Stackelberg duopolists, Firm 1 and Firm 2, face inverse market demand $P = 50 - Q$ . Both have marginal cost, $M C = \$ 2.0$ . Let firm 2, the follower, set its output according to the formula $Q _ { 2 } = 15 - Q _ { 1 } / 2$ . Which of the following outputs maximizes the leader's profit? A) Q₁ = 7.5 B) Q₁ = 10 C) Q₁ = 15 D) Q₁ = 20

Question

Quizplus · Accepted Answer

To maximize its profit, the leader (Firm 1) must first consider the follower's (Firm 2) output decision. Substituting the follower's output formula into the inverse demand equation, we get:

$$P = 50 - Q_2 - (15 - Q_1/2) = 35 - Q_2 + Q_1/2$$

The leader's total revenue is then:

$$\text{Total Revenue} = P \times Q_1 = (35 - Q_2 + Q_1/2) Q_1 = 35 Q_1 - Q_1 Q_2 + 1/2 Q_1^2$$

The leader's profit is its total revenue minus its total cost, which is:

$$\pi_1 = (35 Q_1 - Q_1 Q_2 + 1/2 Q_1^2) - 2 Q_1 = 33 Q_1 - Q_1 Q_2 + 1/2 Q_1^2$$

To maximize profit, we take the derivative of this expression with respect to $Q_1$ and set it equal to zero:

$$\frac{d \pi_1}{d Q_1} = 33 - Q_2 + Q_1 = 0$$
$$Q_1 = Q_2 - 33$$

Substituting the follower's output formula into this expression, we get:

$$Q_1 = 15 - Q_1/2 - 33$$
$$Q_1 = 12$$

Therefore, the optimal output for the leader is $Q_1 = 12$. To verify that this is a maximum, we can take the second derivative of the profit function with respect to $Q_1$:

$$\frac{d^2 \pi_1}{d Q_1^2} = 1/2 > 0$$

Since the second derivative is positive, the profit function is concave up, so $Q_1 = 12$ maximizes the leader's profit.

Stackelberg Duopolists, Firm 1 and Firm 2, Face Inverse Market P=50−QP = 50 - QP=50−Q

Stackelberg Duopolists, Firm 1 and Firm 2, Face Inverse Market $P = 50 - Q$