Static wage posting model and indifference to wage levels Suppose a monopsonistic firm has the following simple profit function:
$$\pi (w) = PL(w) - wL(w) $$
where $L(w)$ is the labor supply, $P$ is the constant output price, and it has a linear production technology.

What is a labor supply schedule $L(w)$ that would make the profit function insensitive to the wage?
The profit function is an optimal relation. So it must satisfy the first-order conditions for a maximum:

$$ \text{f.o.c} : w^*: PL' - L - w^*L' =0 \implies L\left[ (P-w^*) \frac {L'}{L}-1\right] = 0$$

In order for this condition to be satisfied no matter what the wage is, it must be the case that, for all wage-levels we have 

$$(P-w) \frac {L'}{L}-1 = 0 \;\;\forall w \implies L' = \frac{1}{P-w}L \;\;\forall w$$

Of course this looks exactly like the first-order condition - the critical difference is that it now must hold for all $w$, and so it is not any more a condition that can hold for a specific $w$, but a differential equation in $w$, that falls in the general category $y'_x = f(x)\cdot y$. It can be solved (it is "separable").

I guess you can finish this.