Labor market model with subsidies homework I would appreciate some help with my homework in comparative statics.

L is the demand of labor
F(L) is the production function
$ \frac{\partial F}{\partial L} &gt; 0 &gt; \frac{\partial^2 F}{\partial L^2} $
P is the price of goods
W is the wage
S is subsidy per hired worker
AA(W) is the supply of labor
$ \frac{\partial AA}{\partial W} &gt; 0 $
$ max_{L} P*F(L) - (W - S)*L $ is what companies want to maximize

a) Analyze the effect of an increase in subsidies on the equilibriums of W and L.
b) Illustrate your findings in a graph. Chose Quantity of labor and W as axis.
c) Analyze the effect of an increase in subsidies on the slope of L in W.
My ideas for a):
The equilibrium conditions are $ L = AA(W) \Leftrightarrow L - AA(W) = 0 $. So lets call it $ H(W; L) = 0 $ and let´s call the profit function G so we get as the second condition $ \frac{ \partial G}{ \partial L} = P * \frac{\partial F(L)}{\partial L} - (W - S) = 0$, so we have $ g(L; P, W, S) = 0$.
I used the IFT:
$ \frac{\partial G}{\partial S} = - \frac{\frac{\partial g}{\partial S}}{\frac{\partial g}{\partial L}} = - \frac{ S}{P* \frac{\partial^2 F}{\partial L^2}} &gt; 0$. This means L increases if S increases.
Now let´s take a look at the labor market:
$ - \frac{\frac{\partial H}{\partial L}}{\frac{\partial H}{\partial W}} = - \frac{1}{\frac{ - \partial AA(W)}{\partial W}} &gt; 0$. Since L increases the wage also increases.
Regarding c):
I have to find $\frac{\partial L(W)}{\partial W \partial S} $, but I don´t really have an idea how to get it.
Does this make sense?:
$$\frac{\partial L_d}{\partial W} = \frac{-\partial g/\partial W}{\partial g /\partial L}=\frac{1}{PF_{LL}(L_d)}$$
Now from solution of (a) we know that as $S \uparrow$, $L_d \uparrow \implies F_{LL} \downarrow \implies \partial L_d/\partial W \uparrow$
Therefore, $\partial^2 L_d/\partial W\partial S &gt;0$
EDIT: Note that I have assumed above that $F_{LLL} &lt;0$. Clearly this is not given in your question statement. So as per the answer above, what happens to the slope should depend on sign of $F_{LLL}$. If, for example, it is zero, then the curve simply shifts with no change in slope.