<p>Does this make sense?:</p>
<p><span class="math-container">$$\frac{\partial L_d}{\partial W} = \frac{-\partial g/\partial W}{\partial g /\partial L}=\frac{1}{PF_{LL}(L_d)}$$</span></p>
<p>Now from solution of (a) we know that as <span class="math-container">$S \uparrow$</span>, <span class="math-container">$L_d \uparrow \implies F_{LL} \downarrow \implies \partial L_d/\partial W \uparrow$</span></p>
<p>Therefore, <span class="math-container">$\partial^2 L_d/\partial W\partial S &gt;0$</span></p>
<p>EDIT: Note that I have assumed above that <span class="math-container">$F_{LLL} &lt;0$</span>. 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 <span class="math-container">$F_{LLL}$</span>. If, for example, it is zero, then the curve simply shifts with no change in slope.</p>

CC BY-SA 4.0
2020-11-30T13:19:15.320Z
Yes this makes sense. I will make a case by case analysis regarding the third derivative of F. Thank you very much!
JimTrouserHead
CC BY-SA 4.0
2020-11-30T11:42:13.197Z
2020-11-30T12:22:48.390Z
2020-11-30T12:22:48.390Z
Dayne
<p>I would appreciate some help with my homework in comparative statics.</p>
<ul>
<li>L is the demand of labor</li>
<li>F(L) is the production function</li>
<li><span class="math-container">$ \frac{\partial F}{\partial L} &gt; 0 &gt; \frac{\partial^2 F}{\partial L^2} $</span></li>
<li>P is the price of goods</li>
<li>W is the wage</li>
<li>S is subsidy per hired worker</li>
<li>AA(W) is the supply of labor</li>
<li><span class="math-container">$ \frac{\partial AA}{\partial W} &gt; 0 $</span></li>
<li><span class="math-container">$ max_{L} P*F(L) - (W - S)*L $</span> is what companies want to maximize</li>
</ul>
<p>a) Analyze the effect of an increase in subsidies on the equilibriums of W and L.</p>
<p>b) Illustrate your findings in a graph. Chose Quantity of labor and W as axis.</p>
<p>c) Analyze the effect of an increase in subsidies on the slope of L in W.</p>
<p>My ideas for a):</p>
<p>The equilibrium conditions are <span class="math-container">$ L = AA(W) \Leftrightarrow L - AA(W) = 0 $</span>. So lets call it <span class="math-container">$ H(W; L) = 0 $</span> and let´s call the profit function G so we get as the second condition <span class="math-container">$ \frac{ \partial G}{ \partial L} = P * \frac{\partial F(L)}{\partial L} - (W - S) = 0$</span>, so we have <span class="math-container">$ g(L; P, W, S) = 0$</span>.</p>
<p>I used the IFT:</p>
<p><span class="math-container">$ \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}} &gt; 0$</span>. This means L increases if S increases.</p>
<p>Now let´s take a look at the labor market:</p>
<p><span class="math-container">$ - \frac{\frac{\partial H}{\partial L}}{\frac{\partial H}{\partial W}} = - \frac{1}{\frac{ - \partial AA(W)}{\partial W}} &gt; 0$</span>. Since L increases the wage also increases.</p>
<p>Regarding c):</p>
<p>I have to find <span class="math-container">$\frac{\partial L(W)}{\partial W \partial S} $</span>, but I don´t really have an idea how to get it.</p>

CC BY-SA 4.0
2020-11-30T08:28:07.890Z
Are wages fixed/exogenous?
Dayne
CC BY-SA 4.0
2020-11-30T08:41:41.017Z
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
<labor-economics>
Labor market model with subsidies homework
JimTrouserHead
# Labor market model with subsidies homework

(Asked by: username_0 on 2020-11-29)

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.

**Tags:** labor-economics
> **Comments:**
> **username_1**: Are wages fixed/exogenous?  
> **username_0**: 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.  

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## 1 answers

### Answer by username_1 on 2020-11-30. Score: 5

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.

> **Comments:**
> **username_0**: Yes this makes sense. I will make a case by case analysis regarding the third derivative of F. Thank you very much!  

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