Derive the supply function with the production function production function y $=$ l

$1/3$K

$2/3$

and

the price w and for input labor $($L$)$ and capital $($K$)$ respectively.

If the price taking firm can produce output in two different factories with the individual cost

functions c$1($y$1)=$ ay$1$

$2$

and c$2($y$2)=$ by$2$

$2$

and Y$=$y$1+$y$2$