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I am currently struggling trying to find the short-run equilibrium price, output per firm, and profit per firm if $190$ firms supply the market. I am given $p=102-1/2Q$ and $C(q)=5q-6q^2+3q^3$.

First I started off with $P=MC$. So $P=9q^2-12q+5$, thus $9q^2-12q+(5-P)=0$.

Then, I Used the quadratic equation and came out with $q=2/3+1/3\sqrt{p-1}$.

Then I did $Q=nq=190\left(2/3+1/3\sqrt{p-1}\right)$.

Equating demand and supply I obtained $P\approx 2.32$ and then plugged back into $q\approx2/3+1/3\sqrt{2.32-1}$ to get $q\approx1.05$

Finally, I plugged $q$ and $P$ into $\pi=q\cdot P-C(q)$ and came up with $\pi\approx 0.33$. This just seems crazy to me! I have tried for the past few days over and over again to understand, but I am completely lost.

Any help would be greatly appreciated!

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  • $\begingroup$ The policy of this forum is that you should show that you have tried doing it, and be precise on what part is confusing. Otherwise, it looks like you are simply looking for someone to do your homework. Not saying that is your case, but if you update your question people can help you here. $\endgroup$ – Regio Jun 19 at 4:41
  • $\begingroup$ Hi Regio. I didn't know that! Okay well first I started off with P=MC. P=9q^2-12q+5. 9q^2-1212q+(5-p)=0. Used quadratic equation and came out with 2/3+1/3*sqrt(p=1). Then I did Q=nq=190(2/3+1/3*sqrt(p-1)). I solved down to 2.32....=p and then plugged back in to q=2/3+1/3*sqrt(2.32....-1) to get q=1.05.... Then I plugged q and p into q*p-C(q) and came up with .33.... This just seems crazy to me. I have tried for the past few days over and over again to understand but I am completely lost. Thanks for the help! $\endgroup$ – jj99 Jun 19 at 4:57
  • $\begingroup$ I will edit that into the original question for future reference for other people if you don't mind. $\endgroup$ – Regio Jun 19 at 5:04
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I went over your calculations and they seem to be right. In principle, this is a valid answer, but I understand that the numbers seem suspicious. Are you sure this is the right problem?

Update: everything checks. I think your answer is correct, although it looks "weird" for a textbook problem.

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  • $\begingroup$ Here is the screenshot of the homework. tinypic.com/r/10576u1/9 $\endgroup$ – jj99 Jun 19 at 5:42
  • $\begingroup$ Everything seems right. As a sanity check, you can compare the number of firms in this short run equilibrium (190) with the number of firms that would be in the long run. Since profits are positive, you should expect more firms in the long run, but since profits are tiny (0.33) the number of firms, in the long run, should not be a lot more than 190. $\endgroup$ – Regio Jun 19 at 5:52
  • $\begingroup$ Firms in the long run are 200. Does that seem correct? $\endgroup$ – jj99 Jun 19 at 5:56
  • $\begingroup$ I think so. If you think I've provided you with the answer you were looking for, I would appreciate if you could mark it as the accepted answer. I wish I had helped you find a mistake, but I truly believe your calculations are correct. $\endgroup$ – Regio Jun 19 at 6:02

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