Two firms compete as a Stackelberg duopoly.The inverse market demand they face is P = 62 − 4.5Q.The cost function for each firm is C(Q)= 8Q.The outputs of the two firms are: A) Q_L = 48; Q_F = 24. B) Q_L = 35; Q_F = 6. C) Q_L = 6; Q_F = 3. D) None of the preceding answers is correct.

Question

Quizplus · Accepted Answer

In a Stackelberg duopoly, one firm acts as the leader and the other as the follower. In this case, let's assume firm 1 is the leader and firm 2 is the follower.

The profit maximization problem for firm 1 can be written as:

max (π1) = (P - 8)QS1

Substituting the inverse demand function into this equation:

max (π1) = (62 - 4.5Qtotal - 8)QS1

We know that Qtotal = QS1 + QS2, and given that firm 2 is the follower, it will choose its output based on the output of firm 1. The reaction function for firm 2 can be written as:

QS2 = (1/9)(QS1 - 8)

Substituting this into the total output equation:

Qtotal = QS1 + (1/9)(QS1 - 8)

Simplifying and setting Qtotal equal to the demand:

62 - 4.5Qtotal = QS1 + (1/9)(QS1 - 8)

Solving for QS1:

QS1 = 6

Substituting this back into the reaction function for firm 2:

QS2 = (1/9)(6 - 8) = -2/9

Clearly, firm 2 cannot produce a negative quantity, so it will choose to produce 0.

Therefore, the equilibrium outputs are QS1 = 6 and QS2 = 0. This matches choice C.

Two Firms Compete as a Stackelberg Duopoly