Quiz 4

Q: What are the returns to scale and the equation for total cost for a
firm with f(L,K)=L^{1/2}K^{2/3} that faces w=3 and v=4.

A: To measure returns to scale compare 2*f(1,1) and f(2,2).
f(1,1)=1^{1/2}1^{2/3} =1. f(2,2)=2^{1/2}2^{2/3}
=2^{7/6} > 2. Since f(2,2) > 2*f(1,1) this production
function exhibits increasing returns to scale.

The optimal condition for a firm is RTS=w/v since it has CD production.
RTS =(.5L^{-1/2}K^{2/3})/(.66L^{1/2}K^{-1/3})=3K/4L.
Setting RTS=w/v gives 3K/4L=3/4 or K=L. q=L^{1/2}K^{2/3}
=K^{1/2}K^{2/3} =K^{7/6} or K=q^{6/7}.
Thus L=q^{6/7} and TC=wL+vK= 3q^{6/7} +4q^{6/7} = 7q^{6/7}.