<p>Essentially, from a teaching point of view, the reason perfect competition is given in your question is because you can assume p=MR, i.e the firm is a price taker.</p>
<p>As more firms enter the market the price may eventually fall to the point where profits are zero.</p>
<p>So in the question’s premise, we may not be in full competition as more firms could still enter, but the firm in question is small enough relative to the number of firms already there, that is operates as a price taker. So for that firm the market is perfectly competitive / competitive enough.</p>
<p>Admittedly this may be a bit weird, but I wouldn’t worry about it too much. That’s just the nature of these types of questions at this level.</p>

CC BY-SA 4.0
2022-10-10T06:25:05.050Z
2022-10-10T06:25:05.050Z
BB King
<p>Zero profit condition only holds if free entry and long-run equilibrium state are also assumed (and even then it only holds for the marginal entrant).</p>
<p>Here the profits being positive means that <span class="math-container">$p=8$</span> is higher than the long-run equilibrium price. (Which seems to be <span class="math-container">$0.5+\sqrt{8}$</span>.)</p>
<p>No need to worry about it though, not all these exercises assume long-run equilibrium.</p>

CC BY-SA 4.0
2022-10-17T20:58:43.177Z
Did you mean the long run price is $0.5 + \sqrt{8}$? Can you tell me how you calculated it? Thank you
Rick_Morty
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2022-10-17T21:33:01.593Z
It is the minimum value of the average cost function; unless I miscalculated.
Giskard
CC BY-SA 4.0
2022-10-17T21:52:28.853Z
You didn't miscalculate. But I think it's still not a perfect competition model (I could be wrong). Even at this price, $\max[\pi] = \max[(0.5 + \sqrt{8})q - c(q)] \approx 2.3 \cdot 10^{-15} > 0$.
Rick_Morty
CC BY-SA 4.0
2022-10-18T05:43:54.613Z
That looks like a rounding error. Try the multiplication by hand instead of using a computer.
Giskard
CC BY-SA 4.0
2022-10-18T06:32:46.107Z
In fact, since $\pi = p*q - C(q) = p*q - AC(q)*q = (p-AC(q))*q$, you know that if $p = AC(q)$, then the profits are going to be zero.
Giskard
CC BY-SA 4.0
2022-10-18T08:23:45.510Z
Hi @MatteoBulgarelli ! Your comment seems to be the same as my [second comment](https://economics.stackexchange.com/questions/53030/violation-of-the-zero-profit-condition/53031?noredirect=1#comment86759_53031).
Giskard
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2022-10-18T08:27:35.567Z
Hi @Giskard, sorry, I didn't see it!
Matteo Bulgarelli
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2022-10-19T00:46:45.327Z
@Giskard Hm, that makes sense. But the $q$ is unique in this, unlike in the case of constant AC (which is understandable). Thank you.
Rick_Morty
CC BY-SA 4.0
2022-10-10T05:34:32.503Z
2022-10-10T05:39:54.273Z
2022-10-10T05:39:54.273Z
Giskard
<p>Suppose a perfectly competitive firm has cost function <span class="math-container">$C(q) = 40 + 0.5q + 0.05q^2$</span>. If the market price is <span class="math-container">$p = 8$</span>, the profit made is maximized at <span class="math-container">$MR = MC \implies 8 = 0.5 + 0.1q \implies q^{*} = 75$</span>. Further calculation tells us that profit is <span class="math-container">$&gt; 0$</span>.</p>
<p>Doesn't this violate the zero-profit condition in perfect competition? Maybe one can argue that it's the short-run cost function. What if it's actually the long-run cost function?</p>
<p><strong>Edit:</strong> @Giskard mentions that the long-run price is supposed to be <span class="math-container">$0.5 + 2\sqrt{2}$</span> and it appears, after calculation, that <span class="math-container">$P = MC = AC$</span> indeed gives that. A further calculation tells us that <span class="math-container">$\max_q[\pi(q)] = \max[(0.5+8–√)q−c(q)]≈2.3⋅10−15&gt;0$</span>. Does that mean that a competitive market can not have a cost function <span class="math-container">$C(q)$</span> as mentioned?</p>

CC BY-SA 4.0
2022-10-18T05:45:31.187Z
Please do not edit questions substantially after they have already been answered, it is not fair to the people who answered the original question. If you want, post your new query as a new question.
Giskard
<microeconomics><markets><perfect-competition>
Violation of the zero-profit condition
Rick_Morty
# Violation of the zero-profit condition

(Asked by: username_0 on 2022-10-10)

Suppose a perfectly competitive firm has cost function $C(q) = 40 + 0.5q + 0.05q^2$. If the market price is $p = 8$, the profit made is maximized at $MR = MC \implies 8 = 0.5 + 0.1q \implies q^{\*} = 75$. Further calculation tells us that profit is $> 0$.

Doesn't this violate the zero-profit condition in perfect competition? Maybe one can argue that it's the short-run cost function. What if it's actually the long-run cost function?

**Edit:** @username_1 mentions that the long-run price is supposed to be $0.5 + 2\sqrt{2}$ and it appears, after calculation, that $P = MC = AC$ indeed gives that. A further calculation tells us that $\max\_q[\pi(q)] = \max[(0.5+8–√)q−c(q)]≈2.3⋅10−15>0$. Does that mean that a competitive market can not have a cost function $C(q)$ as mentioned?

**Tags:** microeconomics, markets, perfect-competition

> **Comments:**
> **username_1**: Please do not edit questions substantially after they have already been answered, it is not fair to the people who answered the original question. If you want, post your new query as a new question.  

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## 2 answers

### Answer by username_2 on 2022-10-10. Score: 5

Essentially, from a teaching point of view, the reason perfect competition is given in your question is because you can assume p=MR, i.e the firm is a price taker.

As more firms enter the market the price may eventually fall to the point where profits are zero.

So in the question’s premise, we may not be in full competition as more firms could still enter, but the firm in question is small enough relative to the number of firms already there, that is operates as a price taker. So for that firm the market is perfectly competitive / competitive enough.

Admittedly this may be a bit weird, but I wouldn’t worry about it too much. That’s just the nature of these types of questions at this level.


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### Answer by username_1 on 2022-10-10. Score: 4

Zero profit condition only holds if free entry and long-run equilibrium state are also assumed (and even then it only holds for the marginal entrant).

Here the profits being positive means that $p=8$ is higher than the long-run equilibrium price. (Which seems to be $0.5+\sqrt{8}$.)

No need to worry about it though, not all these exercises assume long-run equilibrium.

> **Comments:**
> **username_0**: Did you mean the long run price is $0.5 + \sqrt{8}$? Can you tell me how you calculated it? Thank you  
> **username_1**: It is the minimum value of the average cost function; unless I miscalculated.  
> **username_0**: You didn't miscalculate. But I think it's still not a perfect competition model (I could be wrong). Even at this price, $\max[\pi] = \max[(0.5 + \sqrt{8})q - c(q)] \approx 2.3 \cdot 10^{-15} > 0$.  
> **username_1**: That looks like a rounding error. Try the multiplication by hand instead of using a computer.  
> **username_1**: In fact, since $\pi = p*q - C(q) = p*q - AC(q)*q = (p-AC(q))*q$, you know that if $p = AC(q)$, then the profits are going to be zero.  
> **username_1**: Hi @MatteoBulgarelli ! Your comment seems to be the same as my [second comment](https://economics.stackexchange.com/questions/53030/violation-of-the-zero-profit-condition/53031?noredirect=1#comment86759_53031).  
> **username_3**: Hi @username_1, sorry, I didn't see it!  
> **username_0**: @username_1 Hm, that makes sense. But the $q$ is unique in this, unlike in the case of constant AC (which is understandable). Thank you.  

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