Two firms compete as a Stackelberg duopoly.The demand they face is P = 24 − Q.The cost function for each firm is C(Q)= 4Q.The profits of the two firms are: A)  _L = $100;  _F = $50. B)  _L = $50;  _F = $25. C)  _L = $25;  _F = $12.5. D)  _L = $20;  _F = $10.

Question

Quizplus · Accepted Answer

To solve for the profits of the firms, we first need to determine the quantity produced by the leader (Firm 1) and the follower (Firm 2) in the Stackelberg equilibrium.

Since Firm 1 is the leader, it will determine its optimal quantity first. It will set its quantity at the level where its marginal revenue equals its marginal cost. The marginal revenue for Firm 1 is given by:

MR1 = P - 2Q

where Q is the total quantity produced by both firms. Setting MR1 equal to the marginal cost (4), we get:

P - 2Q = 4

Solving for Q, we get:

Q = 10

Therefore, Firm 1 will produce 10 units of output. Knowing this, we can then determine the best response of Firm 2. The best response of Firm 2 is to set its quantity where its marginal revenue equals its marginal cost, given the quantity produced by Firm 1.

The marginal revenue for Firm 2 is:

MR2 = P - Q - Q1

where Q1 is the quantity produced by Firm 1. Substituting in the values for P and Q1, we get:

MR2 = 24 - Q - 10 = 14 - Q

Setting MR2 equal to the marginal cost (4), we get:

14 - Q = 4

Solving for Q, we get:

Q = 10

Therefore, Firm 2 will produce 10 units of output as well.

Now we can calculate the profits for each firm. The total revenue for each firm is:

TR1 = P1 Q1 = (24 - 10) × 10 = $140

TR2 = P2 Q2 = (24 - 10) × 10 = $140

The total cost for each firm is:

TC1 = 4 × 10 = $40

TC2 = 4 × 10 = $40

Therefore, the profits for each firm are:

π1 = TR1 - TC1 = $100

π2 = TR2 - TC2 = $100

Therefore, choice B is the correct answer.

Two Firms Compete as a Stackelberg Duopoly