Question

Two Cournot firms produce slightly different products. Product prices depend on both firms' outputs and are determined by the following equations P₁ = 70 - 2Q₁ - Q_2, P₂ = 100 - Q₁- 2Q_2. Both Firm 1 and Firm 2 have constant marginal cost of $10 and zero fixed cost. Firm 1 chooses Q₁ and Firm 2 chooses Q_2.

1. (3pts) Find Firm 1's best response as a function of Firm 2's output Q₂.  
2. (3pts) Find Firm 2's best response as a function of Firm 1's output Q₁.
3. (6pts) Use the two best response functions to find the Nash equilibrium. In equilibrium, how much profit will each firm earn?

Answer 1

Firm 1's inverse demand equation is   

Firm 2's inverse demand equation is   

Marginal Cost of firm 1 and firm 2 is $10. Fixed costs are 0 for both firms this imply total cost is for firm 1 and for firm 2.

Let represents profit of firm 1.

Putting in

taking first partial derivative of with respect to :

Setting equal to 0 and solving for imply:

So, firm 1's best response as a function of firm 2's output is

Let represents profit of firm 2.

Putting in

taking first partial derivative of with respect to :

Setting equal to 0 and solving for imply:

So, firm 2's best response as a function of firm 1's output is

Putting best response of firm 2 i.e in :

Put in

When and then:

so,

When and then:

So,

Nash equilibirum : Firm 1 will sell 10 units () at price 30 () and Firm 2 will sell 20 units () at price 50 ().

Firm 1's profit is represented by . Put and in :

In equilibrium, profit of firm 1 is $200.

Firm 2's profit is represented by . Put and in   :

In equilibrium, profit of firm 2 is $800.