Question

A factory produces product A according to the production function QA = 100XA, where XA denotes the amount of input and QA is the quantity produced. It also produces product B according to the production function QB = 200XB - XB2, where XB denotes the amount of input and QB is the quantity produced. The total available amount of input X is 100 units. The firm’s profit-maximizing allocation of input X is: Group of answer choices XA = 60 and XB = 40. XA = 50 and XB = 50. XA = 0 and XB = 100. XA = 100 and XB = 0. XA = 30 and XB = 70.

Answer 1

Answer:

Option (B) is correct.

Explanation:

XA + XB = 100

QA = 100XA

QB = 200XB - XB^2

Use the fact that,

XA = 100 - XB

Now total production is Q = QA + QB

Q = 100XA + 200XB - XB^2

Q = 100 × (100 - XB) + 200XB - XB^2

Q = 10,000 + 100XB - XB^2

Output is maximum when Q'(XB) = 0

100 = 2XB = 0

XB = 50

XA = 50

Therefore, the firm’s profit-maximizing allocation of input X is 50 units of XA and 50 units of XB.