qnCour =
a−c
.
n+1
Equilibrium per-firm profit per period is
µ
¶
¡ Cour ¢2
a−c 2
Cour
πn
= qn
−F =
− F.
n+1
If entry occurs until equilibrium per-firm profit per period is driven to
zero, the Cournot equilibrium number of firms is
¶2
µ
a−c
−F =0
n+1
a−c √
= F.
n+1
a−c
nCour = √ − 1.
F
For this particular problem,
90
nCour = √
− 1 = 2.
900
30
CHAPTER 5. ORGANIZATION
The equilibrium number of firms is two. The equilibrium number of firms
falls as fixed costs rise, and the equilibrium number of firms is not affected
by changes in the extent to which costs are sunk.
(c) Now suppose that the rental cost of capital services rises, the more
are investments in the industry sunk, that is, that the rental cost of capital
services is
ρ = ρ(α), with ρ(1) = r, ρ0 < 0.
The opportunity cost to a firm of investing in an industry is the amount
it must pay to borrow financial capital. The resale value of physical capital
is collateral that secures the value of loans (or that reverts to bondholders,
if a firm should go bankrupt). The more are costs sunk (the lower is α), the
lower the value of this collateral, all else equal, and the greater the interest
rate that financial markets will require to finance investments in the industry.
How do changes in the extent to which an industry’s costs are sunk affect
the equilibrium long-run number of firms in this altered specification?
Write the expression for the long-run Cournot equilibrium number of
firms as
£
¤
k
a
−
ρ(α)p
a
+
wa
a
−
c
K
L
− 1.
nCour (α) = √ − 1 = q
F
ρ(α)pk K + wL
If α falls, c and F both rise, and nCour (α) falls.
5—2 (sunk cost and market structure II) Continuing Problem 5—1, let α =
1/2, so that half of a firm’s investment in physical assets is sunk. Suppose
the firm is supplied by one firm that produces the monopoly output.
(a) What is the firm’s monopoly profit?
If the firm operates efficiently, profit per period is
π m = (100 − 10 − q1 ) q1 − 900.
This is maximized for qm = 45 units of output, resulting in a profit
π m = (100 − 10 − 45) (45) − 900 = 1125
per period.
(b) If a second firm comes into the market, what is the first firm’s marginal
cost? (Hint: calculate the present-discounted value of the first firm’s cost if
it sells its excess capital at the start of the period in which entry occurs.)
Write K = 205 for the first firm’s pre-entry capital stock. If the first
firm permanently reduces its output level to the Cournot equilibrium output
q (which we will determine shortly), it sells excess capital at the start of the
31
period in which entry occurs at price αpk per unit; the present-discounted
value of its cost is
¡
¡
¢
¡
¢
¢
w
w
L
+
a
q
w
L
+
a
q
L
+
a
q
L
L
L
−αpk (K − aK q) +
+
+ ... =
+
1+r
(1 + r)2
(1 + r)3
¡
¢
w L + aL q
−αp K + αp aK q +
=
r
k
k
−αpk K + αpk aK q +
wL + waL q
=
r
−αrpk K + wL αrpk aK q + waL q
+
=
r
r
F (K, α) + cα q
,
r
where the first firm’s marginal cost per period is
cα = αrpk aK + waL .
For this problem
µ ¶µ ¶
1
1
(50)(205) + (5)(20) = −412.5
F (K, α) = −
2
10
µ ¶µ ¶
1
1
(50)(1) + (5)(1) = 7.5.
cα =
2
10
(c) if the post-entry market is a Cournot duopoly, what is the second
firm’s equilibrium profit? How do changes in the extent to which costs are
sunk affect the second firm’s post-entry profit?
If entry occurs, the entrant has marginal cost
c = rpk aK + waL = 10.
The system of equations of the best response functions, written in matrix
form, is
µ
¶µ
¶ µ
¶
2 1
q1
a − cα
=
1 2
q2
a−c
µ
¶ µ
¶µ
¶
2 −1
a − cα
q1
3
=
q2
−1 2
a−c
q2 =
1
[2(a − c) − (a − cα )]
3
32
CHAPTER 5. ORGANIZATION
175
1
1
[2(90) − 92.5] =
= 29 .
3
6
6
The entrant’s equilibrium profit per period is
q2 =
µ
175
6
¶2
q22 − F =
− 900 = −
1775
11
= −49 .
36
36
In general, the entrant’s profit per period is less than or equal to zero for
¡
¢
1
(a + cα − 2c)2 − rpk K + wL ≤ 0
9
¡
¢
(a + cα − 2c)2 ≤ 9 rpk K + wL
q
a + cα − 2c ≤ 3 rpk K + wL
q
2c − cα ≥ a − 3 rpk K + wL
q
¡ k
¢ ¡
¢
k
2 rp aK + waL − αrp aK + waL ≥ a − 3 rpk K + wL
q
(2 − α)rpk aK + waL ≥ a − 3 rpk K + wL.
This condition is more likely to be met, the smaller is α (the more costs
are sunk), the smaller is a (an indicator of the size of the market), and the
larger are the entrant’s fixed costs rpk K + wL.
5—3 Consider a market with linear inverse demand function
p(Q) = a − bQ,
where Q is total output. Let the firm-level cost function be cubic,
C(q) = F + cq − dq2 + eq 3 .
Here F , a, b, c, d, e ≥ 0. Assume also that a − c > 0 and d > b.
Find the long-run equilibrium number of firms if the market is a Cournot
oligopoly and entry occurs until profit per firm is zero.
Firm 1’s payoff function is
π 1 = [a − b (q1 + Q−1 )] q1 − F − cq1 + dq12 − eq13
= [a − c − b (q1 + Q−1 )] q1 − F + dq12 − eq13
where Q−1 is the combined output of all other firms.
33
The first-order condition to maximize firm 1’s profit is
a − c − b (2q1 + Q−1 ) + 2dq1 − 3eq12 ≡ 0.
Note that the first-order condition implies
a − c − b (q1 + Q−1 ) = (b − 2d + 3eq1 ) q1 ,
so that along its first order condition, and in particular in equilibrium, firm
1’s payoff is
π 1 = (b − 2d + 3eq1 ) q12 + dq12 − eq13 − F
= [b − 2d + 3eq1 + d − eq1 ] q12 − F
= (b − d + 2eq1 ) q12 − F.
Given the symmetry that characterizes this problem, in equilibrium all
firms produce the same output. Substitute q1 = q, Q−1 = (n − 1) q in the
equation of firm 1’s best response function and solve for equilibrium output
with n firms in the market:
a − c − b (n + 1) q + 2dq − 3eq2 = 0
q(n) =
3eq 2 + [(n + 1) b − 2d] q − (a − c) = 0
q
− [(n + 1) b − 2d] + [(n + 1) b − 2d]2 + 12e (a − c)
6e
(5.1)
The equilibrium payoff per firm is
π = (b − d + 2eq) q 2 − F.
and the number of firms adjusts until π = 0:
(b − d + 2eq) q 2 − F = 0.
For F > 0, this is a cubic equation with one real root. To find the general
solution, substitute the analytic expression for this root on the left in (5.1)
and solve the resulting expression for n.
If F = 0 and d > b, long-run equilibrium output per firm is
q=
d−b
.
2e
34
CHAPTER 5. ORGANIZATION
Substituting in (5.1)
q
− [(n + 1) b − 2d] + [(n + 1) b − 2d]2 + 12e (a − c)
=
d−b
2e
6e
q
− [(n + 1) b − 2d] + [(n + 1) b − 2d]2 + 12e (a − c) = 3 (d − b)
q
[(n + 1) b − 2d]2 + 12e (a − c) = [(n + 1) b − 2d] + 3 (d − b)
[(n + 1) b − 2d]2 + 12e (a − c) =
[(n + 1) b − 2d]2 + 6 (d − b) [(n + 1) b − 2d] + 9 (d − b)2
12e (a − c) = 6 (d − b) [(n + 1) b − 2d] + 9 (d − b)2
µ
¶
a−c
4e
= 2 [(n + 1) b − 2d] + 3 (d − b)
d−b
¶
µ
a−c
− 3 (d − b)
2 [(n + 1) b − 2d] = 4e
d−b
µ
¶
a−c
3
(n + 1) b − 2d = 2e
− (d − b)
d−b
2
µ
¶
a−c
3
+ 2d − (d − b)
(n + 1) b = 2e
d−b
2
µ
¶
a−c
1
3
+ d+ b
(n + 1) b = 2e
d−b
2
2
µ
¶
e a−c
1d 3
+
+
n+1=2
b d−b
2b 2
µ
¶
e a−c
1d
1
+
.
n= +2
2
b d−b
2b
(5.2)
With F = 0, the long-run Cournot equilibrium number of firms is the
greatest integer less than the right-hand side of (5.2).
5—4 (Equilibrium number of firms, Cournot oligopoly, differentiated
products) For a price-setting oligopoly with product differentiation, let the
equations of the inverse demand curves be
for i = 1, 2, ..., n and Q−i
function
pi = 100 − (qi + θQ−i ) ,
P
= nj6=i qj , with the equation of the firm-level cost
c(q) = F + 10q + dq 2 .
35
Find the equilibrium number of firms if the long-run equilibrium number of
firms adjusts until Cournot equilibrium profit per firm is zero. How does
the equilibrium number of firms change as θ changes?
Firm 1’s profit function is
¡
¢
π 1 = p1 q1 − F + 10q1 + dq12
= (p1 − 10) q1 − F − dq12
"
#
n
X
= 90 − (1 + d)q1 − θ
qj q1 − F.
2
The first-order condition to maximize π 1 is
90 − 2(1 + d)q1 − θ
from which
90 − (1 + d)q1 − θ
and
n
X
2
n
X
2
qj ≡ 0,
qj ≡ (1 + d)q1
π 1 = (1 + d)q12 − F
when the first-order condition holds, and in particular in equilibrium.
Since firms are identical, they produce the same output in equilibrium.
From the first-order condition, this output is
90 − 2(1 + d)q − (n − 1)θq = 0
[2(1 + d) + (n − 1)θ] q = 90
90
q=
.
2(1 + d) + (n − 1)θ
Equilibrium profit per firm is
·
¸2
90
π = (1 + d)
−F
2(1 + d) + (n − 1)θ
and this is zero for
·
90
2(1 + d) + (n − 1)θ
¸2
90
=
2(1 + d) + (n − 1)θ
=
F
1+d
r
F
1+d
36
CHAPTER 5. ORGANIZATION
r
1+d
2(1 + d) + (n − 1)θ = 90
F
#
" r
1
1+d
nCour = 1 +
90
− 2(1 + d)
θ
F
#
"
r
1 90
1
=1+
1 + − 2(1 + d) ,
θ qmes
d
q
for qmes = Fd . The Cournot long-run equilibrium number of firms rises as
θ falls – as products become more differentiated – and falls as qmes or d
rise.
5—5 (Equilibrium number of firms, Bertrand oligopoly, differentiated products)
For a price-setting oligopoly with product differentiation, let the equations of the demand curves be
P
90(1 − θ) − [1 + (n − 2)θ](pi − 10) + θ nj6=i (pj − 10)
qi =
,
(1 − θ)[1 + (n − 1)θ]
with the equation of the firm-level cost function
c(q) = F + 10q + dq 2 .
Find the equilibrium number of firms if the long-run equilibrium number of
firms adjusts until Bertrand equilibrium profit per firm is zero. How does
the equilibrium number of firms change as θ changes?
Firm 1’s profit function is
¡
¢
π 1 = p1 q1 − F + 10q1 + dq12
= (p1 − 10 − dq1 ) q1 − F
For notational simplicity, write
xi = pi − 10.
P
¾
90(1 − θ) − [1 + (n − 2)θ]x1 + θ n2 xj
π 1 = x1 − d
×
(1 − θ)[1 + (n − 1)θ]
P
½
¾
90(1 − θ) − [1 + (n − 2)θ]x1 + θ n2 xj
− F.
(1 − θ)[1 + (n − 1)θ]
Collect the terms in x1 within the first set of braces on the right:
P
90(1 − θ) − [1 + (n − 2)θ]x1 + θ n2 xj
x1 − d
=
(1 − θ)[1 + (n − 1)θ]
½
37
·
Then
½·
P
¸
90(1 − θ) + θ n2 xj
1 + (n − 2)θ
x1 − d
1+d
(1 − θ)[1 + (n − 1)θ]
(1 − θ)[1 + (n − 1)θ]
π1 =
P
¸
¾
1 + (n − 2)θ
90(1 − θ) + θ n2 xj
1+d
x1 − d
×
(1 − θ)[1 + (n − 1)θ]
(1 − θ)[1 + (n − 1)θ]
P
½
¾
90(1 − θ) + θ n2 xj
1 + (n − 2)θ
−
x1 − F.
(1 − θ)[1 + (n − 1)θ] (1 − θ)[1 + (n − 1)θ]
Again for notational compactness, write this as
π 1 = [(1 + dA1 )x1 − dB1 ] [B1 − A1 x1 ] − F
for
1 + (n − 2)θ
(1 − θ)[1 + (n − 1)θ]
P
90(1 − θ) + θ n2 xj
B1 =
(1 − θ)[1 + (n − 1)θ]
A1 =
For future reference, note that in this notation
q1 = B1 − A1 x1
π 1 = [(1 + dA1 )x1 − dB1 ] [B1 − A1 x1 ] − F
−(1 + dA1 )A1 x21 + (1 + 2dA1 )B1 x1 − dB12 − F
The first-order condition to maximize π 1 is
−2(1 + dA1 )A1 x1 + (1 + 2dA1 )B1 ≡ 0.
Note that if the first-order condition holds, then
q1 = B1 − A1 x1 = A1 [(1 + 2dA1 ) x1 − 2dB1 ] .
The first-order condition can be solved for the equation of firm 1’s price
best response function, although that is not of immediate interest in the
present context. Rather, write the first-order condition as
2(1 + dA1 )A1 x1 = (1 + 2dA1 )B1
P
90(1 − θ) + θ n2 xj
2(1 + dA1 )A1 x1 = (1 + 2dA1 )
(1 − θ)[1 + (n − 1)θ]
38
CHAPTER 5. ORGANIZATION
In equilibrium, all firms will set the same price; let xi = x for all i; then
the first-order condition becomes
¸
·
(n − 1)θ
90
2(1 + dA1 )A1 x = (1 + 2dA1 )
+
x
1 + (n − 1)θ (1 − θ)[1 + (n − 1)θ]
or
2(1 + dA1 )A1 x = (1 + 2dA1 ) (C1 + D1 x)
for
C1 =
D1 =
Note that in equilibrium
90
1 + (n − 1)θ
(n − 1)θ
(1 − θ)[1 + (n − 1)θ]
P
90(1 − θ) + θ n2 xj
B1 =
= C1 + D1 x.
(1 − θ)[1 + (n − 1)θ]
Solve the condensed first-order condition for x:
2(1 + dA1 )A1 x = (1 + 2dA1 ) (C1 + D1 x)
2(1 + dA1 )A1 x = (1 + 2dA1 )C1 + (1 + 2dA1 )D1 x
[2(1 + dA1 )A1 − (1 + 2dA1 )D1 ] x = (1 + 2dA1 )C1
£
¤
2A1 − D1 − 2dA1 D1 + 2dA21 x = (1 + 2dA1 )C1
x=
Numerator:
(1 + 2dA1 )C1
[2A1 − D1 − 2dA1 D1 + 2dA21 ]
½
(1 + 2dA1 )C1 = 1 +
2d [1 + (n − 2)θ]
(1 − θ)[1 + (n − 1)θ]
¾
90
1 + (n − 1)θ
Denominator:
2A1 − D1 − 2dA1 D1 + 2dA21 =
D1 + 2(A1 − D1 ) + 2dA1 (dA1 − D1 ) =
µ
¶
(n − 1)θ
2
1 + (n − 2)θ
+
1+d
=
(1 − θ)[1 + (n − 1)θ] [1 + (n − 1)θ]
(1 − θ)[1 + (n − 1)θ]
½
·
¸¾
1
(n − 1)θ
1 + (n − 2)θ
+2 1+d
[1 + (n − 1)θ] (1 − θ)
(1 − θ)[1 + (n − 1)θ]
39
x=
n
1+
n
2d[1+(n−2)θ]
(1−θ)[1+(n−1)θ]
(n−1)θ
(1−θ)
1
[1+(n−1)θ]
o
90
1+(n−1)θ
h
io
1+(n−2)θ
+ 2 1 + d (1−θ)[1+(n−1)θ]
o
n
2d[1+(n−2)θ]
90 1 + (1−θ)[1+(n−1)θ]
h
i
=
(n−1)θ
1+(n−2)θ
+
2
1
+
d
(1−θ)
(1−θ)[1+(n−1)θ]
Equilibrium profit per firm is
π = [(1 + dA1 )x − dB1 ] [B1 − A1 x] − F
= [(1 + dA1 )x − d (C1 + D1 x)] [C1 + D1 x − A1 x] − F
= [(1 + d(A1 − D1 ))x − dC1 ] [C1 − (A1 − D1 )x] − F
= [x − d (C1 − (A1 − D1 )x)] [C1 − (A1 − D1 )x] − F
C1 − (A1 − D1 )x =
·
¸
90
1 + (n − 2)θ
(n − 1)θ
−
−
x=
1 + (n − 1)θ
(1 − θ)[1 + (n − 1)θ] (1 − θ)[1 + (n − 1)θ]
·
¸
1−θ
90
−
x=
1 + (n − 1)θ
(1 − θ)[1 + (n − 1)θ]
90 − x
1 + (n − 1)θ
·
¸
90 − x
90 − x
π = x−d
−F
1 + (n − 1)θ 1 + (n − 1)θ
Expressing x in terms of the number of firms, the long-run equilibrium
number of firms satisfies the equation:


2d[1+(n−2)θ]
n
o
1+ (1−θ)[1+(n−1)θ]
2d[1+(n−2)θ]
1 − (n−1)θ
1+(n−2)θ
90 1 + (1−θ)[1+(n−1)θ]

+2[1+d (1−θ)[1+(n−1)θ]
]
(1−θ)

×
h
i
90 
− 90d

(n−1)θ
1+(n−2)θ
1
+
(n
−
1)θ
+ 2 1 + d (1−θ)[1+(n−1)θ]
(1−θ)
1−
2d[1+(n−2)θ]
1+ (1−θ)[1+(n−1)θ]
(n−1)θ
1+(n−2)θ
+2 1+d (1−θ)[1+(n−1)θ]
(1−θ)
[
]
= F.
1 + (n − 1)θ
This can be reduced to a quartic equation in n.
If d = 0, the equation that determines n becomes
Ã
! 1−
1
(n−1)θ
+2
90
(1−θ)
90 (n−1)θ
−F =0
+ 2 1 + (n − 1)θ
(1−θ)
40
CHAPTER 5. ORGANIZATION
Ã
1
(n−1)θ
(1−θ)
+2
!
(n−1)θ
(1−θ)
µ (n−1)θ
+2−1
(1−θ)
(n−1)θ
+2
(1−θ)
¶
1 + (n − 1)θ
+1
1 + (n − 1)θ
=
=
F
(90)2
F
.
(90)2
The long-run equilibrium number of firms is
n=
(90)2 (2θ − 1) + (1 − θ)2 F
£
¤
θ (90)2 − (1 − θ)F
5—6 (Merger in a linear Cournot model)
Let the market demand curve of a market initially supplied by 3 firms be
p = 100 − Q.
(5.3)
Let all firms have the cost function
c(q) = 10q.
(5.4)
(a) Find equilibrium price, outputs, and profits if the three firms act as
Cournot oligopolists.
(b) Find the same results if firms 1 and 2 merge and the combined
firm competes with firm three, all firms in the post-merger market acting
as Cournot oligopolists.
The equation of the residual demand function of firm 1 is
p = (100 − q2 − q3 ) − q1 .
(5.5)
To find the equation of firm 1’s best response function,
M R1 = (100 − q2 − q3 ) − 2q1 = 10 = mc1
1
q1 = (90 − q2 − q3 )
(5.6)
2
where qc = 90 is the quantity demanded in perfectly competitive long-run
equilibrium. (5.6) is the equation of a plane in (q1 , q2 , q3 )-space: the plane
connecting the three points (45, 0, 0), (0, 90, 0), and (0, 0, 90).
As a way of squeezing three dimensions into two, consider the case in
which q1 = q2 . This amounts to looking at the intersection of the place
defined by (5.6) and a plane defined by the vertical (q3 ) axis and the 45degree line in the (q1 , q2 ) plane.
41
In the particular case of this problem, the three firms are identical, so
firm 1 and firm 2 will produce the same output in equilibrium. There is no
loss of generality in restricting q1 to be equal to q2 outside of equilibrium.
If q1 = q2 = q12 , (5.6) becomes
1
q12 = (90 − q12 − q3 )
2
3
1
q12 = (90 − q3 )
2
2
1
q12 = (90 − q3 )
(5.7)
3
In the same way, the equation of firm 3’s best response function when
q1 = q2 = q12 becomes
1
q3 = (90 − q1 − q2 )
2
1
(5.8)
q3 = (90 − 2q12 ) = 45 − q12 .
2
Cournot equilibrium output with three identical firms is
90
= 22.5;
4
(5.9)
price is
90
= 32.5;
4
(5.10)
(22.5)2 = 506.25,
(5.11)
10 +
profit per firm is
so that before the merger firms 1 and 2 together have a profit of 1012.5.
If firms 1 and 2 merge, the profit of the post-merger firm is
π 12 = (p − c)q1 + (p − c)q2
= (100 − 10 − q1 − q2 − q3 )(q1 + q2 ).
(5.12)
There are a couple of ways to obtain the equation of the post-merger
firm’s best response function. One is simply to calculate the first-order
condition to maximize π 12 with respect to q1 :
90 − 2q1 − 2q2 − q3 ≡ 0
1
q1 = (90 − 2q2 − q3 ).
2
(5.13)
42
CHAPTER 5. ORGANIZATION
Alternatively, and perhaps with a more direct economic interpretation,
consider the post-merger firm’s marginal revenue if division 1 produces an
additional unit of output:
M R1 = p − q1 − q2 .
(5.14)
If the post-merger firm sells an extra unit of output by way of division
1, it gains the revenue from sale of that unit (p), but price falls by 1. This
1 price reduction lowers the firm’s revenue on its sales from division 1 and
from division 2.
Setting division 1’s marginal revenue equal to its marginal cost, we obtain
100 − 2q1 − 2q2 − q3 = 10.
(5.15)
With a little rearrangement of terms, this leads to (5.13).
Once again relying on symmetry to move from three dimensions to two,
set q1 = q2 = q12 in the equation of division 1’s best response function:
100 − 10 − 2q12 − 2q12 − q3 ≡ 0
1
q12 = (90 − q3 ).
(5.16)
4
Firm 3’s best response function has not changed; it continues to have
equation (5.8).
Find post-merger equilibrium outputs by solving the equations of the best
response functions, (5.8) and (5.16).
q3 = 45 − q12
1
q12 = (90 − q3 ).
4
1
q3 = 45 − (90 − q3 )
4
3
90
180 − 90
90
q3 = 45 −
=
=
4
4
4
4
q3 = 30
1
q12 = (90 − 30) = 15.
4
First, the post-merger market is a Cournot duopoly. Firm 3 produces
30 units of output in equilibrium, and divisions 1 and 2 together produce 30
units of output. Total postmerger output is 60, price is 40. The postmerger
43
firm 1/2 earns a profit of 900, less than the combined premerger profit of the
two divisions.
Firm 3 also earns a profit of 900, greater than its premerger profit of
506.25.
There are some aspects of this way of modelling mergers that are not satisfactory: we do not expect a post-merger firm to restrict output so much after
the merger that it is as if it has shut down one of its pre-merger components.
What is realistic about this model is the idea that firms in a merger cannot
control the behavior of firms outside the merger, and that the reactions of
those firms may reduce the profitability of the merger.
Other remarks:
if there are many firms in the premerger market, and the merger combines a large number of them (roughly, 80% or more), the merger will
be profitable for the firms that carry it out;
if products are differentiated and firms set prices, mergers are generally
profitable (with linear demand and constant marginal cost); we will not
consider this kind of model formally.
q1 = q2 Firm 3
....
45 •....... .........
...........
Firm 1/Firm
2 (pre-merger)
.....
....
.
.....
.
.
.
.....
....
.
.
.
.
.
.
.
Division 1/Division 2
.
.
.
.
30 •................. .....
..... (post-merger)
............. ..... . .......
.
.
..... .. ....
..
...
22.5 •........................ ........•..........................
.
.
.
................ .... ............ ...
.................... ..................
.•.................... .............
15
..... ................... .............
................. .............
.....
..............................
.....
.............................
.....
......................
.....
...........
.
•
q3
22.5 30
45
90
Figure 5.1: Pre- and post-merger best response curves, Cournot quantitysetting oligopoly.
44
CHAPTER 5. ORGANIZATION
Chapter 6
Innovation
This chapter intentionally left blank.
45
46
CHAPTER 6. INNOVATION
Chapter 7
Imperfect Competition and
International Trade: I
22 November 2002: On pages 151-3 there is an argument that trade improves welfare for countries of equal size. Professor David Collie of the
Cardiff Business School, to whom I am grateful, writes to point out that this
argument is correct only if transportation cost is zero. With sufficiently
great transportation cost, the opening up of trade may leave each country
worse off. A formal demonstration now appears at the end of the answer to
Problem 7—1 .
7—1 Let there be two countries, each home to one widget producer. The
subscript 1 denotes both country 1 and its widget company; the subscript
2 denotes both country 2 and its widget company. Let the inverse demand
curves in the two countries be
p1 = a1 − b1 (q11 + q21 )
,
p2 = a2 − b2 (q12 + q22 )
(7.1)
where p1 is the price in country 1, p2 is the price in country 2, and qij is the
quantity of widgets sold by firm i in country j, for i, j = 1, 2. The a and
b parameters are respectively the price-axis intercept and the absolute value
of the slope of the inverse demand curves.
Suppose also that widgets are produced at a constant marginal cost c per
unit, and that there is a transportation cost t per unit to ship a widget from
one country to another.
(a) write out the payoff functions of the two firms.
π1 = [a1 − c − b1 (q11 + q21 )]q11 + [a2 − (c + t) − b2 (q12 + q22 )q12
(7.2)
π 2 = [a1 − (c + t) − b1 (q11 + q21 )]q21 + [a2 − c − b2 (q12 + q22 )]q22
(7.3)
47
48
CHAPTER 7. INTERNATIONAL TRADE I
(b) show that the amounts the firms sell in one country are independent of
the amounts they sell in the other country.
This follows from the fact that the derivative of π 1 with respect to q11
depends only on country 1 variables, similarly for ∂π 1 /∂q12 , and similarly for
the derivatives of π2 with respect to firm 2’s sales in the two countries.
(c) Find the equations of the best response functions for country 1.
Take the derivative of (7.15) with respect to q11 and the derivative of
(7.16) with respect to q21 to obtain
a1 − c
b1
(7.4)
a1 − c − t
b1
(7.5)
2q11 + q21 =
q11 + 2q21 =
or equivalently
1
q11 = (a1 − c − q21 )
(7.6)
2
1
q21 = (a1 − c − t − q11 )
(7.7)
2
Neither firm would sell below its marginal cost. This means that the
equation of firm 1’s best response function is valid only for combinations of
q11 and q21 that result in prices greater than or equal to c and the equation
of firm 2’s best response function is valid only for combinations of q11 and q21
that result in prices greater than or equal to c + t. This implies restrictions
that are not worked out here.
(d) Solve the equations of the best response functions for equilibrium outputs
in country 1.
µ
¶µ
¶
µ ¶
µ ¶
2 1
q11
1
0
b1
= (a1 − c)
−t
1 2
q21
1
1
µ
¶
µ
¶µ ¶
µ
¶µ ¶
q11
2 −1
1
2 −1
0
3b1
= (a1 − c)
−t
q21
−1 2
1
−1 2
1
µ
¶
µ ¶
µ
¶µ ¶
q11
1
−1
0
3b1
= (a1 − c)
−t
q21
1
2
1
∗
q11
=
a1 − c
t
+
3b1
3b1
(7.8)
49
∗
=
q21
a1 − c
2t
−
3b1
3b1
(7.9)
(e) What restriction on transportation cost applies if firm 2 is to sell in
country 1?
∗
q21
must be nonnegative,
∗
q21
=
a1 − c
t
−2
> 0,
3b1
3b1
and this is the case if
1
t < (a1 − c).
(7.10)
2
That is, transportation cost cannot exceed the monopoly price-cost margin if the country 2 firm is to sell in country 1. A corresponding condition
must hold if the country 1 firm is to sell in country 2.
(f) What is equilibrium price in country 1? Compare the equilibrium price
with each firm’s marginal cost of supplying country 1. (This part of the
exercise relates to the analysis of dumping.)
Sales in country 1 are
2(a1 − c) − t
3b1
∗
∗
q11
+ q21
=
(7.11)
Hence equilibrium price is
p1 = a1 − b1
2(a1 − c) − t
3b1
1
1
= c + (a1 − c) + t
3
3
µ
¶
2 a1 − c
= (c + t) +
−t
3
2
Price is greater than c, firm 1’s cost of serving its home market.
is greater than c + t, firm 2’s cost of serving country 1, if condition
(which is the condition for firm 2 to sell in country 1) is met.
22 November 2002
With the opening up of trade, firm 1’s profit in country 1 is
∗
(p1 − c) q11
=
1
(a1 − c + t)2
9b1
(7.12)
(7.13)
Price
(7.10)
50
CHAPTER 7. INTERNATIONAL TRADE I
To write out an expression for firm 1’s profit in country 2, first write out
an expression for firm 2’s profit in country 1, then reverse the subscripts.
With the opening up of trade, firm 2’s profit in country 1 is
µ
¶µ
¶
a1 − c
2 a1 − c
t
∗
−t
−2
[p1 − (c + t)] q21 =
3
2
3b1
3b1
µ
¶2
4
a1 − c
=
−t .
9b1
2
Hence with the opening up of trade, firm 1’s profit in country 2 is
µ
¶2
4
a2 − c
−t .
9b2
2
Firm 1’s total profit with the opening up of trade is
¶2
µ
4
1
a2 − c
2
−t .
(a1 − c + t) +
9b1
9b2
2
We limit attention to the case in which markets are of the same size; drop
the country-specific subscripts:
µ
¶2
1
4 a−c
2
(a − c + t) +
−t .
9b
9b
2
Without trade, firm 1 was a monopolist in country 1; price, output, and
profit were
1
p = c + (a − c)
2
1a−c
q=
2 b
1
(a − c)2 .
4b
The reduction in profit with the opening up of trade is
µ
¶2
1
1
4 a−c
2
2
(a − c) − (a − c + t) −
−t =
4b
9b
9b
2
1
(a − c + 10t) (a − c − 2t) =
36b
µ
¶
1
a−c
(a − c + 10t)
− t > 0.
18b
2
51
Trade leaves each firm with lower profit.
Before trade, consumer surplus in country 1 would be
1
1
(a − p) q = (a − a + bq) q =
2
2
µ
¶
a−c 2
1 2 1
bq = b
=
2
2
2
1
(a − c)2 .
8b
With trade, consumer surplus in country 1 is (using the expressions for
post-trade price and output in country 1, and eliminating the country-specific
subscripts)
½
·
¸¾
2(a − c) − t
2(a − c) − t
1
a− a−
=
2
3
3b
·
¸
1
2(a − c) − t 2(a − c) − t
a−a+
=
2
3
3b
·
¸
1 2(a − c) − t 2(a − c) − t
=
2
3
3b
·
¸2
1 2(a − c) − t
=
2b
3
µ
¶2
2
1
a−c− t .
9b
2
The change in consumer surplus with the opening up of trade is
µ
¶2
2
1
1
a − c − t − (a − c)2 =
9b
2
8b
[7 (a − c) − 2t] (a − c − 2t)
72b
µ
¶µ
¶
7
2
a−c
a−c− t
− t > 0.
36b
7
2
Trade leaves country 1 consumers better off.
The net change in country 1 welfare is the gain in consumer surplus minus
the loss of firm 1 profit
µ
¶µ
¶
µ
¶
7
2
a−c
1
a−c
a−c− t
−t −
(a − c + 10t)
−t =
36b
7
2
18b
2
52
CHAPTER 7. INTERNATIONAL TRADE I
· µ
¶
¸µ
¶
2
a−c
1 7
a − c − t − (a − c + 10t)
−t =
18b 2
7
2
·
¸µ
¶
a−c
1 7
(a − c) − t − (a − c) − 10t
−t =
18b 2
2
·
¸µ
¶
a−c
1 5
(a − c) − 11t
−t =
18b 2
2
·
¸µ
¶
5 a − c 11
a−c
− t
−t .
18b
2
5
2
Welfare falls for
a − c 11
− t<0
2
5
5 a−c
5
t>
=
(a − c) .
11 2
22
7—2 Analyze the Cournot duopoly trade model for general demand curves,
p1 = p1 (q11 + q21 )
.
p2 = p2 (q12 + q22 )
(7.14)
Payoffs are
π 1 = [p1 (q11 + q21 ) − c]q11 + [p2 (q12 + q22 ) − (c + t)]q12 − F
(7.15)
π 2 = [p1 (q11 + q21 ) − (c + t)]q21 + [p2 (q12 + q22 ) − c)]q22 − F
(7.16)
The first-order conditions for profit-maximization are
Firm 1, country 1:
dp1
∂π 1
= p1 (q1 ) − c + q11
=0
∂q11
dq1
(7.17)
∂π 1
dp2
= p2 (q2 ) − (c + t) + q12
=0
∂q12
dq2
(7.18)
Firm 1, country 2:
Firm 2, country 1:
∂π 2
dp1
= p1 (q1 ) − (c + t) + q21
=0
∂q21
dq1
(7.19)
53
Firm 2, country 2:
dp2
∂π 2
= p2 (q2 ) − c + q22
=0
∂q22
dq2
(7.20)
Second-order conditions must be satisfied, as well as conditions to ensure
stability. These are not dealt with here.
Equation (7.17), the first-order condition for the domestic firm in its home
market can be rewritten
p1 − c
s11
=
,
(7.21)
p1
ε1
where s11 = q11 /q1 is firm 1’s market share in its home market. This will
be recognized from Chapter 4 as the generalization of the Lerner index of
monopoly power to the case of different production costs.
In the same way, for firm 2 in country 1 one obtains
s21
p1 − (c + t)
=
p1
ε1
(7.22)
Solving (7.21) and (7.22) for p1 gives
ε1
c
ε1 − s11
(7.23)
ε1
(c + t),
ε1 − s21
(7.24)
p1 =
and
p1 =
respectively.
(7.23) and (7.24) can be solved for s21 and p1 ,
s21
1 + ct (1 − ε1 )
1 + ct (1 − ε1 )
=
=
2 + ct
2 + ct
(7.25)
and
ε1
(2c + t).
(7.26)
2ε1 − 1
From the numerator on the right in (7.25), the condition for firm 2 to have
a positive market share in country 1 – this is the condition for intra-industry
trade to occur – is
t
1
<
(7.27)
c
ε1 − 1
(transportation cost cannot be too high) or
p1 =
ε1 < 1 +
1
c+t
=
t/c
t
(7.28)
54
CHAPTER 7. INTERNATIONAL TRADE I
(the price elasticity of demand cannot be too great).
From (7.26),
¶ µ
¶
µ
1
t
ε1 − 1
c
−
.
p1 − (c + t) =
2ε1 − 1
ε1 − 1 c
(7.29)
Examining the final term in parentheses on the right, the condition for
firm 2 to have a positive market share in country 1, (7.27), is also the condition for the country 1 price to exceed firm 2’s marginal cost of supplying
country 1, c + t.
7-3 (a) Answer Problem 7—1 if firms set prices rather than quantities. Suppose that products are differentiated, with demand curves in country i given
by equations
p1i = ai − bi (q1i + θq2i )
,
(7.30)
p2i = ai − bi (θq1i + q2i )
where the first subscript denotes the firm and the second, i = 1, 2, denotes
the country, 0 ≤ θ < 1, with average and marginal cost c and transportation
cost t per unit as in Problem 7—1.
First solve the equations of the inverse demand curves to obtain equations
for the demand curves, expressing the quantity demanded of each variety as
a function of the prices of both varieties.
Because these demand equations will be used to write down expressions
for profit on the country 1 market,
π11 = (p11 − c)q11
(7.31)
π 21 = (p21 − c − t)q21
(7.32)
it is convenient to rewrite (7.30) so that prices are expressed as deviations
from marginal cost,
p11 − c = a1 − c − b1 (q11 + θq21 )
p21 − c − t = a1 − c − t − b1 (θq11 + q21 )
(7.33)
There equations must be solved for q11 and q21 as functions of p11 − c and
p21 − c − t. There are several ways to do this; the method presented here uses
linear algebra. Write the equations of the inverse demand curves in matrix
form as
µ
¶ µ
¶
µ
¶µ
¶
p11 − c
a1 − c
1 θ
q11
=
−b
,
(7.34)
p21 − c − t
a1 − c − t
θ 1
q21
55
from which
¶ µ
¶ µ
¶
µ
¶µ
a1 − c
p11 − c
1 θ
q11
=
−
.
b
q21
a1 − c − t
p21 − c − t
θ 1
(7.35)
Using the formula for the inverse of a 2 × 2 matrix,
µ
α β
γ δ
¶−1
1
=
αδ − βγ
µ
δ −β
−γ α
¶
(7.36)
,
(which is valid provided the determinant αδ − βγ 6= 0), one obtains expressions for the quantities demanded,
µ
¶
q11
2
b(1 − θ )
=
q21
µ
1 −θ
−θ 1
¶µ
a1 − c
a1 − c − t
¶
µ
1 −θ
−θ 1
¶µ
p11 − c
−
p21 − c − t
·
¸ ·
¸
(a1 − c) − θ(a1 − c − t)
p11 − c − θ(p21 − c − t)
=
−
.
(a1 − c − t) − θ(a1 − c)
p21 − c − t − θ(p11 − c)
¶
.
(7.37)
Writing each equation separately,
q11 =
(1 − θ)(a1 − c) + θt − (p11 − c) + θ(p21 − c − t)
b(1 − θ2 )
q21 =
(1 − θ)(a1 − c) − t − (p21 − c − t) + θ(p11 − c)
b(1 − θ2 )
(7.38)
(7.39)
First examine firm 1’s behavior. Substituting from (7.38) into (7.31), π 11
satisfies
b(1 − θ2 )π 11 =
= (p11 − c)[(1 − θ)(a1 − c) + θt − (p11 − c) + θ(p21 − c − t)]
(7.40)
The first-order condition to maximize π 11 with respect to p11 is
2(p11 − c) − θ(p21 − c − t) = (1 − θ)(a1 − c) + θt.
(7.41)
Solving for p11 − c, this can be written as the equation of firm 1’s price
best response function for country 1,
1
p11 − c = [(1 − θ)(a1 − c) + θ(p21 − c)]
2
(7.42)
56
CHAPTER 7. INTERNATIONAL TRADE I
Note that the term θt drops out: transportation cost, paid by firm 2, does
not directly affect firm 1; it affects firm 1 only insofar as it affects firm 2’s
price.
This is the equation of a straight line with slope θ/2. (Actually, if drawn
on a graph with p21 on the vertical axis and p11 on the horizontal axis, the
slope is 2/θ.)
Proceeding in the same way for firm 2,
b(1 − θ2 )π 21 =
= (p21 − c − t)[(1 − θ)(a1 − c) − t − (p21 − c − t) + θ(p11 − c)]
(7.43)
−θ(p11 − c) + 2(p21 − c − t) = (1 − θ)(a1 − c) − t
(7.44)
1
1
p21 − c = [(1 − θ)(a1 − c) + θ(p11 − c)] + t,
2
2
(7.46)
1
p21 − c − t = [(1 − θ)(a1 − c) − t + θ(p11 − c)]
(7.45)
2
This is the equation of a straight line with positive slope θ/2. Collecting
terms in t on the right-hand side,
the greater is unit transportation cost, the greater the price firm 2 will charge
for any price set by firm 1.
Firms cannot sell negative quantities. This means that the equation of
firm 1’s best response function is valid only for combinations of p11 and p21
that imply q11 ≥ 0 and the equation of firm 2’s best response function is
valid only for combinations of p11 and p21 that imply q21 ≥ 0. This implies
restrictions that are not worked out here.
The price best response functions are graphed in Figure 7.1.
The equations of the best response functions (7.41) and (7.44) can be
written as a system of equations
¶
µ
¶µ
µ ¶
µ ¶
2 −θ
p11 − c
1
θ
= (1 − θ)(a1 − c)
+t
. (7.47)
p21 − c − t
−θ 2
1
1
This can be solved for equilibrium prices,
µ
¶
µ
¶µ ¶ µ
¶µ ¶
p11 − c
2 θ
1
2 θ
θ
2
= (1−θ)(a1 −c)
+t
(4−θ )
p21 − c − t
θ 2
1
θ 2
1
= (1 − θ)(2 + θ)(a1 − c)
µ
2 θ
θ 2
¶µ
1
1
¶
+t
·
θ
−(2 − θ2 )
¸
,
57
p21
Firm 2’s best response
function, country 1 @
(1−θ)(a1 −c)+t
2
¤
¤
»
¤ »»»»
»
r
»
» ¤
@
»»» ¤
R
@
»
»
»»
¤
»»»
»
¤
»
»
»»
¤
»»»
¤
¤
¤
¤
¤
¤
¤
¤
¤
Firm 1’s best response
¤
function, country 1 @
¤
@ ¤
R
@
¤
¤
¤
¤
¤
¤
1−θ
(a1 − c)
2
p11
Figure 7.1: Price best response functions, country 1, Bertrand duopoly trade
model
58
CHAPTER 7. INTERNATIONAL TRADE I
from which
p∗11 − c =
(1 − θ)(2 + θ)(a1 − c) + θt
1−θ
θ
=
t
(a1 − c) +
2
2−θ
4−θ
4 − θ2
(7.48)
(1 − θ)(2 + θ)(a1 − c) + 2
1−θ
2 − θ2
−c−t=
=
(a1 − c) −
t. (7.49)
2−θ
4 − θ2
4 − θ2
Firm 1’s equilibrium price-cost margin rises, and firm 2’s equilibrium price
falls, as transportation cost t rises.
Substituting from the equations of the best response functions into the
expressions for the demand curves (7.38) and (7.39), equilibrium quantities
demanded satisfy
p∗ − c
∗
= 11 2
(7.50)
q11
b(1 − θ )
p∗21
∗
q21
=
p∗21 − c − t
.
b(1 − θ2 )
(7.51)
It follows from (7.51) that the condition for firm 2 to sell in country 1 is
that the price of firm 2’s variety in country 1 exceed firm 2’s cost of selling
in country 1, p∗21 > c − t. From (7.49), this translates into
(2 − θ 2 )(a1 − c − t) − θ(a1 − c) > 0
2 − θ − θ2
(a1 − c) > t
2 − θ2
(1 − θ)(2 + θ)
(a1 − c) > t
(7.52)
2 − θ2
As expected, and as for the quantity-setting model, the condition for
two-way trade is that unit transportation cost be not too great.
The derivative of the fraction on the left in (7.52) with respect to θ
is negative. Hence as θ falls, so that product differentiation increases and
varieties 1 and 2 become poorer substitutes one for the other, firm 2 can bear
higher transportation cost and still profitably sell in country 1.
A representative consumer utility function that produces the demand
curves (7.30) is
1
2
2
U = m + ai (q1i + q2i ) − bi (q1i
+ 2θq1i q2i + q2i
),
2
where m represents consumption on other goods.
Analyze the welfare effects of trade.
(7.53)
59
For simplicity, suppose transportation cost is zero and that the inverse
demand curves in the two markets have the same intercept and slopes, so
that the two markets are identical.
Concentrate on country 1. Without trade, firm 1 is a monopolist in
country 1. Net social welfare generated in the industry for any output level
q1 is the sum of consumers’ surplus and economic profit,
1
NSW1 = aq1 − bq12 − p11 q11 + (p11 − c)q21
2
1
= (a − c)q1 − bq12 ,
2
(7.54)
as shown in Figure 7.2:
Evaluating this at the monopoly output
qm =
1a−c
,
2 b
1 (a − c)2
.
8
b
With trade, consumers’ surplus in country 1 is
N SW1m =
1 2
2
CS1 = a(q1i + q2i ) − b(q1i
+ 2θq1i q2i + q2i
) − (p11 q11 + p21 q21 ).
2
(7.55)
(7.56)
(7.57)
Firm 1’s profit is
π 1 = (p11 − c)q11 + (p12 − c)q12 ,
(7.58)
where the first term is firm 1’s profit in country 1 and the second term is firm
1’s profit in country 2. Given the symmetry assumptions we have made, in
equilibrium firm 1’s sales and profit in country 2 are equal to firm 2’s sales
and profit in country 1. Equilibrium net social welfare in country 1 is then
1 2
2
NSW1 = (a − c)(q1i + q2i ) − b(q1i
+ 2θq1i q2i + q2i
).
2
(7.59)
With product differentiation, net social welfare is not susceptible to straightforward graphical illustration: additional output of one variety to some extent
reduces welfare generated by other varieties.
Once again in equilibrium, each firm sells the same output in country 1,
say q1 . (7.59) becomes
·
¸
a−c
2
N SW1 = 2(a − c)q1 − b(1 + θ)q1 = b 2
− (1 + θ)q1 q1 .
(7.60)
b
60
CHAPTER 7. INTERNATIONAL TRADE I
p1
a1 @
q
@
@
@
@
@
@
@
CS
@
@
@
π
c
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
Figure 7.2: Net social welfare, monopoly, country 1
q1
61
Evaluating (7.47) for the case of zero transportation cost, equilibrium
price is
1−θ
(a − c).
(7.61)
p1 − c =
2−θ
Using (7.50) to find equilibrium output per firm gives
q1 =
p1 − c
1
a−c
.
2 =
(1 − θ)(2 − θ) b
b(1 − θ )
(7.62)
Then substituting (7.62) into (7.60), in equilibrium country 1 net social
welfare with trade is
·
¸
a−c
1+θ
a−c
1
a−c
−
NSW1 = b 2
b
(1 − θ)(2 − θ) b
(1 − θ)(2 − θ) b
·
=b 2−
¸
µ
¶2
1+θ
1
a−c
(1 − θ)(2 − θ) (1 − θ)(2 − θ)
b
·
¸
µ
¶2
2−θ
1
a−c
=b
(1 − θ)(2 − θ) (1 − θ)(2 − θ)
b
=
1
(a − c)2
.
(1 − θ)2 (2 − θ)
b
(7.63)
1
1
− .
2
(1 − θ) (2 − θ) 8
(7.64)
Comparing (7.56) and (7.63), the change in net social welfare with the
opening up of trade is proportional to
This is positive for θ = 0. The first term on the right rises as θ approaches
1. Trade is beneficial when products are differentiated and firms set price,
and the benefit rises as varieties are closer substitutes.
7—4 What are the direct and indirect labor input requirements to produce
one unit of food and one unit of machinery implied by the input-output table
IO-1 (page 9)? If there are 100 units of labor in the economy, what is the
equation of its consumption possibility frontier?
Use the notation given in Table 7.1.
The balance inequalities for food, machinery, and labor are
FF F + FF M + CF ≤ F
(7.65)
62
CHAPTER 7. INTERNATIONAL TRADE I
total food output
total machinery output
food used as input in the food industry
food used as input in the machinery industry
machinery used as input in the food industry
machinery used as input in the machinery industry
food available for consumption
machinery available for consumption
labor
labor used as input in the food industry
labor used as input in the machinery industry
F
M
FF F
FF M
MMF
MMM
CF
CM
L
LF
LM
Table 7.1: Notation
MMF + MMM + CM ≤ M
(7.66)
LF + LM ≤ L
(7.67)
and
respectively.
Using the input-output coefficients in Table IO-2, (7.65) and (7.66) become
3
1
F + M + CF ≤ F
(7.68)
8
8
3
1
F + M + CM ≤ M
(7.69)
8
4
µ
Collecting terms, (7.68) and (7.69) hold with equality for
µ
¶µ
¶ µ
¶
5/8 −1/8
F
CF
=
CM
−3/8 3/4
M
F
M
¶
64
=
27
µ
3/4 1/8
3/8 5/8
¶µ
CF
CM
¶
=
µ
16/9 8/27
8/9 40/27
¶µ
(7.70)
CF
CM
¶
(7.71)
This gives the total outputs F and M of food and machinery needed for
any menu CF and CM of food and machinery available for final consumption.
Then using the labor-output coefficients of Table IO-2, the balance inequality of labor (7.67) becomes
µ
¶µ
¶
3
F
,3
≤ L = 100.
(7.72)
M
4
63
CF
25 e
e
4CF + 14
C ≤ 100
e
3 M
e
¡
e
¡
e
ª
e¡
e
e
e
e
e
e
e
e
CM
21.43
Figure 7.3: Consumption possiblity frontier, Problem 10-8
If this holds with equality, substituting (7.71) gives the equation of the
consumption possibility frontier,
¶
µ
¶µ
¶µ
3
16/9 8/27
CF
,3
= 100.
(7.73)
CM
8/9 40/27
4
4CF +
14
CM = 100,
3
(7.74)
shown in Figure 7.3.
Producing a unit of food for final consumption demand requires, directly
and indirectly, 4 units of labor. Producing a unit of machinery for final
consumption demand requires, directly and indirectly, 4 2/3 units of labor.
64
CHAPTER 7. INTERNATIONAL TRADE I
Chapter 8
Imperfect Competition and
International Trade II
8—1 Firm 1, based in country 1, and firm 2, based in country 2, sell quantities
q1 and q2 , respectively, in country 3. The demand curve in country 3 is
p = a − (q1 + q2 ),
(8.1)
and sales in country 3 have no impact on the country 1 and country 2 markets. The marginal and average cost of production and transportation, c, is
constant and the same for both firms.
(a) Find equilibrium outputs and profits in country 3 if there are no export
subsidies.
This is a standard Cournot duopoly model. Firm 1’s profit is
π 1 = [a − c − (q1 + q2 )]q1
(8.2)
Taking the derivative of (8.2) with respect to q1 , the equation of firm 1’s
best response function is
2q1 + q2 = a − c
(8.3)
Substituting (8.3) into the term in brackets on the right in (8.2), it follows
that firm 1’s payoff anywhere along its best response function, and particular
in equilibrium, is
π 1 = q12
(8.4)
The equations of the best response functions of the two firms form the
system of equations
µ
¶µ
¶
µ ¶
2 1
q1
1
= (a − c)
,
(8.5)
1 2
q2
1
65
66
CHAPTER 8. INTERNATIONAL TRADE II
with solution
1
q1 = q2 = (a − c).
3
From (8.4), equilibrium firm profits are
1
π 1 = π 2 = (a − c)2 .
9
(8.6)
(8.7)
(b) Find equilibrium outputs and profits in country 3 if country 1 grants its
firm a subsidy s1 per unit sold in country 3. What subsidy is best for country
1?
With an export subsidy, firm 1’s unit cost of supply country 3 is c − s1 .
The system of equations of best response functions becomes
µ
¶µ
¶
µ ¶
µ ¶
2 1
q1
1
1
= (a − c)
+ s1
,
(8.8)
1 2
q2
1
0
with solution
1
q1 = (a − c + 2s1 )
3
1
q2 = (a − c − s1 )
3
With an export subsidy, firm 1’s profit is
1
π 1 = (a − c + 2s1 )2 .
9
(8.9)
(8.10)
(8.11)
The change in country 1’s net social welfare is
1
1
1
∆NSW1 = (a − c + 2s1 )2 − (a − c + 2s1 )s1 − (a − c)2
9
3
9
(8.12)
Taking the derivative of (8.12) with respect to s1 and rearranging terms
gives the subsidy level that maximizes1 ∆NSW1 :
1
s1 = (a − c).
4
Substituting (8.13) into (8.12), country 1’s optimal ∆N SW1 is
µ
¶
1
1 a−c 2
2
∆N SW1 = (a − c) =
.
72
8
3
1
is.
(8.13)
(8.14)
One must verify that the second-order condition for a maximum is satisfied, which it
67
With a country 1 subsidy given by (8.13), the change in country 2’s net
social welfare is
µ
¶2
1
1
7 a−c
2
2
.
(8.15)
∆N SW2 = (a − c − s1 ) − (a − c) = −
9
9
16
3
(c) Find equilibrium outputs and profits in country 3 if country 1 grants its
firm a subsidy s1 per unit sold and country 2 grants its firm a subsidy s2
per unit sold in country 3. What are the equilibrium subsidies if the two
countries set subsidy levels noncooperatively (that is, if each country sets
the best possible subsidy level for itself, taking the subsidy level of the other
country as given)?
If each country grants its own firm an export subsidy, the system of
equations of the best response functions is
¶
µ ¶ µ
¶
µ
¶µ
1
s1
2 1
q1
= (a − c)
+
,
(8.16)
q2
1
s2
1 2
with solution
1
q1 = (a − c + 2s1 − s2 )
3
1
q2 = (a − c + 2s2 − s1 ).
3
The change in country 1’s net social welfare is
(8.17)
(8.18)
1
∆NSW1 = π 1 − s1 q1 − (a − c)2
9
1
= q12 − s1 q1 − (a − c)2
9
1
1
= (a − c + 2s2 − s1 )(a − c − s2 − s1 ) − (a − c)2 .
(8.19)
9
9
Taking the derivative of (8.20) with respect to s1 , the equation of country
1’s subsidy best response function is
4s1 + s2 = a − c.
(8.20)
This is the equation of a downward-sloping curve in (s1 , s2 )—space (Figure 8.1). There is a similar equation for country 2’s subsidy best response
function.
68
CHAPTER 8. INTERNATIONAL TRADE II
s2
Country 1’s subsidy best response function
a−c
a−c
4
s∗2
A
A
A
A
A
¢
¢
¢
¢
¢
¢
¢
¢
A ¢
A ¢®
A
A
A
A
HH
A
HH
A
Country 2’s subsidy best response curve
HH A
E
0
HH
As
¢
AHH
¢
A HH
¢
HH
A
¢
HH ¢
A
HH
¢®
A
HH
A
HH
A
s
A
H
s∗1
a−c
4
a−c
1
Figure 8.1: Subsidy best response functions
Either by symmetry or solving the equations of the subsidy best response
functions, equilibrium subsidies are
1
s1 = s2 = (a − c).
5
Evaluating ∆N SW1 for the equilibrium subsidies,
µ
¶2
7 a−c
∆NSW1 = −
.
16
3
(8.21)
(8.22)
7—2 Answer question 7—1 if products are differentiated, with inverse demand
curves
p1 = a − (q1 + θq2 )
(8.23)
p2 = a − (θq1 + q2 ),
(8.24)
69
with 0 ≤ θ < 1, and firms set prices rather than quantities.
(a) Writing the system of equations of the inverse demand curves in matrix
form as
¶
µ ¶ µ
¶µ
¶
µ
1
1 θ
q1
p1 − c
= (a − c)
−
,
(8.25)
p2 − c
1
θ 1
q2
they can be inverted to obtain expressions for the demand curves,
µ
¶
µ ¶ ·
¸
q1
1
p1 − c − θ(p2 − c)
2
(1 − θ )
= (1 − θ)(a − c)
−
,
q2
1
p2 − c − θ(p1 − c)
(8.26)
expressions which are valid provided both quantities are nonnegative.
Firm 1’s profit as a function of the prices of both firms is
(1 − θ2 )π 1 = (p1 − c)[(1 − θ)(a − c) − (p1 − c) + θ(p2 − c)].
(8.27)
The first-order condition to maximize π 1 with respect to p1 is
2(p1 − c) − θ(p2 − c) = (1 − θ)(a − c);
(8.28)
this is also the equation of firm 1’s price best response function.
Substituting (8.27) into (8.28), firm 1’s payoff anywhere along its best
response function, and in particular in equilibrium, is
π1 =
(p1 − c)2
.
1 − θ2
(8.29)
Either by symmetry or by solving the system of equations of the price
best response functions, equilibrium prices without subsidies are
p1 − c = p2 − c =
1−θ
(a − c).
2−θ
Using (8.30), equilibrium payoffs without subsidies are
·
¸2
1
1−θ
1−θ
π1 = π2 =
(a − c) =
(a − c)2 .
2
(1 + θ)(2 − θ)
1−θ 2−θ
(8.30)
(8.31)
(b) If country 1 grants its firm a subsidy, firm 1’s profit becomes
(1 − θ2 )π 1 = (p1 − c + s1 )[(1 − θ)(a − c) − (p1 − c) + θ(p2 − c)].
The system of equations of the best response functions is
µ
¶µ
¶
µ ¶
µ ¶
2 −θ
p11 − c
1
1
= (1 − θ)(a1 − c)
− s1
.
−θ 2
p21 − c
1
0
(8.32)
(8.33)
70
CHAPTER 8. INTERNATIONAL TRADE II
Solving for equilibrium prices gives
p1 − c =
2s1
1−θ
(a − c) −
2−θ
4 − θ2
(8.34)
θs1
1−θ
(a − c) −
.
(8.35)
2−θ
4 − θ2
A positive subsidy lowers the equilibrium prices of both varieties. However
1−θ
2 − θ2
s1 ,
(8.36)
(a − c) +
p1 − c + s1 =
2−θ
4 − θ2
so that a positive subsidy is privately beneficial for firm 1.
With a subsidy, firm 1’s equilibrium profit is
p2 − c =
π1 =
1
(p1 − c + s1 )2 .
1 − θ2
(8.37)
The change in firm 1’s net social welfare is
·
¸2
1
1−θ
2 − θ2
(a − c) +
∆NSW1 =
s1
1 − θ2 2 − θ
4 − θ2
·
¸
·
¸2
1
1−θ
2 − θ2
1
1−θ
(a − c) +
(a − c)
−
s1 s1 −
1 − θ2 2 − θ
4 − θ2
1 − θ2 2 − θ
(8.38)
Taking the derivative of this with respect to s1 and solving the resulting
first-order condition gives an expression for the “subsidy” that maximizes
the change in firm 1’s net social welfare,
(1 − θ)(2 + θ)θ2
s1 = −
(a − c) ≤ 0.
4(2 − θ2 )
(8.39)
The optimal subsidy, being negative, is in fact an export tax. An export
tax increases welfare if it induces both firms to raise price. If θ = 0, an
export tax by country 1 induces firm 1 to raise its price, but firm 2 does not
alter its price; hence the optimal tax is zero.
(c) If country 1 imposes an export tax t1 on firm 1 and country 2 imposes an
export tax t2 on firm 2, the system of equations of best response functions is
µ
¶µ
¶
µ ¶ µ
¶
2 −θ
p11 − c
1
t1
= (1 − θ)(a1 − c)
+
.
(8.40)
−θ 2
p21 − c
1
t2
71
Firm 1’s equilibrium price is
p1 − c =
1−θ
2t1 + θt2
.
(a − c) +
2−θ
4 − θ2
(8.41)
There is a similar expression for firm 2. The effect of the export taxes is to
increase equilibrium prices.
Firm 1’s margin after the export tax is
p1 − c − t1 =
(2 − θ2 )t1 − θt2
1−θ
(a − c) −
.
2−θ
4 − θ2
(8.42)
Firm 1’s equilibrium payoff with the export taxes is
π1 =
1
(p1 − c − t1 )2
1 − θ2
The change in country 1’s net social welfare is
·
¸2
1
1−θ
∆N SW1 = π 1 + t1 q1 −
(a − c) .
1 − θ2 2 − θ
(8.43)
(8.44)
Evaluating this at the equilibrium values and taking the derivative of
the resulting expression with respect to t1 yields the equation of country 1’s
export tax best response function,
t1 = θ2
(1 − θ)(2 + θ)(a − c) + θt2
.
4(2 − θ2 )
(8.45)
This is the equation of an upward-sloping line in (t1 , t2 )-space.
By symmetry, the equilibrium export tax is
t1 = t2 =
θ2 (1 − θ)(2 + θ)
(a − c).
8 − 4θ2 − θ 3
If (8.46) is used to evaluate (8.44), the equilibrium ∆N SW1 is
·
¸2
1 6 − 3θ2 − θ3 1 − θ
∆N SW1 = −
(a − c) .
1 − θ2 8 − 4θ2 − θ3 2 − θ
(8.46)
(8.47)
7—3 Return to Problem 8-1. Initially, let transportation cost t equal 0.
(a) Analyze the impact of a quota q that restricts firm 2’s sales in country
1 below the Cournot equilibrium level on outputs, prices, profits, and net
social welfare in country 1.
72
CHAPTER 8. INTERNATIONAL TRADE II
Without transportation cost, this is a Cournot duopoly model in which
the two firms have identical constant unit costs. Equilibrium outputs and
profits in country 1 are
a1 − c
0
0
= q12
=
(8.48)
q11
3b1
and
µ
¶
a1 − c 2
0
0
π 11 = π 12 = b1
(8.49)
3b1
respectively.
A quota is binding if it holds firm 2’s sales below the Cournot equilibrium
level, i.e., if
a1 − c
q<
.
(8.50)
3b1
Firm 1’s profit-maximizing output choice is described by its best response
function,
¶
µ
1 a1 − c
q11 =
− q21 .
(8.51)
2
b1
An aside: suppose the demand structure is altered to introduce some
product differentiation. As long as there is only one domestic firm, it does
not matter whether the firm is thought of as a price-setter or a quantity
setter. The quota fixes the domestic firm’s residual demand curve, and there
is one profit-maximizing (output, price) combination on that residual demand
curve. If there is more than one domestic firm, then the price/quantitysetting distinction becomes important, since the nature of interaction among
domestic firms is different for the two cases.
With a quota q, firm 1’s sales in country 1 are
µ
¶
µ
¶
1 a1 − c
a1 − c 1 a1 − c
1
q11 =
−q =
+
−q .
(8.52)
2
b1
3b1
2
3b1
Firm 1 increases its output by half the amount that firm 2 is prevented from
selling by the quota.
Total sales in country 1 with the quota are
µ
¶
2 a1 − c 1 a1 − c
1
q11 + q =
−
−q ,
(8.53)
3 b1
2
3b1
which is less then total sales without the quota. This output reduction implies
that price rises because of the quota,
µ
¶
1
b1 a1 − c
1
1
p11 = c + (a1 − c) − b1 (q11 + q) = c + (a1 − c) +
− q . (8.54)
3
2
3b1
73
Firm 1’s profit with the quota is
µ
·
¶¸2
¡ 1 ¢2
a1 − c 1 a1 − c
1
+
−q
,
π 11 = b1 q11 = b1
3b1
2
3b1
(8.55)
which is greater than its profit without the quota.
Firm 2’s profit under the quota is
µ
·
¶¸
1
b1 a1 − c
1
π 12 = (a1 − c) +
−q q
3
2
3b1
µ
µ
¶2
¶µ
¶
b1 a1 − c
a1 − c
a1 − c
= b1
−
−q
2
−q ,
3b1
2
3b1
3b1
(8.56)
which is less than profit without the quota (given that both terms in parentheses on the right are positive).
Consumer surplus is proportional to one-half the square of output. Since
the quota causes price to rise, it causes consumers’ surplus to fall.
The quota therefore makes the domestic firm better off and consumers
worse off. The change in net social welfare is
(·
¶¸2 µ
¶2 )
µ
a1 − c
b1
a1 − c 1 a1 − c
∆N SW =
2
−
−q
− 2
2
3b1
2
3b1
3b1
+b1
(·
a1 − c 1
+
3b1
2
µ
a1 − c
−q
3b1
¶¸2
−
µ
a1 − c
3b1
¶2 )
,
(8.57)
where the first term in braces is the reduction in consumers’ surplus and the
second is the increase in profit. (8.57) simplifies to
µ
¶2
3
a1 − c
∆NSW = b1
− q > 0.
(8.58)
8
3b1
Hence the quota increases domestic welfare.
In a more general model, if the number of domestic firms is large relative
to the number of domestic firms, and the quota is sufficiently small, this
welfare impact of a quota may be negative. The welfare loss of consumers,
as the competition of a large number of foreign producers is constrained, can
exceed the profit increase of domestic producers.
(b) Return to the model without a quota and with t > 0, but now interpret
t as a tariff collected by country 1 on each unit of output sold by firm 2 in
country 1. What is the impact of the tariff on country 1’s net social welfare?
74
CHAPTER 8. INTERNATIONAL TRADE II
Equilibrium outputs and price are given by equations (7.21), (7.22), and
(7.26), reproduced here for convenience:
∗
=
q11
a1 − c
t
+
3b1
3b1
a1 − c
2t
−
3b1
3b1
µ
¶
2 a1 − c
p1 = (c + t) +
−t .
3
2
The change in net social welfare with a tariff is
"µ
¶2 µ
¶2 #
b1
t
a1 − c
a1 − c
∆N SW =
−
− 2
2
2
3b1
3b1
3b1
∗
q21
=
"µ
¶2 µ
¶2 #
a1 − c + t
a1 − c
(8.59)
+b1
−
3b1
3b1
µ
¶
a1 − c
2t
+t
−
.
3b1
3b1
The first term in brackets on the right is the loss in consumers’ surplus,
the second the increase in profit, and the third tariff revenue collected by the
government. (8.60) simplifies to
µ
¶
a1 − c
t
∆N SW = t
−
> 0.
(8.60)
3b1
2b1
7—4 Let there be two countries with identical demand curves, each home to
one widget producer. The subscript 1 denotes both country 1 and its widget
company; similarly for the subscript 2. Let the inverse demand curves in the
two countries be
p1 = a − (q11 + q21 )
,
(8.61)
p2 = a − (q12 + q22 )
where p1 is the price in country 1, p2 is the price in country 2, and qij is the
quantity of widgets sold by firm i in country j, for i, j = 1, 2. The parameter
a is the price-axis intercept of the inverse demand curves, which are the same
in both countries. The slope of the inverse demand curves is −1.
Let the cost function be
1
c(qi1 + qi2 ) = α(qi1 + qi2 ) − β(qi1 + qi2 )2 ,
2
(8.62)
75
where α and β are both positive and β is sufficiently small that marginal
cost remains positive over the relevant output range. Assume there are no
transportation costs or tariffs.
(a) write out the payoff functions of the two firms.
The payoff functions are
·
¸
1
2
π 1 = [a − (q11 + q21 )]q11 + [a − (q12 + q22 )]q12 − α(q11 + q12 ) − β(q11 + q12 )
2
(8.63)
·
¸
1
2
π 2 = [a − (q11 + q21 )]q21 + [a − (q12 + q22 )]q22 − α(q21 + q22 ) − β(q21 + q22 )
2
(8.64)
(b) find the first-order conditions to maximize the payoffs and solve them for
equilibrium outputs.
The first-order conditions for firm 1 are
∂π 1
= a − 2q11 − q21 − α + β(q11 + q12 ) = 0
∂q11
(8.65)
∂π 1
= a − 2q12 − q22 − α + β(q11 + q12 ) = 0
∂q12
(8.66)
Note that equation (8.65), the first-order condition for firm 1’s sales in
country 1, includes not only the two sales levels for country 1, q11 and q21 ,
but also firm 1’s sales in country 2, q12 . This is a consequence of the fact
that firm 1’s marginal cost depends on its total output, the sum of its sales
in both markets. In contrast to the model of Problem 7-1, it is not possible
to analyze the two markets separately.
The equations (8.65) and (8.66) of the firm 1’s first-order conditions can
be rewritten
(2 − β)q11 + q21 − βq12 = a − α
(8.67)
−βq11 + (2 − β)q12 + q12 = a − α.
(8.68)
Going through the same steps for firm 2, the equations of the first-order
conditions to maximize (8.64) can be written
q11 + (2 − β)q21 − βq22 = a − α
(8.69)
q12 − βq21 + (2 − β)q22 = a − α.
(8.70)
76
CHAPTER 8. INTERNATIONAL TRADE II
The system of equations of the
matrix form as

2 − β −β
1
0
 −β 2 − β
0
1

 1
0
2 − β −β
0
1
−β 2 − β
first-order conditions can be written in


q11

  q12 
 = (a − α) 


  q21 
q22


1
1 

1 
1
(8.71)
One way to solve this is to use the inverse of the coefficient matrix on the
left, which satisfies

−1
2 − β −β
1
0
 −β 2 − β
0
1 
 =
3(1 − 2β)(3 − 2β) 
(8.72)
 1
0
2 − β −β 
0
1
−β 2 − β


−2β(2 − β)
(2 − β)(3 − 4β)
β(5 − 4β)
−(3 − 4β + 2β 2 )

β(5 − 4β)
(2 − β)(3 − 4β)
−2β(2 − β)
−(3 − 4β + 2β 2 ) 


 −(2β 2 + 3 − 4β)

−2β(2 − β)
(2 − β)(3 − 4β)
β(5 − 4β)
2
−2β(2 − β)
−(3 − 4β + 2β )
β(5 − 4β)
(2 − β)(3 − 4β)
This would be necessary if there were transportation cost or tariffs in
the model. In the present case, we can observe that firms and markets are
identical, implying that equilibrium is symmetric; substituting q11 = q12 =
q21 = q22 in any one of the equations of the first-order conditions gives
0
0
0
0
q11
= q12
= q21
= q22
=
a−α
,
3 − 2β
(8.73)
where the superscript 0 denotes initial equilibrium values.
If marginal cost were constant and equal to α, equilibrium sales per firm
in each market would be (a − α)/3. Since
a−α
a−α
2 β
−
=
>0
3 − 2β
3
3 3 − 2β
(8.74)
(for β > 0), economies of scale result imply an increase in equilibrium output
compared with the constant returns to scale case.
For the cost function 8.62 to lead to sensible results, it must be that
3
β< ,
2
(8.75)
otherwise the equilibrium sales given in (8.73) are negative. As indicted in
the statement of the problem, economies of scale cannot be too great if the
quadratic cost function is to be a suitable approximation.
77
There is actually a stricter limit on the range of β. The second-order
condition for firm 1’s profit-maximization requires that the Hessian matrix
! ·
à ∂2π
¸
∂ 2 π1
1
2
−(2 − β)
β
∂q
q
∂q11
12 11
=
(8.76)
∂ 2 π1
∂ 2 π2
β
−(2 − β)
∂q q
∂q 2
12 11
22
have a positive determinant, i.e., that
[(2 − β)2 − β 2 ] = 4(1 − β) > 0,
(8.77)
β < 1.
(8.78)
or
A further restriction will be derived below.
(c) substitute equilibrium outputs for country 2 in the equations of the firstorder conditions for country 1 outputs, and interpret the resulting expressions
as equilibrium best response functions for country 1.
0
Substitute q12
= (a − α)/(3 − 2β) into (8.67) to obtain
(2 − β)q11 + q21 =
3−β
(a − α)
3 − 2β
(8.79)
3−β
(a − α)
3 − 2β
(8.80)
0
and q22
= (a − α)/(3 − 2β) into (8.69) to obtain
q11 + (2 − β)q21 =
For expositional purposes, (8.79) and (8.81) can be interpreted as the
equations of equilibrium best response functions for country 1. There is a
corresponding set of equations for the equilibrium best response functions for
country 1. It should be kept in mind, however, that equilibrium sales in the
two markets are simultaneously determined.
(d) Suppose now that country 1 imposes a quota that restricts the country
2 firm’s sales in country 1 to a level q that is below the equilibrium level
from (b). Find the new equilibrium outputs; describe the new equilibrium in
terms of movements in the equilibrium best response functions.
The firms’ payoff functions become
·
¸
1
2
π 1 = [a − (q11 + q)]q11 + [a − (q12 + q22 )]q12 − α(q11 + q12 ) − β(q11 + q12 )
2
(8.81)
78
CHAPTER 8. INTERNATIONAL TRADE II
·
¸
1
2
π2 = [a−(q11 +q)]q +[a−(q12 +q22 )]q22 − α(q + q22 ) − β(q + q22 ) (8.82)
2
The first-order conditions for firm 1 are
∂π1
= a − 2q11 − q − α + β(q11 + q12 ) = 0
∂q11
(8.83)
∂π 1
= a − 2q12 − q22 − α + β(q11 + q12 ) = 0
∂q12
(8.84)
and that for firm 2 is
∂π2
= a − q12 − 2q22 − α + β(q + q22 ) = 0
∂q22
(8.85)
The system of equations of the first-order conditions can be written in
matrix form as

 




2 − β −β
0
1
−1
q11
 −β 2 − β
1   q12  = (a − α)  1  + q  0  . (8.86)
1
β
q22
0
1
2−β
This can be solved directly using the expression for the inverse of the
coefficient matrix on the left,

−1
2 − β −β
0
 −β 2 − β
1  =
0
1
2−β


(1 − β)(3 − β) β(2 − β)
−β
1

β(2 − β)
(2 − β)2 −(2 − β)  .
=
(3 − 4β)(2 − β)
−β
−(2 − β) 4(1 − β)
(8.87)
Before proceeding, however, it is useful to express all the variables in
terms of deviations from the no-quota equilibrium values. Hence let
∗
q11
= q11 −
a−α
3 − 2β
(8.88)
∗
q12
= q12 −
a−α
3 − 2β
(8.89)
∗
= q22 −
q22
a−α
3 − 2β
(8.90)
79
and substitute in (8.86) to obtain a revised version of the system of equations
of the first-order conditions,



 ∗ 
µ
¶
1
2−β
β
0
q11
a−α
∗ 
 β
2−β
1   q12
=
(8.91)
−q  0 
3 − 2β
∗
q22
−β
0
1
2−β
Then

∗
q11
∗ 
(3 − 4β)(2 − β)  q12
=
(8.92)
∗
q22



µ
¶
(1 − β)(3 − β) β(2 − β)
−β
1
a
−
α
2

β(2 − β)
(2 − β) −(2 − β)   0 
−q
3 − 2β
−β
−(2 − β) 4(1 − β)
−β

 ∗ 

¶
3 − 4β + 2β 2 µ
q11
1
a
−
α
∗
 2β(2 − β) 
 q12  =
−q
(8.93)
(3 − 4β)(2 − β)
3 − 2β
∗
−β(5 − 4β)
q22
Hence

¶
µ
3 − 4β + 2β 2
a−α
=
−q
(3 − 4β)(2 − β) 3 − 2β
µ
¶
(1 − β)(3 − β) + β 2 a − α
=
−q > 0
(3 − 4β)(2 − β)
3 − 2β
¶
µ
2β
a−α
∗
−q >0
q12 =
(3 − 4β) 3 − 2β
¶
µ
β(5 − 4β)
a−α
∗
q22 = −
− q < 0,
(3 − 4β)(2 − β) 3 − 2β
∗
q11
where
(8.94)
(8.95)
(8.96)
3
4
is a necessary condition for the indicated signs to be valid.
8—5 (Concentration effect of trade) Suppose there are two identical countries,
each with demand curve
p = a − bQ,
(8.97)
β<
(where Q is total sales in the country) and that firms in each country operate
with the cost function
c(q) = cq + F,
(8.98)
where c is constant marginal cost, q is firm output, and F is fixed and sunk
cost. Assume that firms behave as Cournot oligopolists.
80
CHAPTER 8. INTERNATIONAL TRADE II
(a) What is the long-run equilibrium number of firms in each country if
trade between the two countries is not possible?
Let n0 be the long-run number of firms in each country without trade.
The profit of a single firm in (say) country 1 is
π 1 = (a − c − bQ1 )q1 − F.
(8.99)
The equation of the firm’s quantity best response function is
2q1 + Q−1 =
a−c
= S,
b
(8.100)
where Q−1 is the combined output of all other firms, with corresponding
profit
π 1 = bq12 − F.
(8.101)
In equilibrium, all firms will produce the same output level, say q0 . Substituting q1 = q0 , Q−1 = (n − 1)q0 in (8.100) and rearranging terms gives
q0 =
1
S
p
,
n + 1 F/b
(8.102)
which gives short-run equilibrium output as a function of the market size S,
fixed cost F , and the number of firms n.
Substituting (8.102) in (8.101), setting the resulting expression for firm
profit equal to zero, and solving for the number of firms n gives
S
n0 = p
− 1.
F/b
(8.103)
(b) Suppose trade opens up between the two countries (and for simplicity,
assume there are no transportation costs or tariffs). What is the profit of
each firm, after trade, if the number of firms in each country is the long-run
equilibrium number of firms from (a)?
With trade, there are 2n0 firms selling in each country. Evaluating (8.102)
with n = 2n0 gives equilibrium firm output immediately after the opening
up of trade,
1
S
p
q=
,
(8.104)
2n0 + 1 F/b
Using the equations of the best response functions, and taking into account the fact that each firm sells in both countries, profit per firm is
Ã
!2
1
S
p
π = 2b
− F.
(8.105)
2n0 + 1 F/b
81
But n0 is determined so that
F =
b
S 2.
n0 + 1
(8.106)
Substituting (8.106) in (8.105) to F eliminate and rearranging terms gives
π=−
n0 − 1 2
bS ,
2n0 + 1
(8.107)
which is negative for n0 > 1. The opening up of trade causes firms to lose
money in the short run.
(c) What is the long-run equilibrium number of firms (in both countries)
after trade opens up?
Write m for the long-run number of firms after trade opens up. Taking
into account the fact that each firm sells in both countries, m satisfies
π = 2b
µ
1
S
2m + 1
yielding
m+1 =
m=
¶2
− F ≡ 0,
√
S
2p
F/b
√
2(n0 − 1) − 1.
(8.108)
(8.109)
(8.110)
The equilibrium number of firms rises approximately in proportion to the
square root to the number of equally-sized trading countries.
(b) What is the long-run number of firms if the two countries form a
single market?
The results are the same as above. If the two countries form a single
market, the equation of the implied demand curve (obtained by horizontally
summing demand in the two counties is)
1
p = a − bQ.
2
(8.111)
The long-run Cournot equilibrium number of firms for a market with this
demand curve is given by (8.110).
8—6 (Exchange rate passthrough, quantity-setting firms) Let markets for the
same product in two different countries have the inverse demand curves
p1 = a1 − b1 (q11 + q21 )
.
p2 = a2 − b2 (q12 + q22 )
(8.112)
82
CHAPTER 8. INTERNATIONAL TRADE II
Let firm 1 be based in country 1 and firm 2 in country 2. Call the constant
unit cost of firm 1 c1 dollars and the constant unit cost of firm 2 c2 euros.
Let the exchange rate e be the number of euros required to buy a dollar on
world currency markets. Assume firms compete by selecting outputs, and
each firm exports to the other market if it is profitable to do so.
(a) Find the equations of the quantity best response functions of each
firm for each country.
Country 1: payoffs are
π 11 = [a1 − c1 − b1 (q11 + q21 )]q11
(8.113)
h³
i
c2 ´
(8.114)
π 21 = e a1 −
− b1 (q11 + q21 ) q21
e
Maximizing payoffs with respect to own price gives the first-order conditions, which are the equations of the quantity best response functions,
a1 − c1
b1
(8.115)
a1 − ce2
=
b1
(8.116)
2q11 + q21 =
q11 + 2q21
Country 2: payoffs are
1
π 11 = [a2 − ec1 − b2 (q12 + q22 )]q12
e
(8.117)
π 22 = [a1 − c2 − b2 (q12 + q22 )]q22
(8.118)
Maximizing payoffs with respect to own price gives the first-order conditions, which are the equations of the quantity best response functions,
2q12 + q22 =
a2 − ec1
b2
(8.119)
q12 + 2q22 =
a2 − c2
.
b2
(8.120)
(b) Find equilibrium outputs in each country and discuss the way they
are affected by changes in e.
Country 1: solving (8.115) and (8.116), equilibrium country 1 outputs
are
³c
´i
1 h
2
q11 =
(a1 − c1 ) +
− c1
(8.121)
3b1
e
³c
´i
1 h
2
q21 =
(a1 − c1 ) − 2
− c1 .
(8.122)
3b1
e
83
These expressions are valid only if both are nonnegative.
From (8.121) and (8.122),
1 c2
∂q11
=−
<0
∂e
3b1 e2
(8.123)
∂q21
2 c2
=
> 0.
(8.124)
∂e
3b1 e2
A euro depreciation decreases firm 1’s sales in country 1 and increases
firm 2’s sales in country 1.
Country 2: solving (8.119) and (8.120), equilibrium country 2 outputs
are
µ
¶
1 a2 − c2
ec1 − c2
q12 =
−2
(8.125)
3
b2
b2
¶
µ
1 a2 − c2 ec1 − c2
q22 =
+
.
(8.126)
3
b2
b2
From (8.125) and (8.126),
∂q12
2c1
=−
<0
∂e
3b2
(8.127)
∂q22
c1
=
> 0.
(8.128)
∂e
3b1
A euro depreciation decreases firm 1’s sales in country 2 and increases
firm 2’s sales in country 2.
(c) Find equilibrium price in each country and discuss how they are affected by changes in e.
Country 1: adding (8.121) and (8.122), total output in country 1 is2
³c
´i
1 h
2
q11 + q21 =
− c1 .
2(a1 − c1 ) −
(8.129)
3b1
e
Substituting this expression for total output in the equation of the inverse
demand curve, equilibrium Country 1 price is
´
1
1 ³ c2
− c1
(8.130)
p1 = c1 + (a1 − c1 ) +
3
3 e
Taking the derivative of (8.130) with respect to e,
1 c2
dp1
= − 2.
de
3e
2
(8.131)
Total output can also be obtained by adding (8.115) and (8.116) and dividing both
sides by 3.
84
CHAPTER 8. INTERNATIONAL TRADE II
An increase in e – a euro depreciation – increases firm 2’s sales and
total sales in country 1, leading to a reduction in the country 1 price. The
reduction in price is proportional to firm 2’s marginal cost.
But from (8.130) we also obtain
c2 ´ 2 c2
d ³
p1 −
=
> 0.
de
e
3 e2
(8.132)
Although a euro depreciation leads to a lower price level in country 1, it also
leads to a high price-cost margin for the foreign firm.
The equation of the inverse demand curve and the equations of the best
response functions imply
p1 = c1 + b1 q11 =
c2
+ b1 q21 ,
e
(8.133)
so price is above marginal cost for both firms provided equilibrium outputs
are positive.
Country 2: adding (8.125) and (8.126), total output in country 2 is
q12 + q22 =
2 a2 − c2 1 ec1 − c2
−
.
3 b2
3 b2
(8.134)
Substituting this expression for total output in the equation of the inverse
demand curve, equilibrium country 2 price is
1
1
p2 = c2 + (a2 − c2 ) + (ec1 − c2 ),
3
3
(8.135)
from which
dp2
1
= c1 .
(8.136)
de
3
A euro depreciation decreases firm 1’s sales and total sales in country
2, leading to an increase in the country 2 price. The increase in price is
proportional to firm 1’s marginal cost.
(d) Compare the impact of exchange rate fluctuations with changes in
(i) a specific tariff t per unit paid by the country 2 firm on each unit of
output sold in country 1;
(ii) an ad valorem tariff τ , a fraction of the country 1 price paid by the
country 2 firm on each unit of output sold in country 1.
Note: it is sufficient to write out the expression for firm 2’s payoff in
country 1 with a specific and alternatively an ad valorem tariff.
85
With a specific tariff, firm 2’s payoff on sales in country 1 is
(8.137)
π 21 = e(p1 − t)q21 − c2
h
i
c2
= e a1 − − t − b1 (q11 + q21 )
(8.138)
e
Changes of the same magnitude in t and changes in c2 /e have the same
effect on firm 2’s payoff; it is the sum of t and c2 /e that enters expressions
for best response functions and equilibrium payoffs.
With an ad valorem tariff, firm 2’s payoff on sales in country 1 is
π 21 = e(1 − τ )p1 q21 − c2 .
(8.139)
It is the product e(1 − τ ) that enters expressions for best response functions and equilibrium payoffs. Changes in either e or 1 − τ that lead to the
same change in e(1 − τ ) have the same effect on equilibrium outputs and
price.
8-7 (Exchange rate passthrough, price-setting firms) Suppose that products are differentiated, with inverse demand curves
p11 = a1 − b1 (q11 + θ 1 q21 )
p21 = a1 − b1 (θ1 q12 + q22 )
(8.140)
p12 = a2 − b2 (q11 + θ 2 q21 )
p22 = a2 − b2 (θ2 q12 + q22 )
(8.141)
in country 1 and
in country 2. Assume firms compete by selecting prices, and that each firm
exports to the other market if it is profitable to do so. Let other aspects of
the model be as in Problem 8-6.
(a) Find the equations of the price best response functions of each firm
for each country.
The equations of the price best response functions are found as in Problem
7-3. They satisfy
³
³
c2 ´
c2 ´
2
b1 (1 − θ1 )q11 = (a1 − c1 ) − θ1 a1 −
− (p11 − c1 ) + θ1 p21 −
(8.142)
e
e
³
³
c2 ´
c2 ´
b1 (1 − θ21 )q21 = −θ1 (a1 − c1 ) + a1 −
+ θ1 (p11 − c1 ) − p21 −
(8.143)
e
e
for country 1 and
b2 (1 − θ22 )q12 = (a2 − ec1 ) − θ2 (a2 − c2 ) − (p12 − ec1 ) + θ2 (p22 − c2 ) (8.144)
b2 (1 − θ22 )q22 = −θ 2 (a2 − ec1 ) + (a2 − c2 ) + θ2 (p11 − c1 ) − (p21 − c2 ) (8.145)
86
CHAPTER 8. INTERNATIONAL TRADE II
p21
p22
Firm 1’s best response curve
Firm 1’s best response curves
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p11
p12
(a) Country 1
(b) Country 2
Figure 8.2: Exchange rate fluctuations and Bertrand equilibrium
Country 1: profits satisfy
(8.146)
b1 (1 − θ21 )π11 =
h
³
³
c2 ´
c2 ´i
− (p11 − c1 ) + θ1 p21 −
(p11 − c1 ) (a1 − c1 ) − θ1 a1 −
e
e
b1
(8.147)
(1 − θ21 )π 21 =
e
³
³
³
c2 ´ h
c2 ´
c2 ´i
p21 −
−θ 1 (a1 − c1 ) + a1 −
+ θ1 (p11 − c1 ) − p21 −
e
e
e
Maximizing payoffs with respect to own price, the first-order condition
give the equations of the best response functions,
³
³
c2 ´
c2 ´
2(p11 − c1 ) − θ 1 p21 −
= (a1 − c1 ) − θ1 a1 −
(8.148)
e
e
³
³
c2 ´
c2 ´
−θ1 (p11 − c1 ) + 2 p21 −
= −θ 1 (a1 − c1 ) + a1 −
(8.149)
e
e
Country 2: profits satisfy
eb2 (1 − θ22 )π 12 =
(8.150)
(p12 − ec1 ) [(a2 − ec1 ) − θ2 (a2 − c2 ) − (p12 − ec1 ) + θ2 (p22 − c2 )]
b2 (1 − θ22 )π22 =
(8.151)
87
(p22 − c2 ) [−θ2 (a2 − ec1 ) + (a2 − c2 ) + θ2 (p12 − ec1 ) − (p22 − c2 )]
Maximizing payoffs with respect to own price, the first-order condition
give the equations of the best response functions,
³
³
c2 ´
c2 ´
2(p11 − c1 ) − θ1 p21 −
= (a1 − c1 ) − θ 1 a1 −
(8.152)
e
e
³
³
c2 ´
c2 ´
−θ 1 (p11 − c1 ) + 2 p21 −
= −θ1 (a1 − c1 ) + a1 −
(8.153)
e
e
for country 1 and
2(p12 − ec1 ) − θ 2 (p22 − c2 ) = (a2 − ec1 ) − θ1 (a2 − c2 )
(8.154)
−θ2 (p12 − ec1 ) + 2(p22 − c2 ) = −θ 2 (a2 − ec1 ) + (a2 − c2 )
(8.155)
for country 2.
In each case, the equations of the best response functions are valid only
if the implied quantities are nonnegative.
(b) Find equilibrium prices in each country and discuss the way they are
affected by changes in e.
Country 1: the system of equations of the price best response functions
is
µ
¶µ
¶ µ
¶µ
¶
p11 − c1
a1 − c1
2 −θ 1
1 −θ1
=
.
(8.156)
−θ 1 2
−θ1 1
p21 − ce2
a1 − ce2
This can be solved for equilibrium country 1 prices,
1 − θ1
θ1 ³
c2 ´
∗
p11 − c1 =
(a1 − c1 ) −
c1 −
2 − θ1
e
4 − θ21
p∗21 −
This implies
c2
1 − θ1
2 − θ 21 ³
c2 ´
=
(a1 − c1 ) +
c
−
.
1
e
2 − θ1
e
4 − θ 21
∂p∗11
θ1 c2
=−
<0
∂e
4 − θ 21 e
∂p∗21
2 c2
=−
<0
∂e
4 − θ 21 e
c2 ´ 2 − θ21 c2
∂ ³ ∗
p21 −
=
>0
∂e
e
4 − θ21 e
(8.157)
(8.158)
(8.159)
(8.160)
(8.161)
A euro depreciation reduces both country 1 prices, but increases the country 2 firm’s price-cost margin.
88
CHAPTER 8. INTERNATIONAL TRADE II
Country 2: the system of equations of the price best response functions
is
µ
2 −θ2
−θ2 2
¶µ
p12 − ec1
p22 − c2
¶
=
µ
1 −θ 2
−θ2 1
¶µ
a1 − ec1
a1 − c2
¶
.
(8.162)
This can be solved for equilibrium country 1 prices,
1 − θ2
2 − θ22
(a2 − c2 ) +
(c2 − ec1 )
2 − θ2
4 − θ22
(8.163)
1 − θ2
θ2
(a2 − c2 ) −
(c2 − ec2 ).
2 − θ2
4 − θ22
(8.164)
p∗12 − ec1 =
p∗22 − c2 =
This implies
∂p∗12
2
=
c2 > 0
∂e
4 − θ22
∂p∗22
θ2
=
c2 > 0
∂e
4 − θ22
∂ ∗
2 − θ22
(p12 − ec1 ) = −
c2 < 0.
∂e
4 − θ22
(8.165)
(8.166)
(8.167)
A euro depreciation raises both country 2 prices, but reduces the country
1 firm’s price-cost margin.
Chapter 9
Imperfect Competition and
International Trade III
Stephen Martin, 1998.
9—1 Analyze the impact of an export cartel on national welfare
(a) if two domestic firms are the only suppliers in a third market of a good
which is not consumed on their home market.
Let the demand curve in the third market be
p = a − b(q1 + q2 ),
(9.1)
suppose both firms produce with constant average and marginal cost c per
unit, and let transportation cost is zero.
If the two firms act as quantity-setting duopolists, the Cournot equilibrium profit per firm is
µ
¶2
a−c
(a − c)2
duo
duo
π1 = π2 = b
=
(9.2)
3b
9b
If the two firms form an export cartel and maximize joint profit, each
earns half of monopoly profit in equilibrium,
µ
¶2
1
a−c
(a − c)2
car
car
π1 = π2 = b
=
(9.3)
2
2b
8b
Firm profits rise under an export cartel. Since there are (by assumption)
no home market effects, home market welfare rises as well.
89
90
CHAPTER 9. INTERNATIONAL TRADE III
(b) if there is a third firm supplying the product, based in the export
market, that competes as a quantity-setting firm with the two domestic firms.
Without an export cartel, the foreign market is a Cournot triopoly. Equilibrium firm profit is
µ
¶2
a−c
(a − c)2
tri
=
,
(9.4)
πi = b
4b
16b
for i = 1, 2, 3. Total profit for the country 1 firms is
tri
π tri
1 + π2 =
(a − c)2
.
8b
(9.5)
With an export cartel, the foreign market becomes a Cournot duopoly.
Total profit of the cartel is (9.2), less than (9.5).
(c) if the product is consumed in both countries, if there are two firms based
in each country, and if firms in each country are allowed to form an export
cartel.
For simplicity, consider the case of equal-sized markets, with inverse demand curve (9.1) in each.
In the absence of an export cartel, there are 4 Cournot suppliers in each
market. Profit per firm in each market is
µ
¶2
a−c
(a − c)2
4
π =b
.
(9.6)
=
5b
25b
Total sales in country 1 are
Q1 =
4a−c
,
5 5b
(9.7)
and consumers’ surplus is
1 2 1
bQ =
2 1 2
µ ¶2
4 (a − c)2
8 (a − c)2
=
.
5
b
25
b
(9.8)
Recalling that country 1 firms each profit in both countries, country 1
net social welfare without export cartels is
NSW1Cour =
8 (a − c)2
(a − c)2
12 (a − c)2
+4
=
.
25
b
25b
25
b
(9.9)
91
If firms in each country form an export cartel and the cartels compete as
Cournot duopolists, consumers’ surplus in country 1 falls to
·
¸2
2 (a − c)2
1 2 (a − c)
b
.
=
2 3 b
9
b
(9.10)
In each market, each firm earns half of duopoly profit,
µ
¶
1
a − c 2 (a − c)2
=
b
:
2
3b
18b
(9.11)
firm profit rises with two cartels.
Country 1 net social welfare with export cartels is
NSW1Car =
2 (a − c)2
(a − c)2
4 (a − c)2
+4
=
.
9
b
18b
9
b
(9.12)
Since 4/9 = 0.44 < 12/25 = 0.48, country 1 net social welfare falls with
duelling export cartels.
(d) if formation of an export cartel allows domestic firms to tacitly collude
on the home market.
Suppose the two country 1 firms are the only suppliers of the product.
If they compete as Cournot duopolists in both markets, country 1 net social
welfare is
µ
¶2
µ
¶2
1
2a−c
a−c
2 (a − c)2
b
.
(9.13)
+ 4b
=
2
3 b
3b
3
b
If the two firms maximize joint profit in both markets, country 1 net
social welfare is
µ
¶2
µ
¶2
1
1a−c
a−c
5 (a − c)2
b
+ 2b
=
.
(9.14)
2
2 b
2b
8
b
Firm profit rises but net social welfare falls if the export cartel allows
firms to collude in the home market.
9—2 (VERs and direct foreign investment; see Flam (1994)).
There are three markets, each with a linear inverse demand curve
pi = a − Qi , i = 1, 2, 3
for a homogeneous product.
(9.15)
92
CHAPTER 9. INTERNATIONAL TRADE III
Countries 1 and 2 form a custom union, which has aggregate inverse
demand curve
1
(9.16)
pU = a − QU
2
Countries 1 and 3 are each home to one automobile manufacturer, which
we will call firm 1 and firm 3 respectively.
Only firm 3 sells in country 3; firm 3’s cost function for its operations in
country 3 is
C33 (x3 ) = F3 + c3 x3
(9.17)
Firm 1’s cost function for its operations in the custom union is
C1 (x1 ) = F1 + c1 x1
(9.18)
The country 3 firm has lower marginal cost in country 3:
c3 < c1
(9.19)
If firm 3 opens a plant in the customs union, its cost function at that
plant is
C3U (x3U ) = F3 + c1 x3U
(9.20)
If firm 3 opens a plant in the customs union, it must pay an extra set
of fixed costs. Its marginal cost is the same as firm 1: marginal cost is
country specific. This is an assumption that simplifies the analysis of market
equilibrium if there is foreign direct investment.
(a) find Cournot equilibrium profits if there is free trade and firm 3 exports
from country 3 to the customs union; find equilibrium consumers’ surplus and
net social welfare in the customs union.
If there is free trade, firm 3 would never open a plant in the customs
union: to do so, it would have to pay fixed cost and then it would have
higher marginal cost in the U market. This conclusion need not hold if there
are transportation costs and if the country 3 firm can retain some of its
marginal cost advantage at a customs union plant.
The customs union is a Cournot duopoly. Firm 1 maximizes
·
¸
1
π 1U = a − c1 − (q1U + q3U ) q1U − F1 ;
(9.21)
2
the first-order condition is
1
a − c1 − (2q1U + q3U ) = 0,
2
(9.22)
93
so that
1
1
a − c1 − (q1U + q3U ) = q1U
2
2
(9.23)
1 2
π 1U = q1U
− F1
2
(9.24)
and
in equilibrium.
The equation of firm 1’s best response function is
2q1U + q3U = 2(a − c1 )
(9.25)
In the same way, if firm 3 produces only at a plant in country 3, it
maximizes
·
¸
1
(9.26)
π 3 = (a − c3 − Q3 )Q3 + a − c3 − (q1U + q3U ) q3U − F3 .
2
Firm 3 is a monopolist in its home market; it sells the monopoly output
1
Q3 = (a − c3 )
2
(9.27)
1
Q23 = (a − c3 )2
4
(9.28)
and earns monopoly profit
The first-order condition for firm 3’s sales in the customs union is
1
a − c3 − (q1U + 2q3U ) = 0,
2
(9.29)
so that
1
1
a − c3 − (q1U + q3U ) = q3U
2
2
and firm 3’s equilibrium profit on sales in the customs union is
1 2
q
2 3U
(9.30)
(9.31)
The equation of firm 3’s export best response function for the customs
union is
q1U + 2q3U = 2(a − c3 )
(9.32)
Find equilibrium outputs by solving the system of equations of the best
response functions:
µ
¶µ
¶
µ
¶
2 1
q1U
a − c1
=2
1 2
q3U
a − c3
94
CHAPTER 9. INTERNATIONAL TRADE III
3
µ
q1U
q3U
¶
=2
µ
2
−1
−1 2
¶µ
a − c1
a − c3
¶
2
2
q1U = [2(a − c1 ) − (a − c3 )] = (a + c3 − 2c1 )
3
3
2
2
q3U = [2(a − c3 ) − (a − c1 )] = (a + c1 − 2c3 )
3
3
Assume that both of these output levels are positive.
Firm 1’s equilibrium profit is
π 1U =
2
(a + c3 − 2c1 )2 − F1
9
(9.33)
(9.34)
(9.35)
Firm 3’s equilibrium profit is
1
2
π 3 = (a − c3 )2 + (a + c1 − 2c3 )2 − F3 .
4
9
Total sales in the customs union are
¸
·
4
1
q1U + q3U =
a − (c1 + c3 )
3
2
(9.36)
(9.37)
This illustrates a general characteristic of linear demand, constant marginal cost Cournot models: total output depends on the unweighted average
of marginal cost.
Consumers’ surplus in the customs union is
¸2
·
1
8
1
2
(q1U + q3U ) =
a − (c1 + c3 )
2
9
2
(9.38)
Net social welfare in country 1 is the sum of consumers’ surplus and firm
1’s profit:
·
¸2
8
1
2
a − (c1 + c3 ) + (a + c3 − 2c1 )2 − F1
(9.39)
9
2
9
(b) Suppose firm 3 is persuaded or constrained to limit its exports to a
level v that is below its free trade equilibrium export level. Find Cournot
equilibrium profits under this voluntary export restraint. Also find equilibrium consumers’ surplus and net social welfare in the customs union.
Suppose firm 3 is persuaded or constrained to limit its exports to a level
2
v < q3U = (a + c1 − 2c3 )
3
(9.40)
95
Firm 1 will produce along its best response function:
1
1
q1U v = a − c1 − q3U = a − c1 − v
2
2
(9.41)
Total output is
1
q1U + v = a − c1 + v
(9.42)
2
Total sales must fall if firm 3’s sales fall; moving along its best response
function, firm 1 makes up only 1/2 of firm 3’s output reduction.
The country 1 price rises to
µ
¶
µ
¶
1
1
1
1
1
pU v = a −
a − c1 + v = c1 + q1U v = c1 +
a − c1 − v
(9.43)
2
2
2
2
2
Firm 1’s profit is
1
2
µ
¶2
1
a − c1 − v − F1
2
This falls as v rises.
Consumers’ surplus in the customs union is
µ
¶2
1
1
a − c1 + v
2
2
(9.44)
(9.45)
This rises as v rises.
Net social welfare in the custom union with a VER is the sum of firm 1’s
profit and consumers’ surplus,
µ
¶2
µ
¶2
1
1
1
1
a − c1 − v − F1 +
a − c1 + v
2
2
2
2
µ
¶
1
1 2
2
=
(a − c1 ) + v − F1
(9.46)
2
2
This rises as v rises.
Firm 3’s profit on sales in the customs union is
µ
µ
¶¶
µ
¶
1
1
1
1
c1 − c3 +
a − c1 − v
v=
a + c1 − 2c3 − v v
2
2
2
2
1
1
(9.47)
= (a + c1 − 2c3 )v − v 2
2
4
This rises as v rises so long as the VER actually restricts firm 3’s sales:
·
¸
µ
¶
∂ 1
1 2
1
1
(a + c1 − 2c3 )v − v =
a + c1 − 2c3 − v
∂v 2
4
2
2
96
CHAPTER 9. INTERNATIONAL TRADE III
1
= (3q3U − v)
4
1
(9.48)
[2q3U + (q3U − v)] > 0
4
(c) find Cournot equilibrium profits, consumers’ surplus and net social
welfare in the customs union if firm 3 sets up a plant in country 1. What is
the condition that must be satisfied for direct foreign investment to be the
most profitable choice for firm 3? How is this condition affected by v? How
is this condition affected by F3 ?
If firm 3 engages in foreign direct investment and opens a plant in the
customs union, the post-FDI market is a Cournot duopoly in which the two
firms have identical marginal costs. Equilibrium outputs are
=
2
a − c1
3
per firm, and firm 3’s profit is
·
¸2
2
1 2
(a − c1 ) − F3 = (a − c1 )2 − F3
2 3
9
(9.49)
(9.50)
A VER will make foreign direct investment the most profitable choice for
firm 3 if
µ
¶
1
1
2
2
(a − c1 ) − F3 >
a + c1 − 2c3 − v v
(9.51)
9
2
2
The right-hand side falls as v falls. If the VER is sufficiently restrictive,
and if firm 3’s fixed costs are small enough, the VER will make it privately
optimal for firm 3 to open a plant in country 1.
9—3 (Reciprocal dumping) Consider two firms, firm 1 based in country 1
and firm 2 based in country 2. Markets in the two countries are identical.
The two firms produce differentiated varieties of the same product. Inverse
demand curves are
p11 = a − q11 − θq21
(9.52)
p21 = a − θq11 − q21
(9.53)
in country 1 (p21 is the price of variety 2 in country 1, and so forth) and
p12 = a − q12 − θq22
(9.54)
p22 = a − θq12 − q22
(9.55)
in country 2. The parameter θ lies between 0 and 1 and measures the degree
of substitutability between the two varieties.
97
The cost of production is c per unit. Transportation cost to ship from
one country to another is t per unit.
Calculate equilibrium prices and quantities in both markets. For the
exported varieties, calculate price net of transportation cost (i.e., calculate
p21 − t and p12 − t ). Compare export prices net of transportation cost with
the price of the same variety in its home market.
Because there are constant returns to scale, the two national markets can
be analyzed separately.
Profits in country 1 are
π 11 = (p11 − c)q11 = (a − c − q11 − θq21 )q11
(9.56)
π 21 = (p21 − c − t)q21 = (a − c − t − θq11 − q21 )q21
(9.57)
best response functions are found by taking the derivative of (9.56) with
respect to q11 and the derivative of (9.57) with respect to q21 and setting the
resulting expressions equal to zero:
2q11 + θq21 = a − c
(9.58)
θq11 + 2q21 = a − c − t
(9.59)
Note that the first-order conditions imply that
p11 = a − q11 − θq21 = c + q11
(9.60)
p21 = a − θq11 − q21 = c + t + q21
(9.61)
in equilibrium.
Equilibrium quantities are found by solving the system of equations of
first-order conditions:
¶
µ ¶
µ ¶
µ
¶µ
1
0
2 θ
q11
= (a − c)
−t
q21
1
1
θ 2
µ
¶
µ
¶µ ¶
µ
¶µ ¶
q11
2
−θ
1
2
−θ
0
2
(4 − θ )
= (a − c)
−t
q21
−θ 2
1
−θ 2
1
¶
µ ¶
µ
¶µ ¶
µ
q11
1
−θ
0
= (a − c)(2 − θ)
−t
(4 − θ 2 )
q21
1
2
1
a−c
θ
+
t
2 + θ 4 − θ2
a−c
2
=
−
t
2 + θ 4 − θ2
q11 =
(9.62)
q21
(9.63)
98
CHAPTER 9. INTERNATIONAL TRADE III
Equilibrium prices are then
p11 = c +
θ
a−c
+
t
2 + θ 4 − θ2
(9.64)
2
a−c
−
t,
(9.65)
2 + θ 4 − θ2
where the equilibrium price of variety 2 in country 1 is expressed net of
transportation cost.
Because the countries are identical, in equilibrium p22 = p11 ; then
µ
¶
θ
2
t
=
p22 − (p21 − t) =
(9.66)
2t − −
2t
2−θ
4−θ
4−θ
p21 − t = c +
Market performance: inverse demand curves for country 1 of the indicated
form would be produced by a representative consumer utility function of the
form
U = aq11 + θq11 q21 + aq21 − p11 q11 − p21 q21
(9.67)
With trade, net social welfare in country 1 is
U + π 11 + π 12
(9.68)
Since
p11 = c + q11 ,
Since
a − p11 = a − c − q11 .
p21 = c + t + q21 ,
a − p21 = a − c − t − q21
Equilibrium consumer welfare is
(a − p11 )q11 + θq11 q21 + (a − p21 )q21 =
(a − c − q11 )q11 + θq11 q21 + (a − c − t − q21 )q21 =
2
2
(a − c)q11 + (a − c − t)q21 − q11
+ θq11 q21 − q21
Firm 1’s profit in country 1 is
2
(p11 − c)q11 = q11
Because the model is symmetric, firm 1’s profit in country 2 is the same
as firm 2’s profit in country 1; this is
2
(p21 − c − t)q21 = q21
99
Net social welfare in country 1 is the sum of consumer welfare and firm
1’s profit:
2
2
2
2
+ θq11 q21 − q21
+ q11
+ q21
=
(a − c)q11 + (a − c − t)q21 − q11
(a − c)q11 + (a − c − t)q21 + θq11 q21 .
Writing for x = a − c notational compactness, this is
µ
¶
µ
¶
θ
2
x
x
+
−
x
t + (x − t)
t
2 + θ 4 − θ2
2 + θ 4 − θ2
+θ
=−
µ
¶µ
¶
x
θ
2
x
+
t
−
t
2 + θ 4 − θ2
2 + θ 4 − θ2
−3x2 θ3 + 3xθ3 t − 8xθ2 t + 8x2 θ2 + 4θ2 t2 + 4x2 θ − 4xθt − 16x2 − 8t2 + 16xt
¡
¢2
4 − θ2
−(3θ3 − 8θ2 − 4θ + 16)x2 + (3θ 3 − 8θ2 − 4θ + 16)xt − 4(2 − θ2 )t2
=−
¡
¢2
4 − θ2
=
(4 + 3θ)(2 − θ)2 x2 − (4 + 3θ)(2 − θ)2 xt + 4(2 − θ2 )t2
¡
¢2
4 − θ2
=
(4 + 3θ)x2 − (4 + 3θ)xt + 4t2
(2 + θ)2
(4 + 3θ)(a − c)2 − (4 + 3θ)(a − c)t + 4t2
=
(2 + θ)2
9—4 (Antidumping duties) For the model of 9—3, if country 1 imposes an
antidumping dumping d on firm 2’s sales in country 1, what is the impact
on equilibrium prices in country 1? How great an antidumping duty would
country 1 need to impose to make p22 = p21 − t?
Replace t by t + d in (9.62) and (9.63)
(9.61) becomes
q11 =
a−c
θ
+
(t + d)
2 + θ 4 − θ2
(9.69)
q21 =
a−c
2
−
(t + d)
2 + θ 4 − θ2
(9.70)
p21 = c + t + d + q21 ,
(9.71)
100
CHAPTER 9. INTERNATIONAL TRADE III
so that with an antidumping duty d equilibrium prices are
p11 = c +
θ
a−c
θ
+
d
2t +
2+θ 4−θ
4 − θ2
a−c
2
(t + d)
−
2 + θ 4 − θ2
a−c
2
2 − θ2
=c+
−
t
+
d
2 + θ 4 − θ2
4 − θ2
The antidumping duty increases p11 by
(9.72)
p21 − t = c + d +
and increases p21 − t by
θ
d<d
4 − θ2
(9.73)
(9.74)
2 − θ2
d < d.
(9.75)
4 − θ2
p21 −t increases by less than the amount of the duty because the market is
imperfectly competitive and firm 2 absorbs part of the artificial cost increase
created by the antidumping duty. p11 increases by less than the amount of
the duty because the increase in p11 comes in response to firm 2’s reduction
in output and the slope of firm 1’s best response function is less than 1.
As long as θ < 1 the increase in p11 is less than the increase in p21 − t.
From (9.66), the price increase that is needed to make p21 − t = p22 is
t/(2−θ). From (9.75), the antidumping duty d that will result in this increase
satisfies
t
2 − θ2
2d =
2−θ
4−θ
4 − θ2
2+θ
d=
t=
t
(9.76)
2
(2 − θ )(2 − θ)
2 − θ2
From (9.74), the price of the domestic variety will rise
θ 2+θ
θ
t.
2
2t =
4−θ 2−θ
(2 − θ)(2 − θ2 )
(9.77)
9—5 (Antidumping undertaking) For the model of 9—3, what is firm 2’s profitmaximizing price if it agrees to charge the same price (net of transportation
cost) in both countries? (Assume firm 1 continues to act as a Cournot firm
in both markets.)
If firm 2 agrees to charge the same net price in both countries, it seeks to
maximize
π 2 = (a − c − t − θq11 − q21 )q21 + (a − c − θq12 − q22 )q22
(9.78)
101
subject to the constraint that net prices be the same in both countries:
p21 − t = a − t − θq11 − q21 = a − θq12 − q22 = p22
(9.79)
(9.79) can be rewritten as
θq11 + q21 − θq12 − q22 = −t
(9.80)
To solve this problem, maximize the Lagrangian
L=
(a − c − t − θq11 − q21 )q21 + (a − c − θq12 − q22 )q22 + λ(θq11 + q21 − θq12 − q22 + t)
(9.81)
with respect to q21 , q22 , and λ. The first-order conditions are
∂L
= a − c − t − θq11 − 2q21 + λ = 0
∂q21
(9.82)
θq11 + 2q21 − λ = a − c − t
(9.83)
∂L
= a − c − θq12 − 2q22 − λ = 0
∂q22
(9.84)
θq12 + 2q22 + λ = a − c
(9.85)
θq11 + q21 − θq12 − q22 = −t
(9.86)
or
or
and
(9.82) implies that in equilibrium
p21 − t = a − t − θq11 − q21 = c + q21 − λ
(9.87)
(9.84) implies that in equilibrium
p22 = a − θq12 − q22 = c + q22 + λ
(9.88)
Firm 1’s best response functions are
2q11 + θq21 = a − c
(9.89)
2q12 + θq22 = a − c − t
(9.90)
102
CHAPTER 9. INTERNATIONAL TRADE III
The system of equations of first-order conditions is






2
θ
0
0
θ
θ
2
0
0
1
0
0
2
θ
−θ
0
0
θ
2
−1
0
−1
0
1
0









(a − c) 


1
1
1
1
0
q11
q21
q12
q22
λ



 
 
=
 
 





 −t




0
1
1
0
1
a−c
a−c−t
a−c−t
a−c
−t




=







(9.91)
where the first two rows are the best response functions for country 1, the
third and fourth rows are the best response functions for country 2, and the
fifth row is the equal price constraint.



det 


2
θ
0
0
θ
θ
2
0
0
1
0
0
2
θ
θ
0
0
θ
2
1
0
−1
0
1
0







 = 16 − 12θ2 + 2θ4 = 2(2 − θ2 )(4 − θ2 )





(16 − 12θ + 2θ ) 


2


4
2
θ
0
0
θ
θ
2
0
0
1
0
0
2
θ
−θ
0
0
θ
2
−1
0
−1
0
1
0
−1





=
¡
¢
¡
¢
2
2
2
2
2
8−
3θ
−θ(2
−
θ
)
−θ
−θ
2
−
θ
−θ
4
−
θ
¡
¢
−2θ 3 − θ2
2(2¡ − θ2 ) ¢
2θ
2(2¡ − θ2 ) ¢
2(4
− θ 2 )¢
¡
2
−θ2
−θ 2 − θ2
8−
−θ 2 − θ2
θ 4 − θ2
¡ 3θ 2 ¢
2
2
−2θ 3 − θ
2(2 − θ )
−2(4 − θ2 )
2θ
2(2 − θ )
¡
¢
¡
¢
¡
¢
¡
¢
¡
¢2
−θ 4 − θ2
− 2 − θ2 (4 − θ2 ) θ 4 − θ2
2 − θ2 (4 − θ2 )
4 − θ2



¡
¢
¡
¢
2 2 − θ2 4 − θ2 


q11
q21
q12
q22
λ



=








103












¡
¢
¡
2
2
2
2
8−
3θ
−θ(2
−
θ
)
−θ
−θ
2
−
θ
¡
¢
−2θ 3 − θ2
2(2¡ − θ2 ) ¢
2θ
2(2¡ − θ2 ) ¢
2
−θ2
−θ 2 − θ2
8−
−θ 2 − θ2
¡ 3θ 2 ¢
2
2θ
2(2 − θ )
−2θ 3 − θ
2(2 − θ2 )
¡
¡
¢
¡
¢
¢
¡
¢
−θ 4 − θ2
− 2 − θ2 (4 − θ2 ) θ 4 − θ2
2 − θ2 (4 − θ2 )
 
1
 1 
 

×
 1  (a − c)−
 1 
0
¡
¢
2
2
2
2
8−
3θ
−θ(2
−
θ
)
−θ
−θ
2
−
θ
¡
¢
2θ
2(2¡ − θ2 ) ¢
−2θ 3 − θ2
2(2¡ − θ2 ) ¢
2
−θ2
−θ 2 − θ2
−θ 2 − θ2
8−
¡ 3θ 2 ¢
2
2θ
2(2 − θ )
−2θ 3 − θ
2(2 − θ2 )
¡
¡
¢
¡
¢
¢
¡
¢
−θ 4 − θ2
− 2 − θ2 (4 − θ2 ) θ 4 − θ2
2 − θ2 (4 − θ2 )
 
0
 1 
 

×
 1 t =
 0 
1
 


1
−θ(3 + 2θ)
 1 
 2(3 + 2θ) 




¡
¢
 (a − c) − (2 − θ)  4 + 3θ  t
1
= 2 (2 − θ) 2 − θ2 
 


 1 
 −2(1 + θ)2 
0
(2 + θ)2
2 2−θ
2
¢¡


¢
4−θ 


2

q11
q21
q12
q22
λ
q11
 q21
¡
¢
2 (2 + θ) 2 − θ2 
 q12
 q22
λ



¡
¢
−θ 4 − θ2
2
2(4
¡ − θ 2)¢
θ 4−θ
−2(4 − θ2 )
¡
¢2
4 − θ2

¡
¢
−θ 4 − θ2
2
2(4
¡ − θ 2)¢
θ 4−θ
−2(4 − θ2 )
¡
¢2
4 − θ2


−θ(3 + 2θ)

 
 2(3 + 2θ)


¢ 
¡
 = 2 (2 − θ) 2 − θ2   (a−c)−(2−θ)  4 + 3θ

 


 
 −2(1 + θ)2
(2 + θ)2
(9.92)

 


1
−θ(3 + 2θ)

 
 2(3 + 2θ) 



¡
¢ 1 
 = 2 2 − θ2  1  (a − c) −  4 + 3θ  t

 



 1 
 −2(1 + θ)2 
0
(2 + θ)2
(9.93)
1
1
1
1
0













t


104
CHAPTER 9. INTERNATIONAL TRADE III
q11 =
q21 =
q12 =
q22 =
λ=−
Using (9.87)
θ(3 + 2θ)
1
¡
¢t
(a − c) +
2+θ
2 (2 + θ) 2 − θ2
1
(3 + 2θ)
¡
¢t
(a − c) −
2+θ
(2 + θ) 2 − θ2
4 + 3θ
1
¡
¢t
(a − c) −
2+θ
2 (2 + θ) 2 − θ 2
(1 + θ)2
1
¡
¢t
(a − c) +
2+θ
(2 + θ) 2 − θ2
2+θ
(2 + θ)2
¡
¢
¡
¢t
2 t = −
2 (2 + θ) 2 − θ
2 2 − θ2
(9.94)
(9.95)
(9.96)
(9.97)
(9.98)
p21 − t = c + q21 − λ
=c+
1
(3 + 2θ)
2+θ
¡
¡
¢
¢t
(a − c) −
2 t+
2+θ
(2 + θ) 2 − θ
2 2 − θ2
µ
¶
1
1
a−c− t
=c+
2+θ
2
(9.99)
Using (9.88)
p22 = c + q22 + λ
=c+
2+θ
1
(1 + θ)2
¡
¢
¡
¢t
(a − c) +
2 t−
2+θ
(2 + θ) 2 − θ
2 2 − θ2
µ
¶
1
1
=c+
a−c− t
2+θ
2
(9.100)
Hence the equal net price constraint is satisfied.
What are the changes in firm 2’s prices?
Using (9.64), with an undertaking
p22 = c +
a−c
θ
+
t.
2 + θ 4 − θ2
The change in p22 is
∆p22
1
=c+
2+θ
µ
¶ µ
¶
1
a−c
θ
a−c− t − c+
+
t
2
2 + θ 4 − θ2
=−
1
t<0
2(2 − θ)
(9.101)
105
Because of the antidumping undertaking, firm 2 lowers its price in its
home market.
Using (9.65), without an undertaking
p21 − t = c +
The change in p21 − t is
1
∆(p21 − t) = c +
2+θ
2
a−c
−
t.
2 + θ 4 − θ2
µ
¶
1
a−c
2
t
a−c− t −c−
+
2
2 + θ 4 − θ2
=
1
t > 0.
2(2 − θ)
(9.102)
Because of the antidumping duty, the firm 2 raises its net price in country
1.
Compare sales with and without the antidumping undertaking?
From Problem 9—3, equilibrium outputs without the antidumping undertaking are
a−c
θ
+
t
q11 = q22 =
2 + θ 4 − θ2
a−c
2
q21 = q12 =
−
t
2 + θ 4 − θ2
Changes in quantities sold due to the antidumping undertaking are
¶
µ
1
θ(3 + 2θ)
θ
a−c
¡
¢t −
(a − c) +
+
t
∆q11 =
2+θ
2 + θ 4 − θ2
2 (2 + θ) 2 − θ2
=
θ
¡
¢t > 0
2 (2 − θ) 2 − θ2
(9.103)
Firm 1 increases its sales in its own market.
∆q21
1
(3 + 2θ)
¢t −
¡
=
(a − c) −
2+θ
(2 + θ) 2 − θ 2
µ
¶
a−c
2
−
t
2 + θ 4 − θ2
1
¢
= −¡
t≤0
2
2 − θ (2 − θ)
(9.104)
Firm 2 reduces its sales in country 1.
∆q12
1
4 + 3θ
¡
¢t −
=
(a − c) −
2+θ
2 (2 + θ) 2 − θ2
µ
a−c
2
−
t
2 + θ 4 − θ2
¶
106
CHAPTER 9. INTERNATIONAL TRADE III
θ
¢
t≤0
− ¡
2
2 2 − θ (2 − θ)
(9.105)
Firm 1’s sales in country 2 go down.
∆q22
1
(1 + θ)2
¡
¢t −
=
(a − c) +
2+θ
(2 + θ) 2 − θ2
µ
¶
a−c
θ
t
+
2 + θ 4 − θ2
1
¢
=¡
t≥0
2
2 − θ (2 − θ)
(9.106)
Firm 2’s sales on its own market go up.
The total output of each firm remains the same. There is a reallocation
of the output of each firm toward its home market.
9—6 Answer questions 9—4 and 9—5 if firms’ choice variables are prices
rather than quantities.
It is convenient to work in terms of prices measured as deviations from
a firm’s marginal cost of serving a particular market. This brings out the
essential symmetry in the solution.
Let
(9.107)
c11 = c22 = c
c21 = c21 = c + t
The equations of the demand curves are
µ
¶ µ
¶ µ
¶µ
¶
p11 − c11
a − c11
1 θ
q11
=
−
p21 − c21
a − c21
θ 1
q21
(9.108)
(9.109)
Solving for the equations of the demand curves,
µ
¶µ
¶ µ
¶ µ
¶
1 θ
q11
a − c11
p11 − c11
=
−
θ 1
q21
a − c21
p21 − c21
µ
¶ µ
¶µ
¶ µ
¶µ
¶
q11
1 −θ
a − c11
1 −θ
p11 − c11
2
(1 − θ )
=
−
q21
−θ 1
a − c21
−θ 1
p21 − c21
a − c11 − θ(a − c21 ) − (p11 − c11 ) + θ(p21 − c21 )
1 − θ2
a − c21 − θ(a − c11 ) − (p21 − c21 ) + θ(p11 − c11 )
q21 =
1 − θ2
In like manner, the demand curves for country 2 are
q11 =
q12 =
a − c12 − θ(a − c22 ) − (p12 − c12 ) + θ(p22 − c22 )
1 − θ2
(9.110)
(9.111)
(9.112)
107
a − c22 − θ(a − c12 ) − (p22 − c22 ) + θ(p12 − c12 )
1 − θ2
Payoffs in country 1 satisfy
q22 =
(9.113)
(1−θ 2 )π 11 = (p11 −c11 )[a−c11 −θ(a−c21 )−(p11 −c11 )+θ(p21 −c21 )] (9.114)
(1−θ 2 )π 21 = (p21 −c21 )[a−c21 −θ(a−c11 )−(p21 −c21 )+θ(p11 −c11 )] (9.115)
First-order conditions are
2(p11 − c11 ) − θ(p21 − c21 ) = a − c11 − θ(a − c21 )
(9.116)
−θ(p12 − c12 ) + 2(p22 − c22 ) = −θ(a − c11 ) + (a − c21 )
(9.117)
From the first-order conditions, equilibrium profits are
π 11 =
(p11 − c11 )2
1 − θ2
(9.118)
(p21 − c21 )2
(9.119)
1 − θ2
The system of equations formed by the first-order conditions for country
1 is
µ
¶µ
¶ µ
¶µ
¶
2 −θ
p11 − c11
1 −θ
a − c11
=
.
(9.120)
−θ 2
p12 − c12
−θ 1
a − c12
π 21 =
Equilibrium prices are
µ
¶ µ
¶µ
¶µ
¶
p11 − c11
2 θ
1 −θ
a − c11
2
(4 − θ )
=
p21 − c21
θ 2
−θ 1
a − c21
=
µ
2 − θ2
−θ
−θ
2 − θ2
p11 − c11 =
¶µ
a − c11
a − c21
¶
(2 − θ2 )(a − c11 ) − θ(a − c21 )
4 − θ2
(2 − θ2 )(a − c21 ) − θ(a − c11 )
p21 − c21 =
4 − θ2
In like manner, equilibrium prices in country 2 are
p12 − c12
(2 − θ2 )(a − c12 ) − θ(a − c22 )
=
4 − θ2
(9.121)
(9.122)
(9.123)
108
CHAPTER 9. INTERNATIONAL TRADE III
(2 − θ2 )(a − c22 ) − θ(a − c12 )
(9.124)
p22 − c22 =
4 − θ2
Now express the unit costs in terms of their underlying components:
p11 − c =
(2 − θ2 )(a − c) − θ(a − c − t)
4 − θ2
(2 − θ2 )(a − c − t) − θ(a − c)
p21 − c − t =
4 − θ2
(2 − θ2 )(a − c − t) − θ(a − c)
p12 − c − t =
4 − θ2
(9.125)
(9.126)
(9.127)
(2 − θ2 )(a − c) − θ(a − c − t)
(9.128)
4 − θ2
Then the difference between firm 2’s price in country 2 and firm 2’s price
in country 1, net of transportation cost, is
p22 − c =
p22 − c − (p21 − c − t) =
c21
(2 − θ2 )(a − c) − θ(a − c − t) (2 − θ2 )(a − c − t) − θ(a − c)
−
=
4 − θ2
4 − θ2
1+θ
t>0
(9.129)
2+θ
To see the impact of an antidumping duty d imposed by country 1, let
= c + t + d:
p11 − c =
(2 − θ2 )(a − c) − θ(a − c − t − d)
(2 − θ2 )(a − c) − θ(a − c − t)
θd
=
+
2
2
4−θ
4−θ
4 − θ2
p21 − c − t − d =
(2 − θ2 )(a − c − t − d) − θ(a − c)
(2 − θ2 )(a − c − t) − θ(a − c) 2 − θ2
=
=
−
d
4 − θ2
4 − θ2
4 − θ2
µ
¶
(2 − θ2 )(a − c − t) − θ(a − c)
2 − θ2
p21 − c − t =
+ 1−
d
4 − θ2
4 − θ2
(2 − θ2 )(a − c − t) − θ(a − c)
2
+
d
2
4−θ
4 − θ2
1+θ
2
p22 − c − (p21 − c − t) =
t−
d.
2+θ
4 − θ2
=
(9.130)
109
To eliminate dumping entirely, d must be
t
4 − θ2 1 + θ
t = (1 + θ)(2 − θ)
d=
2 2+θ
2
(9.131)
If there is an antidumping undertaking, firm 2 maximizes
π2 = (p21 − c12 )
a − c21 − θ(a − c11 ) − (p21 − c21 ) + θ(p11 − c11 )
1 − θ2
a − c22 − θ(a − c12 ) − (p22 − c22 ) + θ(p12 − c12 )
1 − θ2
subject to the constraint that
+(p22 − c22 )
p21 − t = p22 .
(9.132)
(9.133)
The Lagrangian for this problem is
(1 − θ2 )L = (p21 − c12 )[a − c21 − θ(a − c11 ) − (p21 − c21 ) + θ(p11 − c11 )]
+(p22 − c22 )[a − c22 − θ(a − c12 ) − (p22 − c22 ) + θ(p12 − c12 )]
+(1 − θ2 )λ(p21 − t − p22 )
(9.134)
The first-order conditions are
a − c21 − θ(a − c11 ) − 2(p21 − c21 ) + θ(p11 − c11 ) + (1 − θ2 )λ = 0
(9.135)
a − c22 − θ(a − c12 ) − 2(p22 − c22 ) + θ(p12 − c12 ) − (1 − θ2 )λ = 0
(9.136)
(1 − θ 2 )p21 − (1 − θ2 )t − (1 − θ2 )p22 = 0
(9.137)
These can be rewritten
−θ(p11 − c11 ) + 2(p21 − c21 ) − (1 − θ2 )λ = −θ(a − c11 ) + a − c21
(9.138)
−θ(p12 − c12 ) + 2(p22 − c22 ) + (1 − θ2 )λ = −θ(a − c12 ) + a − c22
(9.139)
−(1 − θ2 )(a − c21 ) − (1 − θ 2 )t + (1 − θ2 )(a − c22 )
(9.140)
−(1 − θ2 )(p21 − c21 ) + (1 − θ 2 )(p22 − c22 ) =
Firm 1’s first-order conditions are
2(p11 − c11 ) − θ(p21 − c21 ) = a − c11 − θ(a − c21 )
(9.141)
2(p12 − c12 ) − θ(p22 − c22 ) = a − c12 − θ(a − c22 )
(9.142)
110
CHAPTER 9. INTERNATIONAL TRADE III
The system of equations formed by the first-order conditions is



2
−θ
0
0
0
p11 − c11


 −θ
2
0
0
−(1 − θ 2 ) 
  p21 − c21 




 0
0
2
−θ
0
  p12 − c12  =

2
 0
0
−θ
2
(1 − θ )   p22 − c22 
2
2
λ
0 −(1 − θ ) 0 (1 − θ )
0






1
−θ
0
0
0
−θ
1
0
0
0
0
0
1
−θ
0
0
0
−θ
1
0
2
2
0 −(1 − θ ) 0 (1 − θ ) −(1 − θ 2 )
The determinant of the coefficient matrix is

2
−θ
0
0
0
 −θ
2
0
0
−(1 − θ2 )

0
2
−θ
0
det 
 0
 0
0
−θ
2
(1 − θ2 )
0 −(1 − θ2 ) 0 (1 − θ 2 )
0






a − c11
a − c21
a − c12
a − c22
t






(9.143)



 = −4(4 − θ2 )(1 − θ2 )2


(9.144)
The inverse of the coefficient matrix is
1
−4(4 − θ )(1 − θ2 )2
2
(9.145)
times the 5 × 5 matrix the first three columns of which are
 ¡

¢¡
¢2
¡
¢2
¡
¢2
− 8 − θ2 1 − θ2
−2θ 1 − θ2
−θ2 1 − θ2
¡
¢2
¡
¢2
¡
¢2


−2θ 1 − θ2
−4 1 − θ 2
−2θ 1 − θ2

¡
¢2
¡
¢2
¡
¢¡
¢2 


2
2
2
2
2


−θ 1 − θ
−2θ 1 − θ
− 8−θ
1−θ


¡
¢
¡
¢
¡
¢
2 2
2 2
2 2


−2θ
1
−
θ
−4
1
−
θ
−2θ
1
−
θ
¡
¢¡
¢
¡
¢¡
¢
¡
¢¡
¢
2
2
2
2
2
2
θ 1−θ
4−θ
2 1−θ
4−θ
−θ 1 − θ
4−θ
(9.146)
and the last two columns of which are

¡
¢2
¡
¢¡
¢ 
−2θ 1 − θ 2
θ 1 − θ2 4 − θ 2
¡
¢2
¡
¢¡
¢

−4 1 − θ2
2 1 − θ2 4 − θ2 

¡
¢
¡
¢¡
¢ 

2 2
2
2 
(9.147)

−2θ 1 − θ
−θ 1 − θ
4 −θ .


¡
¢
¡
¢
¡
¢
2

−4 1 − θ2
−2 1 − θ 2 4 − θ 2 
¡
¢
¡
¢
¡
¢2
−2 1 − θ2 4 − θ2
4 − θ2
111
The solution of the system of equations formed by the first-order conditions is


p11 − c11
 p21 − c21 


=
p
−
c
−4(4 − θ2 )(1 − θ2 )2 
12
12


 p22 − c22 
λ



1
−θ
0
0
0
a − c11
 −θ
  a − c21 
1
0
0
0



  a − c12  ,
0
0
1
−θ
0
(A, B) × 



 0
  a − c22 
0
−θ
1
0
t
0 −(1 − θ2 ) 0 (1 − θ2 ) −(1 − θ2 )
(9.148)
where
 ¡

¡
¢¡
¢2
¢2
¡
¢2
−2θ 1 − θ2
−θ2 1 − θ2
− 8 − θ 2 1 − θ2
¡
¢2
¡
¢2
¡
¢2


−2θ 1 − θ2
−4 1 − θ2
−2θ 1 − θ 2

¡
¢¡
¡
¢2
¢2
¡ 2
¢2 


2
2
2
2
A=

−θ
−2θ
1
−
θ
−
8
1
−
θ
1
−
θ
θ


¡
¡
¡
¢
¢
¢
2 2
2 2
2 2


−2θ
1 ¢−¡θ
¡−2θ 21¢−
¡ θ 2¢
¡ −4 12 ¢−¡ θ
¢
¡
¢
θ 1−θ 4−θ
2 1 − θ 4 − θ2 −θ 1 − θ2 4 − θ2
and
¢2
¡
¢¡
¢ 
¡
−2θ 1 − θ2
θ 1 − θ2 4 − θ2
¡
¢2
¡
¢¡
¢

−4 1 − θ2
2 1 − θ2 4 − θ2 

¡
¡
¢2
¢¡
¢ 


B=
−2θ 1 − θ2
−θ 1 − θ 2 4 − θ 2 


¡
¢
¢
¡
¢
¡
2

−2 1 − θ2 4 − θ2 
−4 1 − θ2
¡
¢¡
¢
¡
¢2
−2 1 − θ2 4 − θ 2
4 − θ2
¡
¢
The product of the two coefficient matrices is 1 − θ2 times
¢
¡
¢¡
¢ 
¡

2θ 1 − θ 2
−θ ¡1 − θ2 ¢ ¡4 − θ2 ¢
2

4(1
−2¡ 1 − θ2¢ ¡ 4 − θ2¢ 
¡ − θ 2)¢


2
2


2θ
1
−
θ
θ
1
−
θ
4
−
θ
¡
¢¡
¢
¡
¢¡
¢ 

2
2
2
2
 −4 1 − θ

3−θ
2 1−θ 4−θ
¡
¢
2
2 2
2(4 − θ )
− 4−θ

(9.149)
Equilibrium values are
¡
8 − 3θ
¢
2
p11 − c11 =
¢
¡
(1 + θ) (a − c11 ) − 2θ(a − c21 ) − θ2 (a − c12 ) − 2θ(a − c22 ) + θ 4 − θ2 t
4(4 − θ2 )
(9.150)
112
CHAPTER 9. INTERNATIONAL TRADE III
¡
¢
2
¡
¢
2
p21 − c21 =
¡
¢
(a − c21 ) − 2θ(a − c12 ) − 4(a − c22 ) + 2 4 − θ2 t
4(4 − θ 2 )
(9.151)
p12 − c12 =
¡
¢
¡
¢
−θ2 (a − c11 ) − 2θ(a − c21 ) + 8 − 3θ2 (1 + θ) (a − c12 ) − 2θ(a − c22 ) − θ 4 − θ2 t
4(4 − θ2 )
(9.152)
p22 − c22 =
¡
¢
¡
¢
−2θ(a − c11 ) − 4(a − c21 ) − 2θ(a − c12 ) + 4 3 − θ2 (a − c22 ) − 2 4 − θ2 t
4(4 − θ 2 )
(9.153)
¡
¢
θ(a − c11 ) − 2(a − c21 ) − θ(a − c12 ) − 2(a − c22 ) + 4 − θ2 t
λ=
(9.154)
4(1 − θ2 )
Check the price undertaking constraint:
−2θ(a − c11 ) + 4 3 − θ
p21 − c21 − (p22 − c22 )
¡
¢
(a − c21 ) − 2θ(a − c12 ) − 4(a − c22 ) + 2 4 − θ2 t
4(4 − θ 2 )
Ã
¡
¢
¡
¢ !
−2θ(a − c11 ) − 4(a − c21 ) − 2θ(a − c12 ) + 4 3 − θ2 (a − c22 ) − 2 4 − θ 2 t
−
4(4 − θ 2 )
−2θ(a − c11 ) + 4 3 − θ
or
= −c21 + t + c22
p21 − t = p22
What are the changes in firm 2’s prices?
Firm 2 raises its price in country 1 :
¡
¡
¢
¢
−2θ(a − c11 ) + 4 3 − θ2 (a − c21 ) − 2θ(a − c12 ) − 4(a − c22 ) + 2 4 − θ2 t
4(4 − θ 2 )
(2 − θ2 )(a − c21 ) − θ(a − c11 )
−
=
4 − θ2
¡
¢
2θ(a − c11 ) + 4(a − c21 ) − 2θ(a − c12 ) − 4(a − c22 ) + 2 4 − θ2 t
=
4(4 − θ2 )
¡
¢
2θ(c12 − c11 ) + 4(c22 − c21 ) + 2 4 − θ2 t
=
4(4 − θ2 )
113
1+θ
t>0
2(2 + θ)
(9.155)
Firm 2 lowers its price in country 2:
¡
¢
¡
¢
−2θ(a − c11 ) − 4(a − c21 ) − 2θ(a − c12 ) + 4 3 − θ2 (a − c22 ) − 2 4 − θ2 t
4(4 − θ2 )
−
(2 − θ2 )(a − c22 ) − θ(a − c12 )
=
4 − θ2
¡
¢
−2θ(a − c11 ) − 4(a − c21 ) + 2θ(a − c12 ) + 4(a − c22 ) − 2 4 − θ2 t
=
4(4 − θ2 )
¡
¢
2θ(c11 − c12 ) + 4(c21 − c22 ) − 2 4 − θ2 t
=
4(4 − θ2 )
¡
¢
−2θ + 4 − 2 4 − θ2
t=
4(4 − θ 2 )
−
(1 + θ)
t<0
2(2 + θ)
(9.156)
114
CHAPTER 9. INTERNATIONAL TRADE III
Chapter 10
Market Integration in the
European Union
10—1 Evaluate the consequences of market integration using the model of
Section 10.2.2, that is, two countries i = 1, 2 each with an inverse demand
curve with equation
pi = 100 − Qi ,
n1 firms in market 1 and n2 firms in market 2, and all firms operating with
the cost function,
c(q) = 10q,
if country i collects a per-unit tax ti on each unit sold within its territory.
The pre-integration Cournot equilibrium prices are
p1 = 10 + t1 +
100 − 10 − t1
n1 + 1
p2 = 10 + t2 +
100 − 10 − t2
.
n2 + 1
If firms continue to treat each country as a separate market, because taxes
are collected separately in each market, then the post-integration markets are
Cournot oligopoly markets with n1 + n2 firms supplying each market; in the
usual way, post-integration prices are
p1 = 10 + t1 +
100 − 10 − t1
n1 + n2 + 1
p2 = 10 + t2 +
100 − 10 − t2
.
n1 + n2 + 1
115
116
CHAPTER 10. MARKET INTEGRATION
If firms treat both countries as a single market after integration, the payoff
of (say) firm 1 in country is
¶
µ
1
π 11 = 100 − 10 − Q (q11 + q22 ) − t1 q11 − t2 q22 .
2
In such a scenario, all firms would sell, and all customers purchase, in the
lower-tax country. The post-integration price is
p = 10 + tmin + 2
100 − 10 − tmin
,
n1 + n2 + 1
where tmin is the smaller of t1 , t2 .
10—2 Suppose n1 Cournot oligopolists supply the market in country 1, where
the equation of the inverse demand curve is
p1 = 100 − Q1 ,
while n2 different Cournot oligopolists supply the market in country 2, where
the equation of the inverse demand curve is
1
p2 = 100 − Q2 .
2
Compare equilibrium prices and outputs before and after market integration, holding the number of firms fixed.
Country 1:
p1 = 100 − Q1
Q1 = 100 − p1
Equilibrium output per firm:
q1 =
90
n1 + 1
Equilibrium price:
p1 = 10 +
Country 2:
90
n1 + 1
1
p2 = 100 − Q2
2
Q2 = 200 − 2p1
Equilibrium output per firm:
q2 =
190
n2 + 1
117
Equilibrium price:
p2 = 10 +
190
n2 + 1
Post integration:
Q = 300 − 3p
1
p = 100 − Q
3
Payoff of a typical firm:
¶
µ
1
π i = 100 − 10 − Q qi
3
µ
¶
1
π i = 90 − Q qi
3
Condensed first-order condition for profit maximization:
270 = (n1 + n2 + 1) q
Equilibrium output per firm:
q=
270
n1 + n2 + 1
Equilibrium price:
p = 10 +
270
n1 + n2 + 1
Compare equilibrium prices:
µ
¶
90
270
−2n1 + n2 − 2
10 +
− 10 +
= 90
n1 + 1
n1 + n2 + 1
(n1 + 1) (n1 + n2 + 1)
µ
¶
190
270
19n1 − 8n2 − 8
10 +
− 10 +
= 10
n2 + 1
n1 + n2 + 1
(n2 + 1) (n1 + n2 + 1)
Loosely: if n2 is large, prices fall in country 1 but rise in country 2; if n1
is large, prices fall in country 2 but rise in country 1.
If the number of firms is equal:
µ
¶
90
270
n+2
10 +
− 10 +
= −90
n+1
2n + 1
(n + 1) (2n + 1)
µ
¶
190
270
11n − 8
10 +
− 10 +
= 10
n+1
2n + 1
(n + 1) (2 + 1)
118
CHAPTER 10. MARKET INTEGRATION
Price rises in the low elasticity market, falls in the high elasticity market.
10—3 (Market integration, Cournot, tax differences.) Verify (10.8).
Equation (10.8) gives equilibrium prices if country 1 charges a tax t1 on
every unit sold in its territory, country 2 charges a tax t2 on every unit sold
in its territory, the number of firms is the same in each country, and (because
of the tax policy), firms continue to treat each country as a separate market
after integration. Equilibrium prices in each country are then the standard
Cournot equilibrium prices with 2n firms operating in each country. See
Problem 10—1.
Suppose instead country 1 collects a tax t1 on every unit sold by a country
1 firm, no matter where that unit is sold, and country 2 collects a tax t2 on
every unit sold by a country 2 firm, no matter where that unit is sold.
The payoff function of a typical country 1 firm is
µ
¶
1
π 1i = 100 − 10 − t1 − Q q1i ,
2
and there is a similar expression for the payoff of a country 2 firm.
From the first order conditions for payoff maximization, we obtain the
equations of the best response functions,
(n1 + 1)q1 + n2 q2 = 2 (90 − t1 )
n1 q1 + (n2 + 1)q2 = 2 (90 − t2 )
Solve for equilibrium output per firm:
µ
¶µ
¶
µ
¶
n1 + 1
n2
q1
90 − t1
=2
n1
n2 + 1
q2
90 − t2
µ
¶
µ
¶−1 µ
¶
q1
n1 + 1
n2
90 − t1
=2
q2
n1
n2 + 1
90 − t2
¶
µ
¶
µ
n2 +1
n2
(90
−
t
)
−
(90
−
t
)
q1
1
2
n1 +n2 +1
n1 +n2 +1
=2
n1
n1 +1
q2
− n1 +n
(90
−
t
)
+
(90 − t2 )
1
+1
n1 +n2 +1
2
µ
¶
2
90 − (n2 + 1) t1 + n2 t2
=
n1 + n2 + 1 90 + n1 t1 − (n1 + 1) t2
Find total output:
¡
¢
2
n1 n2
Q=
n1 + n2 + 1
=
µ
90 − (n2 + 1) t1 + n2 t2
90 + n1 t1 − (n1 + 1) t2
¶
2
2
n1 (90 − (n2 + 1) t1 + n2 t2 )+
n2 (90 + n1 t1 − (n1 + 1) t2 )
n1 + n2 + 1
n1 + n2 + 1
119
=2
90 (n1 + n2 ) − n1 t1 − n2 t2
n1 + n2 + 1
Find equilibrium price:
1
p = 100 −
2
µ
90 (n1 + n2 ) − n1 t1 − n2 t2
2
n1 + n2 + 1
¶
90 (n1 + n2 ) − n1 t1 − n2 t2
n1 + n2 + 1
Express equilibrium price in terms of deviations from marginal production
cost plus the sales:
p − (10 + t1 ) =
= 100 −
100 −
90 (n1 + n2 ) − n1 t1 − n2 t2
− (10 + t1 )
n1 + n2 + 1
=
90 − (n2 + 1) t1 + n2 t2
n1 + n2 + 1
p − (10 + t2 ) =
100 −
90 (n1 + n2 ) − n1 t1 − n2 t2
− (10 + t2 )
n1 + n2 + 1
=
90 + n1 t1 − (n1 + 1) t2
n1 + n2 + 1
p − t1 = 10 +
90 − (n2 + 1) t1 + n2 t2
n1 + n2 + 1
p − t2 = 10 +
90 + n1 t1 − (n1 + 1) t2
n1 + n2 + 1
If n1 = n2 = n, these become
p − t1 = 10 +
90 − (n + 1) t1 + nt2
2n + 1
p − t2 = 10 +
90 + nt1 − (n + 1) t2
.
2n + 1