3 Find out the cost minimizing labor capital and output for

3. Find out the cost minimizing labor, capital, and output for C (v, w) = w K + r L subject to f (K, L) = Q = LK/(L+K)

We have Profit function Z that needs to be maximized

Z=P*Q-wK-rL

Z=P(L^-1+K^-1)-wL-rK

dZ/dL=-P/L^2-w=0

dZ/dK=-P/K^2-r=0

wL^2=rK^2

L=sqrt(r/w)K

C=sqrt(wr)K+rK

K=C/(r+sqrt(wr))

L=C/(w+sqrt(wr))

L+K=C(r+w+2sqrt(wr))/A where A=(r+sqrt(wr))*(w+sqrt(wr))

LK=C^2/A

(C^2/A)*/C(r+w+2sqrt(wr))/A=C/(r+w+2sqrt(wr))=Q

C/(r+w+2sqrt(wr))=Q.... output