# 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
• $$\frac{\partial F}{\partial L} > 0 > \frac{\partial^2 F}{\partial L^2}$$
• 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.

• Are wages fixed/exogenous? Nov 30 '20 at 8:28
• The text does not explicitly say what is endogenous or exogenous. But a) wants us to look what happens to the equilibirium wages so wages must be endogenous. Nov 30 '20 at 8:41

$$\frac{\partial L_d}{\partial W} = \frac{-\partial g/\partial W}{\partial g /\partial L}=\frac{1}{PF_{LL}(L_d)}$$
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.