<p>By Hotelling's lemma, the partial derivative of the profit function with respect to the input price gives the factor demand. As such, the demand for labour for firm <span class="math-container">$i$</span> equals:
<span class="math-container">$$
\frac{\partial \pi_i(i,w)}{\partial w}.
$$</span>
The requirement that <span class="math-container">$\sum_i \frac{\partial \pi_i(i,w)}{\partial w} = L$</span> requires that total labor demand (over all firms) equals the total labor available in the economy. You need to check but probably in this model, labour is inelastically supplied by the households so this equation simply states that aggregate labour demand equals aggregate labour supply.</p>

CC BY-SA 4.0
2024-05-06T13:26:53.563Z
2024-05-06T13:26:53.563Z
tdm
<p>I am given the following full-employment condition, where sectors of the economy are indexed by <span class="math-container">$i$</span>.</p>
<p><span class="math-container">$L+\sum_i\frac{\partial \pi_i(p_i,w)}{\partial w}=0$</span></p>
<p>Why, if all workers are employed, should how profit changes given a wage increase equal the <em>number</em> of workers <span class="math-container">$L$</span>? That doesn't seem at all related, number of workers and profits.</p>
<hr />
<p>I have added the context below, but I really can't see what in the text it follows from.</p>
<p><a href="https://i.sstatic.net/oTw4C71A.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oTw4C71A.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-05-06T11:02:44.440Z
That condition is clearly result from some derivation. Depending on the objective function this can make perfect sense. Unless you share that there is nothing more that can be said there.
1muflon1
CC BY-SA 4.0
2024-05-06T12:39:17.940Z
@1muflon1 I added the full context, I can't see that it follows from anything.
1989-a-number
CC BY-SA 4.0
2024-05-06T13:33:06.950Z
You already got an answer, as the answer says it’s just equating demand for labor to supply of labor. It makes intuitive sense that demand will equal supply in equilibrium.
1muflon1
<general-equilibrium>
What is the intuition behind this full-employment condition?
1989-a-number
# What is the intuition behind this full-employment condition?

(Asked by: username_0 on 2024-05-06)

I am given the following full-employment condition, where sectors of the economy are indexed by $i$.

$L+\sum\_i\frac{\partial \pi\_i(p\_i,w)}{\partial w}=0$

Why, if all workers are employed, should how profit changes given a wage increase equal the *number* of workers $L$? That doesn't seem at all related, number of workers and profits.

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I have added the context below, but I really can't see what in the text it follows from.

[![enter image description here](https://i.sstatic.net/oTw4C71A.png)](https://i.sstatic.net/oTw4C71A.png)

**Tags:** general-equilibrium

> **Comments:**
> **username_1**: That condition is clearly result from some derivation. Depending on the objective function this can make perfect sense. Unless you share that there is nothing more that can be said there.  
> **username_0**: @username_1 I added the full context, I can't see that it follows from anything.  
> **username_1**: You already got an answer, as the answer says it’s just equating demand for labor to supply of labor. It makes intuitive sense that demand will equal supply in equilibrium.  

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## 1 answers

### Answer by username_2 on 2024-05-06. Score: 2 | Accepted answer

By Hotelling's lemma, the partial derivative of the profit function with respect to the input price gives the factor demand. As such, the demand for labour for firm $i$ equals:
$$
\frac{\partial \pi\_i(i,w)}{\partial w}.
$$
The requirement that $\sum\_i \frac{\partial \pi\_i(i,w)}{\partial w} = L$ requires that total labor demand (over all firms) equals the total labor available in the economy. You need to check but probably in this model, labour is inelastically supplied by the households so this equation simply states that aggregate labour demand equals aggregate labour supply.


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