ECO 5341 Cournot Quantity Competition with
Homogenous Products
Saltuk Ozerturk (SMU)
January 2016
Saltuk Ozerturk (SMU)
Cournot Competition
Cournot Quantity Competition
Cournot Quantity Competition
Two firms are competing by choosing quantities of an
identical good to place on the market;
Both have identical, constant marginal cost c = 0 (for
simplicity).
Both firms face a linear demand curve: P = A − Q. As there
are only two firms in the market, we have Q = q1 + q2 ;
Profit function of firm i, i ∈ {1, 2}, is:
π1(q1, q2) = Pq1 = (A − Q) q1 = (A − q1 − q2) q1;
π2(q1, q2) = Pq2 = (A − Q) q2 = (A − q1 − q2) q2.
Saltuk Ozerturk (SMU)
Cournot Competition
Cournot Quantity Competition
How to predict the decisions of the firms? What is the Nash
equilibrium of this game?
Saltuk Ozerturk (SMU)
Cournot Competition
Cournot Quantity Competition
First-Order Condition
To maximize firm i’s profit, we differentiate the profit function
πi = Pqi with respect to qi , and then set the first-order
differentiation equal to zero;
Setting the first-order differentiation to be zero gives the
”First-Order Condition”;
Solving the First-Order Condition yields the optimal quantity
qi∗ that maximizes firm i’s profit for any given qj by the other
firm.
The Nash equilibrium corresponds to a pair of optimal
quantities (q1∗ , q2∗ ) that are best responses to each other for
the two firms.
Saltuk Ozerturk (SMU)
Cournot Competition
Cournot Quantity Competition
Deriving the best response function of Firm 1.
Given any q2 by Firm 2, Firm 1 chooses q1 to maximize
π1 (q1 , q2 ) = Pq1 = (A − q1 − q2 ) q1
= Aq1 − q12 − q2 q1
First Order Condition yields
A − 2q1 − q2 = 0
Hence we have the best response
q1∗ (q2 ) =
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A − q2
.
2
Cournot Competition
Cournot Quantity Competition
Deriving the best response function of Firm 2.
Given any q1 by Firm 1, Firm 2 chooses q2 to maximize
π2 (q1 , q2 ) = Pq2 = (A − q1 − q2 ) q2
= Aq2 − q22 − q1 q2
First Order Condition yields
A − 2q2 − q1 = 0
Hence we have the best response
q2∗ (q1 ) =
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A − q1
.
2
Cournot Competition
Cournot Quantity Competition
The best response functions are
q1∗ (q2 ) =
A − q2
A − q1
and q2∗ (q1 ) =
2
2
The Nash Equilibrium pair (q1∗ , q2∗ ) solves the above pair of
equations.
A 1 A − q1∗
A − q2
∗
∗
⇒ q1 = −
q1 (q2 ) =
2
2
2
2
2A − A + q1∗
A
⇒ q1∗ =
4
3
A
A− 3
A
A − q1
=
⇒ q2∗ =
q2∗ =
2
2
3
Equilibrium price in Cournot Duopoly
⇒ q1∗ =
P ∗ = A − Q = A − q1∗ − q2∗ =
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Cournot Competition
A
3
Cournot Quantity Competition
Comparison with Monopoly
Note that in a monopoly there is only one firm and the other
firm produces zero. Suppose firm 1 is the monopoly. What is
the monopoly output?
q1∗ (q2 ) =
A − q2
A−0
A
⇒ q1M =
= (monopoly output)
2
2
2
Recall that total Cournot duopoly output is
q1∗ + q2∗ =
2A
A
> = q1M (monopoly output)
3
2
Monopoly price is
P M = A − q1M = A −
PM =
A
A
=
2
2
A
A
> P ∗ = (eqb Cournot duopoly price)
2
3
Saltuk Ozerturk (SMU)
Cournot Competition