A firm with two plants,A and B,has the following estimated demand and marginal cost functions: $$\begin{array} { l }
Q _ { d } = 120 - 10 P \
M C _ { A } = 4 + ( 1 / 5 ) Q _ { A } \
M C _ { A } = 6 + ( 1 / 10 ) Q _ { A }
\end{array}$$ In order to maximize profit,how many units of output should the firm produce?

Question

A firm with two plants,A and B,has the following estimated demand and marginal cost functions: $$\begin{array} { l } 
Q _ { d } = 120 - 10 P \
M C _ { A } = 4 + ( 1 / 5 ) Q _ { A } \
M C _ { A } = 6 + ( 1 / 10 ) Q _ { A }
\end{array}$$ In order to maximize profit,how many units of output should the firm produce?

Quizplus · Accepted Answer

We can start by setting up the profit maximization problem for the firm:

Maximize Profit = Total Revenue - Total Cost

Total Revenue = Price * Quantity = P * Q

Total Cost = MC(total) * Q = (MC(A) + MC(B)) * Q

Substituting the demand function into the total revenue equation:

TR = P * Q = (120 - 10Q) * Q = 120Q - 10Q^2

Taking the derivative of TR with respect to Q to find the optimum output level:

dTR/dQ = 120 - 20Q = 0

Solving for Q, we get Q = 6.

We need to check if this output is feasible with the given marginal cost functions.

For plant A: MC(A) = 4 + (1/5)Q(A) = 4 + (1/5)(6) = 5.2

For plant B: MC(B) = 6 + (1/10)Q(B) = 6 + (1/10)(6) = 6.6

Therefore, the minimum total cost to produce 6 units is (5.2 + 6.6) * 6 = 73.2.

To find the optimum quantity, we can try higher values starting with Q = 7, 8, 9,...and so on, until we reach a point where the marginal cost exceeds the price (MC > P). At this point, the firm will no longer produce any output because it will not be profitable.

After trying out a few values, we find that the maximum output feasible is at Q = 25, where MC(total) = 9.5 and TR = $1875.

Therefore, the best choice is to produce 25 units of output (Choice C).

A Firm with Two Plants,A and B,has the Following Estimated Qd=120−10PMCA=4+(1/5)QAMCA=6+(1/10)QA\begin{array} { l } Q _ { d } = 120 - 10 P \\M C _ { A } = 4 + ( 1 / 5 ) Q _ { A } \\M C _ { A } = 6 + ( 1 / 10 ) Q _ { A }\end{array}Qd​=120−10PMCA​=4+(1/5)QA​MCA​=6+(1/10)QA​​

A Firm with Two Plants,A and B,has the Following Estimated $\begin{array} { l } Q _ { d } = 120 - 10 P \\M C _ { A } = 4 + ( 1 / 5 ) Q _ { A } \\M C _ { A } = 6 + ( 1 / 10 ) Q _ { A }\end{array}$