University of Pittsburgh-Pittsburgh Campus - ECON 1100

Intermediate Micro Problem Set 2

Fall 2014

1)Consider the production function

. Does this exhibit constant, increasing, or decreasing returns to scale?

 

2. Consider the production function

   

   . Does this exhibit constant, increasing, or decreasing returns to scale?
3.  The Cobb-Douglas production function is given by

   

   . It turns out that the type of returns to scale of this function will depend on the magnitude of a + b. Which values of a + b will be associated with the different kinds of returns to scale?
4. The technical rate of substitution between factors x₂ and x₁ and is - If you desire to produce the same amount of output but cut your use of x₁ by 3 units, how many more units of x₂ will you need?

 

 

5. If a firm had decreasing returns to scale at all levels of output and it divided up into two equal-size smaller firms, what would happen to its overall profits?
6. If pMP₁ > w₁, then should the firm increase or decrease the amount of factor 1 in order to increase profits?
7. Suppose a firm is maximizing profits in the short run with variable factor x₁ and fixed factor x₂. If the price of x₂ goes down, what happens to the firm’s use of x₁? What happens to the firm’s level of profits?
8. Prove that a profit maximizing firm will always minimize costs.
9. If a firm is producing where MP₁/w₁ > MP₂/w₂, what can it do to reduce costs but maintain the same output?
10. A firm has a cost function given by c(y) = 10y²+1000. What is the supply curve?

 

11. A firm has a cost function given by c(y) = 10y² +1000. At what output is average cost minimized?

 

12. If a supply curve is given by S(p) = 100+20p, what is the formula for the inverse supply curve?

 

 

13. Below are descriptions of three production processes. In each case draw the corresponding isoquant. When possible calculate the marginal rate of technical substitution, that is, if marginal products are given.
    1. Each unit of output requires exactly 1 unit of labor and 1 unit of capital
    2. For any unit of output, the marginal product of capital is 4 and the marginal product of labor is 1.
    3. For any unit of output, the marginal product of capital is 3L and the marginal product of labor is 9K.

Solution: (a) Perfect complements/ Leontief with corners along the 45 degree line

(b) Perfect complements with downsloping lines, MRTS = 4 (c) Cobb-Douglas with

MRTS =

14. For a production function described by Q = L^1/3K^1/6. A. Find the marginal product of labor.
    2. Find the marginal product of capital.
    3. Find the MRTS.
    4. Determine whether the production technology exhibits increasing, decreasing, or constant returns to scale.
15. For a production function described by Q = L^1/3K^1/6 while the market wage w = 2 and rental rate of capital is r = 1.
    1. Find the cost minimizing input bundle when Q = 8 units are produced.
    2. Find the minimum cost.

 

16. For a production function described by Q = min{L,K} while the market wage w = 1 and rental rate of capital is r = 1.
    1. What type of production technology is this?
    2. Find the cost minimizing input bundle when Q = 8 units are produced.
    3. Find the minimum cost.
17. For a production function described by Q = L^1/2K^1/2 while the market wage w and rental rate of capital is r
    1. Find the firm’s conditional factor demands for labor and capital.
    2. Find the long run minimum cost function.

 

18. For a production function described by Q = L^1/3K^1/6 while the market wage w and

rental rate of capital is r

1. 1. 
      Find the firm’s conditional factor demands for labor and capital.
   2. Find the long run minimum cost function.