\"The amount of product a firm can produce in one week as a function of its capi

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 14 machines. The only expenses are the cost to operate the machines and wages for the labor. The operating cost per machine is $580 per week and the hourly wage is $54. Assume the firm can sell everything that it produces at a per-unit price of $506. 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."

Explanation / Answer

Revenue (R) = 506x

Given that x = (KL) or, x = 14L = 3.74L. So, R = 506*3.74L = 1893.38L1/2

dR/dL = (1/2)*1893.38*L-1/2 = 946.64L-1/2

Total Cost (C) = Cost of machine operation + labor hrs. cost = 14 x 580 + 54L = 8120 + 54L

dC/dL = 54

For max profit, dR/dL = dC/dL

or, 946.64L-1/2 = 54

or, L-1/2 = 0.057

or, L = 307.31

So, x = 3.74L = 65.56 (this should be production level to maximize the profit)

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