# Violation of the zero-profit condition

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: @Giskard 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?

• 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. Oct 18, 2022 at 5:45

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.

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.

• Did you mean the long run price is $0.5 + \sqrt{8}$? Can you tell me how you calculated it? Thank you Oct 17, 2022 at 20:58
• It is the minimum value of the average cost function; unless I miscalculated. Oct 17, 2022 at 21:33
• You didn't miscalculate. But I think it's still not a perfect competition model (I could be wrong). Even at this price, $\max[\pi] = \max[(0.5 + \sqrt{8})q - c(q)] \approx 2.3 \cdot 10^{-15} > 0$. Oct 17, 2022 at 21:52
• 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. Oct 18, 2022 at 6:32
• Hi @Giskard, sorry, I didn't see it! Oct 18, 2022 at 8:27