# 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? – Dayne 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. – JimTrouserHead 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.