Quiz 4

Q:  What are the returns to scale and the equation for total cost for a firm with f(L,K)=L^1/2K^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)=1^1/21^2/3 =1.  f(2,2)=2^1/22^2/3 =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^-1/2K^2/3)/(.66L^1/2K^-1/3)=3K/4L.  Setting RTS=w/v gives 3K/4L=3/4 or K=L.   q=L^1/2K^2/3 =K^1/2K^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.