Question

Suppose that the production function of the firm is:
Q = 100L1/2.K1/2
K= 100, P = $1, w = $30 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

The production function of the firm is given by;

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

Where Q is the total output produced by the firm, L is the amount of labor employed in the firm, K is the amount of capital employed in the firm and $k=100$

The cost function for the given firm is;

$C=wL+rK$

Where $w$ is the wage rate, $\$30$ and $r$ is the interest rate $\$40$ .

So now the cost function is given by;

$C= 30L+40K$

$C=30L+40(100)$

$C=30L+4000$

Where $K=100$

The market price is given as $p=\$1$

So the revenue function of the firm is given as;

$R=P\times Q$

$=1\times100L ^{0.5}(100)^{0.5}$

Put the value of 

$K(100)$

$R=100L^{0.5}(10)^{\frac{2}{2}}$

$\implies R=100L \times L^{0.5} ​$

So now the profit function of the firm is given by;

$π=R−C$

Put the values of the revenue function (R) and the cost (c)

$π=100L^{0.5}−30L−4000$

Firm has to maximize its profit

To maximize profit, the First Order Condition (F.O.C) must be satisfied. For the F.O.C, take the differentiation with respect to 'L'

$\frac{δπ}{\delta L}=\frac{1000}{2}L^-{\frac{1}{2}}-30$

F.O.C: $\frac{δL}{δπ} =0$

$500L^-{\frac{1}{2}} −30=0$

$⟹\frac{500}{L^{0.5}}=30 ⟹30L^{0.5}=500$

Squaring both sides of the equation;

$900L=250,000⟹L=277.7777≈278$

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

$L=278\ units$

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

$L=278\ units$

$\pi =1,000L^{0.5}-30L-4,000$

put the value of $L=278$

$\pi = 139,000 - 8,340-4,000 \implies \pi = 126,660$

$\pi = 126,660$

The maximum profit of the firm is

$\pi = 126,660$