Q: What are the returns to scale and the equation for total cost for a firm with f(L,K)=L1/2K2/3 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)=11/212/3 =1. f(2,2)=21/222/3 =27/6 > 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-1/2K2/3)/(.66L1/2K-1/3)=3K/4L. Setting RTS=w/v gives 3K/4L=3/4 or K=L. q=L1/2K2/3 =K1/2K2/3 =K7/6 or K=q6/7. Thus L=q6/7 and TC=wL+vK= 3q6/7 +4q6/7 = 7q6/7.