ECO 5341 Cournot Competition and Collusion

Saltuk Ozerturk (SMU)

Saltuk Ozerturk (SMU) Cournot Cournot Competition

Cournot Quantity Competition Suppose that two firms (Firm 1 and Firm 2) face an industry demand P = 150 − Q where Q = q1 + q2 is the total industry output. Both firms have the same unit production cost c = 30. The firms are competing by simultaneously setting their quantities to maximize own profits.

Saltuk Ozerturk (SMU) Cournot Cournot Competition

Deriving Firm 1’s best response

For any given q2, Firm 1 chooses q1 to maximize

π1 (q1, q2) = (150 − q1 − q2)q1 − 30q1

First order condition:

150 − 2q1 − q2 − 30 = 0

which yields the best response function: q q∗(q ) = 60 − 2 1 2 2

Saltuk Ozerturk (SMU) Cournot Cournot Competition

Deriving Firm 2’s best response

For any given q1, Firm 2 chooses q2 to maximize

π2 (q1, q2) = (150 − q1 − q2)q2 − 30q2

First order condition:

150 − 2q2 − q1 − 30 = 0

which yields the best response function: q q∗(q ) = 60 − 1 2 1 2

Saltuk Ozerturk (SMU) Cournot Cournot Competition

c c Cournot Nash Equilibrium Pair Solves q1 and q2 solve qc qc (qc ) = 60 − 2 1 2 2 qc qc (qc ) = 60 − 1 2 1 2 which yields c c q1 = q2 = 40

Saltuk Ozerturk (SMU) Cournot Cournot Competition

Cournot Equilibrium continued The equilibrium market price in the Cournot NE is given by

c c c P = 150 − Q = 150 − (q1 + q2 ) = 70

Saltuk Ozerturk (SMU) Cournot Cournot Competition

Cournot Profits of Each Firm

c c c c c c c π1 (q1 , q2 ) = (150 − q1 − q2 )q1 − 30q1 ⇒ π1 (q1, q2) = (150 − 80) ∗ 40 − 30 ∗ 40 = 1600

c c c c c c c π2 (q1 , q2 ) = (150 − q1 − q2 )q2 − 30q2 ⇒ π2 (q1, q2) = (150 − 80) ∗ 40 − 30 ∗ 40 = 1600

Saltuk Ozerturk (SMU) Cournot Monopoly

Monopoly Output and Price in this market is (how do I know?) qm = 60 Pm = 150 − Q = 150 − 60 = 90 which yields a monopoly profit

πm = (Pm − c) qm = (90 − 30) ∗ 60 = 3600

Saltuk Ozerturk (SMU) Cournot Monopoly

In principle, these two firms could decide to collude and act like a monopolist, each agreeing to produce half of the m monopoly output (q1 = q2 = q /2 = 30) and sell at the monopoly price Pm = 90, making a profit of 1800 each, which c c is better than Cournot profit of π1 = π2 = 1600. m But is collusion q1 = q2 = q /2 = 30 a Nash Equilibrium?

Saltuk Ozerturk (SMU) Cournot Collusion

m Collusion outcome q1 = q2 = q /2 = 30 is not a Nash Equilibrium. Suppose Firm 2 produces at

m q2 = q /2 = 30.

m What is Firm 1’s best response to q2 = q /2 = 30? q 30 q∗(q ) = 60 − 2 = 60 − = 45 1 2 2 2

Saltuk Ozerturk (SMU) Cournot Collusion

A Finite (Baby) version of Collusion Game. This is Exercise 1.5 in the textbook (page 49) Suppose each of the two firms can either produce qm/2 = 30 or the Cournot equilibrium quantity qc = 40. No other quantity is feasible. The game matrix look like this (you should verify all the payoffs)

qm/2 = 30 qc = 40 qm/2 = 30 (1800, 1800) (1500, 2000) qc = 40 (2000, 1500) (1600, 1600)

But note that this game is a Prisoners’ Dilemma! qc = 40 is a strictly dominant action for both players.

Saltuk Ozerturk (SMU) Cournot