Question

a. A cost minimizing firm’s production is given by Q=L^(1/2)K^(1/2)

. Suppose the desired output is

Q=10. Let w=12 and r=4. What is this firm’s cost minimizing combination of K & L? What it the

total cost of producing this output?

b. Suppose the firm wishes to increase its output to Q=12. In the short run, the firm’s K is fixed

at the amount found in (a), but L is variable. How much labor will the firm use? What will the

total cost be?

Answer 1

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

Maximize function Z s.t. C=wL+rK

Z=Q-(wL+rK-C)

finding FOC

dZ/dL=1/2L^(-1/2)*K^(1/2)-w=0

dZ/dK=1/2K^(-1/2)*L^(1/2)-r=0

dZ/d=C-wL-rK=0

(K/L)^(1/2)*(1/w)=(L/K)^(1/2)*(1/r)

r/w=L/K

w=rK/L

C=wL+rK=L(rK/L)+rk=2rk

C=8k

Q=L^(1/2)K^(1/2)=10

10=L^(1/2)*(wL/r)^(1/2)=L*(w/r)^(1/2)

10=L*3^(1/2)

L=10/1.74=5.75 & K=wL/r=3*10/3^(1/2)=3^(1/2)*10=17.32

C=wL+rK=12(5.75)+4(17.32)=138.28

If Q=12

12=(LK)^(0.5)=(17.32*L)^(0.5)

144=17.32L

New L=8.314 This will be used in later case & old L=5.75

Cost =C=wL+rK=12*8.314+4*17.32=169.05