Question

"The amount of product a firm can produce in one week as a function of its capital investment K and its labor L and is given by
x = √(KL)
where x is the number of units the firm produces in one week, K is the number of machines, and L is the number of man-hours per week. Assume that K is fixed at 7 machines. The only expenses are the cost to operate the machines and wages for the labor. The operating cost per machine is $680 per week and the hourly wage is $46. Assume the firm can sell everything that it produces at a per-unit price of $635. How much should the firm produce in one week if it wants to maximize its profit?
The answer does not have to be an integer."

Answer 1

Production function is given by x = √(KL). K is fixed at 7 machines. Hence production function becomes x = √(7L).

Cost structure comprises of operating cost per machine written as rental price of capital as 'r' = $680 per week and the hourly wage written as 'w' = $46.There is a per-unit price of its product = $635.

Profit is given by revenue - cost. Here cost is C = 680*7 + 46L. From the production function, we find L = x^2/7. Hence profit function is PR = 635x - (680*7 + 46*x^2/7) = 635x - 4760 - (46/7)x^2

Profit is maximum when marginal profit is 0

dPr/dx = 0

635 - (46*2/7)x = 0

This gives x* = 48.31