Industrial Organization: A European Perspective Answers to ... · Figure 2.2: Sales maximization...

Industrial Organization: A EuropeanPerspective

Answers to Problems

Stephen MartinDepartment of EconomicsKrannert School of Business

Purdue University403 State Street West

West Lafayette, Indiana 47907-2056USA

[email protected]

November 2001; updated November 2002

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 Lernerindex 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 = 55πm = (55− 10)(45) = 2025

DWL =1

2(90− 45)(55− 10) = 1

2(2025)

L =p− 10

p=55− 1055

=45

55=9

11= 0.8181

1—2 (a) Graph the average variable, average, and marginal cost curves if thecost function is

C(q) = 1 + 9q.

The equations of the cost curves are

AV C(q) =MC (q) = 9

AC(q) =1

q+ 9.

(b) Graph the average variable, average, and marginal cost curves if the cost

5

6 CHAPTER 1. BACKGROUND

p

Q

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• • • • • •

100

55

10

10045 90

p

Q

Demand curve...........................................................................................................................................................................................................................................

Marginalrevenue curve.........................................................................................................................

Marginalcost

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Deadweightloss

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Figure 1.1: Monopolist’s output decision, p = 100−Q, marginal cost = 10

7

q7

9

costunit......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

AC(q) = 9 + 1q.........................................................................................................................................................................................

............................

AV C(q) =MC(q) = 9...........................................................................................................................................................................................

Figure 1.2: Firm cost curves, C(q) = 1 + 9q

function isC(q) = 1 + 9q − q2 + q3.

The equations of the cost curves are

AV C(q) = 9− q + q2

MC(q) = 9− 2q + 3q2

AC(q) =1

q+ 9− q + q2.

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 minimumvalue of average variable cost:

AV C(q) = 9− q + q2 = 9− 2q + 3q2 =MC(q)

9− q + q2 = 9− 2q + 3q2

2q2 − q = 0⇒ q = 0,1

2


and the common value of average variable cost and marginal cost for q = 1/2is

9− 12+1

4= 9− 2

µ1

2

¶+ 3

µ1

4

¶=35

4= 8.75.

Average cost and marginal cost have same value for the output level thatgives the minimum value of average cost:

AC(q) =1

q+ 9− q + q2 = 9− 2q + 3q2 =MC(q)

1

q= 2q2 − q

2q3 − q2 − 1 = 0.(q − 1) ¡2q2 + q + 1

¢= 0

The minimum value of average cost thus occurs for q = 1:

AC(1) =1

1+ 9− 1 + 1 = 10.

Find the minimum value of marginal cost:

MC(q) = 9− 2q + 3q2

dMC(q)

dq= −2 + 6q = 0⇔ q =

1

3

and the minimum value of marginal cost is

MC

µ1

3

¶= 9− 2

µ1

3

¶+ 3

µ1

3

¶2=26

3= 4.67.

9

q

costunit

• •

•

• •

•

7

98.75

10

0.5 1

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

MC(q) = 9− 2q + 3q2

AV C(q) = 9− q + q2........................................................

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AC(q) = 1q+ 9− q + q2......................................................................................................................................................................................................

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Figure 1.3: Firm cost curves, C(q) = 1 + 9q − q2 + q3

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 ifeach firm maximizes a weighted average of profit and sales,

gi = (1− σ)πi + σpiqi. (2.2)

Illustrate noncooperative equilibrium on a reaction curve diagram.For this example

π1 = [100− 10− (q1 + q2)] q1 = [90− (q1 + q2)] q1

p1q1 = [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

12 CHAPTER 2. OLIGOPOLY I

Given the symmetry of the model, in equilibrium firms produce the sameoutput:

3q = 90 + 10σ.

q = 30 +10

3σ.

The greater the weight given to revenue in the objective function, thegreater is equilibrium output. Figure 2.1 shows best response functions forσ = 0 and σ = 4/5.

q2

q1

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• • • • • • • • •

90

45 •

30

9045•

30

E0

E4/5,4/5

1’s best response curve, σ = 0..........................................................................................................................................................................................................

1’s best response curve, σ = 45.......................................................................................................................................................................................................................

............................

2’s best response curve, σ = 0..........................................................................................................................................................................................................

2’s best responsecurve, σ = 4/5

.......................................................................................................................................................................

••

•

•

Figure 2.1: Cournot best response curves, partial sales maximization by bothfirms, σ = 4

5

2—2 For a price-setting duopoly with product differentiation, let the equa-tions of the inverse demand curves be

p1 = 100−µq1 +

1

2q2

¶(2.3)

13

p2 = 100−µ1

2q1 + q1

¶, (2.4)

with corresponding demand functions

q1 =2

3(100− 2p1 + p2) (2.5)

q2 =2

3(100 + p1 − 2p2). (2.6)

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)π1 + σp1q1 (2.7)

g1 = [(1− σ)(p1 − 10) + σp1] q1

= [p1 − (1− σ)10] q1

=2

3[p1 − (1− σ)10] (100− 2p1 + p2). (2.8)

Set the derivative of g1 with respect to p1 equal to zero:

∂g1∂p1

=2

3{[p1 − (1− σ)10] (−2) + 100− 2p1 + p2} = 0. (2.9)

Rearrangement of terms gives the equation of firm 1’s price best responsefunction:

4p1 − p2 = 120− 20σ. (2.10)

Firm 2’s profit is

π2 =2

3(p2 − 10)(100 + p1 − 2p2) (2.11)

Setting the derivative of π2 with respect to p2 equal to zero gives theequation of firm 2’s price best response function:

−p1 + 4p2 = 120. (2.12)

Note that the first-order condition implies

100 + p1 − 2p2 − 2(p2 − 10) = 0,so that firm 2’s equilibrium profit is

π2 =4

3(p2 − 10)2


Write the equations of the first-order conditions as a system:µ4 −1−1 4

¶µp1p2

¶= 120

µ11

¶− 20σ

µ10

¶. (2.13)

The solution is

15

µp1p2

¶= 120

µ4 11 4

¶µ11

¶− 20σ

µ4 11 4

¶µ10

¶

15

µp1p2

¶= 600

µ11

¶− 20σ

µ41

¶3

µp1p2

¶= 120

µ11

¶− 4σ

µ41

¶p1 = 40− 16

3σ (2.14)

p2 = 40− 43σ. (2.15)

Equilibrium outputs are

q1 = 40 +56

9σ (2.16)

q2 = 40− 169σ. (2.17)

Profits are

π1 = (p1 − 10)q1 =µ30− 16

3σ

¶µ40 +

56

9σ

¶, (2.18)

which falls as σ rises from 0 to 1 (Figure 2.2), and

π2 = (p2 − 10)q2 =µ30− 4

3σ

¶µ40− 16

9σ

¶=4

3

µ30− 4

3σ

¶2(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

pi = 100− (qi + θQ−i) , (2.20)

for i = 1, 2, ..., n and Q−i =Pn

j 6=i qj.

15

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• • • • • • • • • •

1200

1150

•

•

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

πσ

σ

πσ................................................................................................................................................................................................................................................................................................

Figure 2.2: Sales maximization and equilibrium firm profit, price-settingfirms

The equations of the corresponding demand curves are

qi =90(1− θ)− [1 + (n− 2)θ](pi − 10) + θ

Pnj 6=i(pj − 10)

(1− θ)[1 + (n− 1)θ] (2.21)

If marginal cost is constant at 10 per unit, show that when firms set pricesto 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

p =

p1p2..pn

= 100Jn−[(1−θ)In+θJnJ 0n]

q1q2..qn

= 100Jn−[(1−θ)In+θJnJ 0n]q,

(2.23)or

p = 100Jn − [(1− θ)In + θJnJ0n]q (2.24)

where Jn is an n-element column vector of 1s and In is an n × n identifymatrix.It proves to be convenient to express prices in terms of deviations from

marginal cost; (2.24) becomes

p− 10Jn = 90Jn − [(1− θ)In + θJnJ0n]q. (2.25)


Rewrite (2.25) as

[(1− θ)In + θJnJ0n]q = 90Jn − (p− 10Jn). (2.26)

We need to find the inverse of the coefficient matrix

(1− θ)In + θJnJ0n. (2.27)

Suppose the inverse takes the form

1

1− θIn + kJnJ

0n, (2.28)

where the value of the parameter k is to be determined.Then it must be that

[(1− θ)In + θJnJ0n]

µ1

1− θIn + kJnJ

0n

¶= In. (2.29)

Carrying out the multiplication,

In + (1− θ)kJnJ0n +

θ

1− θJnJ

0n + kθJnJ

0nJnJ

0n = In. (2.30)

In +

·(1− θ)k +

θ

1− θ+ nkθ

¸JnJ

0n = In (2.31)

(using J 0nJn = n).

In +

½[1 + (n− 1)θ]k + θ

1− θ

¾JnJ

0n = In, (2.32)

from which it follows that

k = − 1

1− θ

θ

1 + (n− 1)θ , (2.33)

so that the inverse in question is

[(1− θ)In + θJnJ0n]−1 =

1

1− θ

·In − θ

1 + (n− 1)θJnJ0n

¸(2.34)

Substituting (2.34) in (2.26) shows that the equations of the demandcurves satisfy

(1− θ)[1 + (n− 1)θ]q = (2.35)

(1− θ)90Jn − {[1 + (n− 1)θ]In − θJnJ0n} (p− 10Jn).

17

These expressions are valid provided all quantities are nonnegative.For example, the quantity demanded of variety 1 is

q1 =90(1− θ)− [1 + (n− 1)θ](p1 − 10) + θ

Pnj=1(pj − 10)

(1− θ)[1 + (n− 1)θ]

=90(1− θ)− [1 + (n− 2)θ](p1 − 10) + θ

Pnj=2(pj − 10)

(1− θ)[1 + (n− 1)θ] . (2.36)

Firm 1’s profit as a function of prices satisfies

(1− θ)[1 + (n− 1)θ]π1 = (2.37)

(p1 − 10)(90(1− θ)− [1 + (n− 2)θ](p1 − 10) + θ

nXj=2

(pj − 10)).

The first-order condition to maximize π1 with respect to p1 is

90(1−θ)−[1+(n−2)θ](p1−10)+θnX

j=2

(pj−10)+(p1−10) {−[1 + (n− 2)θ]} = 0

(2.38)

2[1 + (n− 2)θ](p1 − 10)− θnX

j=2

(pj − 10) = 90(1− θ). (2.39)

Because firms in this example hold identical beliefs and have identicalcost functions, in equilibrium, all firms will charge the same price. Settingp1 = 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)

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)

the expression for the Lerner index of market power that corresponds to (??)is

p1 − c0(q1)p1

=s1

εQ1p1, (3.3)

wheres1 =

q1q1 + θq2

≡ q1Q1

(3.4)

is firm 1’s market share, taking account of the imperfect substitutability ofvariety 2 for variety 1, and

εQ1p1 ≡ −Q1

p1

dp1dQ1

. (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 firm1’s profit is

π1 = f(q1 + θq2)q1 − c(q1).

The first-order condition to maximize π1 with respect to q1 is

p1 + q1dp1dQ1

= c0(q1)

19

20 CHAPTER 3. COLLUSION AND TACIT COLLUSION

p1 − c0(q1) = −q1 dp1dQ1

p1 − c0(q1)p1

=q1Q1

µ−Q1

p1

dp1dQ1

¶p1 − c0(q1)

p1=

s1εQ1p1

Each firm’s market share is measured relative to the size of its ownmarket,which (because of product differentiation) differs from the markets of otherfirms. The size of firm 1’s market is q1 + θq2, the size of firm 2’s market isθq1 + q2.

Chapter 4

Dominance

4—1 (Price leadership with product differentiation) For a price-setting duopolywith product differentiation, let the equations of the inverse demand curvesbe

p1 = 100−µq1 +

1

2q2

¶, (4.1)

p2 = 100−µ1

2q1 + q1

¶, (4.2)

with corresponding demand functions

q1 =2

3(100− 2p1 + p2) (4.3)

q2 =2

3(100 + p1 − 2p2). (4.4)

Let marginal cost be constant at 10 per unit.Find equilibrium prices and profits if firm 2 sets its price p2 noncoopera-

tively to maximize its own profit, if firm 1 knows this, and if firm 1 maximizesits own profit, taking firm 2’s behavior into account.Rewrite the equations of the demand curves as

q1 =2

3[90− 2 (p1 − 10) + (p2 − 10)] (4.5)

q2 =2

3[90 + (p1 − 10)− 2(p2 − 10)] . (4.6)

Firm 2’s payoff function is

π2 = (p2 − 10)q2 = 2

3(p2 − 10) [90 + (p1 − 10)− 2(p2 − 10)] . (4.7)

21

22 CHAPTER 4. DOMINANCE

The equation of the first-order condition to maximize π2 with respect top2 is

90 + (p1 − 10)− 4(p2 − 10) ≡ 0 (4.8)

p2 − 10 = 1

4[90 + (p1 − 10)] (4.9)

Substituting the equation of firm 2’s best response function in the equa-tion of firm 1’s demand curve and collecting terms gives the equation of firm1’s residual demand curve:

q1 =2

3(90)− 4

3(p1 − 10) + 2

3(p2 − 10)

=2

3(90)− 4

3(p1 − 10) + 2

3

µ1

4

¶[90 + (p1 − 10)]

=2

3(90) +

2

3

µ1

4

¶(90) +

·2

3

µ1

4

¶− 43

¸(p1 − 10)

= 75− 76(p1 − 10) . (4.10)

Firm 1’s payoff along its residual demand curve is

π1 = (p1 − 10) q1 = (p1 − 10)·75− 7

6(p1 − 10)

¸(4.11)

and this is maximized for

75− 2µ7

6

¶(p1 − 10) ≡ 0

p1 = 10 +225

7= 10 + 32

1

7. (4.12)

The follower’s price satisfies

p2 − 10 = 1

4

·90 +

225

7

¸=855

28= 30

15

28. (4.13)

Quantities demanded are

q1 =2

3

·90− 2

µ225

7

¶+

µ855

28

¶¸=75

2= 37

1

2(4.14)

q2 =2

3

·90 +

µ225

7

¶− 2

µ855

28

¶¸=285

7= 40

5

7. (4.15)

23

Payoffs are

π1 = (p1 − 10) q1 =µ225

7

¶µ75

2

¶=16 875

14= 1205

5

14(4.16)

π2 = (p2 − 10)q2 =µ855

28

¶µ285

7

¶=243 675

196= 1243

47

196. (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 firm1 be an incumbent and firm 2 a potential entrant that must pay a fixed andsunk entry cost e if it comes into the market. If firm 1 can commit to apost-entry price, what price must it set to make entry unprofitable? Underwhat circumstances (for what values of re, where r is the interest rate usedto discount income) would firm 1 prefer to deter entry (a) if the post-entrymarket would be a Bertrand (noncooperative) duopoly and (b) if firm 1 wouldbe a Stackelberg price leader in the post-entry market?From (4.8), on firm 2’s best response function

q2 =2

3[90 + (p1 − 10)− 2(p2 − 10)] = 4

3(p2 − 10) (4.18)

and its payoff per period is

π2 = (p2 − 10)q2 = 4

3(p2 − 10)2 = 1

12[90 + (p1 − 10)]2 . (4.19)

The entrant’s present discounted value if the post-entry market is aBertrand duopoly is

V2 =[90 + (p1 − 10)]2

12r− e (4.20)

and this is zero or negative for

[90 + (p1 − 10)]212r

− e ≤ 0. (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)]212r

− e ≤ 0

re ≥ [90 + (55− 10)]2

12=(135)2

12=6075

4= 1518.75. (4.22)


[90 + (p1 − 10)]2 ≤ 12re90 + (p1 − 10) ≤ 2

√3re

p1 − 10 ≤ 2√3re− 90. (4.23)

If the incumbent commits to a price

pL = 2√3re− 80, (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 is³2√3re− 90

´³180− 2

√3re´= 540

√3re− 12re− 16 200 (4.25)

and its value is

VL =4¡√3re− 45¢ ¡90−√3re¢

r. (4.26)

The incumbent’s value in a Bertrand duopoly is

1200

r, (4.27)

and if the alternative is Bertrand duopoly the incumbent will have at leastas great a value committing to price pL if

4¡√3re− 45¢ ¡90−√3re¢

r≥ 1200

r³√3re− 45

´³90−

√3re´≥ 300

135√3re− 3re− 4050 ≥ 300

135√3re− 3re− 4350 ≥ 0

45√3re− re− 1450 ≥ 0 (4.28)

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, theincumbent’s value in the post-entry market is, from (4.16),

16 875

14r. (4.29)

25

re

∆VVPL − VBert.................................................................................................................................................................................................................. ..................

.....

• •1000

•1100

•1200

•1300

•1400

•1500

•1600

•−10

•

•10

20

30

40

50

60

70

•

•

•

•

•

•

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Figure 4.1: Entry cost and VPL − VBert

The incumbent’s value if it deters entry is at least as great as if it letsthe entrant into the market if

4¡√3re− 45¢ ¡90−√3re¢

r≥ 16 875

14r

4³√3re− 45

´³90−

√3re´≥ 16 875

14³√3re− 45

´³90−

√3re´≥ 16 875

56³√3re− 45

´³90−

√3re´≥ 16 875

56

135√3re− 3re− 4050 ≥ 16 875

56

135√3re− 3re− 243 675

56≥ 0

45√3re− re− 81225

56≥ 0. (4.30)

45√3re−re− 81225

56= 0 for re = 942.89, and firm 1’s value as a Stackelberg

price leader rises with re from this value.

Chapter 5

Organization

14/12/01: I now believe that there is an additional avenue through whichsunk cost may affect equilibrium market structure. When some part of costsare sunk, entry creates excess capacity and makes the shadow value of fixedassets, at least for a time, equal to zero. This reduction in incumbents’ unitcost 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 operatewith production function

q = min

·K −K

aK,L− L

aL

¸for K ≥ K, L ≥ L, and q = 0 otherwise, where K is a minimum amountof physical capital needed to produce at all, L is a minimum amount oflabor needed to produce at all, and aK, aL are capital and labor input-output coefficients, respectively. Thus if production is efficient in the senseof minimizing cost, so that a firm employs no excess capital or labor,

q =K −K

aK=

L− L

aL

and input levels are

K = K + aKq L = L+ aLq.

Firms hire labor at wage rate w per period and purchase physical capitalat price pk; for simplicity, assume both input prices are constant over time,and assume also that physical capital does not depreciate. The rental rateof 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 capitalin the firm). If the firm wishes to resell a unit of physical capital, it cando so at price αpk, where the cost-sunkenness parameter α is a number thatlies between 0 and 1. If α = 0, investments in the industry are completelysunk, in the sense that if the firm should wish to exit the industry, it wouldnot 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

q = min

·K − 160

1,L− 201

¸so that to produce at all requires hiring at least one hundred and eighty unitsof capital and twenty units of labor, and that each unit of output requiresone additional unit of capital and one additional unit of labor over theseminimum 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 isrpk + 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 berecovered 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 iffirms produce efficiently? How does the level of fixed cost affect the long-run equilibrium number of firms? How does the level of sunk cost affect thelong-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 = rpkK + wL+¡rpkaK + 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,

qCourn =a− c

n+ 1.

Equilibrium per-firm profit per period is

πCourn =¡qCourn

¢2 − F =

µa− c

n+ 1

¶2− F.

If entry occurs until equilibrium per-firm profit per period is driven tozero, the Cournot equilibrium number of firms isµ

a− c

n+ 1

¶2− F = 0

a− c

n+ 1=√F.

nCour =a− c√

F− 1.

For this particular problem,

nCour =90√900− 1 = 2.


The equilibrium number of firms is two. The equilibrium number of firmsfalls as fixed costs rise, and the equilibrium number of firms is not affectedby 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 capitalservices is

ρ = ρ(α), with ρ(1) = r, ρ0 < 0.

The opportunity cost to a firm of investing in an industry is the amountit must pay to borrow financial capital. The resale value of physical capitalis 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), thelower the value of this collateral, all else equal, and the greater the interestrate 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

nCour(α) =a− c√

F− 1 = a− £ρ(α)pkaK + waL

¤qρ(α)pkK + wL

− 1.

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. Supposethe 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

πm = (100− 10− q1) q1 − 900.This is maximized for qm = 45 units of output, resulting in a profit

πm = (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 ifit 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 outputq (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-discountedvalue of its cost is

−αpk(K − aKq) +w¡L+ aLq

¢1 + r

+w¡L+ aLq

¢(1 + r)2

+w¡L+ aLq

¢(1 + r)3

+ . . . =

−αpkK + αpkaKq +w¡L+ aLq

¢r

=

−αpkK + αpkaKq +wL+ waLq

r=

−αrpkK + wL

r+

αrpkaKq + waLq

r=

F (K,α) + cαq

r,

where the first firm’s marginal cost per period is

cα = αrpkaK + waL.

For this problem

F (K,α) = −µ1

2

¶µ1

10

¶(50)(205) + (5)(20) = −412.5

cα =

µ1

2

¶µ1

10

¶(50)(1) + (5)(1) = 7.5.

(c) if the post-entry market is a Cournot duopoly, what is the secondfirm’s equilibrium profit? How do changes in the extent to which costs aresunk affect the second firm’s post-entry profit?If entry occurs, the entrant has marginal cost

c = rpkaK + waL = 10.

The system of equations of the best response functions, written in matrixform, is µ

2 11 2

¶µq1q2

¶=

µa− cαa− c

¶3

µq1q2

¶=

µ2 −1−1 2

¶µa− cαa− c

¶q2 =

1

3[2(a− c)− (a− cα)]


q2 =1

3[2(90)− 92.5] = 175

6= 29

1

6.

The entrant’s equilibrium profit per period is

q22 − F =µ175

6

¶2− 900 = −1775

36= −4911

36.

In general, the entrant’s profit per period is less than or equal to zero for

1

9(a+ cα − 2c)2 −

¡rpkK + wL

¢ ≤ 0(a+ cα − 2c)2 ≤ 9

¡rpkK + wL

¢a+ cα − 2c ≤ 3

qrpkK + wL

2c− cα ≥ a− 3qrpkK + wL

2¡rpkaK + waL

¢− ¡αrpkaK + waL¢ ≥ a− 3

qrpkK + wL

(2− α)rpkaK + waL ≥ a− 3qrpkK + wL.

This condition is more likely to be met, the smaller is α (the more costsare sunk), the smaller is a (an indicator of the size of the market), and thelarger are the entrant’s fixed costs rpkK + 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 + eq3.

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

π1 = [a− b (q1 +Q−1)] q1 − F − cq1 + dq21 − eq31

= [a− c− b (q1 +Q−1)] q1 − F + dq21 − eq31

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 − 3eq21 ≡ 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, firm1’s payoff is

π1 = (b− 2d+ 3eq1) q21 + dq21 − eq31 − F

= [b− 2d+ 3eq1 + d− eq1] q21 − F

= (b− d+ 2eq1) q21 − F.

Given the symmetry that characterizes this problem, in equilibrium allfirms produce the same output. Substitute q1 = q, Q−1 = (n− 1) q in theequation of firm 1’s best response function and solve for equilibrium outputwith n firms in the market:

a− c− b (n+ 1) q + 2dq − 3eq2 = 0

3eq2 + [(n+ 1) b− 2d] q − (a− c) = 0

q(n) =− [(n+ 1) b− 2d] +

q[(n+ 1) b− 2d]2 + 12e (a− c)

6e(5.1)

The equilibrium payoff per firm is

π = (b− d+ 2eq) q2 − F.

and the number of firms adjusts until π = 0:

(b− d+ 2eq) q2 − F = 0.

For F > 0, this is a cubic equation with one real root. To find the generalsolution, 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.


Substituting in (5.1)

− [(n+ 1) b− 2d] +q[(n+ 1) b− 2d]2 + 12e (a− c)

6e=

d− b

2e

− [(n+ 1) b− 2d] +q[(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

4e

µa− c

d− b

¶= 2 [(n+ 1) b− 2d] + 3 (d− b)

2 [(n+ 1) b− 2d] = 4eµa− c

d− b

¶− 3 (d− b)

(n+ 1) b− 2d = 2eµa− c

d− b

¶− 32(d− b)

(n+ 1) b = 2e

µa− c

d− b

¶+ 2d− 3

2(d− b)

(n+ 1) b = 2e

µa− c

d− b

¶+1

2d+

3

2b

n+ 1 = 2e

b

µa− c

d− b

¶+1

2

d

b+3

2

n =1

2+ 2

e

b

µa− c

d− b

¶+1

2

d

b. (5.2)

With F = 0, the long-run Cournot equilibrium number of firms is thegreatest integer less than the right-hand side of (5.2).5—4 (Equilibrium number of firms, Cournot oligopoly, differentiatedproducts) For a price-setting oligopoly with product differentiation, let theequations of the inverse demand curves be

pi = 100− (qi + θQ−i) ,

for i = 1, 2, ..., n and Q−i =Pn

j 6=i qj, with the equation of the firm-level costfunction

c(q) = F + 10q + dq2.

35

Find the equilibrium number of firms if the long-run equilibrium number offirms adjusts until Cournot equilibrium profit per firm is zero. How doesthe equilibrium number of firms change as θ changes?Firm 1’s profit function is

π1 = p1q1 −¡F + 10q1 + dq21

¢= (p1 − 10) q1 − F − dq21

=

"90− (1 + d)q1 − θ

nX2

qj

#q1 − F.

The first-order condition to maximize π1 is

90− 2(1 + d)q1 − θnX2

qj ≡ 0,

from which

90− (1 + d)q1 − θnX2

qj ≡ (1 + d)q1

andπ1 = (1 + d)q21 − 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 = 90q =

90

2(1 + d) + (n− 1)θ .

Equilibrium profit per firm is

π = (1 + d)

·90

2(1 + d) + (n− 1)θ¸2− F

and this is zero for ·90

2(1 + d) + (n− 1)θ¸2=

F

1 + d

90

2(1 + d) + (n− 1)θ =r

F

1 + d


2(1 + d) + (n− 1)θ = 90r1 + d

F

nCour = 1 +1

θ

"90

r1 + d

F− 2(1 + d)

#

= 1 +1

θ

"90

qmes

r1 +

1

d− 2(1 + d)

#,

for qmes =q

Fd. The Cournot long-run equilibrium number of firms rises as

θ falls – as products become more differentiated – and falls as qmes or drise.5—5 (Equilibrium number of firms, Bertrand oligopoly, differenti-ated products)For a price-setting oligopoly with product differentiation, let the equa-

tions of the demand curves be

qi =90(1− θ)− [1 + (n− 2)θ](pi − 10) + θ

Pnj 6=i(pj − 10)

(1− θ)[1 + (n− 1)θ] ,

with the equation of the firm-level cost function

c(q) = F + 10q + dq2.

Find the equilibrium number of firms if the long-run equilibrium number offirms adjusts until Bertrand equilibrium profit per firm is zero. How doesthe equilibrium number of firms change as θ changes?Firm 1’s profit function is

π1 = p1q1 −¡F + 10q1 + dq21

¢= (p1 − 10− dq1) q1 − F

For notational simplicity, write

xi = pi − 10.

π1 =

½x1 − d

90(1− θ)− [1 + (n− 2)θ]x1 + θPn

2 xj(1− θ)[1 + (n− 1)θ]

¾×½

90(1− θ)− [1 + (n− 2)θ]x1 + θPn

2 xj(1− θ)[1 + (n− 1)θ]

¾− F.

Collect the terms in x1 within the first set of braces on the right:

x1 − d90(1− θ)− [1 + (n− 2)θ]x1 + θ

Pn2 xj

(1− θ)[1 + (n− 1)θ] =

37·1 + d

1 + (n− 2)θ(1− θ)[1 + (n− 1)θ]

¸x1 − d

90(1− θ) + θPn

2 xj(1− θ)[1 + (n− 1)θ]

Thenπ1 =½·

1 + d1 + (n− 2)θ

(1− θ)[1 + (n− 1)θ]¸x1 − d

90(1− θ) + θPn

2 xj(1− θ)[1 + (n− 1)θ]

¾×½

90(1− θ) + θPn

2 xj(1− θ)[1 + (n− 1)θ] −

1 + (n− 2)θ(1− θ)[1 + (n− 1)θ]x1

¾− F.

Again for notational compactness, write this as

π1 = [(1 + dA1)x1 − dB1] [B1 −A1x1]− F

for

A1 =1 + (n− 2)θ

(1− θ)[1 + (n− 1)θ]

B1 =90(1− θ) + θ

Pn2 xj

(1− θ)[1 + (n− 1)θ]For future reference, note that in this notation

q1 = B1 − A1x1

π1 = [(1 + dA1)x1 − dB1] [B1 −A1x1]− F

−(1 + dA1)A1x21 + (1 + 2dA1)B1x1 − dB2

1 − F

The first-order condition to maximize π1 is

−2(1 + dA1)A1x1 + (1 + 2dA1)B1 ≡ 0.

Note that if the first-order condition holds, then

q1 = B1 −A1x1 = A1 [(1 + 2dA1) x1 − 2dB1] .

The first-order condition can be solved for the equation of firm 1’s pricebest response function, although that is not of immediate interest in thepresent context. Rather, write the first-order condition as

2(1 + dA1)A1x1 = (1 + 2dA1)B1

2(1 + dA1)A1x1 = (1 + 2dA1)90(1− θ) + θ

Pn2 xj

(1− θ)[1 + (n− 1)θ]


In equilibrium, all firms will set the same price; let xi = x for all i; thenthe first-order condition becomes

2(1 + dA1)A1x = (1 + 2dA1)

·90

1 + (n− 1)θ +(n− 1)θ

(1− θ)[1 + (n− 1)θ]x¸

or2(1 + dA1)A1x = (1 + 2dA1) (C1 +D1x)

forC1 =

90

1 + (n− 1)θ

D1 =(n− 1)θ

(1− θ)[1 + (n− 1)θ]Note that in equilibrium

B1 =90(1− θ) + θ

Pn2 xj

(1− θ)[1 + (n− 1)θ] = C1 +D1x.

Solve the condensed first-order condition for x:

2(1 + dA1)A1x = (1 + 2dA1) (C1 +D1x)

2(1 + dA1)A1x = (1 + 2dA1)C1 + (1 + 2dA1)D1x

[2(1 + dA1)A1 − (1 + 2dA1)D1] x = (1 + 2dA1)C1£2A1 −D1 − 2dA1D1 + 2dA

21

¤x = (1 + 2dA1)C1

x =(1 + 2dA1)C1

[2A1 −D1 − 2dA1D1 + 2dA21]

Numerator:

(1 + 2dA1)C1 =

½1 +

2d [1 + (n− 2)θ](1− θ)[1 + (n− 1)θ]

¾90

1 + (n− 1)θDenominator:

2A1 −D1 − 2dA1D1 + 2dA21 =

D1 + 2(A1 −D1) + 2dA1 (dA1 −D1) =

(n− 1)θ(1− θ)[1 + (n− 1)θ] +

2

[1 + (n− 1)θ]µ1 + d

1 + (n− 2)θ(1− θ)[1 + (n− 1)θ]

¶=

1

[1 + (n− 1)θ]½(n− 1)θ(1− θ)

+ 2

·1 + d

1 + (n− 2)θ(1− θ)[1 + (n− 1)θ]

¸¾

39

x =

n1 + 2d[1+(n−2)θ]

(1−θ)[1+(n−1)θ]o

901+(n−1)θ

1[1+(n−1)θ]

n(n−1)θ(1−θ) + 2

h1 + d 1+(n−2)θ

(1−θ)[1+(n−1)θ]io

=90n1 + 2d[1+(n−2)θ]

(1−θ)[1+(n−1)θ]o

(n−1)θ(1−θ) + 2

h1 + d 1+(n−2)θ

(1−θ)[1+(n−1)θ]i

Equilibrium profit per firm is

π = [(1 + dA1)x− dB1] [B1 − A1x]− F

= [(1 + dA1)x− d (C1 +D1x)] [C1 +D1x−A1x]− 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− 1)θ −·

1 + (n− 2)θ(1− θ)[1 + (n− 1)θ] −

(n− 1)θ(1− θ)[1 + (n− 1)θ]

¸x =

90

1 + (n− 1)θ −·

1− θ

(1− θ)[1 + (n− 1)θ]¸x =

90− x

1 + (n− 1)θ

π =

·x− d

90− x

1 + (n− 1)θ¸

90− x

1 + (n− 1)θ − F

Expressing x in terms of the number of firms, the long-run equilibriumnumber of firms satisfies the equation:

90

90n1 + 2d[1+(n−2)θ]

(1−θ)[1+(n−1)θ]o

(n−1)θ(1−θ) + 2

h1 + d 1+(n−2)θ

(1−θ)[1+(n−1)θ]i − 90d1−

1+2d[1+(n−2)θ]

(1−θ)[1+(n−1)θ](n−1)θ(1−θ) +2[1+d

1+(n−2)θ(1−θ)[1+(n−1)θ] ]

1 + (n− 1)θ

×1− 1+ 2d[1+(n−2)θ]

(1−θ)[1+(n−1)θ](n−1)θ(1−θ) +2[1+d

1+(n−2)θ(1−θ)[1+(n−1)θ] ]

1 + (n− 1)θ = F.

This can be reduced to a quartic equation in n.If d = 0, the equation that determines n becomes

90

Ã90

(n−1)θ(1−θ) + 2

! 1− 1(n−1)θ(1−θ) +2

1 + (n− 1)θ − F = 0


Ã1

(n−1)θ(1−θ) + 2

! µ (n−1)θ(1−θ) +2−1(n−1)θ(1−θ) +2

¶1 + (n− 1)θ =

F

(90)2

(n−1)θ(1−θ) + 1

1 + (n− 1)θ =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 asCournot 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 actingas 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,

MR1 = (100− q2 − q3)− 2q1 = 10 = mc1

q1 =1

2(90− q2 − q3) (5.6)

where qc = 90 is the quantity demanded in perfectly competitive long-runequilibrium. (5.6) is the equation of a plane in (q1, q2, q3)-space: the planeconnecting 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 placedefined by (5.6) and a plane defined by the vertical (q3) axis and the 45-degree line in the (q1, q2) plane.

41

In the particular case of this problem, the three firms are identical, sofirm 1 and firm 2 will produce the same output in equilibrium. There is noloss of generality in restricting q1 to be equal to q2 outside of equilibrium.If q1 = q2 = q12, (5.6) becomes

q12 =1

2(90− q12 − q3)

3

2q12 =

1

2(90− q3)

q12 =1

3(90− q3) (5.7)

In the same way, the equation of firm 3’s best response function whenq1 = q2 = q12 becomes

q3 =1

2(90− q1 − q2)

q3 =1

2(90− 2q12) = 45− q12. (5.8)

Cournot equilibrium output with three identical firms is

90

4= 22.5; (5.9)

price is

10 +90

4= 32.5; (5.10)

profit per firm is(22.5)2 = 506.25, (5.11)

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

π12 = (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-mergerfirm’s best response function. One is simply to calculate the first-ordercondition to maximize π12 with respect to q1:

90− 2q1 − 2q2 − q3 ≡ 0

q1 =1

2(90− 2q2 − q3). (5.13)


Alternatively, and perhaps with a more direct economic interpretation,consider the post-merger firm’s marginal revenue if division 1 produces anadditional unit of output:

MR1 = p− q1 − q2. (5.14)

If the post-merger firm sells an extra unit of output by way of division1, it gains the revenue from sale of that unit (p), but price falls by 1. This1 price reduction lowers the firm’s revenue on its sales from division 1 andfrom 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

q12 =1

4(90− q3). (5.16)

Firm 3’s best response function has not changed; it continues to haveequation (5.8).Find post-merger equilibrium outputs by solving the equations of the best

response functions, (5.8) and (5.16).

q3 = 45− q12

q12 =1

4(90− q3).

q3 = 45− 14(90− q3)

3

4q3 = 45− 90

4=180− 90

4=90

4

q3 = 30

q12 =1

4(90− 30) = 15.

First, the post-merger market is a Cournot duopoly. Firm 3 produces30 units of output in equilibrium, and divisions 1 and 2 together produce 30units 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 thetwo 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 satis-

factory: we do not expect a post-merger firm to restrict output so much afterthe 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 cannotcontrol the behavior of firms outside the merger, and that the reactions ofthose firms may reduce the profitability of the merger.Other remarks:

if there are many firms in the premerger market, and the merger com-bines a large number of them (roughly, 80% or more), the merger willbe profitable for the firms that carry it out;

if products are differentiated and firms set prices, mergers are generallyprofitable (with linear demand and constant marginal cost); we will notconsider this kind of model formally.

Firm 3......................................................................................................................................... Firm 1/Firm 2 (pre-merger)........................................................................................................................................................................................

............................

Division 1/Division 2(post-merger)....................................................................................................................................................................

............................

•

•

•

• •

•

q1 = q2

q322.5 30 45 90

45

30

22.5

15

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 5.1: Pre- and post-merger best response curves, Cournot quantity-setting oligopoly.

Chapter 6

Innovation

This chapter intentionally left blank.

45

46 CHAPTER 6. INNOVATION

Chapter 7

Imperfect Competition andInternational Trade: I

22 November 2002: On pages 151-3 there is an argument that trade im-proves welfare for countries of equal size. Professor David Collie of theCardiff Business School, to whom I am grateful, writes to point out that thisargument is correct only if transportation cost is zero. With sufficientlygreat transportation cost, the opening up of trade may leave each countryworse off. A formal demonstration now appears at the end of the answer toProblem 7—1 .7—1 Let there be two countries, each home to one widget producer. Thesubscript 1 denotes both country 1 and its widget company; the subscript2 denotes both country 2 and its widget company. Let the inverse demandcurves 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 thequantity of widgets sold by firm i in country j, for i, j = 1, 2. The a andb parameters are respectively the price-axis intercept and the absolute valueof 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 fromone 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)

π2 = [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 ofthe amounts they sell in the other country.

This follows from the fact that the derivative of π1 with respect to q11depends only on country 1 variables, similarly for ∂π1/∂q12, and similarly forthe 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

2q11 + q21 =a1 − c

b1(7.4)

q11 + 2q21 =a1 − c− t

b1(7.5)

or equivalently

q11 =1

2(a1 − c− q21) (7.6)

q21 =1

2(a1 − c− t− q11) (7.7)

Neither firm would sell below its marginal cost. This means that theequation of firm 1’s best response function is valid only for combinations ofq11 and q21 that result in prices greater than or equal to c and the equationof firm 2’s best response function is valid only for combinations of q11 and q21that result in prices greater than or equal to c + t. This implies restrictionsthat are not worked out here.

(d) Solve the equations of the best response functions for equilibrium outputsin country 1.

b1

µ2 11 2

¶µq11q21

¶= (a1 − c)

µ11

¶− t

µ01

¶

3b1

µq11q21

¶= (a1 − c)

µ2 −1−1 2

¶µ11

¶− t

µ2 −1−1 2

¶µ01

¶3b1

µq11q21

¶= (a1 − c)

µ11

¶− t

µ −12

¶µ01

¶q∗11 =

a1 − c

3b1+

t

3b1(7.8)

49

q∗21 =a1 − c

3b1− 2t

3b1(7.9)

(e) What restriction on transportation cost applies if firm 2 is to sell incountry 1?

q∗21 must be nonnegative,

q∗21 =a1 − c

3b1− 2 t

3b1> 0,

and this is the case ift <

1

2(a1 − c). (7.10)

That is, transportation cost cannot exceed the monopoly price-cost mar-gin if the country 2 firm is to sell in country 1. A corresponding conditionmust hold if the country 1 firm is to sell in country 2.

(f) What is equilibrium price in country 1? Compare the equilibrium pricewith each firm’s marginal cost of supplying country 1. (This part of theexercise relates to the analysis of dumping.)

Sales in country 1 are

q∗11 + q∗21 =2(a1 − c)− t

3b1(7.11)

Hence equilibrium price is

p1 = a1 − b12(a1 − c)− t

3b1

= c+1

3(a1 − c) +

1

3t (7.12)

= (c+ t) +2

3

µa1 − c

2− t

¶(7.13)

Price is greater than c, firm 1’s cost of serving its home market. Priceis greater than c + t, firm 2’s cost of serving country 1, if condition (7.10)(which is the condition for firm 2 to sell in country 1) is met.22 November 2002With the opening up of trade, firm 1’s profit in country 1 is

(p1 − c) q∗11 =1

9b1(a1 − c+ t)2


To write out an expression for firm 1’s profit in country 2, first write outan 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

[p1 − (c+ t)] q∗21 =2

3

µa1 − c

2− t

¶µa1 − c

3b1− 2 t

3b1

¶

=4

9b1

µa1 − c

2− t

¶2.

Hence with the opening up of trade, firm 1’s profit in country 2 is

4

9b2

µa2 − c

2− t

¶2.

Firm 1’s total profit with the opening up of trade is

1

9b1(a1 − c+ t)2 +

4

9b2

µa2 − c

2− t

¶2.

We limit attention to the case in which markets are of the same size; dropthe country-specific subscripts:

1

9b(a− c+ t)2 +

4

9b

µa− c

2− t

¶2.

Without trade, firm 1 was a monopolist in country 1; price, output, andprofit were

p = c+1

2(a− c)

q =1

2

a− c

b1

4b(a− c)2 .

The reduction in profit with the opening up of trade is

1

4b(a− c)2 − 1

9b(a− c + t)2 − 4

9b

µa− c

2− t

¶2=

1

36b(a− c+ 10t) (a− c− 2t) =

1

18b(a− c+ 10t)

µa− c

2− t

¶> 0.

51

Trade leaves each firm with lower profit.Before trade, consumer surplus in country 1 would be

1

2(a− p) q =

1

2(a− a+ bq) q =

1

2bq2 =

1

2b

µa− c

2

¶2=

1

8b(a− c)2 .

With trade, consumer surplus in country 1 is (using the expressions forpost-trade price and output in country 1, and eliminating the country-specificsubscripts)

1

2

½a−

·a− 2(a− c)− t

3

¸¾2(a− c)− t

3b=

1

2

·a− a+

2(a− c)− t

3

¸2(a− c)− t

3b=

1

2

·2(a− c)− t

3

¸2(a− c)− t

3b=

1

2b

·2(a− c)− t

3

¸2=

2

9b

µa− c− 1

2t

¶2.

The change in consumer surplus with the opening up of trade is

2

9b

µa− c− 1

2t

¶2− 1

8b(a− c)2 =

[7 (a− c)− 2t] (a− c− 2t)72b

7

36b

µa− c− 2

7t

¶µa− c

2− t

¶> 0.

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

7

36b

µa− c− 2

7t

¶µa− c

2− t

¶− 1

18b(a− c+ 10t)

µa− c

2− t

¶=


1

18b

·7

2

µa− c− 2

7t

¶− (a− c+ 10t)

¸µa− c

2− t

¶=

1

18b

·7

2(a− c)− t− (a− c)− 10t

¸µa− c

2− t

¶=

1

18b

·5

2(a− c)− 11t

¸µa− c

2− t

¶=

5

18b

·a− c

2− 115t

¸µa− c

2− t

¶.

Welfare falls fora− c

2− 115t < 0

t >5

11

a− c

2=5

22(a− c) .

7—2 Analyze the Cournot duopoly trade model for general demand curves,

p1 = p1(q11 + q21)p2 = p2(q12 + q22)

. (7.14)

Payoffs are

π1 = [p1(q11 + q21)− c]q11 + [p2(q12 + q22)− (c+ t)]q12 − F (7.15)

π2 = [p1(q11 + q21)− (c+ t)]q21 + [p2(q12 + q22)− c)]q22 − F (7.16)

The first-order conditions for profit-maximization areFirm 1, country 1:

∂π1∂q11

= p1(q1)− c + q11dp1dq1

= 0 (7.17)

Firm 1, country 2:

∂π1∂q12

= p2(q2)− (c+ t) + q12dp2dq2

= 0 (7.18)

Firm 2, country 1:

∂π2∂q21

= p1(q1)− (c+ t) + q21dp1dq1

= 0 (7.19)

53

Firm 2, country 2:

∂π2∂q22

= p2(q2)− c+ q22dp2dq2

= 0 (7.20)

Second-order conditions must be satisfied, as well as conditions to ensurestability. These are not dealt with here.Equation (7.17), the first-order condition for the domestic firm in its home

market can be rewrittenp1 − c

p1=

s11ε1

, (7.21)

where s11 = q11/q1 is firm 1’s market share in its home market. This willbe recognized from Chapter 4 as the generalization of the Lerner index ofmonopoly power to the case of different production costs.In the same way, for firm 2 in country 1 one obtains

p1 − (c + t)

p1=

s21ε1

(7.22)

Solving (7.21) and (7.22) for p1 gives

p1 =ε1

ε1 − s11c (7.23)

andp1 =

ε1ε1 − s21

(c+ t), (7.24)

respectively.(7.23) and (7.24) can be solved for s21 and p1,

s21 =1 + t

c(1− ε1)

2 + tc

=1 + t

c(1− ε1)

2 + tc

(7.25)

andp1 =

ε12ε1 − 1(2c+ t). (7.26)

From the numerator on the right in (7.25), the condition for firm 2 to havea positive market share in country 1 – this is the condition for intra-industrytrade to occur – is

t

c<

1

ε1 − 1 (7.27)

(transportation cost cannot be too high) or

ε1 < 1 +1

t/c=

c + t

t(7.28)


(the price elasticity of demand cannot be too great).From (7.26),

p1 − (c+ t) =

µε1 − 12ε1 − 1

¶c

µ1

ε1 − 1 −t

c

¶. (7.29)

Examining the final term in parentheses on the right, the condition forfirm 2 to have a positive market share in country 1, (7.27), is also the con-dition for the country 1 price to exceed firm 2’s marginal cost of supplyingcountry 1, c+ t.

7-3 (a) Answer Problem 7—1 if firms set prices rather than quantities. Sup-pose that products are differentiated, with demand curves in country i givenby equations

p1i = ai − bi(q1i + θq2i)p2i = ai − bi(θq1i + q2i)

, (7.30)

where the first subscript denotes the firm and the second, i = 1, 2, denotesthe country, 0 ≤ θ < 1, with average and marginal cost c and transportationcost t per unit as in Problem 7—1.

First solve the equations of the inverse demand curves to obtain equationsfor the demand curves, expressing the quantity demanded of each variety asa 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)

π21 = (p21 − c− t)q21 (7.32)

it is convenient to rewrite (7.30) so that prices are expressed as deviationsfrom 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 q21as functions of p11− c andp21−c− t. There are several ways to do this; the method presented here useslinear algebra. Write the equations of the inverse demand curves in matrixform asµ

p11 − cp21 − c− t

¶=

µa1 − c

a1 − c− t

¶− b

µ1 θθ 1

¶µq11q21

¶, (7.34)

55

from which

b

µ1 θθ 1

¶µq11q21

¶=

µa1 − c

a1 − c− t

¶−µ

p11 − cp21 − c− t

¶. (7.35)

Using the formula for the inverse of a 2× 2 matrix,µα βγ δ

¶−1=

1

αδ − βγ

µδ −β−γ α

¶, (7.36)

(which is valid provided the determinant αδ − βγ 6= 0), one obtains expres-sions for the quantities demanded,

b(1− θ2)

µq11q21

¶=

µ1 −θ−θ 1

¶µa1 − c

a1 − c− t

¶−µ

1 −θ−θ 1

¶µp11 − c

p21 − c− t

¶.

=

·(a1 − c)− θ(a1 − c− t)(a1 − c− t)− θ(a1 − c)

¸−·p11 − c− θ(p21 − c− t)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)(7.38)

q21 =(1− θ)(a1 − c)− t− (p21 − c− t) + θ(p11 − c)

b(1− θ2)(7.39)

First examine firm 1’s behavior. Substituting from (7.38) into (7.31), π11satisfies

b(1− θ2)π11 =

= (p11 − c)[(1− θ)(a1 − c) + θt− (p11 − c) + θ(p21 − c− t)] (7.40)

The first-order condition to maximize π11with 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 pricebest response function for country 1,

p11 − c =1

2[(1− θ)(a1 − c) + θ(p21 − c)] (7.42)


Note that the term θt drops out: transportation cost, paid by firm 2, doesnot directly affect firm 1; it affects firm 1 only insofar as it affects firm 2’sprice.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, theslope is 2/θ.)Proceeding in the same way for firm 2,

b(1− θ2)π21 =

= (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)

p21 − c− t =1

2[(1− θ)(a1 − c)− t+ θ(p11 − c)] (7.45)

This is the equation of a straight line with positive slope θ/2. Collectingterms in t on the right-hand side,

p21 − c =1

2[(1− θ)(a1 − c) + θ(p11 − c)] +

1

2t, (7.46)

the greater is unit transportation cost, the greater the price firm 2 will chargefor 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 p21that imply q11 ≥ 0 and the equation of firm 2’s best response function isvalid only for combinations of p11 and p21 that imply q21 ≥ 0. This impliesrestrictions 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 equationsµ2 −θ−θ 2

¶µp11 − c

p21 − c− t

¶= (1− θ)(a1 − c)

µ11

¶+ t

µθ1

¶. (7.47)

This can be solved for equilibrium prices,

(4−θ2)µ

p11 − cp21 − c− t

¶= (1−θ)(a1−c)

µ2 θθ 2

¶µ11

¶+t

µ2 θθ 2

¶µθ1

¶

= (1− θ)(2 + θ)(a1 − c)

µ2 θθ 2

¶µ11

¶+ t

·θ

−(2− θ2)

¸,

57

p21

p11

(1−θ)(a1−c)+t2

1− θ

2(a1 − c)

¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤

»»»»»»

»»»»»»

»»»»»»

»»»»»»

»»r

Firm 1’s best responsefunction, country 1 @

@@R

Firm 2’s best responsefunction, country 1 @

@@R

Figure 7.1: Price best response functions, country 1, Bertrand duopoly trademodel


from which

p∗11 − c =(1− θ)(2 + θ)(a1 − c) + θt

4− θ2=1− θ

2− θ(a1 − c) +

θ

4− θ2t (7.48)

p∗21 − c− t =(1− θ)(2 + θ)(a1 − c) + 2

4− θ2=1− θ

2− θ(a1 − c)− 2− θ2

4− θ2t. (7.49)

Firm 1’s equilibrium price-cost margin rises, and firm 2’s equilibrium pricefalls, 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 quantitiesdemanded satisfy

q∗11 =p∗11 − c

b(1− θ2)(7.50)

q∗21 =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 isthat the price of firm 2’s variety in country 1 exceed firm 2’s cost of sellingin country 1, p∗21 > c− t. From (7.49), this translates into

(2− θ2)(a1 − c− t)− θ(a1 − c) > 0

2− θ − θ2

2− θ2(a1 − c) > t

(1− θ)(2 + θ)

2− θ2(a1 − c) > t (7.52)

As expected, and as for the quantity-setting model, the condition fortwo-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 andvarieties 1 and 2 become poorer substitutes one for the other, firm 2 can bearhigher transportation cost and still profitably sell in country 1.A representative consumer utility function that produces the demand

curves (7.30) is

U = m+ ai(q1i + q2i)− 12bi(q

21i + 2θq1iq2i + q22i), (7.53)

where m represents consumption on other goods.Analyze the welfare effects of trade.

59

For simplicity, suppose transportation cost is zero and that the inversedemand curves in the two markets have the same intercept and slopes, sothat 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 levelq1 is the sum of consumers’ surplus and economic profit,

NSW1 = aq1 − 12bq21 − p11q11 + (p11 − c)q21

= (a− c)q1 − 12bq21, (7.54)

as shown in Figure 7.2:Evaluating this at the monopoly output

qm =1

2

a− c

b, (7.55)

NSWm1 =

1

8

(a− c)2

b. (7.56)

With trade, consumers’ surplus in country 1 is

CS1 = a(q1i + q2i)− 12b(q21i + 2θq1iq2i + q22i)− (p11q11 + p21q21). (7.57)

Firm 1’s profit is

π1 = (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 firm1’s profit in country 2. Given the symmetry assumptions we have made, inequilibrium firm 1’s sales and profit in country 2 are equal to firm 2’s salesand profit in country 1. Equilibrium net social welfare in country 1 is then

NSW1 = (a− c)(q1i + q2i)− 12b(q21i + 2θq1iq2i + q22i). (7.59)

With product differentiation, net social welfare is not susceptible to straight-forward graphical illustration: additional output of one variety to some extentreduces welfare generated by other varieties.Once again in equilibrium, each firm sells the same output in country 1,

say q1. (7.59) becomes

NSW1 = 2(a− c)q1 − b(1 + θ)q21 = b

·2a− c

b− (1 + θ)q1

¸q1. (7.60)


p1

q1

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

a1

CS

q

π

c

Figure 7.2: Net social welfare, monopoly, country 1

61

Evaluating (7.47) for the case of zero transportation cost, equilibriumprice is

p1 − c =1− θ

2− θ(a− c). (7.61)

Using (7.50) to find equilibrium output per firm gives

q1 =p1 − c

b(1− θ2)=

1

(1− θ)(2− θ)

a− c

b. (7.62)

Then substituting (7.62) into (7.60), in equilibrium country 1 net socialwelfare with trade is

NSW1 = b

·2a− c

b− 1 + θ

(1− θ)(2− θ)

a− c

b

¸1

(1− θ)(2− θ)

a− c

b

= b

·2− 1 + θ

(1− θ)(2− θ)

¸1

(1− θ)(2− θ)

µa− c

b

¶2= b

·2− θ

(1− θ)(2− θ)

¸1

(1− θ)(2− θ)

µa− c

b

¶2=

1

(1− θ)2(2− θ)

(a− c)2

b. (7.63)

Comparing (7.56) and (7.63), the change in net social welfare with theopening up of trade is proportional to

1

(1− θ)2(2− θ)− 18. (7.64)

This is positive for θ = 0. The first term on the right rises as θ approaches1. 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 produceone unit of food and one unit of machinery implied by the input-output tableIO-1 (page 9)? If there are 100 units of labor in the economy, what is theequation of its consumption possibility frontier?

Use the notation given in Table 7.1.The balance inequalities for food, machinery, and labor are

FFF + FFM + CF ≤ F (7.65)


F total food outputM total machinery outputFFF food used as input in the food industryFFM food used as input in the machinery industryMMF machinery used as input in the food industryMMM machinery used as input in the machinery industryCF food available for consumptionCM machinery available for consumptionL laborLF labor used as input in the food industryLM labor used as input in the machinery industry

Table 7.1: Notation

MMF +MMM + CM ≤M (7.66)

andLF + LM ≤ L (7.67)

respectively.Using the input-output coefficients in Table IO-2, (7.65) and (7.66) be-

come3

8F +

1

8M + CF ≤ F (7.68)

3

8F +

1

4M + CM ≤M (7.69)

Collecting terms, (7.68) and (7.69) hold with equality forµ5/8 −1/8−3/8 3/4

¶µFM

¶=

µCF

CM

¶(7.70)

µFM

¶=64

27

µ3/4 1/83/8 5/8

¶µCF

CM

¶=

µ16/9 8/278/9 40/27

¶µCF

CM

¶(7.71)

This gives the total outputs F and M of food and machinery needed forany menu CF and CM of food and machinery available for final consumption.

Then using the labor-output coefficients of Table IO-2, the balance in-equality of labor (7.67) becomesµ

3

4, 3

¶µFM

¶≤ L = 100. (7.72)

63

CF

CM

25

21.43

eeeeeeeeeeeeeee

4CF +143CM ≤ 100

¡¡

¡ª

Figure 7.3: Consumption possiblity frontier, Problem 10-8

If this holds with equality, substituting (7.71) gives the equation of theconsumption possibility frontier,µ

3

4, 3

¶µ16/9 8/278/9 40/27

¶µCF

CM

¶= 100. (7.73)

4CF +14

3CM = 100, (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 finalconsumption demand requires, directly and indirectly, 4 2/3 units of labor.

Chapter 8

Imperfect Competition andInternational Trade II

8—1 Firm 1, based in country 1, and firm 2, based in country 2, sell quantitiesq1and 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 mar-kets. The marginal and average cost of production and transportation, c, isconstant and the same for both firms.

(a) Find equilibrium outputs and profits in country 3 if there are no exportsubsidies.

This is a standard Cournot duopoly model. Firm 1’s profit is

π1 = [a− c− (q1 + q2)]q1 (8.2)

Taking the derivative of (8.2) with respect to q1, the equation of firm 1’sbest response function is

2q1 + q2 = a− c (8.3)

Substituting (8.3) into the term in brackets on the right in (8.2), it followsthat firm 1’s payoff anywhere along its best response function, and particularin equilibrium, is

π1 = q21 (8.4)

The equations of the best response functions of the two firms form thesystem of equations µ

2 11 2

¶µq1q2

¶= (a− c)

µ11

¶, (8.5)

65

66 CHAPTER 8. INTERNATIONAL TRADE II

with solutionq1 = q2 =

1

3(a− c). (8.6)

From (8.4), equilibrium firm profits are

π1 = π2 =1

9(a− c)2. (8.7)

(b) Find equilibrium outputs and profits in country 3 if country 1 grants itsfirm a subsidy s1 per unit sold in country 3. What subsidy is best for country1?

With an export subsidy, firm 1’s unit cost of supply country 3 is c− s1.The system of equations of best response functions becomesµ

2 11 2

¶µq1q2

¶= (a− c)

µ11

¶+ s1

µ10

¶, (8.8)

with solutionq1 =

1

3(a− c+ 2s1) (8.9)

q2 =1

3(a− c− s1) (8.10)

With an export subsidy, firm 1’s profit is

π1 =1

9(a− c+ 2s1)

2. (8.11)

The change in country 1’s net social welfare is

∆NSW1 =1

9(a− c+ 2s1)

2 − 13(a− c+ 2s1)s1 − 1

9(a− c)2 (8.12)

Taking the derivative of (8.12) with respect to s1 and rearranging termsgives the subsidy level that maximizes1 ∆NSW1:

s1 =1

4(a− c). (8.13)

Substituting (8.13) into (8.12), country 1’s optimal ∆NSW1 is

∆NSW1 =1

72(a− c)2 =

1

8

µa− c

3

¶2. (8.14)

1One must verify that the second-order condition for a maximum is satisfied, which itis.

67

With a country 1 subsidy given by (8.13), the change in country 2’s netsocial welfare is

∆NSW2 =1

9(a− c− s1)

2 − 19(a− c)2 = − 7

16

µa− c

3

¶2. (8.15)

(c) Find equilibrium outputs and profits in country 3 if country 1 grants itsfirm a subsidy s1 per unit sold and country 2 grants its firm a subsidy s2per unit sold in country 3. What are the equilibrium subsidies if the twocountries set subsidy levels noncooperatively (that is, if each country setsthe best possible subsidy level for itself, taking the subsidy level of the othercountry as given)?

If each country grants its own firm an export subsidy, the system ofequations of the best response functions isµ

2 11 2

¶µq1q2

¶= (a− c)

µ11

¶+

µs1s2

¶, (8.16)

with solution

q1 =1

3(a− c+ 2s1 − s2) (8.17)

q2 =1

3(a− c + 2s2 − s1). (8.18)


∆NSW1 = π1 − s1q1 − 19(a− c)2

= q21 − s1q1 − 19(a− c)2

=1

9(a− c+ 2s2 − s1)(a− c− s2 − s1)− 1

9(a− c)2. (8.19)

Taking the derivative of (8.20) with respect to s1, the equation of country1’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 (Fig-ure 8.1). There is a similar equation for country 2’s subsidy best responsefunction.


s2

s1

s∗2

s∗1

E0

AAAAAAAAAAAAAAAAAAAAAAA

HHHHHHHHHHHHHHHHHHHHHHH

Country 1’s subsidy best response function¢¢¢¢¢¢¢¢¢¢®

Country 2’s subsidy best response curve¢¢¢¢¢¢®

s

a−c4

a− c

a−c4

a− c

Figure 8.1: Subsidy best response functions

Either by symmetry or solving the equations of the subsidy best responsefunctions, equilibrium subsidies are

s1 = s2 =1

5(a− c). (8.21)

Evaluating ∆NSW1 for the equilibrium subsidies,

∆NSW1 = − 716

µa− c

3

¶2. (8.22)

7—2 Answer question 7—1 if products are differentiated, with inverse demandcurves

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 matrixform as µ

p1 − cp2 − c

¶= (a− c)

µ11

¶−µ1 θθ 1

¶µq1q2

¶, (8.25)

they can be inverted to obtain expressions for the demand curves,

(1− θ2)

µq1q2

¶= (1− θ)(a− c)

µ11

¶−·p1 − c− θ(p2 − c)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)π1 = (p1 − c)[(1− θ)(a− c)− (p1 − c) + θ(p2 − c)]. (8.27)

The first-order condition to maximize π1with 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 pricebest response functions, equilibrium prices without subsidies are

p1 − c = p2 − c =1− θ

2− θ(a− c). (8.30)

Using (8.30), equilibrium payoffs without subsidies are

π1 = π2 =1

1− θ2

·1− θ

2− θ(a− c)

¸2=

1− θ

(1 + θ)(2− θ)(a− c)2. (8.31)

(b) If country 1 grants its firm a subsidy, firm 1’s profit becomes

(1− θ2)π1 = (p1 − c+ s1)[(1− θ)(a− c)− (p1 − c) + θ(p2 − c)]. (8.32)

The system of equations of the best response functions isµ2 −θ−θ 2

¶µp11 − cp21 − c

¶= (1− θ)(a1 − c)

µ11

¶− s1

µ10

¶. (8.33)


Solving for equilibrium prices gives

p1 − c =1− θ

2− θ(a− c)− 2s1

4− θ2(8.34)

p2 − c =1− θ

2− θ(a− c)− θs1

4− θ2. (8.35)

A positive subsidy lowers the equilibrium prices of both varieties. How-ever

p1 − c+ s1 =1− θ

2− θ(a− c) +

2− θ2

4− θ2s1, (8.36)

so that a positive subsidy is privately beneficial for firm 1.With a subsidy, firm 1’s equilibrium profit is

π1 =1

1− θ2(p1 − c+ s1)

2. (8.37)

The change in firm 1’s net social welfare is

∆NSW1 =1

1− θ2

·1− θ

2− θ(a− c) +

2− θ2

4− θ2s1

¸2

− 1

1− θ2

·1− θ

2− θ(a− c) +

2− θ2

4− θ2s1

¸s1 − 1

1− θ2

·1− θ

2− θ(a− c)

¸2(8.38)

Taking the derivative of this with respect to s1 and solving the resultingfirst-order condition gives an expression for the “subsidy” that maximizesthe change in firm 1’s net social welfare,

s1 = −(1− θ)(2 + θ)θ2

4(2− θ2)(a− c) ≤ 0. (8.39)

The optimal subsidy, being negative, is in fact an export tax. An exporttax increases welfare if it induces both firms to raise price. If θ = 0, anexport tax by country 1 induces firm 1 to raise its price, but firm 2 does notalter its price; hence the optimal tax is zero.

(c) If country 1 imposes an export tax t1 on firm 1 and country 2 imposes anexport tax t2 on firm 2, the system of equations of best response functions isµ

2 −θ−θ 2

¶µp11 − cp21 − c

¶= (1− θ)(a1 − c)

µ11

¶+

µt1t2

¶. (8.40)

71

Firm 1’s equilibrium price is

p1 − c =1− θ

2− θ(a− c) +

2t1 + θt2

4− θ2. (8.41)

There is a similar expression for firm 2. The effect of the export taxes is toincrease equilibrium prices.Firm 1’s margin after the export tax is

p1 − c− t1 =1− θ

2− θ(a− c)− (2− θ2)t1 − θt2

4− θ2. (8.42)

Firm 1’s equilibrium payoff with the export taxes is

π1 =1

1− θ2(p1 − c− t1)

2 (8.43)


∆NSW1 = π1 + t1q1 − 1

1− θ2

·1− θ

2− θ(a− c)

¸2. (8.44)

Evaluating this at the equilibrium values and taking the derivative ofthe resulting expression with respect to t1 yields the equation of country 1’sexport 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)

8− 4θ2 − θ3(a− c). (8.46)

If (8.46) is used to evaluate (8.44), the equilibrium ∆NSW1 is

∆NSW1 = − 1

1− θ26− 3θ2 − θ3

8− 4θ2 − θ3

·1− θ

2− θ(a− c)

¸2. (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 country1 below the Cournot equilibrium level on outputs, prices, profits, and netsocial welfare in country 1.


Without transportation cost, this is a Cournot duopoly model in whichthe two firms have identical constant unit costs. Equilibrium outputs andprofits in country 1 are

q011 = q012 =a1 − c

3b1(8.48)

and

π011 = π012 = b1

µa1 − c

3b1

¶2(8.49)

respectively.A quota is binding if it holds firm 2’s sales below the Cournot equilibrium

level, i.e., if

q <a1 − c

3b1. (8.50)

Firm 1’s profit-maximizing output choice is described by its best responsefunction,

q11 =1

2

µa1 − c

b1− q21

¶. (8.51)

An aside: suppose the demand structure is altered to introduce someproduct differentiation. As long as there is only one domestic firm, it doesnot matter whether the firm is thought of as a price-setter or a quantitysetter. The quota fixes the domestic firm’s residual demand curve, and thereis one profit-maximizing (output, price) combination on that residual demandcurve. If there is more than one domestic firm, then the price/quantity-setting distinction becomes important, since the nature of interaction amongdomestic firms is different for the two cases.With a quota q, firm 1’s sales in country 1 are

q111 =1

2

µa1 − c

b1− q

¶=

a1 − c

3b1+1

2

µa1 − c

3b1− q

¶. (8.52)

Firm 1 increases its output by half the amount that firm 2 is prevented fromselling by the quota.Total sales in country 1 with the quota are

q111 + q =2

3

a1 − c

b1− 12

µa1 − c

3b1− q

¶, (8.53)

which is less then total sales without the quota. This output reduction impliesthat price rises because of the quota,

p111 = c+ (a1 − c)− b1(q111 + q) = c+

1

3(a1 − c) +

b12

µa1 − c

3b1− q

¶. (8.54)

73

Firm 1’s profit with the quota is

π111 = b1¡q111¢2= b1

·a1 − c

3b1+1

2

µa1 − c

3b1− q

¶¸2, (8.55)

which is greater than its profit without the quota.Firm 2’s profit under the quota is

π112 =

·1

3(a1 − c) +

b12

µa1 − c

3b1− q

¶¸q

= b1

µa1 − c

3b1

¶2− b12

µa1 − c

3b1− q

¶µ2a1 − c

3b1− q

¶, (8.56)

which is less than profit without the quota (given that both terms in paren-theses 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

∆NSW =b12

(·2a1 − c

3b1− 12

µa1 − c

3b1− q

¶¸2−µ2a1 − c

3b1

¶2)

+b1

(·a1 − c

3b1+1

2

µa1 − c

3b1− q

¶¸2−µa1 − c

3b1

¶2), (8.57)

where the first term in braces is the reduction in consumers’ surplus and thesecond is the increase in profit. (8.57) simplifies to

∆NSW =3

8b1

µa1 − c

3b1− q

¶2> 0. (8.58)

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, thiswelfare impact of a quota may be negative. The welfare loss of consumers,as the competition of a large number of foreign producers is constrained, canexceed the profit increase of domestic producers.

(b) Return to the model without a quota and with t > 0, but now interprett as a tariff collected by country 1 on each unit of output sold by firm 2 incountry 1. What is the impact of the tariff on country 1’s net social welfare?


Equilibrium outputs and price are given by equations (7.21), (7.22), and(7.26), reproduced here for convenience:

q∗11 =a1 − c

3b1+

t

3b1

q∗21 =a1 − c

3b1− 2t

3b1

p1 = (c+ t) +2

3

µa1 − c

2− t

¶.

The change in net social welfare with a tariff is

∆NSW =b12

"µ2a1 − c

3b1− t

3b1

¶2−µ2a1 − c

3b1

¶2#

+b1

"µa1 − c+ t

3b1

¶2−µa1 − c

3b1

¶2#(8.59)

+t

µa1 − c

3b1− 2t

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 thegovernment. (8.60) simplifies to

∆NSW = t

µa1 − c

3b1− t

2b1

¶> 0. (8.60)

7—4 Let there be two countries with identical demand curves, each home toone widget producer. The subscript 1 denotes both country 1 and its widgetcompany; similarly for the subscript 2. Let the inverse demand curves in thetwo countries be

p1 = a− (q11 + q21)p2 = a− (q12 + q22)

, (8.61)

where p1 is the price in country 1, p2 is the price in country 2, and qij is thequantity of widgets sold by firm i in country j, for i, j = 1, 2. The parametera is the price-axis intercept of the inverse demand curves, which are the samein both countries. The slope of the inverse demand curves is −1.Let the cost function be

c(qi1 + qi2) = α(qi1 + qi2)− 12β(qi1 + qi2)

2, (8.62)

75

where α and β are both positive and β is sufficiently small that marginalcost remains positive over the relevant output range. Assume there are notransportation costs or tariffs.

(a) write out the payoff functions of the two firms.

The payoff functions are

π1 = [a− (q11+q21)]q11+[a− (q12+q22)]q12−·α(q11 + q12)− 1

2β(q11 + q12)

2

¸(8.63)

π2 = [a− (q11+q21)]q21+[a− (q12+q22)]q22−·α(q21 + q22)− 1

2β(q21 + q22)

2

¸(8.64)

(b) find the first-order conditions to maximize the payoffs and solve them forequilibrium outputs.

The first-order conditions for firm 1 are

∂π1∂q11

= a− 2q11 − q21 − α+ β(q11 + q12) = 0 (8.65)

∂π1∂q12

= a− 2q12 − q22 − α+ β(q11 + q12) = 0 (8.66)

Note that equation (8.65), the first-order condition for firm 1’s sales incountry 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 factthat firm 1’s marginal cost depends on its total output, the sum of its salesin both markets. In contrast to the model of Problem 7-1, it is not possibleto 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-orderconditions 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)


The system of equations of the first-order conditions can be written inmatrix form as

2− β −β 1 0−β 2− β 0 11 0 2− β −β0 1 −β 2− β

q11q12q21q22

= (a− α)

1111

(8.71)

One way to solve this is to use the inverse of the coefficient matrix on theleft, which satisfies

3(1− 2β)(3− 2β)

2− β −β 1 0−β 2− β 0 11 0 2− β −β0 1 −β 2− β

−1

= (8.72)

(2− β)(3− 4β) β(5− 4β) −(3− 4β + 2β2) −2β(2− β)

β(5− 4β) (2− β)(3− 4β) −2β(2− β) −(3− 4β + 2β2)−(2β2 + 3− 4β) −2β(2− β) (2− β)(3− 4β) β(5− 4β)−2β(2− β) −(3− 4β + 2β2) 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 areidentical, implying that equilibrium is symmetric; substituting q11 = q12 =q21 = q22 in any one of the equations of the first-order conditions gives

q011 = q012 = q021 = q022 =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

3− 2β −a− α

3=2

3

β

3− 2β > 0 (8.74)

(for β > 0), economies of scale result imply an increase in equilibrium outputcompared with the constant returns to scale case.For the cost function 8.62 to lead to sensible results, it must be that

β <3

2, (8.75)

otherwise the equilibrium sales given in (8.73) are negative. As indicted inthe statement of the problem, economies of scale cannot be too great if thequadratic cost function is to be a suitable approximation.

77

There is actually a stricter limit on the range of β. The second-ordercondition for firm 1’s profit-maximization requires that the Hessian matrixÃ

∂2π1∂q211

∂2π1∂q12q11

∂2π1∂q12q11

∂2π2∂q222

!=

· −(2− β) ββ −(2− β)

¸(8.76)

have a positive determinant, i.e., that

[(2− β)2 − β2] = 4(1− β) > 0, (8.77)

orβ < 1. (8.78)

A further restriction will be derived below.

(c) substitute equilibrium outputs for country 2 in the equations of the first-order conditions for country 1 outputs, and interpret the resulting expressionsas equilibrium best response functions for country 1.

Substitute q012 = (a− α)/(3− 2β) into (8.67) to obtain

(2− β)q11 + q21 =3− β

3− 2β (a− α) (8.79)

and q022 = (a− α)/(3− 2β) into (8.69) to obtain

q11 + (2− β)q21 =3− β

3− 2β (a− α) (8.80)

For expositional purposes, (8.79) and (8.81) can be interpreted as theequations of equilibrium best response functions for country 1. There is acorresponding set of equations for the equilibrium best response functions forcountry 1. It should be kept in mind, however, that equilibrium sales in thetwo markets are simultaneously determined.

(d) Suppose now that country 1 imposes a quota that restricts the country2 firm’s sales in country 1 to a level q that is below the equilibrium levelfrom (b). Find the new equilibrium outputs; describe the new equilibrium interms of movements in the equilibrium best response functions.

The firms’ payoff functions become

π1 = [a− (q11+ q)]q11+ [a− (q12 + q22)]q12−·α(q11 + q12)− 1

2β(q11 + q12)

2

¸(8.81)


π2 = [a−(q11+q)]q+[a−(q12+q22)]q22−·α(q + q22)− 1

2β(q + q22)

2

¸(8.82)

The first-order conditions for firm 1 are

∂π1∂q11

= a− 2q11 − q − α+ β(q11 + q12) = 0 (8.83)

∂π1∂q12

= a− 2q12 − q22 − α+ β(q11 + q12) = 0 (8.84)

and that for firm 2 is

∂π2∂q22

= a− q12 − 2q22 − α+ β(q + q22) = 0 (8.85)

The system of equations of the first-order conditions can be written inmatrix form as 2− β −β 0

−β 2− β 10 1 2− β

q11q12q22

= (a− α)

111

+ q

−10β

. (8.86)

This can be solved directly using the expression for the inverse of thecoefficient matrix on the left, 2− β −β 0

−β 2− β 10 1 2− β

−1 =

=1

(3− 4β)(2− β)

(1− β)(3− β) β(2− β) −ββ(2− β) (2− β)2 −(2− β)−β −(2− β) 4(1− β)

. (8.87)

Before proceeding, however, it is useful to express all the variables interms of deviations from the no-quota equilibrium values. Hence let

q∗11 = q11 − a− α

3− 2β (8.88)

q∗12 = q12 − a− α

3− 2β (8.89)

q∗22 = q22 − a− α

3− 2β (8.90)

79

and substitute in (8.86) to obtain a revised version of the system of equationsof the first-order conditions, 2− β β 0

β 2− β 10 1 2− β

q∗11q∗12q∗22

=

µa− α

3− 2β − q

¶ 10−β

(8.91)

Then

(3− 4β)(2− β)

q∗11q∗12q∗22

= (8.92)

(1− β)(3− β) β(2− β) −ββ(2− β) (2− β)2 −(2− β)−β −(2− β) 4(1− β)

10−β

µ a− α

3− 2β − q

¶ q∗11

q∗12q∗22

=1

(3− 4β)(2− β)

3− 4β + 2β22β(2− β)−β(5− 4β)

µ a− α

3− 2β − q

¶(8.93)

Hence

q∗11 =3− 4β + 2β2(3− 4β)(2− β)

µa− α

3− 2β − q

¶=(1− β)(3− β) + β2

(3− 4β)(2− β)

µa− α

3− 2β − q

¶> 0 (8.94)

q∗12 =2β

(3− 4β)µ

a− α

3− 2β − q

¶> 0 (8.95)

q∗22 = −β(5− 4β)

(3− 4β)(2− β)

µa− α

3− 2β − q

¶< 0, (8.96)

whereβ <

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)

(where Q is total sales in the country) and that firms in each country operatewith the cost function

c(q) = cq + F, (8.98)

where c is constant marginal cost, q is firm output, and F is fixed and sunkcost. Assume that firms behave as Cournot oligopolists.


(a) What is the long-run equilibrium number of firms in each country iftrade 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

π1 = (a− c− bQ1)q1 − F. (8.99)

The equation of the firm’s quantity best response function is

2q1 +Q−1 =a− c

b= S, (8.100)

where Q−1 is the combined output of all other firms, with correspondingprofit

π1 = bq21 − F. (8.101)

In equilibrium, all firms will produce the same output level, say q0. Sub-stituting q1 = q0, Q−1 = (n− 1)q0 in (8.100) and rearranging terms gives

q0 =1

n+ 1

SpF/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

n0 =SpF/b− 1. (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 ofeach firm, after trade, if the number of firms in each country is the long-runequilibrium 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 openingup of trade,

q =1

2n0 + 1

SpF/b

, (8.104)

Using the equations of the best response functions, and taking into ac-count the fact that each firm sells in both countries, profit per firm is

π = 2b

Ã1

2n0 + 1

SpF/b

!2− F. (8.105)

81

But n0 is determined so that

F =b

n0 + 1S2. (8.106)

Substituting (8.106) in (8.105) to F eliminate and rearranging terms gives

π = − n0 − 12n0 + 1

bS2, (8.107)

which is negative for n0 > 1. The opening up of trade causes firms to losemoney 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

µ1

2m+ 1S

¶2− F ≡ 0, (8.108)

yielding

m+ 1 =√2

SpF/b

(8.109)

m =√2(n0 − 1)− 1. (8.110)

The equilibrium number of firms rises approximately in proportion to thesquare 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 horizontallysumming demand in the two counties is)

p = a− 12bQ. (8.111)

The long-run Cournot equilibrium number of firms for a market with thisdemand curve is given by (8.110).8—6 (Exchange rate passthrough, quantity-setting firms) Let markets for thesame product in two different countries have the inverse demand curves

p1 = a1 − b1(q11 + q21)p2 = a2 − b2(q12 + q22)

. (8.112)


Let firm 1 be based in country 1 and firm 2 in country 2. Call the constantunit 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 onworld currency markets. Assume firms compete by selecting outputs, andeach 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

π11 = [a1 − c1 − b1(q11 + q21)]q11 (8.113)

π21 = eh³

a1 − c2e

´− b1(q11 + q21)

iq21 (8.114)

Maximizing payoffs with respect to own price gives the first-order condi-tions, which are the equations of the quantity best response functions,

2q11 + q21 =a1 − c1

b1(8.115)

q11 + 2q21 =a1 − c2

eb1

(8.116)

Country 2: payoffs are

π11 =1

e[a2 − ec1 − b2(q12 + q22)]q12 (8.117)

π22 = [a1 − c2 − b2(q12 + q22)]q22 (8.118)

Maximizing payoffs with respect to own price gives the first-order condi-tions, 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 theyare affected by changes in e.Country 1: solving (8.115) and (8.116), equilibrium country 1 outputs

areq11 =

1

3b1

h(a1 − c1) +

³c2e− c1

´i(8.121)

q21 =1

3b1

h(a1 − c1)− 2

³c2e− c1

´i. (8.122)

83

These expressions are valid only if both are nonnegative.From (8.121) and (8.122),

∂q11∂e

= − 1

3b1

c2e2

< 0 (8.123)

∂q21∂e

=2

3b1

c2e2

> 0. (8.124)

A euro depreciation decreases firm 1’s sales in country 1 and increasesfirm 2’s sales in country 1.Country 2: solving (8.119) and (8.120), equilibrium country 2 outputs

are

q12 =1

3

µa2 − c2

b2− 2ec1 − c2

b2

¶(8.125)

q22 =1

3

µa2 − c2

b2+

ec1 − c2b2

¶. (8.126)

From (8.125) and (8.126),

∂q12∂e

= −2c13b2

< 0 (8.127)

∂q22∂e

=c13b1

> 0. (8.128)

A euro depreciation decreases firm 1’s sales in country 2 and increasesfirm 2’s sales in country 2.(c) Find equilibrium price in each country and discuss how they are af-

fected by changes in e.Country 1: adding (8.121) and (8.122), total output in country 1 is2

q11 + q21 =1

3b1

h2(a1 − c1)−

³c2e− c1

´i. (8.129)

Substituting this expression for total output in the equation of the inversedemand curve, equilibrium Country 1 price is

p1 = c1 +1

3(a1 − c1) +

1

3

³c2e− c1

´(8.130)

Taking the derivative of (8.130) with respect to e,

dp1de

= −13

c2e2. (8.131)

2Total output can also be obtained by adding (8.115) and (8.116) and dividing bothsides by 3.


An increase in e – a euro depreciation – increases firm 2’s sales andtotal sales in country 1, leading to a reduction in the country 1 price. Thereduction in price is proportional to firm 2’s marginal cost.But from (8.130) we also obtain

d

de

³p1 − c2

e

´=2

3

c2e2

> 0. (8.132)

Although a euro depreciation leads to a lower price level in country 1, it alsoleads 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 + b1q11 =c2e+ b1q21, (8.133)

so price is above marginal cost for both firms provided equilibrium outputsare positive.Country 2: adding (8.125) and (8.126), total output in country 2 is

q12 + q22 =2

3

a2 − c2b2

− 13

ec1 − c2b2

. (8.134)

Substituting this expression for total output in the equation of the inversedemand curve, equilibrium country 2 price is

p2 = c2 +1

3(a2 − c2) +

1

3(ec1 − c2), (8.135)

from whichdp2de

=1

3c1. (8.136)

A euro depreciation decreases firm 1’s sales and total sales in country2, leading to an increase in the country 2 price. The increase in price isproportional 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 ofoutput sold in country 1;

(ii) an ad valorem tariff τ , a fraction of the country 1 price paid by thecountry 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 incountry 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

π21 = e(p1 − t)q21 − c2 (8.137)

= eha1 − c2

e− t− b1(q11 + q21)

i(8.138)

Changes of the same magnitude in t and changes in c2/e have the sameeffect on firm 2’s payoff; it is the sum of t and c2/e that enters expressionsfor best response functions and equilibrium payoffs.With an ad valorem tariff, firm 2’s payoff on sales in country 1 is

π21 = e(1− τ )p1q21 − c2. (8.139)

It is the product e(1− τ) that enters expressions for best response func-tions and equilibrium payoffs. Changes in either e or 1− τ that lead to thesame change in e(1 − τ) have the same effect on equilibrium outputs andprice.8-7 (Exchange rate passthrough, price-setting firms) Suppose that prod-

ucts are differentiated, with inverse demand curves

p11 = a1 − b1(q11 + θ1q21)p21 = a1 − b1(θ1q12 + q22)

(8.140)

in country 1 andp12 = a2 − b2(q11 + θ2q21)p22 = a2 − b2(θ2q12 + q22)

(8.141)

in country 2. Assume firms compete by selecting prices, and that each firmexports to the other market if it is profitable to do so. Let other aspects ofthe 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

b1(1− θ21)q11 = (a1− c1)− θ1³a1 − c2

e

´− (p11− c1)+ θ1

³p21 − c2

e

´(8.142)

b1(1−θ21)q21 = −θ1(a1− c1)+³a1 − c2

e

´+θ1(p11−c1)−

³p21 − c2

e

´(8.143)

for country 1 and

b2(1− θ22)q12 = (a2 − ec1)− θ2(a2 − c2)− (p12 − ec1) + θ2(p22 − c2) (8.144)

b2(1− θ22)q22 = −θ2(a2 − ec1) + (a2 − c2) + θ2(p11 − c1)− (p21 − c2) (8.145)


p21

p11

p22

p12(a) Country 1 (b) Country 2

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E0

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©

Firm 1’s best response curve@@@R

Firm 2’s bestresponse curves

@@

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@@I

ss E1E0

¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢

¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢

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Firm 1’s best response curves@@@R

@@@@R

Firm 2’s bestresponse curve

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Higher e@@I

Higher e@@I

Figure 8.2: Exchange rate fluctuations and Bertrand equilibrium

Country 1: profits satisfy

b1(1− θ21)π11 = (8.146)

(p11 − c1)h(a1 − c1)− θ1

³a1 − c2

e

´− (p11 − c1) + θ1

³p21 − c2

e

´ib1e(1− θ21)π21 = (8.147)³

p21 − c2e

´ h−θ1(a1 − c1) +

³a1 − c2

e

´+ θ1(p11 − c1)−

³p21 − c2

e

´iMaximizing payoffs with respect to own price, the first-order condition

give the equations of the best response functions,

2(p11 − c1)− θ1

³p21 − c2

e

´= (a1 − c1)− θ1

³a1 − c2

e

´(8.148)

−θ1(p11 − c1) + 2³p21 − c2

e

´= −θ1(a1 − c1) +

³a1 − c2

e

´(8.149)

Country 2: profits satisfy

eb2(1− θ22)π12 = (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 conditiongive the equations of the best response functions,

2(p11 − c1)− θ1

³p21 − c2

e

´= (a1 − c1)− θ1

³a1 − c2

e

´(8.152)

−θ1(p11 − c1) + 2³p21 − c2

e

´= −θ1(a1 − c1) +

³a1 − c2

e

´(8.153)

for country 1 and

2(p12 − ec1)− θ2(p22 − c2) = (a2 − ec1)− θ1(a2 − c2) (8.154)

−θ2(p12 − ec1) + 2(p22 − c2) = −θ2(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 µ2 −θ1−θ1 2

¶µp11 − c1p21 − c2

e

¶=

µ1 −θ1−θ1 1

¶µa1 − c1a1 − c2

e

¶. (8.156)

This can be solved for equilibrium country 1 prices,

p∗11 − c1 =1− θ12− θ1

(a1 − c1)− θ1

4− θ21

³c1 − c2

e

´(8.157)

p∗21 −c2e=1− θ12− θ1

(a1 − c1) +2− θ214− θ21

³c1 − c2

e

´. (8.158)

This implies∂p∗11∂e

= − θ1

4− θ21

c2e< 0 (8.159)

∂p∗21∂e

= − 2

4− θ21

c2e< 0 (8.160)

∂

∂e

³p∗21 −

c2e

´=2− θ214− θ21

c2e> 0 (8.161)

A euro depreciation reduces both country 1 prices, but increases the coun-try 2 firm’s price-cost margin.


Country 2: the system of equations of the price best response functionsis µ

2 −θ2−θ2 2

¶µp12 − ec1p22 − c2

¶=

µ1 −θ2−θ2 1

¶µa1 − ec1a1 − c2

¶. (8.162)

This can be solved for equilibrium country 1 prices,

p∗12 − ec1 =1− θ22− θ2

(a2 − c2) +2− θ224− θ22

(c2 − ec1) (8.163)

p∗22 − c2 =1− θ22− θ2

(a2 − c2)− θ2

4− θ22(c2 − ec2). (8.164)

This implies∂p∗12∂e

=2

4− θ22c2 > 0 (8.165)

∂p∗22∂e

=θ2

4− θ22c2 > 0 (8.166)

∂

∂e(p∗12 − ec1) = −2− θ22

4− θ22c2 < 0. (8.167)

A euro depreciation raises both country 2 prices, but reduces the country1 firm’s price-cost margin.

Chapter 9

Imperfect Competition andInternational 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 goodwhich 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 perunit, and let transportation cost is zero.If the two firms act as quantity-setting duopolists, the Cournot equilib-

rium profit per firm is

πduo1 = πduo2 = b

µa− c

3b

¶2=(a− c)2

9b(9.2)

If the two firms form an export cartel and maximize joint profit, eachearns half of monopoly profit in equilibrium,

πcar1 = πcar2 =1

2b

µa− c

2b

¶2=(a− c)2

8b(9.3)

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 exportmarket, that competes as a quantity-setting firm with the two domestic firms.

Without an export cartel, the foreign market is a Cournot triopoly. Equi-librium firm profit is

πtrii = b

µa− c

4b

¶2=(a− c)2

16b, (9.4)

for i = 1, 2, 3. Total profit for the country 1 firms is

πtri1 + πtri2 =(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 basedin each country, and if firms in each country are allowed to form an exportcartel.

For simplicity, consider the case of equal-sized markets, with inverse de-mand 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

π4 = b

µa− c

5b

¶2=(a− c)2

25b. (9.6)

Total sales in country 1 are

Q1 =4

5

a− c

5b, (9.7)

and consumers’ surplus is

1

2bQ2

1 =1

2

µ4

5

¶2 (a− c)2

b=8

25

(a− c)2

b. (9.8)

Recalling that country 1 firms each profit in both countries, country 1net social welfare without export cartels is

NSWCour1 =

8

25

(a− c)2

b+ 4

(a− c)2

25b=12

25

(a− c)2

b. (9.9)

91

If firms in each country form an export cartel and the cartels compete asCournot duopolists, consumers’ surplus in country 1 falls to

1

2b

·2

3

(a− c)

b

¸2=2

9

(a− c)2

b. (9.10)

In each market, each firm earns half of duopoly profit,

1

2b

µa− c

3b

¶2=(a− c)2

18b: (9.11)

firm profit rises with two cartels.Country 1 net social welfare with export cartels is

NSWCar1 =

2

9

(a− c)2

b+ 4

(a− c)2

18b=4

9

(a− c)2

b. (9.12)

Since 4/9 = 0.44 < 12/25 = 0.48, country 1 net social welfare falls withduelling export cartels.

(d) if formation of an export cartel allows domestic firms to tacitly colludeon 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 socialwelfare is

1

2b

µ2

3

a− c

b

¶2+ 4b

µa− c

3b

¶2=2

3

(a− c)2

b. (9.13)

If the two firms maximize joint profit in both markets, country 1 netsocial welfare is

1

2b

µ1

2

a− c

b

¶2+ 2b

µa− c

2b

¶2=5

8

(a− c)2

b. (9.14)

Firm profit rises but net social welfare falls if the export cartel allowsfirms 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 (9.15)

for a homogeneous product.


Countries 1 and 2 form a custom union, which has aggregate inversedemand curve

pU = a− 12QU (9.16)

Countries 1 and 3 are each home to one automobile manufacturer, whichwe 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 isC33(x3) = F3 + c3x3 (9.17)

Firm 1’s cost function for its operations in the custom union is

C1(x1) = F1 + c1x1 (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 thatplant is

C3U(x3U) = F3 + c1x3U (9.20)

If firm 3 opens a plant in the customs union, it must pay an extra setof fixed costs. Its marginal cost is the same as firm 1: marginal cost iscountry specific. This is an assumption that simplifies the analysis of marketequilibrium 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 andnet 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 havehigher marginal cost in the U market. This conclusion need not hold if thereare transportation costs and if the country 3 firm can retain some of itsmarginal cost advantage at a customs union plant.The customs union is a Cournot duopoly. Firm 1 maximizes

π1U =

·a− c1 − 1

2(q1U + q3U)

¸q1U − F1; (9.21)

the first-order condition is

a− c1 − 12(2q1U + q3U) = 0, (9.22)

93

so thata− c1 − 1

2(q1U + q3U) =

1

2q1U (9.23)

andπ1U =

1

2q21U − F1 (9.24)

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, itmaximizes

π3 = (a− c3 −Q3)Q3 +

·a− c3 − 1

2(q1U + q3U)

¸q3U − F3. (9.26)

Firm 3 is a monopolist in its home market; it sells the monopoly output

Q3 =1

2(a− c3) (9.27)

and earns monopoly profit

Q23 =

1

4(a− c3)

2 (9.28)

The first-order condition for firm 3’s sales in the customs union is

a− c3 − 12(q1U + 2q3U) = 0, (9.29)

so thata− c3 − 1

2(q1U + q3U) =

1

2q3U (9.30)

and firm 3’s equilibrium profit on sales in the customs union is

1

2q23U (9.31)

The equation of firm 3’s export best response function for the customsunion is

q1U + 2q3U = 2(a− c3) (9.32)

Find equilibrium outputs by solving the system of equations of the bestresponse functions: µ

2 11 2

¶µq1Uq3U

¶= 2

µa− c1a− c3

¶


3

µq1Uq3U

¶= 2

µ2 −1−1 2

¶µa− c1a− c3

¶q1U =

2

3[2(a− c1)− (a− c3)] =

2

3(a+ c3 − 2c1) (9.33)

q3U =2

3[2(a− c3)− (a− c1)] =

2

3(a+ c1 − 2c3) (9.34)

Assume that both of these output levels are positive.Firm 1’s equilibrium profit is

π1U =2

9(a+ c3 − 2c1)2 − F1 (9.35)

Firm 3’s equilibrium profit is

π3 =1

4(a− c3)

2 +2

9(a+ c1 − 2c3)2 − F3. (9.36)

Total sales in the customs union are

q1U + q3U =4

3

·a− 1

2(c1 + c3)

¸(9.37)

This illustrates a general characteristic of linear demand, constant mar-ginal cost Cournot models: total output depends on the unweighted averageof marginal cost.Consumers’ surplus in the customs union is

1

2(q1U + q3U)

2 =8

9

·a− 1

2(c1 + c3)

¸2(9.38)

Net social welfare in country 1 is the sum of consumers’ surplus and firm1’s profit:

8

9

·a− 1

2(c1 + c3)

¸2+2

9(a+ c3 − 2c1)2 − F1 (9.39)

(b) Suppose firm 3 is persuaded or constrained to limit its exports to alevel v that is below its free trade equilibrium export level. Find Cournotequilibrium profits under this voluntary export restraint. Also find equilib-rium consumers’ surplus and net social welfare in the customs union.Suppose firm 3 is persuaded or constrained to limit its exports to a level

v < q3U =2

3(a+ c1 − 2c3) (9.40)

95

Firm 1 will produce along its best response function:

q1Uv = a− c1 − 12q3U = a− c1 − 1

2v (9.41)

Total output is

q1U + v = a− c1 +1

2v (9.42)

Total sales must fall if firm 3’s sales fall; moving along its best responsefunction, firm 1 makes up only 1/2 of firm 3’s output reduction.The country 1 price rises to

pUv = a− 12

µa− c1 +

1

2v

¶= c1 +

1

2q1Uv = c1 +

1

2

µa− c1 − 1

2v

¶(9.43)

Firm 1’s profit is1

2

µa− c1 − 1

2v

¶2− F1 (9.44)

This falls as v rises.Consumers’ surplus in the customs union is

1

2

µa− c1 +

1

2v

¶2(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,

1

2

µa− c1 − 1

2v

¶2− F1 +

1

2

µa− c1 +

1

2v

¶2=1

2

µ(a− c1)

2 +1

2v2¶− F1 (9.46)

This rises as v rises.Firm 3’s profit on sales in the customs union isµ

c1 − c3 +1

2

µa− c1 − 1

2v

¶¶v =

1

2

µa+ c1 − 2c3 − 1

2v

¶v

=1

2(a+ c1 − 2c3)v − 1

4v2 (9.47)

This rises as v rises so long as the VER actually restricts firm 3’s sales:

∂

∂v

·1

2(a+ c1 − 2c3)v − 1

4v2¸=1

2

µa+ c1 − 2c3 − 1

2v

¶


=1

4(3q3U − v)

=1

4[2q3U + (q3U − v)] > 0 (9.48)

(c) find Cournot equilibrium profits, consumers’ surplus and net socialwelfare in the customs union if firm 3 sets up a plant in country 1. What isthe condition that must be satisfied for direct foreign investment to be themost profitable choice for firm 3? How is this condition affected by v? Howis 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 twofirms have identical marginal costs. Equilibrium outputs are

2a− c13

(9.49)

per firm, and firm 3’s profit is

1

2

·2

3(a− c1)

¸2− F3 =

2

9(a− c1)

2 − F3 (9.50)

A VER will make foreign direct investment the most profitable choice forfirm 3 if

2

9(a− c1)

2 − F3 >1

2

µa+ c1 − 2c3 − 1

2v

¶v (9.51)

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 privatelyoptimal for firm 3 to open a plant in country 1.9—3 (Reciprocal dumping) Consider two firms, firm 1 based in country 1and firm 2 based in country 2. Markets in the two countries are identical.The two firms produce differentiated varieties of the same product. Inversedemand 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 degreeof substitutability between the two varieties.

97

The cost of production is c per unit. Transportation cost to ship fromone 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., calculatep21 − t and p12 − t ). Compare export prices net of transportation cost withthe price of the same variety in its home market.

Because there are constant returns to scale, the two national markets canbe analyzed separately.Profits in country 1 are

π11 = (p11 − c)q11 = (a− c− q11 − θq21)q11 (9.56)

π21 = (p21 − c− t)q21 = (a− c− t− θq11 − q21)q21 (9.57)

best response functions are found by taking the derivative of (9.56) withrespect to q11 and the derivative of (9.57) with respect to q21 and setting theresulting 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:µ2 θθ 2

¶µq11q21

¶= (a− c)

µ11

¶− t

µ01

¶

(4− θ2)

µq11q21

¶= (a− c)

µ2 −θ−θ 2

¶µ11

¶− t

µ2 −θ−θ 2

¶µ01

¶(4− θ2)

µq11q21

¶= (a− c)(2− θ)

µ11

¶− t

µ −θ2

¶µ01

¶q11 =

a− c

2 + θ+

θ

4− θ2t (9.62)

q21 =a− c

2 + θ− 2

4− θ2t (9.63)


Equilibrium prices are then

p11 = c+a− c

2 + θ+

θ

4− θ2t (9.64)

p21 − t = c+a− c

2 + θ− 2

4− θ2t, (9.65)

where the equilibrium price of variety 2 in country 1 is expressed net oftransportation cost.Because the countries are identical, in equilibrium p22 = p11; then

p22 − (p21 − t) =θ

4− θ2t−

µ− 2

4− θ2t

¶=

t

2− θ(9.66)

Market performance: inverse demand curves for country 1 of the indicatedform would be produced by a representative consumer utility function of theform

U = aq11 + θq11q21 + aq21 − p11q11 − p21q21 (9.67)

With trade, net social welfare in country 1 is

U + π11 + π12 (9.68)

Sincep11 = c + q11,

a− p11 = a− c− q11.

Sincep21 = c+ t+ q21,

a− p21 = a− c− t− q21

Equilibrium consumer welfare is

(a− p11)q11 + θq11q21 + (a− p21)q21 =

(a− c− q11)q11 + θq11q21 + (a− c− t− q21)q21 =

(a− c)q11 + (a− c− t)q21 − q211 + θq11q21 − q221

Firm 1’s profit in country 1 is

(p11 − c)q11 = q211

Because the model is symmetric, firm 1’s profit in country 2 is the sameas firm 2’s profit in country 1; this is

(p21 − c− t)q21 = q221

99

Net social welfare in country 1 is the sum of consumer welfare and firm1’s profit:

(a− c)q11 + (a− c− t)q21 − q211 + θq11q21 − q221 + q211 + q221 =

(a− c)q11 + (a− c− t)q21 + θq11q21.

Writing for x = a− c notational compactness, this is

x

µx

2 + θ+

θ

4− θ2t

¶+ (x− t)

µx

2 + θ− 2

4− θ2t

¶

+θ

µx

2 + θ+

θ

4− θ2t

¶µx

2 + θ− 2

4− θ2t

¶= −−3x

2θ3 + 3xθ3t− 8xθ2t+ 8x2θ2 + 4θ2t2 + 4x2θ − 4xθt− 16x2 − 8t2 + 16xt¡4− θ2

¢2= −−(3θ

3 − 8θ2 − 4θ + 16)x2 + (3θ3 − 8θ2 − 4θ + 16)xt− 4(2− θ2)t2¡4− θ2

¢2=(4 + 3θ)(2− θ)2x2 − (4 + 3θ)(2− θ)2xt+ 4(2− θ2)t2¡

4− θ2¢2

=(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 anantidumping dumping d on firm 2’s sales in country 1, what is the impacton equilibrium prices in country 1? How great an antidumping duty wouldcountry 1 need to impose to make p22 = p21 − t?Replace t by t+ d in (9.62) and (9.63)

q11 =a− c

2 + θ+

θ

4− θ2(t+ d) (9.69)

q21 =a− c

2 + θ− 2

4− θ2(t+ d) (9.70)

(9.61) becomesp21 = c+ t+ d+ q21, (9.71)


so that with an antidumping duty d equilibrium prices are

p11 = c+a− c

2 + θ+

θ

4− θ2t+

θ

4− θ2d (9.72)

p21 − t = c+ d+a− c

2 + θ− 2

4− θ2(t+ d)

= c+a− c

2 + θ− 2

4− θ2t+

2− θ2

4− θ2d (9.73)

The antidumping duty increases p11 by

θ

4− θ2d < d (9.74)

and increases p21 − t by2− θ2

4− θ2d < d. (9.75)

p21−t increases by less than the amount of the duty because the market isimperfectly competitive and firm 2 absorbs part of the artificial cost increasecreated by the antidumping duty. p11 increases by less than the amount ofthe duty because the increase in p11 comes in response to firm 2’s reductionin 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 increasesatisfies

2− θ2

4− θ2d =

t

2− θ

d =4− θ2

(2− θ2)(2− θ)t =

2 + θ

2− θ2t (9.76)

From (9.74), the price of the domestic variety will rise

θ

4− θ22 + θ

2− θ2t =

θ

(2− θ)(2− θ2)t. (9.77)

9—5 (Antidumping undertaking) For the model of 9—3, what is firm 2’s profit-maximizing price if it agrees to charge the same price (net of transportationcost) in both countries? (Assume firm 1 continues to act as a Cournot firmin both markets.)If firm 2 agrees to charge the same net price in both countries, it seeks to

maximize

π2 = (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

∂q21= a− c− t− θq11 − 2q21 + λ = 0 (9.82)

orθq11 + 2q21 − λ = a− c− t (9.83)

∂L

∂q22= a− c− θq12 − 2q22 − λ = 0 (9.84)

orθq12 + 2q22 + λ = a− c (9.85)

andθq11 + q21 − θq12 − q22 = −t (9.86)

(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)


The system of equations of first-order conditions is2 θ 0 0 0θ 2 0 0 −10 0 2 θ 00 0 θ 2 1θ 1 −θ −1 0

q11q21q12q22λ

=

a− ca− c− ta− c− ta− c−t

=

(a− c)

11110

− t

01101

(9.91)

where the first two rows are the best response functions for country 1, thethird and fourth rows are the best response functions for country 2, and thefifth row is the equal price constraint.

det

2 θ 0 0 0θ 2 0 0 −10 0 2 θ 00 0 θ 2 1θ 1 θ 1 0

= 16− 12θ2 + 2θ4 = 2(2− θ2)(4− θ2)

(16− 12θ2 + 2θ4)

2 θ 0 0 0θ 2 0 0 −10 0 2 θ 00 0 θ 2 1θ 1 −θ −1 0

−1

=

8− 3θ2 −θ(2− θ2) −θ2 −θ ¡2− θ2

¢ −θ ¡4− θ2¢

−2θ ¡3− θ2¢

2(2− θ2) 2θ 2(2− θ2) 2(4− θ2)−θ2 −θ ¡2− θ2

¢8− 3θ2 −θ ¡2− θ2

¢θ¡4− θ2

¢2θ 2(2− θ2) −2θ ¡3− θ2

¢2(2− θ2) −2(4− θ2)

−θ ¡4− θ2¢ − ¡2− θ2

¢(4− θ2) θ

¡4− θ2

¢ ¡2− θ2

¢(4− θ2)

¡4− θ2

¢2

2¡2− θ2

¢ ¡4− θ2

¢

q11q21q12q22λ

=

1038− 3θ2 −θ(2− θ2) −θ2 −θ ¡2− θ2

¢ −θ ¡4− θ2¢

−2θ ¡3− θ2¢

2(2− θ2) 2θ 2(2− θ2) 2(4− θ2)−θ2 −θ ¡2− θ2

¢8− 3θ2 −θ ¡2− θ2

¢θ¡4− θ2

¢2θ 2(2− θ2) −2θ ¡3− θ2

¢2(2− θ2) −2(4− θ2)

−θ ¡4− θ2¢ − ¡2− θ2

¢(4− θ2) θ

¡4− θ2

¢ ¡2− θ2

¢(4− θ2)

¡4− θ2

¢2

×

11110

(a− c)−

8− 3θ2 −θ(2− θ2) −θ2 −θ ¡2− θ2

¢ −θ ¡4− θ2¢

−2θ ¡3− θ2¢

2(2− θ2) 2θ 2(2− θ2) 2(4− θ2)−θ2 −θ ¡2− θ2

¢8− 3θ2 −θ ¡2− θ2

¢θ¡4− θ2

¢2θ 2(2− θ2) −2θ ¡3− θ2

¢2(2− θ2) −2(4− θ2)

−θ ¡4− θ2¢ − ¡2− θ2

¢(4− θ2) θ

¡4− θ2

¢ ¡2− θ2

¢(4− θ2)

¡4− θ2

¢2

×

01101

t =

= 2 (2− θ)¡2− θ2

¢11110

(a− c)− (2− θ)

−θ(3 + 2θ)2(3 + 2θ)4 + 3θ−2(1 + θ)2

(2 + θ)2

t

2¡2− θ2

¢ ¡4− θ2

¢

q11q21q12q22λ

= 2 (2− θ)¡2− θ2

¢11110

(a−c)−(2−θ)−θ(3 + 2θ)2(3 + 2θ)4 + 3θ−2(1 + θ)2

(2 + θ)2

t

(9.92)

2 (2 + θ)¡2− θ2

¢

q11q21q12q22λ

= 2¡2− θ2

¢11110

(a− c)−

−θ(3 + 2θ)2(3 + 2θ)4 + 3θ−2(1 + θ)2

(2 + θ)2

t

(9.93)


q11 =1

2 + θ(a− c) +

θ(3 + 2θ)

2 (2 + θ)¡2− θ2

¢t (9.94)

q21 =1

2 + θ(a− c)− (3 + 2θ)

(2 + θ)¡2− θ2

¢t (9.95)

q12 =1

2 + θ(a− c)− 4 + 3θ

2 (2 + θ)¡2− θ2

¢t (9.96)

q22 =1

2 + θ(a− c) +

(1 + θ)2

(2 + θ)¡2− θ2

¢t (9.97)

λ = − (2 + θ)2

2 (2 + θ)¡2− θ2

¢t = − 2 + θ

2¡2− θ2

¢t (9.98)

Using (9.87)p21 − t = c+ q21 − λ

= c+1

2 + θ(a− c)− (3 + 2θ)

(2 + θ)¡2− θ2

¢t+ 2 + θ

2¡2− θ2

¢t= c+

1

2 + θ

µa− c− 1

2t

¶(9.99)

Using (9.88)p22 = c+ q22 + λ

= c+1

2 + θ(a− c) +

(1 + θ)2

(2 + θ)¡2− θ2

¢t− 2 + θ

2¡2− θ2

¢t= c+

1

2 + θ

µa− c− 1

2t

¶(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

2 + θ+

θ

4− θ2t.

The change in p22 is

∆p22 = c+1

2 + θ

µa− c− 1

2t

¶−µc+

a− c

2 + θ+

θ

4− θ2t

¶

= − 1

2(2− θ)t < 0 (9.101)

105

Because of the antidumping undertaking, firm 2 lowers its price in itshome market.Using (9.65), without an undertaking

p21 − t = c+a− c

2 + θ− 2

4− θ2t.

The change in p21 − t is

∆(p21 − t) = c+1

2 + θ

µa− c− 1

2t

¶− c− a− c

2 + θ+

2

4− θ2t

=1

2(2− θ)t > 0. (9.102)

Because of the antidumping duty, the firm 2 raises its net price in country1.Compare sales with and without the antidumping undertaking?From Problem 9—3, equilibrium outputs without the antidumping under-

taking are

q11 = q22 =a− c

2 + θ+

θ

4− θ2t

q21 = q12 =a− c

2 + θ− 2

4− θ2t

Changes in quantities sold due to the antidumping undertaking are

∆q11 =1

2 + θ(a− c) +

θ(3 + 2θ)

2 (2 + θ)¡2− θ2

¢t−µa− c

2 + θ+

θ

4− θ2t

¶

=θ

2 (2− θ)¡2− θ2

¢t > 0 (9.103)

Firm 1 increases its sales in its own market.

∆q21 =1

2 + θ(a− c)− (3 + 2θ)

(2 + θ)¡2− θ2

¢t− µa− c

2 + θ− 2

4− θ2t

¶

= − 1¡2− θ2

¢(2− θ)

t ≤ 0 (9.104)

Firm 2 reduces its sales in country 1.

∆q12 =1

2 + θ(a− c)− 4 + 3θ

2 (2 + θ)¡2− θ2

¢t−µa− c

2 + θ− 2

4− θ2t

¶


− θ

2¡2− θ2

¢(2− θ)

t ≤ 0 (9.105)

Firm 1’s sales in country 2 go down.

∆q22 =1

2 + θ(a− c) +

(1 + θ)2

(2 + θ)¡2− θ2

¢t−µa− c

2 + θ+

θ

4− θ2t

¶

=1¡

2− θ2¢(2− θ)

t ≥ 0 (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 pricesrather 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 theessential symmetry in the solution.Let

c11 = c22 = c (9.107)

c21 = c21 = c+ t (9.108)

The equations of the demand curves areµp11 − c11p21 − c21

¶=

µa− c11a− c21

¶−µ1 θθ 1

¶µq11q21

¶(9.109)

Solving for the equations of the demand curves,µ1 θθ 1

¶µq11q21

¶=

µa− c11a− c21

¶−µ

p11 − c11p21 − c21

¶

(1− θ2)

µq11q21

¶=

µ1 −θ−θ 1

¶µa− c11a− c21

¶−µ

1 −θ−θ 1

¶µp11 − c11p21 − c21

¶q11 =

a− c11 − θ(a− c21)− (p11 − c11) + θ(p21 − c21)

1− θ2(9.110)

q21 =a− c21 − θ(a− c11)− (p21 − c21) + θ(p11 − c11)

1− θ2(9.111)

In like manner, the demand curves for country 2 are

q12 =a− c12 − θ(a− c22)− (p12 − c12) + θ(p22 − c22)

1− θ2(9.112)

107

q22 =a− c22 − θ(a− c12)− (p22 − c22) + θ(p12 − c12)

1− θ2(9.113)

Payoffs in country 1 satisfy

(1−θ2)π11 = (p11−c11)[a−c11−θ(a−c21)−(p11−c11)+θ(p21−c21)] (9.114)

(1−θ2)π21 = (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

π11 =(p11 − c11)

2

1− θ2(9.118)

π21 =(p21 − c21)

2

1− θ2(9.119)

The system of equations formed by the first-order conditions for country1 is µ

2 −θ−θ 2

¶µp11 − c11p12 − c12

¶=

µ1 −θ−θ 1

¶µa− c11a− c12

¶. (9.120)

Equilibrium prices are

(4− θ2)

µp11 − c11p21 − c21

¶=

µ2 θθ 2

¶µ1 −θ−θ 1

¶µa− c11a− c21

¶

=

µ2− θ2 −θ−θ 2− θ2

¶µa− c11a− c21

¶p11 − c11 =

(2− θ2)(a− c11)− θ(a− c21)

4− θ2(9.121)

p21 − c21 =(2− θ2)(a− c21)− θ(a− c11)

4− θ2(9.122)

In like manner, equilibrium prices in country 2 are

p12 − c12 =(2− θ2)(a− c12)− θ(a− c22)

4− θ2(9.123)


p22 − c22 =(2− θ2)(a− c22)− θ(a− c12)

4− θ2(9.124)

Now express the unit costs in terms of their underlying components:

p11 − c =(2− θ2)(a− c)− θ(a− c− t)

4− θ2(9.125)

p21 − c− t =(2− θ2)(a− c− t)− θ(a− c)

4− θ2(9.126)

p12 − c− t =(2− θ2)(a− c− t)− θ(a− c)

4− θ2(9.127)

p22 − c =(2− θ2)(a− c)− θ(a− c− t)

4− θ2(9.128)

Then the difference between firm 2’s price in country 2 and firm 2’s pricein country 1, net of transportation cost, is

p22 − c− (p21 − c− t) =

(2− θ2)(a− c)− θ(a− c− t)

4− θ2− (2− θ2)(a− c− t)− θ(a− c)

4− θ2=

1 + θ

2 + θt > 0 (9.129)

To see the impact of an antidumping duty d imposed by country 1, letc21 = c+ t+ d:

p11 − c =

(2− θ2)(a− c)− θ(a− c− t− d)

4− θ2=(2− θ2)(a− c)− θ(a− c− t)

4− θ2+

θd

4− θ2

p21 − c− t− d =

=(2− θ2)(a− c− t− d)− θ(a− c)

4− θ2=(2− θ2)(a− c− t)− θ(a− c)

4− θ2−2− θ2

4− θ2d

p21 − c− t =(2− θ2)(a− c− t)− θ(a− c)

4− θ2+

µ1− 2− θ2

4− θ2

¶d

=(2− θ2)(a− c− t)− θ(a− c)

4− θ2+

2

4− θ2d

p22 − c− (p21 − c− t) =1 + θ

2 + θt− 2

4− θ2d. (9.130)

109

To eliminate dumping entirely, d must be

d =4− θ2

2

1 + θ

2 + θt = (1 + θ)(2− θ)

t

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

+(p22 − c22)a− c22 − θ(a− c12)− (p22 − c22) + θ(p12 − c12)

1− θ2(9.132)

subject to the constraint that

p21 − t = p22. (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− θ2)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)(p21 − c21) + (1− θ2)(p22 − c22) =

−(1− θ2)(a− c21)− (1− θ2)t+ (1− θ2)(a− c22) (9.140)

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)


The system of equations formed by the first-order conditions is2 −θ 0 0 0−θ 2 0 0 −(1− θ2)0 0 2 −θ 00 0 −θ 2 (1− θ2)0 −(1− θ2) 0 (1− θ2) 0

p11 − c11p21 − c21p12 − c12p22 − c22

λ

=

1 −θ 0 0 0−θ 1 0 0 00 0 1 −θ 00 0 −θ 1 00 −(1− θ2) 0 (1− θ2) −(1− θ2)

a− c11a− c21a− c12a− c22

t

(9.143)

The determinant of the coefficient matrix is

det

2 −θ 0 0 0−θ 2 0 0 −(1− θ2)0 0 2 −θ 00 0 −θ 2 (1− θ2)0 −(1− θ2) 0 (1− θ2) 0

= −4(4− θ2)(1− θ2)2

(9.144)The inverse of the coefficient matrix is

1

−4(4− θ2)(1− θ2)2(9.145)

times the 5× 5 matrix the first three columns of which are− ¡8− θ2

¢ ¡1− θ2

¢2 −2θ ¡1− θ2¢2 −θ2 ¡1− θ2

¢2−2θ ¡1− θ2

¢2 −4 ¡1− θ2¢2 −2θ ¡1− θ2

¢2−θ2 ¡1− θ2

¢2 −2θ ¡1− θ2¢2 − ¡8− θ2

¢ ¡1− θ2

¢2−2θ ¡1− θ2

¢2 −4 ¡1− θ2¢2 −2θ ¡1− θ2

¢2θ¡1− θ2

¢ ¡4− θ2

¢2¡1− θ2

¢ ¡4− θ2

¢ −θ ¡1− θ2¢ ¡4− θ2

¢

(9.146)

and the last two columns of which are−2θ ¡1− θ2

¢2θ¡1− θ2

¢ ¡4− θ2

¢−4 ¡1− θ2

¢22¡1− θ2

¢ ¡4− θ2

¢−2θ ¡1− θ2

¢2 −θ ¡1− θ2¢ ¡4− θ2

¢−4 ¡1− θ2

¢2 −2 ¡1− θ2¢ ¡4− θ2

¢−2 ¡1− θ2

¢ ¡4− θ2

¢ ¡4− θ2

¢2

. (9.147)

111

The solution of the system of equations formed by the first-order condi-tions is

−4(4− θ2)(1− θ2)2

p11 − c11p21 − c21p12 − c12p22 − c22

λ

=

(A,B)×

1 −θ 0 0 0−θ 1 0 0 00 0 1 −θ 00 0 −θ 1 00 −(1− θ2) 0 (1− θ2) −(1− θ2)

a− c11a− c21a− c12a− c22

t

,

(9.148)where

A =

− ¡8− θ2

¢ ¡1− θ2

¢2 −2θ ¡1− θ2¢2 −θ2 ¡1− θ2

¢2−2θ ¡1− θ2

¢2 −4 ¡1− θ2¢2 −2θ ¡1− θ2

¢2−θ2 ¡1− θ2

¢2 −2θ ¡1− θ2¢2 ¡

θ2 − 8¢ ¡1− θ2¢2

−2θ ¡1− θ2¢2 −4 ¡1− θ2

¢2 −2θ ¡1− θ2¢2

θ¡1− θ2

¢ ¡4− θ2

¢2¡1− θ2

¢ ¡4− θ2

¢ −θ ¡1− θ2¢ ¡4− θ2

¢

and

B =

−2θ ¡1− θ2

¢2θ¡1− θ2

¢ ¡4− θ2

¢−4 ¡1− θ2

¢22¡1− θ2

¢ ¡4− θ2

¢−2θ ¡1− θ2

¢2 −θ ¡1− θ2¢ ¡4− θ2

¢−4 ¡1− θ2

¢2 −2 ¡1− θ2¢ ¡4− θ2

¢−2 ¡1− θ2

¢ ¡4− θ2

¢ ¡4− θ2

¢2

The product of the two coefficient matrices is

¡1− θ2

¢times

2θ¡1− θ2

¢ −θ ¡1− θ2¢ ¡4− θ2

¢4(1− θ2) −2 ¡1− θ2

¢ ¡4− θ2

¢2θ¡1− θ2

¢θ¡1− θ2

¢ ¡4− θ2

¢−4 ¡1− θ2

¢ ¡3− θ2

¢2¡1− θ2

¢ ¡4− θ2

¢2(4− θ2) − ¡4− θ2

¢2

(9.149)

Equilibrium values arep11 − c11 =¡

8− 3θ2¢ (1 + θ) (a− c11)− 2θ(a− c21)− θ2(a− c12)− 2θ(a− c22) + θ¡4− θ2

¢t

4(4− θ2)(9.150)


p21 − c21 =

−2θ(a− c11) + 4¡3− θ2

¢(a− c21)− 2θ(a− c12)− 4(a− c22) + 2

¡4− θ2

¢t

4(4− θ2)(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− θ2)(9.153)

λ =θ(a− c11)− 2(a− c21)− θ(a− c12)− 2(a− c22) +

¡4− θ2

¢t

4(1− θ2)(9.154)

Check the price undertaking constraint:

p21 − c21 − (p22 − c22)

−2θ(a− c11) + 4¡3− θ2

¢(a− c21)− 2θ(a− c12)− 4(a− c22) + 2

¡4− θ2

¢t

4(4− θ2)

−Ã−2θ(a− c11)− 4(a− c21)− 2θ(a− c12) + 4

¡3− θ2

¢(a− c22)− 2

¡4− θ2

¢t

4(4− θ2)

!= −c21 + t+ c22

orp21 − 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− θ2)

−(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

2(2 + θ)t > 0 (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¢

4(4− θ2)t =

− (1 + θ)

2(2 + θ)t < 0 (9.156)

Chapter 10

Market Integration in theEuropean Union

10—1 Evaluate the consequences of market integration using the model ofSection 10.2.2, that is, two countries i = 1, 2 each with an inverse demandcurve with equation

pi = 100−Qi,

n1 firms in market 1 and n2 firms in market 2, and all firms operating withthe 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 taxesare collected separately in each market, then the post-integration markets areCournot oligopoly markets with n1 + n2 firms supplying each market; in theusual way, post-integration prices are

p1 = 10 + t1 +100− 10− t1n1 + n2 + 1

p2 = 10 + t2 +100− 10− t2n1 + n2 + 1

.

115

116 CHAPTER 10. MARKET INTEGRATION

If firms treat both countries as a single market after integration, the payoffof (say) firm 1 in country is

π11 =

µ100− 10− 1

2Q

¶(q11 + q22)− t1q11 − t2q22.

In such a scenario, all firms would sell, and all customers purchase, in thelower-tax country. The post-integration price is

p = 10 + tmin + 2100− 10− tminn1 + n2 + 1

,

where tmin is the smaller of t1, t2.10—2 Suppose n1 Cournot oligopolists supply the market in country 1, wherethe equation of the inverse demand curve is

p1 = 100−Q1,

while n2 different Cournot oligopolists supply the market in country 2, wherethe equation of the inverse demand curve is

p2 = 100− 12Q2.

Compare equilibrium prices and outputs before and after market integra-tion, 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 +90

n1 + 1

Country 2:

p2 = 100− 12Q2

Q2 = 200− 2p1Equilibrium output per firm:

q2 =190

n2 + 1

117

Equilibrium price:

p2 = 10 +190

n2 + 1

Post integration:Q = 300− 3pp = 100− 1

3Q

Payoff of a typical firm:

πi =

µ100− 10− 1

3Q

¶qi

πi =

µ90− 1

3Q

¶qi

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:

10 +90

n1 + 1−µ10 +

270

n1 + n2 + 1

¶= 90

−2n1 + n2 − 2(n1 + 1) (n1 + n2 + 1)

10 +190

n2 + 1−µ10 +

270

n1 + n2 + 1

¶= 10

19n1 − 8n2 − 8(n2 + 1) (n1 + n2 + 1)

Loosely: if n2 is large, prices fall in country 1 but rise in country 2; if n1is large, prices fall in country 2 but rise in country 1.If the number of firms is equal:

10 +90

n+ 1−µ10 +

270

2n+ 1

¶= −90 n+ 2

(n+ 1) (2n+ 1)

10 +190

n+ 1−µ10 +

270

2n+ 1

¶= 10

11n− 8(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 soldin its territory, the number of firms is the same in each country, and (becauseof the tax policy), firms continue to treat each country as a separate marketafter integration. Equilibrium prices in each country are then the standardCournot equilibrium prices with 2n firms operating in each country. SeeProblem 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 onevery unit sold by a country 2 firm, no matter where that unit is sold.The payoff function of a typical country 1 firm is

π1i =

µ100− 10− t1 − 1

2Q

¶q1i,

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 + n2q2 = 2 (90− t1)

n1q1 + (n2 + 1)q2 = 2 (90− t2)

Solve for equilibrium output per firm:µn1 + 1 n2n1 n2 + 1

¶µq1q2

¶= 2

µ90− t190− t2

¶µ

q1q2

¶= 2

µn1 + 1 n2n1 n2 + 1

¶−1µ90− t190− t2

¶µ

q1q2

¶= 2

µ n2+1n1+n2+1

(90− t1)− n2n1+n2+1

(90− t2)

− n1n1+n2+1

(90− t1) +n1+1

n1+n2+1(90− t2)

¶=

2

n1 + n2 + 1

µ90− (n2 + 1) t1 + n2t290 + n1t1 − (n1 + 1) t2

¶Find total output:

Q =2

n1 + n2 + 1

¡n1 n2

¢µ 90− (n2 + 1) t1 + n2t290 + n1t1 − (n1 + 1) t2

¶=

2

n1 + n2 + 1n1 (90− (n2 + 1) t1 + n2t2)+

2

n1 + n2 + 1n2 (90 + n1t1 − (n1 + 1) t2)

119

= 290 (n1 + n2)− n1t1 − n2t2

n1 + n2 + 1

Find equilibrium price:

p = 100− 12

µ290 (n1 + n2)− n1t1 − n2t2

n1 + n2 + 1

¶

= 100− 90 (n1 + n2)− n1t1 − n2t2n1 + n2 + 1

Express equilibrium price in terms of deviations frommarginal productioncost plus the sales:

p− (10 + t1) =

100− 90 (n1 + n2)− n1t1 − n2t2n1 + n2 + 1

− (10 + t1)

=90− (n2 + 1) t1 + n2t2

n1 + n2 + 1

p− (10 + t2) =

100− 90 (n1 + n2)− n1t1 − n2t2n1 + n2 + 1

− (10 + t2)

=90 + n1t1 − (n1 + 1) t2

n1 + n2 + 1

p− t1 = 10 +90− (n2 + 1) t1 + n2t2

n1 + n2 + 1

p− t2 = 10 +90 + n1t1 − (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.

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