Application question on Marginal

A firm can manufacture a product according to the production function:
Q = F(K,L) = K3/4L1/4.
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.
Instruction: Round your responses to 3 decimal places.
3.375

What is the average product of labor when the firm uses 256 units of labor?
.422

b. Find an expression for the marginal product of labor, MPL, when the amount of capital is fixed at 81 units.
Instruction: The second response is the exponent on L in the expression. Round your responses to 2 decimal places.
MPL =
*L2*L^3
^
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.
Instruction: Round your responses to 3 decimal places.
MPL when L = 16:
.25

MPL when L = 81:
.075

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

A firm’s product sells for $4 per unit in a highly competitive market. The firm produces output using capital (which it rents at $25
per hour) and labor (which is paid a wage of $30 per hour under a contract for 20 hours of labor services). Complete the
following table and use that information to answer the questions that follow.
Instruction: Round your answers for Average Product of Capital and Average Product of Labor to 2 decimal places.
-1

-2

Capital

Labor

0
1
2
3
4
5
6
7
8
9
10
11

20
20
20
20
20
20
20
20
20
20
20
20

-3

-4

-5

-6

-7

Value
Marginal Average Average Marginal
Product Product Product Product
Output of Capital of Capital of Labor of Capital
MPK
APK
APL
VMPK
0
50
150
300
400
450
475
475
450
400
300
150

0

50

2.5

–

a. Identify the fixed and variable inputs.

Capital and labor are fixed inputs.

Capital is the fixed input and labor is the
variable input.

Capital and labor are variable inputs.

Labor is the fixed output and capital is the
variable input.

b. What are the firm’s fixed costs?
$
c. What is the variable cost of producing 475 units of output?
$
d. How many units of the variable input should be used to maximize profits?

e. What are the maximum profits this firm can earn?
$
f. Over what range of the variable input usage do increasing marginal returns exist?
From
to

To

g. Over what range of the variable input usage do decreasing marginal returns exist?
From
to

To

h. Over what range of input usage do negative marignal returns exist?