Consider $N$ firms each with the constant$-$returns$-$to$-$scale production function $Y=F(K,AL),$ or $($using the intensive form$)Y=$ALf$(k).$ Assume Assume that all firms can hire labor at wage $wA$ and rent capital at cost $r,$ and that all firms have the same value of $A.$

$($a$)$ Consider the problem of a firm trying to produce $Y$ units of output at minimum cost. Show that the cost$-$minimizing level of $k$ is uniquely defined and is independent of $Y,$ and that all firms therefore choose the same value of $k.$

$($b$)$ Show that the total output of the $N$ cost$-$minimizing firms equals the output that a single firm with the same production function has if it uses all the labor and capital used by the $N$ firms.