Examples and exercises on Nash equilibrium of Cournot's model

To find a Nash equilibrium of Cournot's model for a specific cost function and demand function we follow the general procedure for finding a Nash equilibrium of a game using best response functions.

Example

Each of two firms has the cost function TC(y) = 30y; the inverse demand function for the firms' output is p = 120

Q, where Q is the total output. What are the firms' outputs in a Nash equilibrium of Cournot's model?

First find the firms' best response functions. Firm 1's profit is
y₁(120 y₁ y₂) 30y₁.
Taking the derivative of this profit with respect to y₁ (holding y₂ constant) and setting the derivative equal to zero we obtain
120 2y₁ y₂ 30 = 0,
or
y₁ = (90 y₂)/2.
Thus the best response function of firm 1 is given by b₁(y₂) = (90 y₂)/2. This function is shown in the following figure.

Similarly, we find that the best response function of firm 2 is given by b₂(y₁) = (90 y₁)/2. This function is superimposed on the best response function of firm 1 in the following figure.
We now need to find a pair (y₁, y₂) of outputs with the property that
y₁ = b₁(y₂) and y₂ = b₂(y₁).
That is,
y₁ = (90 y₂)/2 and y₂ = (90 y₁)/2.
Substituting one equation in the other we obtain y₁ = (90 (90 y₁)/2)/2, so that y₁ = 30; substituting in the equation for y₂ we get y₂ = 30.

We conclude that there is a unique Nash equilibrium, in which the output of each firm is 30. Each firm's profit is (30)(120

30)

(30)(30) = 900.

Example

Each of two firms has the cost function TC(y) = y². As in the previous example, the inverse demand function for the firms' output is p = 120

Q, where Q is the total output. What are the firms' outputs in a Nash equilibrium of Cournot's model?

First find the firms' best response functions. If firm 1 chooses the output y₁ its profit is
y₁(120 y₁ y₂) y₁².
Taking the derivative of this profit with respect to y₁ (holding y₂ constant) and setting the derivative equal to zero we obtain
120 2y₁ y₂ 2y₁ = 0,
or
y₁ = (120 y₂)/4.
Thus the best response function of firm 1 is given by b₁(y₂) = (120 y₂)/4.
Similarly, we find that the best response function of firm 2 is given by b₂(y₁) = (120 y₁)/4.
We now need to find a pair (y₁, y₂) of outputs with the property that
y₁ = b₁(y₂) and y₂ = b₂(y₁).
That is,
y₁ = (120 y₂)/4 and y₂ = (120 y₁)/4.
Substituting one equation in the other we obtain y₁ = (120 (120 y₁)/4)/4, so that y₁ = 24; substituting in the equation for y₂ we get y₂ = 24.

We conclude that there is a unique Nash equilibrium, in which the output of each firm is 24. Each firm's profit is (24)(120

24)

(24)² = 1152.

Exercise

An industry contains two firms, one whose cost function is TC(y) = 30y and another whose cost function is TC(y) = y². The inverse demand function for the firms' output is p = 120

Q, where Q is the total output. What are the firms' outputs in a Nash equilibrium of Cournot's model?

[Solution]