PRODUCTION AND COST:
PROBLEM SET 1

1. Given the production function Q = AK^aL^b show that

a + b > 1 implies increasing returns to scale
a + b = 1 implies constant returns to scale
a + b< 1 implies decreasing returns to scale

2. The following expressions show total cost (C) and total revenue (R) as a function of output (x).

      C = x³/3 - 3x² + 9x +16
      R = 21x - x²

Does this producer sell in a monopolistic or a perfectly competitive market ?
At what output is total revenue maximized ? What is the total revenue at this point or output level ?
At what output level is marginal cost at a minimum ?
At what output level is the company's profit maximized ?
Using the above information sketch the total cost and total revenue functions on the same diagram.

3. Find the output where the company maximizes profit if

              TR = 30q - q² 
              TC = 20 + 4q

4. Use the following production function to answer the questions below.

               X = 10 + 12L - L²

Find an expression for the average product of labor.
Find an expression for the marginal product of labor.
How much labor is need to maximize total output ?
What is the value of total output at this point ?
What is the value of the marginal product at this point ?
What is the value of the average product at this point ?

5. The production of output Q is characterized by a production function of the form:

              Q = K^.5 L^.5

Find an expression for the APK and APL .
Show that for this function the marginal product of each input is equal to one-half the average product.
Sketch both the average and marginal product functions for K = 100.
Explain the shapes of the functions.
Sketch both functions for K = 200.
Briefly explain your results for the questions above.

6. For the following production functions, determine the degree of homogeneity and the nature of the returns to scale.

Q = 3L³ +5LK² + K³
Q = 25L⁶ - K²L⁴
Q = 14 / L - 20 / K
Q = 3 / L² + 25 / LK + 6 / K²

7. If the total cost function for a company is as follows:

TC = aq

³ + bq

² + cq + d

What is the corresponding equation for the marginal cost function?
What is the corresponding equation for the average cost function?

8. The American Tool machine Co. has a total cost function represented by the equation

Find an expression for the marginal cost function.
Find an expression for the average cost function.
At what level of output is the average cost at a minimum?
Is this a logical set of cost functions for a company? Why?

9. For each of the following total cost functions, find marginal costs and determine the nature of the marginal cost function.

  TC = 1000q - 180q² + 3q³
  TC =  220 + 55q - 2q³ + q⁴

10. The marginal cost for the production of X is MC = 10 + 24q - 3q².If the total cost of producing 1 unit of X is 25,find the total cost function and the average cost function.

11. The demand function for a particular commodity is p = 28 - 5q and the total cost function for the monopoly firm is TC = q²+ 4q. Determine the maximum profit obtainable by the monopolist.

12. The demand function for a particular commodity is p = 26 - 2q - 4q² and the average cost to the monopolist of producing and distributing the commodity is q + 8 . What is the maximum profit for the monopolist.

13. Suppose that the cost and revenue functions for a company are as shown below in general form.

         MC = aq² - bq + c
         FC = d
         MR = A - Bq

Using this information derive the following functions.

TC
ATC
TVC
AVC
AFC
TR
AR
Total Profit
Output at minimum AVC
Output at minimum TR

14. A company's total cost function is of the following form.

  TC = 1500 + 40q - 6q² +.5q
   q = 50

Use this information to answer the questions below.

What is the AVC ?
What is the AFC ?
What is the ATC ?
What is the MC ?
At what level of output does diminishing returns set in ?
At what level of output does the MC = AVC ?
At what level of output is AVC at a minimum ?
At what level of output is ATC at a minimum ?
At what level of output does the MC = ATC ?

15. For each of the following cost functions determine the AVC, AFC, ATC, and MC and evaluate the cost at the given quantity.

 1400 + 80q , where q = 60

  900 - 10q + 5q², where q =40

 1500 + 40q - 6q² + .4q³ , where q = 50

PRODUCTION AND COST: PROBLEM SET 1

PRODUCTION AND COST:
PROBLEM SET 1