Consider a Cournot duopoly with the following inverse demand function: P = 100 − 2Q₁ − 2Q₂.The firms' marginal costs are identical and are given by MC_i(Q_i)= 2Q_i.Based on this information,firm 1 and 2's marginal revenue functions are: A) MR₁(Q₁,Q₂) = 100 − 2Q₁ − Q₂ and MR₂(Q₁,Q₂) = 100 − Q₁ − 2Q₂. B) MR₁(Q₁,Q₂) = 100 − 4Q₁ − 2Q₂ and MR₂(Q₁,Q₂) = 100 − 2Q₁ − 4Q₂. C) MR₁(Q₁,Q₂) = 100 − 2Q₁ − 4Q₂ and MR₂(Q₁,Q₂) = 100 − 4Q₁ − 2Q₂. D) MR₁(Q₁,Q₂) = 24.5 − 0.5Q₂ and MR₂(Q₁,Q₂) = 24.5 − 0.5Q₁.

Question

Quizplus · Accepted Answer

The marginal revenue functions for firm 1 and firm 2 are given by: 
MRS1(QS1,QS2) = 100 - 4QS1 - 2QS2
MRS2(QS1,QS2) = 100 - 2QS1 - 4QS2
where QS1 and QS2 are the quantities produced by firm 1 and firm 2 respectively. 
To derive the marginal revenue function for each firm, we take the derivative of the inverse demand function with respect to its own output, holding the other firm's output constant: 
MR1(QS1,QS2)=∂P/∂QS1=100−4QS1−2QS2−2
MR2(QS1,QS2)=∂P/∂QS2=100−2QS1−4QS2−2
Setting the marginal revenue equal to the marginal cost (which is the same for both firms) and solving for the output of each firm, we get: 
QS1 = 25 - (1/3)QS2 
QS2 = 25 - (1/3)QS1 
Substituting the second equation into the first equation yields: 
QS1 = 18.75 
QS2 = 18.75 
Therefore, the best choice is B.

Consider a Cournot Duopoly with the Following Inverse Demand Function