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 leader's marginal revenue function is: A) MR(Q_L) = 50 − 2Q_L + c₁/2. B) MR(Q_L) = 50 − 2Q_L + c₂/2. C) MR(Q_F) = 100 − 2Q_F + c₁/2. D) MR(Q_F) = 100 − Q_F + c₂/2.

Question

Quizplus · Accepted Answer

In a Stackelberg duopoly, one firm is the leader and the other firm is the follower. The leader chooses its output first, and then the follower chooses its output. 
In this case, the inverse demand function is given by:
P = 100 - 2QS1U1B11S1U1B0 - 2QS1U1B12S1U1B0
The leader's output is QS1U1B1L, and the follower's output is QS1U1B1F.
The total output is:
Q = QS1U1B1L + QS1U1B1F
The leader's profit-maximizing output is determined by differentiating its profit function with respect to its output and setting the derivative equal to zero:
ΠL = (P - MCS1U1B1L) * QS1U1B1L
ΠL = (100 - 2QS1U1B11S1U1B0 - 2QS1U1B12S1U1B0 - 2QS1U1B1L) * QS1U1B1L
ΠL = (100 - (2QS1U1B11S1U1B0 + 2QS1U1B12S1U1B0 + 2QS1U1B1L)) * QS1U1B1L
MR(QS1U1B1L) = 100 - 2QS1U1B11S1U1B0 - 2QS1U1B12S1U1B0 - 4QS1U1B1L
Setting the derivative of the profit function with respect to QS1U1B1L equal to zero:
dΠL / dQS1U1B1L = MR(QS1U1B1L) - MCS1U1B1L = 0
Substituting the value of MCS1U1B1L: 
MR(QS1U1B1L) - 2 = 0
MR(QS1U1B1L) = 2
Using the inverse demand function, we can find the follower's reaction function, which gives the follower's optimal output as a function of the leader's output:
P = 100 - 2QS1U1B11S1U1B0 - 2QS1U1B12S1U1B0
P = 100 - 2Q + QS1U1B12S1U1B0
Q = (100 - P + QS1U1B12S1U1B0) / 2
QS1U1B1F = (100 - 2QS1U1B11S1U1B0 - QS1U1B12S1U1B0) / 4
Substituting the leader's output into the inverse demand function, we obtain the leader's marginal revenue function:
MR(QS1U1B1L) = 100 - 2QS1U1B11S1U1B0 - 2QS1U1B12S1U1B0 - 4QS1U1B1L
MR(QS1U1B1L) = 100 - 4Q
Substituting Q = QS1U1B1L + QS1U1B1F:
MR(QS1U1B1L) = 100 - 4(QS1U1B1L + QS1U1B1F)
MR(QS1U1B1L) = 100 - 4QS1U1B1L - (100 - 2QS1U1B11S1U1B0 - QS1U1B12S1U1B0) / 2
MR(QS1U1B1L) = 50 - 2QS1U1B1L + QS1U1B12S1U1B0 / 2
Therefore, the correct answer is B) MR(QS1U1B1LS1U1B0) = 50 − 2QS1U1B1LS1U1B0 + cS1U1B12S1U1B0/2.

Consider a Stackelberg Duopoly with the Following Inverse Demand Function