Question

Exercise 4: Production functions and Assume that we have a firm which has the following produ st minimization (20 points) • The wage per unit of labor is given by w = an hour. Let's assume that the firm wants to q=kt. DO is given by w = $4 an hour and the rental cost per unit of capital is assume that the firm wants to produce = 10 units of output. unit of capital is r= $1 1. In the long run, how much labor and Hint: You should use the short-cut method. much labor and capital will the firm use if it wishes to minimize cost 2. What is the total cost of producing 10 units? 3. In the short run, we assume that the firm's capital is fixed at K = 25 units. How many hours of labor services does the firm need to use? 4. What is the total cost of producing the 10 units?

the exponents of a and b are both 0.5. please do all 1-4 thx!

Answer 1

1)

q=K^0.5L^0.5

Marginal Product of labor=MPL=dq/dL=0.5K^0.5L^-0.5

Marginal Product of capital=MPK=dq/dK=0.5K^-0.5L^0.5

For Cost minimization

MPL/MPK=w/r

(0.5K^0.5L^-0.5)/(0.5K^-0.5L^0.5)=4/1

K/L=4/1

K=4L

Set K=4L and q=10 in production function

10=(4L)^0.5L^0.5

10=2L

L=5 (Optimal amount of labor)

K=4L=4*5=20  (Optimal amount of capital)

2)

Total cost=wL+rK=4*5+1*20=$40

3)

q=K^0.5L^0.5

Set q=10, and K=25

10=25^0.5L^0.5

2=L^0.5

L=4

4)

Total cost=wL+rK=4*4+1*25=$41