Consider the following cost minimization problem:
min $w_1x_1 + w_2x_2$
subject to
$f(x_1, x_2) = y$
$x_1, x_2, y \ge 0$
Show that the cost function
$c(w_1, w_2, y) = w_1x_1 + w_2x_2$
where
$x_1^* = x_1(w_1, w_2, y)$
$x_2^* = x_2(w_1, w_2, y)$
are conditional factor demands, is:
1. non-decreasing in $w = (w_1, w_2)$.
$w' = (w_1', w_2') \ge w = (w_1, w_2)$, then $c(w', y) \ge c(w, y)$.
2. homogenous of degree 1 in $w$.
$c(tw, y) = tc(w, y)$ for $t > 0$.
