Labor market model with subsidies homework

Question

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} > 0 > \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} > 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}} > 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}} > 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.

Answer 1

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, $\partial^2 L_d/\partial W\partial S >0$

EDIT: Note that I have assumed above that $F_{LLL} <0$. 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.