<p>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? </p>

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

CC BY-SA 4.0
2019-06-19T05:42:23.383Z
Here is the screenshot of the homework. http://tinypic.com/r/10576u1/9
jj99
CC BY-SA 4.0
2019-06-19T05:52:09.193Z
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.
Regio
CC BY-SA 4.0
2019-06-19T05:56:23.070Z
Firms in the long run are 200. Does that seem correct?
jj99
CC BY-SA 4.0
2019-06-19T05:41:02.033Z
2019-06-19T05:53:56.657Z
2019-06-19T05:53:56.657Z
Regio
<p>I am currently struggling trying to find the short-run equilibrium price, output per firm, and profit per firm if <span class="math-container">$190$</span> firms supply the market. I am given <span class="math-container">$p=102-1/2Q$</span> and <span class="math-container">$C(q)=5q-6q^2+3q^3$</span>. </p>

<p>First I started off with <span class="math-container">$P=MC$</span>. 
So <span class="math-container">$P=9q^2-12q+5$</span>, thus <span class="math-container">$9q^2-12q+(5-P)=0$</span>. </p>

<p>Then, I Used the quadratic equation and came out with <span class="math-container">$q=2/3+1/3\sqrt{p-1}$</span>.</p>

<p>Then I did <span class="math-container">$Q=nq=190\left(2/3+1/3\sqrt{p-1}\right)$</span>. </p>

<p>Equating demand and supply I obtained <span class="math-container">$P\approx 2.32$</span> and then plugged back into <span class="math-container">$q\approx2/3+1/3\sqrt{2.32-1}$</span> to get <span class="math-container">$q\approx1.05$</span> </p>

<p>Finally, I plugged <span class="math-container">$q$</span> and <span class="math-container">$P$</span> into <span class="math-container">$\pi=q\cdot P-C(q)$</span> and came up with <span class="math-container">$\pi\approx 0.33$</span>. 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. </p>

<p>Any help would be greatly appreciated!</p>

CC BY-SA 4.0
2019-06-19T04:41:53.183Z
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.
Regio
CC BY-SA 4.0
2019-06-19T04:57:39.380Z
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!
jj99
CC BY-SA 4.0
2019-06-19T05:04:32.087Z
I will edit that into the original question for future reference for other people if you don't mind.
Regio
<microeconomics><mathematical-economics><industrial-organisation>
Industrial Economics
jj99
# Industrial Economics

(Asked by: username_0 on 2019-06-19)

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!

**Tags:** microeconomics, mathematical-economics, industrial-organisation
> **Comments:**
> **username_1**: 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.  
> **username_0**: 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!  
> **username_1**: I will edit that into the original question for future reference for other people if you don't mind.  

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## 1 answers

### Answer by username_1 on 2019-06-19. Score: 0 | Accepted answer

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.

> **Comments:**
> **username_0**: Here is the screenshot of the homework. http://tinypic.com/r/10576u1/9  
> **username_1**: 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.  
> **username_0**: Firms in the long run are 200. Does that seem correct?  

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