A firm has two factories. One factory has the cost function c1(y1) = 2y21 + 90 and the other has the cost function c2(y2) = 6y22 + 40. If the firm wishes to produce a total of 32 units as cheaply as possible, how many units will be produced in the second factory?
A) 7
B) 2
C) 8
D) 14
E) None of the above.

Question

Quizplus · Accepted Answer

To minimize the cost of producing 32 units, we need to find the point where the marginal cost of each factory is equal. 
The marginal cost for the first factory is MC1 = 4y1, and the marginal cost for the second factory is MC2 = 12y2. 
Setting MC1 = MC2, we get:

4y1 = 12y2
y1 = 3y2

Also, we know that the total number of units produced is 32, so:

y1 + y2 = 32

Substituting y1 = 3y2 into the above equation, we get:

3y2 + y2 = 32
4y2 = 32
y2 = 8

Therefore, the firm should produce 8 units in the second factory and 24 units in the first factory. The total cost will be:

c1(24) + c2(8) = 2(24^2) + 90 + 6(8^2) + 40 = $1,890

So the answer is C) 8.

A Firm Has Two Factories