Question

A firm can manufacture a product according to the production function

$Q=F(K,L)=K^{3/4}{L}^{1/4}$

a. Calculate the average product of labor,$A{P}_L,$when the level of capital is fixed at 81 units and the firm usesunits of labor. How does the average product of labor change when the firm usesunits of labor?

b. Find an expression for the marginal product of labor,$M{P}_L,$, when the amount of capital is fixed atunits. Then, illustrate that the marginal product of labor depends on the amount of labor hired by calculating the marginal product of labor for$16$and$81$units of labor.

c. Suppose capital is fixed at$81$units. If the firm can sell its output at a price of$\backslash (200$per unit and can hire labor at$\backslash )50$per unit, how many units of labor should the firm hire in order to maximize profits?

Short answer

1. The average product of labor for $256$units is 0.42.
2. The marginal product of labor for 16 units is 0.844 and 81 units is 0.25.
3. Profit maximization occurs when firm hires 81 units of labor.

Step-by-step solution

Find the Average Product of Labor: 

$$\begin{gathered} A{P}_L=\frac{Q}{L} \\ =\frac{K^{3/4}{L}^{1/4}}{L} \\ ={\left(\frac{K}{L}\right)}^{3/4} \end{gathered}$$

$A{P}_L$Where K is 81 and L is 16:

$\begin{array}{rcl}A{P}_L& =& {\left(\frac{81}{16}\right)}^{3/4}\\ & =& {\left(\frac{81}{16}\right)}^{0.75}\\ & =& 3.375\\ & & \end{array}$

$A{P}_L$Where K is 81 and L is 256:

$\begin{array}{rcl}A{P}_L& =& {\left(\frac{81}{256}\right)}^{3/4}\\ & =& {\left(\frac{81}{256}\right)}^{0.75}\\ & =& 0.42\\ & & \end{array}$

Hence, the average productivity of labor when 256 units of labor is used is 0.42.

Find the expression of the marginal product 

$\begin{array}{rcl}M{P}_L& =& \frac{\partial Q}{\partial L}\\ & =& K^{3/4}(1/4)L^{\frac{1}{4}-1}\\ & =& \frac{1}{4}{\left(\frac{K}{L}\right)}^{3/4}\\ & & \end{array}$

$M{P}_L$Where K is 81

$\begin{array}{rcl}MPL& =& \frac{1}{4}{\left(\frac{81}{L}\right)}^{3/4}\\ & =& \frac{6.75}{L^{0.75}}\\ & & \end{array}$

$M{P}_L$Where K is 81 and L is 16:

$\begin{array}{rcl}MPL& =& \frac{6.75}{{16}^{0.75}}\\ & =& 0.844\\ & & \end{array}$

$M{P}_L$Where K is 81 and is 81:

$\begin{array}{rcl}MPL& =& \frac{6.75}{{81}^{0.75}}\\ & =& 0.250\\ & & \end{array}$

The marginal productivity of labor when firms use 16 units of labor is 0.844 and when 81 labor units are used is 0.250.

To find the maximize number of profits:

$\begin{array}{rcl}\text{Cost} \text{of}\text{labor}& =& MPL\times TC\\ 50& =& \frac{6.75}{L^{0.75}}\times 200\\ \frac{1350}{L^{0.75}}& =& 50\\ L& =& 81\\ & & \end{array}$

Hence, the cost maximization labor input is 81 units.