1 Cournot Oligopoly with n firms [PDF]

BEE2017, Microeconomics 2, Dieter Balkenborg. 1 Cournot Oligopoly with n firms firm i's output: qi total output: q = q1

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BEE2017, Microeconomics 2, Dieter Balkenborg

1

Cournot Oligopoly with n firms

firm i’s output: qi total output: q = q1 + q2 + · · · + qn opponent’s output: q−i = q − qi = Σj=i qi constant marginal costs of firm i: ci inverse 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 : ∂p ∂Πi = × qi + p − ci = 0 ∂qi ∂qi 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 )

3 2.5

p 2 1.5 1 0.5

0

0.5

1

1.5

2

2.5

3

q ∂p = −B ∂qi FOC: −Bqi + (A − B (q−i + qi )) − ci 2Bqi Reactionfunction qi = ri (q−i ) =

1

= 0 = A − ci − Bq−i

A−c 1 − q−i 2B 2

10

8

q1= r1(q2)

q26

CournotNash

4

2

0

q2= r2(q1) 2

4

q1

6

8

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 −Bq2 + (A − Bq) − c2

−Bqn + (A − Bq) − cn

= 0 = 0 .. . = 0

Summation yields −Bq + n (A − Bq) − n¯ c=0 where c¯ =

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 q

= n (A − c¯) n A − c¯ = n+1 B

p = A − Bq =

1 n A+ c¯ → c¯ for n → ∞ n+1 n+1

2

Each firm produces in the n-firm oligopoly qin =

A − Bq − ci A − ci n A − c¯ 1 A n (¯ c − ci ) − ci = − = + . B B n+1 B n+1B (n + 1) B

Let us now, for simplicity, assume that firms have identical marginal costs ci = c¯ = c. Then n 1 A+ c → c as n → ∞ n+1 n+1 1 A−c = → 0 as n → ∞ n+1 B   (A − c) 2 1 1 A−c n 1 n = (p − c) qi = A+ c−c = 2 n+1 n+1 n+1 B B (n + 1)

p = qin Πni nΠni

=

(A − c)2 → 0 as n → ∞ B (n + 1) n

2

The total profit in the industry decreases with every additional firm entering the market since for all n>1 (n − 1) Πin−1 > 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 and then adapts. If a rational firm 2 observes the quantity q1 it will choose the quantity q2 = r2 (q1 ) = Total output is q1 + q2 =

A−c 1 − q1 2B 2

A−c 1 + q1 2B 2

and the price will be p = A − B (q1 + q2 ) = A −

A−c B A + c − Bq1 − q1 = 2 2 2

Anticipating this, firm 1 expects to make the profit   A + c − Bq1 A − c − Bq1 − c × q1 = × q1 Π1 (q1 , r1 (q2 )) = 2 2 which is maximized for q1 =

A−c 2B 3

yielding the price p=

A + c − B A−c A−c 2B = 2 4

and the profit Π1 = Firm 2 produces q2 =

1 (A − c)2 8 B

A−c 1 A−c − q1 = 2B 2 4B

and makes the profit

1 1 (A − c)2 Π2 = Π1 = 2 16 B Notice that this would not be a Nash equilibrium if firm 2 could not observe the quantity choice because firm 2 reacts optimally while firm 1 should produce q1 = r1 (q2 ) = Total quantity would be

5 A−c 8 B

A−c 1 A−c A−c 3A−c − q2 = − = 2B 2 2B 8B 8 B

and the the price would reduce to p = A−

3A + 5c 5 (A − c) = 8 8

and yield the profit Π1 =



  2 2 3A + 5c 3A−c 1 (A − c) 9 (A − c) −c > = 2 8 8 4B 8 B 8 B

The leader produces in the Stackelberg equilibrium twice as much than the follower and makes twice the 2 profit. In the Cournot duopoly the payoff Π2i = 19 (A−c) which is in between the profit of the leader and B the follower.

3

Bertrand competition with differentiated products

The two firms have the demand functions Q1 Q2

= 100 − 2P1 + P2 = 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 = (+1) × (100 − 2Pi + Pj ) + (Pi − 5) × (−2) = 110 − 4Pi + Pj = 0, i = 1, 2 ∂Pi 2 2×110 = 73 13 units The solution to this system of equations is P1 = P2 = 110 3 = 36 3 . Each firm produces 3 1 2 and makes the profit 73 3 × 36 3 ≈ 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 the two first order conditions

∂ (Π1 + Π2 ) = 110 − 4Pi + Pj + (Pi − 5) = 105 − 3Pi + Pj = 0, i = 1, 2 ∂Pi which havethesolution  P1 = P2 = 52.5. Of each commodity 57.5 units are produced and the total profit is 2 × 47 12 × 57 12 = 5462.5, which is obviously higher than in competition. 4

4

Bertrand “competition” with perfect complements.

Two price-setting firms produce with constant marginal costs c = 3 produce goods which are perfect complements. Consumers therefore buy equal amounts from both firms. The total amount they by of each 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 = 15 − (P1 + P2 ) − (Pi − 3) = 18 − 2Pi − Pj = 0 ∂Pi where j = 3 − i. By symmetry, P1 = P2 in equilibrium, so 3Pi = 18 or P1 = P2 = 6. It follows that Q = 15 − 12 = 3 pairs are sold at the price 6. Each firm makes the profit (6 − 3) × 3 = 9 and the total profit 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+6 = 10.5 where 15 − 10.5 = 4.5 pairs are demanded. Consumer surplus 2 is up in the monopoly because they get more at a lower price. Producer surplus goes up because the monopolist’s profit is 4.52 = 20. 25 > 18.

5

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