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

$$Solution$$

The production is the process that can be transformed the tangible inputs ( labor and capital) in to the final product. But sometimes production includes some intangible factors ( skills, intelligence, idea, knowledge) for producing as well goods and services. There are four factors of production which are used to produce the total output. Each of the firm involves in the production process for maximizing his profit. Firms either wants to maximize his profit or wants to minimize his costs. The above problem constitutes the problem of profit maximizing. The problem of the profit maximization can be solved in the following way.

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;

$C=\$30L+\$40K\\ C=\$30L+\$40(100) \implies C=\$30L+\$4000\\\\$

Where $K=100$

The market price is given as $p=\$1$

So the revenue function of the firm is given as;

$R=P \cdot Q= 1 \cdot 100L^{0.5}(100)^{0.5}$

Put the value of $K(100)$

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

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

$$\pi=R-C$$

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

$$\pi = 100L^{0.5}-30L-4000$$

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'

$\frac{\delta \pi}{\delta L}=\frac{1000}{2}L^{-\frac12}-30\\ F.O.C:\ \frac{\delta \pi}{\delta L}=0\\ 500L^{-\frac{1}{2}}-30=0\\ \implies \frac{500}{L^{0.5}}=30 \implies 30L^{0.5}=500$

Squaring both sides of the equation;

$900L=250,000 \implies L=277.7777 \approx 278$

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

$L=278\ units$

Now, calculate the value of the maximum profit of this firm, put the value of $L=278$,  the maximum profit of this firm can be calculated as;

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

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

$$\pi = 126,660\\$$