ECO 5341 Collusion in In…nitely Repeated Cournot
Competition
Saltuk Ozerturk (SMU)
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
Remember our Cournot Competition Example
Two …rms (Firm 1 and Firm 2) face an industry demand
P = 150
Q
where
Q = q1 + q2
is the total industry output.
Both …rms have the same unit production cost c = 30.
The …rms are competing by simultaneously setting their
quantities to maximize own pro…ts.
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
Cournot Nash Equilibrium Pair q1c and q2c solve
q1c (q2c ) = 60
q2c
2
q2c (q1c ) = 60
q1c
2
which yields
q1c = q2c = 40
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
Cournot Pro…ts of Each Firm
c
1
c
2
(q1c ; q2c )
(q1c ; q2c )
=
(150
)
1
=
(150
)
2
q1c
q2c )q1c
(q1 ; q2 ) = (150
q1c
q2c )q2c
(q1 ; q2 ) = (150
Saltuk Ozerturk (SMU)
30q1c
80) 40
30 40 = 1600
30q2c
80) 40
In…nitely Repeated Cournot
30 40 = 1600
In…nitely Repeated Cournot
Monopoly Output and Price in this market is
q m = 60
P m = 150
Q = 150
60 = 90
which yields a monopoly pro…t
m
= (P m
c) q m = (90
Saltuk Ozerturk (SMU)
30) 60 = 3600
In…nitely Repeated Cournot
In…nitely Repeated Cournot
Collusion outcome q1 = q2 = q m =2 = 30 is not a Nash
Equilibrium in the One-Shot Stage Game
Can we sustain the collusion outcome in the in…nitely
repeated Cournot game?
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
Trigger Strategy for Firm i
Start with Collusion quantity qi = q m =2 = 30.
Continue with qi = q m =2 = 30 as long as everyone has chosen
q m =2 = 30.
Choose the Cournot quantity qic = 40 otherwise.
Is the above trigger strategy a SPE?
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
Trigger Strategy says the following.
Each …rm starts with the collusion quantity qi = q m =2 = 30:
As long as no one deviates from the collusion quantity
qi = q m =2 = 30 continue to produce at qi = q m =2 = 30: This
collusion phase will give a pro…t of $1800 to each …rm.
If any of the two …rms deviate from qi = q m =2 = 30 and
produce some other quentity, then start producing qic = 40
forever. Note that when …rm i produces qic = 40, the other
…rm’s best response is also qjc = 40. In this case, both …rms
will receive their Cournot pro…ts of $1600 until in…nity.
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
First we need to describe the best one-shot deviation. What is
the best quantity that a deviating …rm would produce when
the rival is still producing at q m =2 = 30?
Suppose after any history of collusion (that is after a history
in which both …rms produced at q m =2 = 30); Firm 1 is going
to deviate. What is the best deviation? Remember Firm 1’s
best response
30
= 45
q1 (q2 ) = 60
2
Hence Firm 1’s best deviation is q1D = 45
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
Let’s …nd Firm 1’s deviation pro…t for the period that it
deviates to q1D = 45: Recall that when deviation happens, in
that period Firm 2 still produces q2 = q m =2 = 30: hence we
have
D
1
= (150
45
30) 45
(45 30) = 45 45 = 2025
Hence when Firm 1 deviates, she faces the following pro…t
stream
(2025; 1600; 1600; 1600; 1600; 1600::::::::::::)
If Firm 1 does not deviate and sticks with q1 = q m =2 = 30;
she receives
(1800; 1800; 1800; 1800; 1800; 1800::::::::::::)
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
In…nitely Repeated Cournot
When Firm 1 deviates, she faces the following pro…t stream
(2025; 1600; 1600; 1600; 1600; 1600:::)
which she values at
(2025 + 1600 + 1600
2
+ 1600
= 2025 + 1600 (1 + +
1600
= 2025 +
1
2
+ :::::)
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot
3
+ ::::)
In…nitely Repeated Cournot
If Firm 1 does not deviate and sticks with q1 = q m =2 = 30;
she receives
(1800; 1800; 1800; 1800; 1800; 1800::::::::::::)
which she values at
(1800 + 1800 + 1800
= 1800(1 + +
1800
:
=
1
Saltuk Ozerturk (SMU)
2
2
+ 1800
+ :::::)
In…nitely Repeated Cournot
3
+ ::::)
In…nitely Repeated Cournot
No pro…table one-shot deviation property requires
1800
1
2025 +
)
1600
1
9
:
17
9
Conclusion. The trigger strategy is a SPE for
17 :
Collusion can be sustained if the Firms are patient enough.
Saltuk Ozerturk (SMU)
In…nitely Repeated Cournot