Answer to Question #189163 in Macroeconomics for HASAN ALI KHAN

Question

Suppose that the production function of the firm is:

Q = 100L^1/2.K^1/2

 K= 100, P = $1, w = $50 and r = $40. Determine the quantity of labor that the firm should hire in order to maximize the profits. What is the maximum profit of this firm?

Answer 1

Solution:

Determine the quantity of labor that the firm should hire in order to maximize the profits:

The production function of the firm is given by;

$Q=100L ^{0.5}K ^{0.5}$

Substitute K:

$Q=100L ^{0.5}(100) ^{0.5}$

$Q=100L ^{0.5}(10)$

$Q=1000L ^{0.5}$

R = P$\times$ Q

R = $1\times 1000L ^{0.5} = 1000L ^{0.5}$

R = $1000L ^{0.5}$

Calculate the MPL:

This is the derivative of Labor to Revenue:

MPL = $\frac{\partial L} {\partial R} = \frac{500} {L ^{0.5}}$

Derive Total Cost function:

TC = wL + rK

TC = 50L + 40K

TC = 50L + 40(100)

TC = 50L + 4000

Find the derivative of the cost function with respect to Labor:

= 50

Now, set marginal product equal to zero and solve for L:

=$\frac{500} {L ^{0.5}} -50 = 0$

=$\frac{500} {L ^{0.5}} =50$

= $50L ^{0.5} =500$

Square both sides:

2500L = 250000

L = 100

The quantity of labor that the firm should hire in order to maximize the profits = 100

Therefore:

Profit = TR - TC

Profit = $(1000L ^{0.5}) - (50L + 4000)$

Profit = $(1000(100) ^{0.5}) - (30(100) + 4000)$

Profit = $50,000 - 7,000 = 43,000$

The maximum profit of this firm = $\$43,000$