ECO 5341 in Infinitely Repeated

Saltuk Ozerturk (SMU)

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

Remember our Cournot Competition Example 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) Infinitely Repeated Cournot Infinitely Repeated Cournot

c c Cournot Pair 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) Infinitely Repeated Cournot Infinitely Repeated Cournot

Cournot Profits of Each Firm

πc (qc , qc ) = (150 qc qc )qc 30qc 1 1 2 − 1 − 2 1 − 1 π1 (q1, q2) = (150 80) 40 30 40 = 1600 ⇒ − ∗ − ∗

πc (qc , qc ) = (150 qc qc )qc 30qc 2 1 2 − 1 − 2 2 − 2 π2 (q1, q2) = (150 80) 40 30 40 = 1600 ⇒ − ∗ − ∗

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

Monopoly Output and Price in this market is

qm = 60

Pm = 150 Q = 150 60 = 90 − − which yields a profit

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

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

m Collusion q1 = q2 = q /2 = 30 is not a Nash Equilibrium in the One-Shot Stage Game Can we sustain the collusion outcome in the infinitely repeated Cournot game?

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

Trigger for Firm i m Start with Collusion quantity qi = q /2 = 30. m Continue with qi = q /2 = 30 as long as everyone has chosen qm/2 = 30. c Choose the Cournot quantity qi = 40 otherwise. Is the above a SPE?

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

Trigger Strategy says the following. m Each firm starts with the collusion quantity qi = q /2 = 30. As long as no one deviates from the collusion quantity m m qi = q /2 = 30 continue to produce at qi = q /2 = 30. This collusion phase will give a profit of $1800 to each firm. m If any of the two firms deviate from qi = q /2 = 30 and c produce some other quentity, then start producing qi = 40 c forever. Note that when firm i produces qi = 40, the other c firm’sbest response is also qj = 40. In this case, both firms will receive their Cournot profits of $1600 until infinity.

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

First we need to describe the best one-shot deviation. What is the best quantity that a deviating firm would produce when the rival is still producing at qm/2 = 30? Suppose after any history of collusion (that is after a history in which both firms produced at qm/2 = 30), Firm 1 is going to deviate. What is the best deviation? Remember Firm 1’s 30 q∗(q2) = 60 = 45 1 − 2 D Hence Firm 1’sbest deviation is q1 = 45

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

Let’sfind Firm 1’sdeviation profit for the period that it D deviates to q1 = 45. Recall that when deviation happens, in m that period Firm 2 still produces q2 = q /2 = 30. hence we have

πD = (150 45 30) 45 (45 30) = 45 45 = 2025 1 − − ∗ − ∗ ∗ Hence when Firm 1 deviates, she faces the following profit stream

(2025, 1600, 1600, 1600, 1600, 1600...... )

m If Firm 1 does not deviate and sticks with q1 = q /2 = 30, she receives

(1800, 1800, 1800, 1800, 1800, 1800...... )

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

When Firm 1 deviates, she faces the following profit stream

(2025, 1600, 1600, 1600, 1600, 1600...)

which she values at

(2025 + 1600δ + 1600δ2 + 1600δ3 + ....) = 2025 + 1600δ(1 + δ + δ2 + .....) 1600δ = 2025 + 1 δ −

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

m If Firm 1 does not deviate and sticks with q1 = q /2 = 30, she receives

(1800, 1800, 1800, 1800, 1800, 1800...... )

which she values at

(1800 + 1800δ + 1800δ2 + 1800δ3 + ....) = 1800(1 + δ + δ2 + .....) 1800 = . 1 δ −

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot Infinitely Repeated Cournot

No profitable one-shot deviation property requires 1800 1600δ 2025 + 1 δ ≥ 1 δ − − 9 δ . ⇒ ≥ 17 9 Conclusion. The trigger strategy is a SPE for δ 17 . Collusion can be sustained if the Firms are patient≥ enough.

Saltuk Ozerturk (SMU) Infinitely Repeated Cournot