Answer to Question #309112 in Microeconomics for LALA

Question

A firm can manufacture a product according to the production function (LOI, LO2,

LOS, L06)

Q = F(K, L) = K 3/4 L 1/4

a. 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 average product of labor

change when the firm uses 256 units of labor?

b. Find an expression for the marginal product of labor, MP L, when the amount of capi-

tal 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.

c. 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?

Answer 1

Solution

$APL=F(K,L)$

$=(K/L)^{0.75}$

a)

When

K=81

L=16

$APL=(\frac {81}{16})^{0.75}=3.375$

If

L=256

$APL=(\frac {81}{256})^{0.75}$

$=0.4218$

b)

$MPL=dF(K,L)/dL$

$=0.25(\frac {K}{L})^{0.75}$

When

K=81

$MPL=0.25(\frac {81}{L})^{0.75}$

$=(6.75/L)^{0.75}$

When

L= 16

$MPL=(6.75/16)^{0.75}$

$=6.75\div8=0.84375$

When

L=81

$MPL=(6.75/81)^{0.75}$

$=6.75\div 27=0.25$

c)

At the profit maximization point, Value of marginal product of labour=wage.

$P\times MPL=wage$

$200\times (\frac {6.75}{L})^{0.75}=50$

$4\times 6.75=L^{0.75}$

$L=81$