Suppose a Technology Is Described by the Production Function $f ( \ell )$

Essay

Suppose a technology is described by the production function $f ( \ell )$
a.For a price taking producer who faces output price p and wage w,derive the first order condition and interpret it.
b.Without knowing more about the function f,is the condition you derived in (a)either necessary or sufficient for deriving the profit maximizing production plan? Explain.
c.Suppose
$f ( \ell ) = \ell ^ { \alpha }$ .Derive the first order condition you illustrated in (a)and solve for
$\ell$ .
d.For what values of
$\alpha$ is this first order condition necessary and sufficient for deriving a profit maximizing production plan? Explain.