Averages Product of Labor

A company can manufacture a product by using the product function of

Q=K^(3/4) L^(1/4)

Calculate the average product of labor APL, when the level of capital is fixed at 81 units and the firm uses 16 units of labor.

How does the averages product of labor change when the firm uses 256 units of labor?

The level of the output produced from this function when number of units of K is 81 units and labor units are 16 units, the calculation is shown as;

Q=K^(3/4) L^(1/4)
=(81)^(3/4) (16)^(1/4)
=(3^4 )^(3/4) (2^4 )^(1/4)
=27×2
=54

The mean productivity of labor is shown as the amount of output which is manufactured by use of the unit of the labor. It can be calculated by dividing the total output by the total amount of labor used.

For total output of 54 units the mean product of labor that can be calculated is shown as;

AP_L=Q/L
AP_L=54/16
AP_L=3.375 units

For the 81 units of K and the 256 units of the labor consumed, the level of output produced can be computed as below;

Q=K^(3/4) L^(1/4)
=(81)^(3/4) (256)^(1/4)
=(3^4 )^(3/4) (4^4 )^(1/4)
=27×4
=108

For the total given output of 108 units, the mean product of the labor quantity can be calculated by using the formula as;

AP_L=Q/L
AP_L=108/256
AP_L=0.422 units

Therefore, the mean product of labor can be (AP_L) calculated as 0.422 units of labor.

 

Find an expression for the marginal product of labor MPL, when the amount of capital is fixed at 81 units. 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.

The Marginal product of labor is the amount of extra output which is produced by the hiring of one additional unit of labor. The Marginal product of labor is calculated by the finding of its first derivative of function of production in terms of labor while keeping the capital amount fixed at 81 units.

For the calculation of the computation of the marginal productivity of labor.

MP_L=∂Q/∂L
MP_L=(∂((81)^(3/4) L^(1/4))/∂L
MP_L=27x(1/4)(1/L^(3/4) )
MP_L=27/4 (1/L^(3/4) )

The Marginal productivity of labor is dependent on the no of units of the labor which is hired by the company. At units of labor at 16 units, the MP_L or marginal product of labor is computed as;

MP_L=27/4 (1/L^(3/4) )
MP_L=27/4 (1/16^(3/4) )
MP_L=27/4 (1/(2^4 )^(3/4) )
=27/4 x(1/8)=0.8438

For 81 labor units, the marginal product of the labor; the computations are;

MP_L=27/4 (1/L^(3/4) )
MP_L=27/4 (1/81^(3/4) )
MP_L=27/4 (1/〖3^4〗^(3/4) )
MP_L=27/4 (1/27)
=0.25

Therefore, with the rise in the labor units to 81 units from 16 units, the marginal product of labor declines to 0.25 units from 0.843 units.

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

The value of the additional quantity to be produced by using one additional unit of labor is called as the value of marginal product. It is computed by multiplication of MP_L by the Output Price.

The formula is;

VMP_L=MP_L x P
=27/4 (1/L^(3/4) )x200
=1350(1/L^(3/4) )

The maximization level of output can be achieved at the point in which the marginal product value is equal to the rate of wage as given to the wagers by company.

VMP_L=wage
1350(1/L^(3/4) )=50
L^(3/4)=27
L^(3/4)=3^3
L^(1/4)=3
L=3^4=81

Therefore, this shows that for the maximization of the profit, the company would need to hire 81 laborers.