Question

A firm’s production function is: Q = 6K^1/3 L^2/3. Suppose that capital is fixed at 125 units. If the firm can sell its output at a price of $200 per unit, hire labor at $20 a unit, and rent capital at $40 a unit, how many units of labor should the firm hire to maximize profits?Given your answer, what is the level of output that maximizes profits? Lastly, calculate the level of profits at the profit-maximizing level of labor and output given that capital is fixed at 125 units.

Answer 1

(i)

Profit(Pr) = TR - TC

where TR = Total revenue = P*Q , TC = Total cost = wL + rK

Now P = Price of output = 200, K = 125(Fixed) => Q = 6*125^1/3 L^2/3 = 30L^2/3 , w = wage rate = $20 and r = rent = 40

=> Pr = 200*30L^2/3 - (20L + 40*125)

First order condition:

d(Pr)/dQ = (2/3)*6000/L^1/3 - 20 = 0

=> L = 200³ = 8,000,000 units

(ii)

Q =  6K^1/3 L^2/3. As discussed above K = 125 and L = 8,000,000

=> Q = 6*125^1/38,000,000^1/3 = 1,200,000 units.

Hence, Level of Output that maximizes profit = 1,200,000 units

(iii)

Profit = TR - TC = 200*30L^2/3 - (20L + 40*125) and L = 8,000,000 units

=> Profit = 200*1,200,000 - (20*8,000,000 + 40*125)

=> Profit = $79,995,000

Hence, Level of profits at the profit-maximizing level of labor and output given that capital is fixed at 125 units is $79,995,000.