<p>It depends.</p>
<p>If production has decreasing returns to scale, then there will normally be profits.</p>
<p>Decreasing returns to scale means that marginal product of labour decreases with labor. Let <span class="math-container">$x$</span> be the amount of labour and let <span class="math-container">$f(x)$</span> be the output. Decreasing marginal product of labour implies that <span class="math-container">$f'(x)$</span> decreases if <span class="math-container">$x$</span> increases.</p>
<p>At the optimal level <span class="math-container">$p f'(x^\ast) = w$</span> (<span class="math-container">$w$</span> is wage) but for the levels below <span class="math-container">$x$</span>, <span class="math-container">$p f'(x)$</span> will be higher than <span class="math-container">$f'(x^\ast)$</span>.</p>
<p>As such, we get that:
<span class="math-container">$$
f(x^\ast) = \int_0^{x^\ast} f'(x) dx \ge \int_0^{x^\ast} f'(x^\ast) dx = f'(x^\ast) x^\ast.  
$$</span>
Then,
<span class="math-container">$$
p f(x^\ast) - w x^\ast \ge p f'(x^\ast) x^\ast - w x^\ast = 0.
$$</span></p>
<p>As an example let <span class="math-container">$f(x) = \sqrt{x}$</span>. Then the first order conditions give:
<span class="math-container">$$
\frac{p}{2 \sqrt{x^\ast}} = w \to x^\ast = \left(\frac{p}{2w}\right)^2
$$</span></p>
<p>As such, profits are:
<span class="math-container">$$
p \sqrt{x^\ast} - w x^\ast = \frac{p^2}{2w} - \frac{p^2}{4w} = \frac{p^2}{2w} &gt; 0
$$</span></p>

CC BY-SA 4.0
2023-10-31T16:47:57.670Z
2023-10-31T16:47:57.670Z
tdm
<p>A less mathy version of <a href="https://economics.stackexchange.com/a/56845/1601">tdm's answer</a>: <br>
<em>Marginal</em> means <em>marginal</em>, i.e. based on the last (in this case labor unit's) measurement. Assume a price of 10\$.</p>
<p>If there are a 100 workers, with the first one having an MPL of 100, the second one of 99, etc. until the last one having 1, and you hire the first 50 of them at a wage of 500\$, your profit will not be zero. Your profit from the first worker will be</p>
<p><span class="math-container">$$
100\cdot 10$-50\cdot 10$ = 500$.
$$</span></p>
<p><br> You will similarly profit from all but the last worker.
To have a 'theoretical underpinning' in this case you can assume the production function <span class="math-container">$f(L) = (100 - L/2) \cdot L$</span>.</p>

CC BY-SA 4.0
2023-10-31T21:42:54.383Z
2023-10-31T21:51:00.390Z
2023-10-31T21:51:00.390Z
Giskard
<p>Excuse me if the question is too simple...</p>
<p>Wage = Price x MPL (Marginal Productivity of Labor)</p>
<p>I wonder how we can come to this conclusion as there is nothing left for Profit. I mean let's suppose that in a company A there is 1 worker who produces 5 products monthly. Price of product is 6\$. In this case lets calculate the revenue (let's assume that all products are sold): 6x5=30\$</p>
<p>Now lets calculate wage of the employee which is again 6x5=30\$</p>
<p>So now Revenue - Expense (here employee wage/salary) is 30-30=0</p>
<p>I am wondering where we are getting this assumption. Today in Economy class the whole Specific Factors Model based on this assumption. I asked ChatGPT and it says that in the real world it is not like that as wage is less than revenue.</p>
<p>Please enlighten me.</p>
<p>Thanks in advance for your time.</p>

CC BY-SA 4.0
2023-11-01T04:37:10.393Z
This sounds like the question of why economists say there is zero economic profit in the long run when we observe that firms seem to make a profit. This is because typically firms don't rent their capital, rather they own it so all those factor payments go to them directly.
EconJohn
<profit-maximization>
If wage is equal to P x MPL then where is the profit of the firm?
Heyder Salamov
# If wage is equal to P x MPL then where is the profit of the firm?

(Asked by: username_0 on 2023-10-31)

Excuse me if the question is too simple...

Wage = Price x MPL (Marginal Productivity of Labor)

I wonder how we can come to this conclusion as there is nothing left for Profit. I mean let's suppose that in a company A there is 1 worker who produces 5 products monthly. Price of product is 6\$. In this case lets calculate the revenue (let's assume that all products are sold): 6x5=30\$

Now lets calculate wage of the employee which is again 6x5=30\$

So now Revenue - Expense (here employee wage/salary) is 30-30=0

I am wondering where we are getting this assumption. Today in Economy class the whole Specific Factors Model based on this assumption. I asked ChatGPT and it says that in the real world it is not like that as wage is less than revenue.

Please enlighten me.

Thanks in advance for your time.

**Tags:** profit-maximization
> **Comments:**
> **username_1**: This sounds like the question of why economists say there is zero economic profit in the long run when we observe that firms seem to make a profit. This is because typically firms don't rent their capital, rather they own it so all those factor payments go to them directly.  

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## 2 answers

### Answer by username_2 on 2023-10-31. Score: 2

It depends.

If production has decreasing returns to scale, then there will normally be profits.

Decreasing returns to scale means that marginal product of labour decreases with labor. Let $x$ be the amount of labour and let $f(x)$ be the output. Decreasing marginal product of labour implies that $f'(x)$ decreases if $x$ increases.

At the optimal level $p f'(x^\ast) = w$ ($w$ is wage) but for the levels below $x$, $p f'(x)$ will be higher than $f'(x^\ast)$.

As such, we get that:
$$
f(x^\ast) = \int\_0^{x^\ast} f'(x) dx \ge \int\_0^{x^\ast} f'(x^\ast) dx = f'(x^\ast) x^\ast.
$$
Then,
$$
p f(x^\ast) - w x^\ast \ge p f'(x^\ast) x^\ast - w x^\ast = 0.
$$

As an example let $f(x) = \sqrt{x}$. Then the first order conditions give:
$$
\frac{p}{2 \sqrt{x^\ast}} = w \to x^\ast = \left(\frac{p}{2w}\right)^2
$$

As such, profits are:
$$
p \sqrt{x^\ast} - w x^\ast = \frac{p^2}{2w} - \frac{p^2}{4w} = \frac{p^2}{2w} > 0
$$


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### Answer by username_3 on 2023-10-31. Score: 1

A less mathy version of [tdm's answer](https://economics.stackexchange.com/a/56845/1601):   
*Marginal* means *marginal*, i.e. based on the last (in this case labor unit's) measurement. Assume a price of 10\$.

If there are a 100 workers, with the first one having an MPL of 100, the second one of 99, etc. until the last one having 1, and you hire the first 50 of them at a wage of 500\$, your profit will not be zero. Your profit from the first worker will be

$$
100\cdot 10$-50\cdot 10$ = 500$.
$$

You will similarly profit from all but the last worker.
To have a 'theoretical underpinning' in this case you can assume the production function $f(L) = (100 - L/2) \cdot L$.


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