BEE2017, Microeconomics 2, Dieter Balkenborg

1 Cournot Oligopoly with n firms

firm i’s output: qitotal output: q = q1 + q2 + · · ·+ qnopponent’s output: q−i = q − qi = Σj �=iqiconstant marginal costs of firm i: ciinverse demand function: p (q)firm i′s profit:

Πi (q−i, qi) = p (q)× qi − ci × qi = (p (q−i + qi)− ci) qi

FOC for profit maximum given q−i:

∂Πi∂qi

=∂p

∂qi× qi + p− ci = 0

Solution defines reaction curve qi = ri (q−i) which is often decreasing in q−i.Linear case: p = A−Bq = A−B (q−i + qi)

0

0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5 3

q

p

∂p

∂qi= −B

FOC:

−Bqi + (A−B (q−i + qi))− ci = 0

2Bqi = A− ci −Bq−i

Reactionfunction

qi = ri (q−i) =A− c

2B−1

2q−i

1

0

2

4

6

8

10

2 4 6 8

q1= r1(q2)

q2= r2(q1)

q1

q2Cournot-

Nash

Cournot-Nash equilibrium:

1. Every firm maximizes profit given her expectation of q−i.

2. The expectation is correct.

This yields the simultaneous system of equations

qi = ri (q−i)

for all i = 1, . . . , n. In the linear case the FOC yields, since qi + q−i = q

−Bq1 + (A−Bq)− c1 = 0

−Bq2 + (A−Bq)− c2 = 0

...

−Bqn + (A−Bq)− cn = 0

Summation yields−Bq + n (A−Bq)− nc̄ = 0

where

c̄ =c1 + c2 + · · ·+ cn

n

is the average marginal cost in the market.Thus we can deduce the total quantity produced and the price in the market

(n+ 1)Bq = n (A− c̄)

q =n

n+ 1

A− c̄

B

p = A−Bq =1

n+ 1A+

n

n+ 1c̄→ c̄ for n→∞

2

Each firm produces in the n-firm oligopoly

qni =A−Bq − ci

B=A− ci

B−

n

n+ 1

A− c̄

B=

1

n+ 1

A

B+n (c̄− ci)− ci(n+ 1)B

.

Let us now, for simplicity, assume that firms have identical marginal costs ci = c̄ = c. Then

p =1

n+ 1A+

n

n+ 1c→ c as n→∞

qni =1

n+ 1

A− c

B→ 0 as n→∞

Πni = (p− c) qni =

(1

n+ 1A+

n

n+ 1c− c

)1

n+ 1

A− c

B=

1

(n+ 1)2

(A− c)

B

2

nΠni =n

(n+ 1)2(A− c)2

B→ 0 as n→∞

The total profit in the industry decreases with every additional firm entering the market since for alln > 1

(n− 1)Πn−1i > nΠni

⇐⇒n− 1

(n)2 >

n

(n+ 1)2

⇐⇒ (n− 1) (n+ 1)2 > n3

⇐⇒(n2 − 1

)(n+ 1) > n3

⇐⇒ n3 − n+ n2 − 1 > n3

⇐⇒ n2 > n− 1

which is true since n2 > n for all n > 1.In particular, it always pays for the firms to form a cartel and share the monopolist profit since

nΠni < Π1i .

2 Stackelberg Equilibrium

Two firms with marginal costs 1. Different timing: Firm 1 moves first, firm 2 observes the move andthen adapts.If a rational firm 2 observes the quantity q1 it will choose the quantity

q2 = r2 (q1) =A− c

2B−1

2q1

Total output is

q1 + q2 =A− c

2B+1

2q1

and the price will be

p = A−B (q1 + q2) = A−A− c

2−B

2q1 =

A+ c−Bq12

Anticipating this, firm 1 expects to make the profit

Π1 (q1, r1 (q2)) =

(A+ c−Bq1

2− c

)× q1 =

A− c−Bq1

2× q1

which is maximized for

q1 =A− c

2B

3

yielding the price

p =A+ c−BA−c

2B

2=A− c

4

and the profit

Π1 =1

8

(A− c)2

B

Firm 2 produces

q2 =A− c

2B−1

2q1 =

A− c

4B

and makes the profit

Π2 =1

2Π1 =

1

16

(A− c)2

B

Notice that this would not be a Nash equilibrium if firm 2 could not observe the quantity choice becausefirm 2 reacts optimally while firm 1 should produce

q1 = r1 (q2) =A− c

2B−1

2q2 =

A− c

2B−A− c

8B=3

8

A− c

B

Total quantity would be 58A−cB

and the the price would reduce to

p = A−5

8(A− c) =

3A+ 5c

8

and yield the profit

Π1 =

(3A+ 5c

8− c

)(3

8

A− c

4B

)=9

82(A− c)

2

B>1

8

(A− c)2

B

The leader produces in the Stackelberg equilibrium twice as much than the follower and makes twice the

profit. In the Cournot duopoly the payoff Π2i =19(A−c)2

Bwhich is in between the profit of the leader and

the follower.

3 Bertrand competition with differentiated products

The two firms have the demand functions

Q1 = 100− 2P1 + P2

Q2 = 100− 2P2 + P1

and constant marginal costs c = 5. The profit function for firm i is

Πi (p1, p2) = (Pi − c)Qi = (Pi − 5) (100− 2Pi + Pj)

where j = 3− i. The first order condition for a profit optimum (taking the other firm’s price as given) is

∂Πi∂Pi

= (+1)× (100− 2Pi + Pj) + (Pi − 5)× (−2) = 110− 4Pi + Pj = 0, i = 1, 2

The solution to this system of equations is P1 = P2 =1103 = 3623 . Each firm produces 2×110

3 = 7313 unitsand makes the profit 7313 × 36

23 ≈ 2688× 2 is made. Together they make the profit 5376. If they would

form a cartel they could make the profit Π1 (p1, p2) + Π2 (p1, p2). Maximizing joint profit leads to thetwo first order conditions

∂ (Π1 +Π2)

∂Pi= 110− 4Pi + Pj + (Pi − 5) = 105− 3Pi + Pj = 0, i = 1, 2

which have the solution P1 = P2 = 52.5. Of each commodity 57.5 units are produced and the total profitis 2×

(4712

)×(5712

)= 5462.5, which is obviously higher than in competition.

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4 Bertrand “competition” with perfect complements.

Two price-setting firms produce with constant marginal costs c = 3 produce goods which are perfectcomplements. Consumers therefore buy equal amounts from both firms. The total amount they by ofeach commodity is

Q = Q (P1, P2) = 15− (P1 + P2)

The profit of firm i = 1 or i = 2 is

Πi (P1, P2) = (Pi − 3)Q = (Pi − 3) (15− (P1 + P2))

The first-order condition for a profit maximum is

∂Πi∂Pi

= 15− (P1 + P2)− (Pi − 3) = 18− 2Pi − Pj = 0

where j = 3 − i. By symmetry, P1 = P2 in equilibrium, so 3Pi = 18 or P1 = P2 = 6. It follows thatQ = 15− 12 = 3 pairs are sold at the price 6. Each firm makes the profit (6− 3)× 3 = 9 and the totalprofit in the industry is 18.

If a monopolist takes over both plants and takes the price 2P per pair his profit is

Π(P ) = (2P − 6) (15− 2P )

which is maximized for 2P = 15+62 = 10.5 where 15− 10.5 = 4.5 pairs are demanded. Consumer surplus

is up in the monopoly because they get more at a lower price. Producer surplus goes up because themonopolist’s profit is 4.52 = 20. 25 > 18.

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