Suppose that firms $A$ and $B$ are Cournot duopolists in the salt industry. The market demand curve can be specified as $P = 200 - Q _ { A } - Q _ { B }$. The marginal cost to each firm is $\$ 40$. What is firm $B$ 's profit-maximizing quantity when firm A produces an arbitrary output $Q _ { A }$ ?
A) $Q _ { B } = 160 - Q _ { A }$.
B) $Q _ { B } = 160 - 2 Q _ { A }$.
C) $Q _ { B } = 80 - Q _ { A }$.
D) $Q _ { B } = 80 - \frac { 1 } { 2 } Q _ { A }$.

Question

Quizplus · Accepted Answer

To find firm B's profit-maximizing quantity, we need to use the first-order condition for profit maximization, which is to set marginal revenue equal to marginal cost.

First, we need to find the total quantity supplied in the market by both firms, which is:
Q = QA + QB

Then, we can find the market price by plugging Q into the demand curve:
P = 200 - Q

Each firm's revenue is then:
R = P * Qi

Firm B's profit is:
πB = R - C * Qi

Taking the derivative of πB with respect to QB and setting it equal to zero, we get:
dπB/dQB = d(R - C * QB)/dQB = 0
dR/dQB - C = 0
d/dQB (P * QB) - C = 0
P - C = 0
Solving for QB, we get:
P = C
200 - QA - QB = 40
QB = 160 - QA

Therefore, the answer is D) QB = 80 - 1/2QA We can start by writing the total revenue function for firm B:

$$R _ { B } = P Q _ { B } = (200 - Q _ { A } - Q _ { B }) Q _ { B } = 200 Q _ { B } - Q _ { A } Q _ { B } - Q _ { B } ^ { 2 }$$

Then the profit function for firm B can be written as:

$$ \pi _ { B } = R _ { B } - C _ { B } Q _ { B } = 200 Q _ { B } - Q _ { A } Q _ { B } - Q _ { B } ^ { 2 } - 40 Q _ { B }$$

To maximize profits, we can take the derivative of the profit function with respect to $Q _ { B }$ and set it equal to zero:

$$ \frac { d \pi _ { B } } { d Q _ { B } } = 200 - Q _ { A } - 2 Q _ { B } - 40 = 0$$

Solving for $Q _ { B }$, we get:

$$ Q _ { B } = \frac { 160 - Q _ { A } } { 2 }$$

Therefore, the answer is D.

Suppose That Firms AAA And BBB Are Cournot Duopolists in the Salt Industry

Suppose That Firms $A$ And $B$ Are Cournot Duopolists in the Salt Industry