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.

  • $\begingroup$ Are wages fixed/exogenous? $\endgroup$
    – Dayne
    Nov 30 '20 at 8:28
  • $\begingroup$ 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. $\endgroup$ Nov 30 '20 at 8:41

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.

  • 1
    $\begingroup$ Yes this makes sense. I will make a case by case analysis regarding the third derivative of F. Thank you very much! $\endgroup$ Nov 30 '20 at 13:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.