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.