In: Economics
An industry contains two firms, one whose cost function is C(y) = 30y and another whose cost function is C(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? Graph the best response functions for Firm 1 and Firm 2.
The inverse demand is given to be , and for , we have , where q1 and q2 are quantities of the respective firms.
Firm 1 have the cost function , and its marginal cost would be . The total revenue of firm 1 would be , and the marginal revenue would be . The profit maximization would occur where the MC is equal to MR, ie or or . This is the BR function of firm 1.
Firm 2 have the cost function , and its marginal cost would be . The total revenue of firm 2 would be . and the marginal revenue would be . The profit maximization would occur where the MC is equal to MR, ie or or .
The cournout (nash) equilibrium would occur where the BR function intersects. Putting the first BR function into the second one, we have or or or , and since , we have . These are the required equilibrium output.
The graph is as below.