Consider a Stackelberg duopoly with the following inverse demand function: P = 100 − 2Q₁ − 2Q₂.The firms' marginal costs are identical and are given by MC_i = 2.Based on this information,the Stackelberg follower's reaction function is: A) Q_F = 24.5 − 0.25Q_L. B) Q_F = 49 − 0.25Q_F. C) Q_F = 24.5 − 0.5Q_L. D) Q_F = 24.5 − Q_L.

Question

Quizplus · Accepted Answer

In a Stackelberg duopoly, the leader's quantity is determined first, followed by the follower's quantity. The leader maximizes profit with the knowledge that the follower will react optimally to their choice.

To find the follower's reaction function, we first need to solve for the leader's quantity. The leader's profit-maximizing quantity is found by taking the derivative of the inverse demand function with respect to QS1U1B1 and setting it equal to the leader's marginal cost:

100 − 2QS1U1B11S1U1B0 − 2QS1U1B12S1U1B0 = 2

Simplifying:

QS1U1B11S1U1B0 + QS1U1B12S1U1B0 = 49

Assuming equal production (QS1U1B1 = QS1U1B2), we get:

2QS1U1B1 = 49

QS1U1B1 = 24.5

Now we can find the follower's reaction function by solving for QS1U1B1 in terms of the leader's quantity:

P = 100 − 2QS1U1B11S1U1B0 − 2QS1U1B12S1U1B0

P = 100 − 2QS1U1B1FS1U1B0 − 2QS1U1B12S1U1B0

Solving for QS1U1B1FS1U1B0:

2QS1U1B1FS1U1B0 = 100 − P − 2QS1U1B12S1U1B0

QS1U1B1FS1U1B0 = 50 − 0.5P − QS1U1B12S1U1B0

Plugging in the follower's best response for QS1U1B12S1U1B0:

QS1U1B12S1U1B0 = 24.5 − 0.5QS1U1B1LS1U1B0

QS1U1B1FS1U1B0 = 50 − 0.5P − 24.5 + 0.5QS1U1B1LS1U1B0

Simplifying:

QS1U1B1FS1U1B0 = 24.5 − 0.5QS1U1B1LS1U1B0

Therefore, the correct choice is C.

Consider a Stackelberg Duopoly with the Following Inverse Demand Function