## 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 (y1y2) 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)(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 (y1y2) 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)  (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]