<p>This is not a general result this is a result that holds only for certain market structures.</p>
<p>For example, suppose market is competitive so firms are price takers meaning <span class="math-container">$p(y)=p$</span>. Then when you set up your profit equation:</p>
<p><span class="math-container">$$\pi = R(y)-C(y) = p(y)y - C(y) = py - C(y)$$</span></p>
<p>when you optimize the profit you get:</p>
<p><span class="math-container">$$\pi_y' =  p - C'(y) = 0 \implies p = C'(y) $$</span></p>
<p>As you can see above, in the competitive case benefit/return/profit for the owner is maximized when <span class="math-container">$p=MC = ATC \implies \pi=0$</span>. However, note <span class="math-container">$MC$</span> and <span class="math-container">$ATC$</span> includes opportunity cost so this is not the same as getting zero accounting profit. Here the investor's books  would still be in green, but investor breaks even in terms of <em>economic</em> profit.</p>
<p>Next, suppose that this firm now has some market power so <span class="math-container">$p(y)$</span> will remain some function of <span class="math-container">$y$</span>. Now we have:</p>
<p><span class="math-container">$$\pi = p(y)y - C(y)$$</span></p>
<p>and hence optimum will be found where</p>
<p><span class="math-container">$$\pi_y' = p'(y)y  +  p(y)- C'(y) = 0 \implies  p'(y)y  +  p(y) = C'(y)$$</span></p>
<p>In this case it is trivial to see that (aside from corner solutions)
at the profit maximizing point the profit will be <span class="math-container">$\pi&gt; 0$</span>.</p>

CC BY-SA 4.0
2021-11-22T19:51:18.813Z
What's the difference between accounting profit and economic profit?
Stan Shunpike
CC BY-SA 4.0
2021-11-22T20:06:26.057Z
The explanation of "*the competitive case*" is very brief in this answer, the calculations included do not explain why $\pi = 0$ would hold.
Giskard
CC BY-SA 4.0
2021-11-22T20:14:19.377Z
@Giskard fair point, seeing you already addressed that in your answer i upvoted it instead of changing this one but I will still keep this here because I think other parts of my answer still answer the question even if not fully
1muflon1
CC BY-SA 4.0
2021-11-22T20:15:22.217Z
@1muflon1 I wrote nothing about "the competitive case", assuming you mean free entry; it seemed beside the point. I only wrote about optimization problems of price taking firms.
Giskard
CC BY-SA 4.0
2021-11-22T20:16:27.617Z
@StanShunpike accounting profit is the profit you would find in accounting books following accounting rules. Economic profit is actual economic profit including implicit benefits and costs. For example, opportunity cost is not included in accounting. Or accounting often uses linear depreciation method to determine depreciation costs whether in economics we would actually look at exactly what was the depreciation in each time period and so on.
1muflon1
CC BY-SA 4.0
2021-11-22T19:15:36.987Z
2021-11-22T19:15:36.987Z
1muflon1
<p>You write about two separate optimization problems,
<span class="math-container">$$
1. \max_y \pi(y) \\
2. \max_y r(y)
$$</span>
The first problem's optimum yields <span class="math-container">$MC(y_1) = p_y$</span>, while the second problem's optimum yields <span class="math-container">$MC(y_2) = ATC(y_2)$</span>. Since these belong to solutions of two separate problems, the <span class="math-container">$y$</span> values may differ as well. Thus
<span class="math-container">$$
p_y = MC(y_1) \ \ ?=? \ \ MC(y_2)  = ATC(y_2),
$$</span>
there is no guarantee that <span class="math-container">$p_y = ATC(y_2)$</span> and you are at the break-even point.</p>
<p>On the quoted passage:<br/>
This passage states that if the price <span class="math-container">$p_y$</span> is exactly such that the above equation holds, then even if you try to maximize profits, the best you can do is break even, achieve zero profits. This is only true for a given price level, namely
<span class="math-container">$$
p_y = \min_y ATC(y).
$$</span></p>

CC BY-SA 4.0
2021-11-22T21:34:13.783Z
nicely explained. thanks very much!
Stan Shunpike
CC BY-SA 4.0
2021-11-22T20:03:28.143Z
2023-01-16T14:12:40.917Z
2023-01-16T14:12:40.917Z
Giskard
<p>Let <span class="math-container">$\pi(y) = R(y) - C(y)$</span> be profits where <span class="math-container">$R(y)$</span> is revenue and <span class="math-container">$C(y)$</span> is costs. Let <span class="math-container">$R(y) = p_y y$</span>. Then</p>
<p><span class="math-container">\begin{align*}
\frac{\partial \pi }{\partial y} &amp;= 0\\
\frac{\partial x}{\partial y}(p_y y -C(y))  &amp;= 0\\
C'(y) &amp;= p_y\\
\end{align*}</span></p>
<p>Thus, <span class="math-container">$MR = MC$</span> in this case.</p>
<p>Now define <span class="math-container">$r(y) = \frac{\pi(y)}{C(y)}$</span> to be the return. Intuitively, suppose you own the business. <span class="math-container">$C(y)$</span> is the amount you put in and <span class="math-container">$\pi(y)$</span> is what you get back. You want to maximize your return.</p>
<p><span class="math-container">\begin{align*}
\frac{\partial r}{\partial y} &amp;=0\\
\frac{\partial}{\partial y} \frac{p_y y -C(y)}{C(y)} &amp;=0\\
\frac{C(y)p_y - p_y y C'(y)}{[C(y)]^2}  &amp;=0\\
C(y)p_y - p_y y C'(y) &amp;=0\\
C'(y) &amp;=\frac{C(y)}{y}\\
\end{align*}</span></p>
<p>Thus, <span class="math-container">$MC = ATC$</span>.</p>
<p>But when I googled this, I unexpectedly found this passage:</p>
<blockquote>
<p>The point at which marginal cost equals average total cost (MC =
ATC) is known as the break-even point. When the MR = P line crosses
through this point, as is highlighted by the black circle on the
graph, the product is said to be selling at its break-even price
because the marginal revenue will exactly offset the marginal cost of
production, and total revenue will exactly offset total cost. In this
situation, the firm will break even: it will not be earning any
profits, but it will not be losing money either. If the MR = P line
lies above the break-even point, the firm will be operating at a
profit, since the revenue earned on each unit of output sold will
exceed the average cost of producing a unit of output, and thus total
revenue will exceed total cost. If the MR = P line lies below the
break-even point, the firm will be operating at a loss because the
revenue earned on each unit of output will be less than the average
cost of producing a unit of output, and so total revenue will be less
that total cost.</p>
</blockquote>
<p>I don't understand why, if you are maximizing the return for a founder investing in their business, why they would be getting the best return when they are breaking even. That doesn't make sense to me.</p>
<p>Can someone explain why <span class="math-container">$MC = ATC$</span> is both the point where an investor would be getting their max return and the breakeven point?</p>

<microeconomics><cost-functions>
Why is MC = ATC the same point for both the breakeven point and an investor maximizing return?
Stan Shunpike
# Why is MC = ATC the same point for both the breakeven point and an investor maximizing return?

(Asked by: username_0 on 2021-11-22)

Let $\pi(y) = R(y) - C(y)$ be profits where $R(y)$ is revenue and $C(y)$ is costs. Let $R(y) = p\_y y$. Then

\begin{align\*}
\frac{\partial \pi }{\partial y} &= 0\\
\frac{\partial x}{\partial y}(p\_y y -C(y)) &= 0\\
C'(y) &= p\_y\\
\end{align\*}

Thus, $MR = MC$ in this case.

Now define $r(y) = \frac{\pi(y)}{C(y)}$ to be the return. Intuitively, suppose you own the business. $C(y)$ is the amount you put in and $\pi(y)$ is what you get back. You want to maximize your return.

\begin{align\*}
\frac{\partial r}{\partial y} &=0\\
\frac{\partial}{\partial y} \frac{p\_y y -C(y)}{C(y)} &=0\\
\frac{C(y)p\_y - p\_y y C'(y)}{[C(y)]^2} &=0\\
C(y)p\_y - p\_y y C'(y) &=0\\
C'(y) &=\frac{C(y)}{y}\\
\end{align\*}

Thus, $MC = ATC$.

But when I googled this, I unexpectedly found this passage:

> The point at which marginal cost equals average total cost (MC =
> ATC) is known as the break-even point. When the MR = P line crosses
> through this point, as is highlighted by the black circle on the
> graph, the product is said to be selling at its break-even price
> because the marginal revenue will exactly offset the marginal cost of
> production, and total revenue will exactly offset total cost. In this
> situation, the firm will break even: it will not be earning any
> profits, but it will not be losing money either. If the MR = P line
> lies above the break-even point, the firm will be operating at a
> profit, since the revenue earned on each unit of output sold will
> exceed the average cost of producing a unit of output, and thus total
> revenue will exceed total cost. If the MR = P line lies below the
> break-even point, the firm will be operating at a loss because the
> revenue earned on each unit of output will be less than the average
> cost of producing a unit of output, and so total revenue will be less
> that total cost.

I don't understand why, if you are maximizing the return for a founder investing in their business, why they would be getting the best return when they are breaking even. That doesn't make sense to me.

Can someone explain why $MC = ATC$ is both the point where an investor would be getting their max return and the breakeven point?

**Tags:** microeconomics, cost-functions

---

## 2 answers

### Answer by username_2 on 2021-11-22. Score: 3 | Accepted answer

You write about two separate optimization problems,
$$
1. \max\_y \pi(y) \\
2. \max\_y r(y)
$$
The first problem's optimum yields $MC(y\_1) = p\_y$, while the second problem's optimum yields $MC(y\_2) = ATC(y\_2)$. Since these belong to solutions of two separate problems, the $y$ values may differ as well. Thus
$$
p\_y = MC(y\_1) \ \ ?=? \ \ MC(y\_2) = ATC(y\_2),
$$
there is no guarantee that $p\_y = ATC(y\_2)$ and you are at the break-even point.

On the quoted passage:  
This passage states that if the price $p\_y$ is exactly such that the above equation holds, then even if you try to maximize profits, the best you can do is break even, achieve zero profits. This is only true for a given price level, namely
$$
p\_y = \min\_y ATC(y).
$$

> **Comments:**
> **username_0**: nicely explained. thanks very much!  

---

### Answer by username_1 on 2021-11-22. Score: 0

This is not a general result this is a result that holds only for certain market structures.

For example, suppose market is competitive so firms are price takers meaning $p(y)=p$. Then when you set up your profit equation:

$$\pi = R(y)-C(y) = p(y)y - C(y) = py - C(y)$$

when you optimize the profit you get:

$$\pi\_y' = p - C'(y) = 0 \implies p = C'(y) $$

As you can see above, in the competitive case benefit/return/profit for the owner is maximized when $p=MC = ATC \implies \pi=0$. However, note $MC$ and $ATC$ includes opportunity cost so this is not the same as getting zero accounting profit. Here the investor's books would still be in green, but investor breaks even in terms of *economic* profit.

Next, suppose that this firm now has some market power so $p(y)$ will remain some function of $y$. Now we have:

$$\pi = p(y)y - C(y)$$

and hence optimum will be found where

$$\pi\_y' = p'(y)y + p(y)- C'(y) = 0 \implies p'(y)y + p(y) = C'(y)$$

In this case it is trivial to see that (aside from corner solutions)
at the profit maximizing point the profit will be $\pi> 0$.

> **Comments:**
> **username_0**: What's the difference between accounting profit and economic profit?  
> **username_2**: The explanation of "*the competitive case*" is very brief in this answer, the calculations included do not explain why $\pi = 0$ would hold.  
> **username_1**: @Giskard fair point, seeing you already addressed that in your answer i upvoted it instead of changing this one but I will still keep this here because I think other parts of my answer still answer the question even if not fully  
> **username_2**: @1muflon1 I wrote nothing about "the competitive case", assuming you mean free entry; it seemed beside the point. I only wrote about optimization problems of price taking firms.  
> **username_1**: @StanShunpike accounting profit is the profit you would find in accounting books following accounting rules. Economic profit is actual economic profit including implicit benefits and costs. For example, opportunity cost is not included in accounting. Or accounting often uses linear depreciation method to determine depreciation costs whether in economics we would actually look at exactly what was the depreciation in each time period and so on.  

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