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?
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;
Substitute K:
R = P Q
R =
R =
Calculate the MPL:
This is the derivative of Labor to Revenue:
MPL =
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:
=
=
=
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 =
Profit =
Profit =
The maximum profit of this firm =
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