Cost Minimization?

Question

Q = (KL/25), MP (SMALL L NEXT TO P) = Q/(2L), AND MP(SMALL K NEXT TO P) = Q/(2K). NOW, WAGES = w AND RENTAL RATES = r

a. Derive expressoins for the firm of its long - run demand shedules for labor and capital in terms of w, r, and Q

B. If Q = 1000, w = 10 and r = 20, wht quantities of labor and capital would be demanded by the firm int eh long term?

C. For K= 40, what is the short -term demand of hte firm for Labor in terms of w, r, and Q?

D. For K = 40, what is teh short-term cost schedule of the firm in terms fo w, r, and Q?

E. If Q = 1000, w = 10 and r = 20, what quantities of labor and capital would be demanded by the firm in the short term?

Answer 1

A:

MPL=Q/(2L)

MPL=w

Q/(2L)=w

Long-run demand for labor: L=Q(2w)

MPK=Q/(2K)

MPK=r

Q/(2K)=r

Long-run demand for capital: L=Q(2r)

B:

L=Q/(2w)

w=10 Q=1000

L=1000/(10*2)=100/2=50

K=Q/(2r)

r=20 Q=1000

K=1000/(2*20)=100/4=25

C:

MPL/w=MPK/r

Q/(2Lw)=Q/(2Kr)

L=Kr/w

K=40

L=40r/w

D:

TC=Lw+Kr

K=40

TC=Lw+40r

Q=KL/25

K=25Q/L

MPL/w=MPK/r

Q/(2Lw)=Q/(2Kr)

Kr=Lw

K=Lw/r

K=25Q/L & K=Lw/r

25Q/L=Lw/r

L^2=25Qr/w

L=5(Qr/w)^(1/2)

TC=Lw+40r = 5w(Qr/w)^(1/2) + 40r =

= 5√(Qrw)+40r

E:

...Rest will check later...