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. The equilibrium output of each firm is:
A. 8
B. 16
C. 32
D. 36
16
C(q)=4Q
C(q)'=4
4Q=100-2Q-4
4Q=96-2Q
6Q=96
Q=16
To find the equilibrium output of each firm in a Cournot duopoly, we need to determine the best response function for each firm and then solve for the intersection point.
Step 1: Determine the best response function for each firm.
In a Cournot duopoly, each firm determines its output level based on the assumption that the other firm's output remains constant. The best response function for each firm can be found by maximizing its profit with respect to its own output, taking into account the other firm's output.
The profit function for each firm can be calculated as follows:
Profit (π) = Total Revenue - Total Cost
π = PQ - C(Q)
= (P - MC)Q
= (100 - 2Q - 4Q)Q
= (100 - 6Q)Q
= 100Q - 6Q^2
To find the best response function, take the derivative of the profit function with respect to Q and set it equal to zero:
d(π)/dQ = 100 - 12Q
100 - 12Q = 0
12Q = 100
Q = 100/12
Q = 8.33 (approximately)
So, the best response function for each firm is Q = 8.33.
Step 2: Solve for the equilibrium output of each firm.
To find the intersection point, we need to equate the best response functions of both firms:
Q = 8.33
So, the equilibrium output of each firm is approximately 8.33.
Therefore, the answer is not provided in the options given.
To find the equilibrium output of each firm in a Cournot duopoly, we need to follow these steps:
1. Determine the reaction function of each firm: The reaction function specifies how each firm will react to the quantity produced by its competitor. In a Cournot duopoly, each firm assumes that its competitor's output is fixed and chooses its own output level to maximize its profit.
To find the reaction function, we can start with one firm (let's say firm 1) and solve for its optimal output level. Firm 1's profit is given by:
π1 = (P - C(Q1)) * Q1
= (100 - 2Q1 - 4Q2) * Q1
= (100 - 2Q1 - 4(Q - Q1)) * Q1 (since total quantity Q = Q1 + Q2)
= (100 - 6Q1 + 4Q2) * Q1
= 100Q1 - 6Q1^2 + 4Q2Q1
To maximize its profit, firm 1 takes the derivative of its profit function with respect to its own output Q1 and sets it equal to zero:
dπ1/dQ1 = 100 - 12Q1 + 4Q2 = 0
Solving for Q1, we get:
12Q1 = 100 + 4Q2
Q1 = (100 + 4Q2) / 12
Q1 = (25 + Q2) / 3
This equation represents firm 1's reaction function. Firm 2's reaction function will be symmetric, given by:
Q2 = (25 + Q1) / 3
2. Set the reaction functions equal to each other: Since both firms are identical, their reaction functions should be equal. We can substitute the equations for the reaction functions to find the equilibrium output:
(25 + Q2) / 3 = (25 + Q1) / 3
Simplifying:
25 + Q2 = 25 + Q1
Q2 = Q1
3. Solve for the equilibrium output: Now that we know both firms will produce the same quantity, we can substitute the value of Q2 into either of the reaction functions to solve for Q1:
Q1 = (25 + Q2) / 3
Q1 = (25 + Q1) / 3 (substituting Q2 = Q1)
3Q1 = 25 + Q1
2Q1 = 25
Q1 = 12.5
Thus, the equilibrium output of each firm is Q1 = Q2 = 12.5
Therefore, the correct answer is not provided among the options.