Two identical firms compete as a Cournot duopoly.The demand they face is P = 100 − 2Q.The cost function for each firm is C(Q)= 4Q.Each firm earns equilibrium profits of:
A) $1,024.
B) $2,048.
C) $4,096.
D) $512.

Question

Quizplus · Accepted Answer

In a Cournot duopoly, each firm assumes that its competitor's output will remain constant when making production decisions. The solution involves finding the output that maximizes each firm's profit, given the output of its competitor.

To find the equilibrium output for each firm, we first need to determine the total market output. We start by solving the demand equation for Q:

P = 100 − 2Q
2Q = 100 - P
Q = 50 - 0.5P

Then, we add the two firms' outputs together to get the market output:

Qm = 2Q = 100 - P

Now we can find each firm's best response function. To do this, we assume that the competitor's output is fixed at Qc, and we solve for the output that maximizes each firm's profit.

For Firm 1, the profit function is:

π1 = (100 - (Q1+Q2))Q1 - 4Q1
π1 = (100 - Q1 - Q2)Q1 - 4Q1
π1 = (100 - Q1 - (50 - 0.5Q2))Q1 - 4Q1
π1 = (50 - 0.5Q1 - 0.5Q2)Q1 - 4Q1
π1 = 46.5Q1 - 0.5Q1^2 - 0.5Q1Q2

To find the maximum of this function, we take the derivative with respect to Q1 and set it equal to zero:

dπ1/dQ1 = 46.5 - Q1 - 0.5Q2 = 0
Q1 = 46.5 - 0.5Q2

Similarly, for Firm 2, the profit function is:

π2 = (100 - (Q1+Q2))Q2 - 4Q2
π2 = (100 - Q1 - Q2)Q2 - 4Q2
π2 = (100 - (50 - 0.5Q1) - Q2)Q2 - 4Q2
π2 = (50 - 0.5Q1 - 0.5Q2)Q2 - 4Q2
π2 = 46.5Q2 - 0.5Q1Q2 - 4Q2
π2 = 42.5Q2 - 0.5Q1Q2

Taking the derivative with respect to Q2 and setting it equal to zero gives:

dπ2/dQ2 = 42.5 - 0.5Q1 = 0
Q2 = 85 - Q1

Now we can solve for the equilibrium outputs by setting the two best response functions equal to each other:

46.5 - 0.5Q2 = 85 - Q1
Q1 = 46.5 - 0.5(85 - Q1)
Q1 = 23.25

Q2 = 85 - 23.25
Q2 = 61.75

Finally, we can find each firm's profit by plugging its output into the profit function:

π1 = (100 - (23.25+61.75))23.25 - 4(23.25)
π1 = $512

Therefore, the correct answer is D) $512.

Two Identical Firms Compete as a Cournot Duopoly