Cost Curves?

Question

Q = KL^1/2/100, MP(SMALL L NEXT TO P) = Q/(2L), AND MP(SMALL K NEXT TO P) = Q/K. Now, wages = w and rental rates = r

A. What is long-run total cost curve for output in terms of Q, w, and r?

B. What is the long-run average cost curve for output in terms of Q, w and r?

C. For w = 10 and r = 20, write an expression for the long - run total cost of the firm.

Answer 1

A:

TC= wL+rK

MPL/w=MPK/r

Q/2Lw=Q/Kr

Kr=2Lw

K=2Lw/r

Q=K(L^1/2)/100

K=100Q/L^1/2

100Q/L^1/2=2Lw/r

L^3/2=100Qr/2w=50Qr/w

L^1/2=(50Qr/w)^1/3

TC= wL+rK

K=100Q/L^1/2

TC=wL+100Qr/L^1/2=

=w*(50Qr/w)^2/3+100Qr*(w/50Qr)^1/3=

=(w*w*50*Q*r)^(2/3)) + (20'000*(Q^2)*(r^2)*w)^(1/3)

TC=Q^2/3 *(( w^2*50*r)^2/3 + (20'000*w*r^2)^1/3)

B:

AC=TC/Q

AC= Q^2/3 *(( w^2*50*r)^2/3 + (20'000*w*r^2)^1/3)/Q=

=(( w^2*50*r)^2/3 + (20'000*w*r^2)^1/3) / Q^1/3

C:

w=10

r=20

TC=2585.32 * Q^(2/3)