Answer to Question #189163 in Macroeconomics for HASAN ALI KHAN

Question #189163

Suppose that the production function of the firm is:

Q = 100L1/2.K1/2

 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?


1
Expert's answer
2021-05-05T13:35:03-0400

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=100L0.5K0.5Q=100L ^{0.5}K ^{0.5}

Substitute K:

Q=100L0.5(100)0.5Q=100L ^{0.5}(100) ^{0.5}

Q=100L0.5(10)Q=100L ^{0.5}(10)


Q=1000L0.5Q=1000L ^{0.5}


R = P×\times Q

R = 1×1000L0.5=1000L0.51\times 1000L ^{0.5} = 1000L ^{0.5}


R = 1000L0.51000L ^{0.5}


Calculate the MPL:

This is the derivative of Labor to Revenue:

MPL = LR=500L0.5\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:

=500L0.550=0\frac{500} {L ^{0.5}} -50 = 0


=500L0.5=50\frac{500} {L ^{0.5}} =50


50L0.5=50050L ^{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 = (1000L0.5)(50L+4000)(1000L ^{0.5}) - (50L + 4000)

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


Profit = 50,0007,000=43,00050,000 - 7,000 = 43,000


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

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