<p>Let <span class="math-container">$q$</span> denote the output of the firm, and let <span class="math-container">$\varepsilon_p(q)$</span> denote the elasticity of price w.r.t. quantity sold. We know that when profit is maximized
<span class="math-container">$$
|\varepsilon_p(q)| = \frac{p(q)-MC(q)}{p(q)}.
$$</span>
We also know that in the long-run equilibrium firms have zero economic profit, i.e.
<span class="math-container">$$
AC(q) = p(q).
$$</span>
Productive efficiency is achieved when <span class="math-container">$AC(q)$</span> is minimized, and at that point
<span class="math-container">$$
AC(q) = MC(q)
$$</span>
holds as well. Combining these we get
<span class="math-container">$$
|\varepsilon_p(q)| = \frac{p(q)-MC(q)}{p(q)} = \frac{AC(q)-AC(q)}{p(q)} = 0.
$$</span>
So the only case when all these properties coincide is when the quantity sold has no effect on the market price, i.e. when demand is perfectly elastic.</p>

<hr>

<p>An explanation without the elasticity part: <br>
From the other two properties (long-run equilibrium and productive efficiency) we have 
<span class="math-container">$$
MC(q) = AC(q) = p(q).
$$</span>
Together with profit maximization this implies that firms are price takers, <span class="math-container">$MR(q) = p(q)$</span>, but since
<span class="math-container">$$
MR(q) = \frac{\text{d} \ p(q)q}{\text{d} \ q} = p(q) + p'(q)q
$$</span>
this is only possible if <span class="math-container">$p'(q) = 0$</span>, which again means that demand is perfectly elastic.</p>

<hr>

<p>All the above assumed that the demand and inverse demand functions are continuously differentiable. You could have a situation with not perfectly elastic but kinked demand, where profit maximization occurs exactly at the kink. In this case <span class="math-container">$MR(q)$</span> does not exist at the point of profit maximization.</p>

CC BY-SA 4.0
2019-09-07T02:15:18.550Z
Thanks so much for the response! It's actually a high school textbook for IB economics, so it doesn't go into a lot of detail, but yes perfect competition is introduced first.
rye
CC BY-SA 4.0
2019-09-07T02:18:06.113Z
I think my confusion is in the short run, when a firm (monopolistic competition) can produce abnormal profits, is it possible for the point where AC is minimised to also be the point where MC=MR. Maybe it would be clearer to me if I was working with more detailed text.
rye
CC BY-SA 4.0
2019-09-07T05:06:05.343Z
@rye This is exactly the question I have answered.
Giskard
CC BY-SA 4.0
2019-09-06T06:33:28.967Z
2019-09-06T06:42:15.440Z
2019-09-06T06:42:15.440Z
Giskard
<p>In my textbook, it says that monopolistic competition will never lead to productive efficiency (where <span class="math-container">$MC=AC$</span>) in the short run and long run. I understand that they are meant to be producing at the profit-maximising level of output (where <span class="math-container">$MC=MR$</span>), but all the diagrams showcase scenarios where the <span class="math-container">$AC$</span> curve is always positioned such that the points where MR=MC does not intersect where <span class="math-container">$MC=AC$</span>.</p>

<p>My question is, is there any possibility that the profit-maximising level of output will also happen to be where we've achieved the lowest possible unit cost (lowest <span class="math-container">$AC$</span>)? So that <span class="math-container">$MC=MR=AC$</span>, and the firm can be both productively-efficient whilst also being profit-maximising? Why is this not possible?</p>

CC BY-SA 4.0
2019-09-06T06:25:17.680Z
Which textbook is this? Does it discuss monopolistic competition before it discusses perfect competition?
Giskard
<microeconomics>
Monopolistic Competition- productive efficiency
rye
# Monopolistic Competition- productive efficiency

(Asked by: username_0 on 2019-09-06)

In my textbook, it says that monopolistic competition will never lead to productive efficiency (where $MC=AC$) in the short run and long run. I understand that they are meant to be producing at the profit-maximising level of output (where $MC=MR$), but all the diagrams showcase scenarios where the $AC$ curve is always positioned such that the points where MR=MC does not intersect where $MC=AC$.

My question is, is there any possibility that the profit-maximising level of output will also happen to be where we've achieved the lowest possible unit cost (lowest $AC$)? So that $MC=MR=AC$, and the firm can be both productively-efficient whilst also being profit-maximising? Why is this not possible?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Which textbook is this? Does it discuss monopolistic competition before it discusses perfect competition?  

---

## 1 answers

### Answer by username_1 on 2019-09-06. Score: 1

Let $q$ denote the output of the firm, and let $\varepsilon\_p(q)$ denote the elasticity of price w.r.t. quantity sold. We know that when profit is maximized
$$
|\varepsilon\_p(q)| = \frac{p(q)-MC(q)}{p(q)}.
$$
We also know that in the long-run equilibrium firms have zero economic profit, i.e.
$$
AC(q) = p(q).
$$
Productive efficiency is achieved when $AC(q)$ is minimized, and at that point
$$
AC(q) = MC(q)
$$
holds as well. Combining these we get
$$
|\varepsilon\_p(q)| = \frac{p(q)-MC(q)}{p(q)} = \frac{AC(q)-AC(q)}{p(q)} = 0.
$$
So the only case when all these properties coincide is when the quantity sold has no effect on the market price, i.e. when demand is perfectly elastic.

---

An explanation without the elasticity part:   
From the other two properties (long-run equilibrium and productive efficiency) we have
$$
MC(q) = AC(q) = p(q).
$$
Together with profit maximization this implies that firms are price takers, $MR(q) = p(q)$, but since
$$
MR(q) = \frac{\text{d} \ p(q)q}{\text{d} \ q} = p(q) + p'(q)q
$$
this is only possible if $p'(q) = 0$, which again means that demand is perfectly elastic.

---

All the above assumed that the demand and inverse demand functions are continuously differentiable. You could have a situation with not perfectly elastic but kinked demand, where profit maximization occurs exactly at the kink. In this case $MR(q)$ does not exist at the point of profit maximization.

> **Comments:**
> **username_0**: Thanks so much for the response! It's actually a high school textbook for IB economics, so it doesn't go into a lot of detail, but yes perfect competition is introduced first.  
> **username_0**: I think my confusion is in the short run, when a firm (monopolistic competition) can produce abnormal profits, is it possible for the point where AC is minimised to also be the point where MC=MR. Maybe it would be clearer to me if I was working with more detailed text.  
> **username_1**: @rye This is exactly the question I have answered.  

---

