Question

A firm produces X using the following production function: Q = F(K,L) = 32K^0.76L^0.24. K=6 units. If the firm can sell each unit X at $22.20 and can hire labor at $16.00 per unit. Answer the below questions with the information provided.

The total profits when the firm uses optimal labor is?

How many workers the firm will hire in order to maximize profits?

How many units of X the firm will produce in order to maximize profits?

Answer 1

(i)

Profit = TR - TC

here TC = Total Cost = rK + wL and w = 16 , r = constant and K = 6 => TC = 6r + 16L

TR = Total revenue = PQ = 22Q = 22*32K^0.76L^0.24 and K = 6

=> TR = 22*32*6^0.76L^0.24 = 2747.69L^0.24

Hence , Profit = 2747.69L^0.24 - (6r + 16L)

First order condition

d(Profit)/dL = 0 => 0.24*2747.69/L^0.76 - 16 = 0

=> L = 133(approx)

Profit = 2747.69*133^0.24 - (6r + 16L) and assuming r = rent of capital = 0 we get

Profit = $6757.71

(ii)

As calculated above amount of labor hired in order to maximize Profit = 133 units

(iii)

Amount of X produced in order to maximize profit = 2747.69L^0.24 = 2747.69*133^0.24 =403.89 ~ 404 units