<p>The approach is correct, but it is not true that firms have the same equilibrium production quantities because we are dealing with Cournot competition - it is because the firms have the same profit functions. You can see for yourself with an example that if the two firms had different cost functions, they would not necessarily produce the same outputs as each other.</p>

<p>The step of "substituting one into the other" implicitly uses the concept of Nash equilibrium, i.e. each firm's production quantity is a best response to the other's. Given that $A$ produces some amount $q_A$ and $B$ knows it, $B$ would make the amount $q_B^*$ is profit-maximizing for $B$. Then we fix $q_B^*$ and find what $A$ would make as a profit-maximizing response - call it $q_A^*$. If I then fix $q_A^*$ and find $B$'s profit-maximizing response is the same $q_B^*$, $(q_A^*,q_B^*)$ is a Nash equilibrium: each firm's action is optimal, given the other's. This line of reasoning can be implemented by a solving a system of two equations, given by the profit-maximizing condition $MR=MC$ for each firm that you mentioned.</p>

CC BY-SA 3.0
2016-04-02T18:32:10.793Z
2016-04-02T18:32:10.793Z
manofbear
<p>I have given</p>

<ul>
<li>2 firms in a market with constant marginal costs and no fixed costs</li>
<li>market demand has $D(p)$ </li>
<li>The ﬁrms play a <strong>Cournot</strong> game</li>
</ul>

<p>I'm supposed to Calculate the equilibrium quantity for each ﬁrm and the market equilibrium quantity and price. Also I have to find the implied proﬁt for each ﬁrm.</p>

<p>My approach:</p>

<ul>
<li>Isolate the profit functions as $\pi_i = F(q_A, q_B)$ for $i \in \{A, B\}$</li>
<li>Then I compute marginal costs using that MC = the derivative of total costs w.r.t. Q</li>
<li>Then use that MR = MC for each firm, and solve for $q_A, q_B$</li>
<li>Then using this I was able to find the equilibrium quantity for each firm by substituting one into the other</li>
<li>Finally I subbed these quantities into $D(p)$ to get the equilibrium price </li>
</ul>

<p>Among my results, I get that $q_A = q_B$. I'm not sure if I've solved this correctly but it seems to make sense to me that the firms have the same equilibrium quantities due to it being a Cournot game. </p>

<p>Was the approach correct?</p>

CC BY-SA 3.0
2016-03-29T17:27:41.010Z
I did have a look and I believe that I did try the problem sufficiently myself and that others could learn from it as well. Am I missing something?
Astag
CC BY-SA 3.0
2016-03-29T17:31:37.473Z
In my opinion, the question is unnecessarily specific right now. Let me show what I mean and you can reverse if you want to.
FooBar
CC BY-SA 3.0
2016-03-29T17:39:44.873Z
Generally, we don't do homework corrections here, so don't expect others to redo everything with your numbers. If you have a specific issue, point that out. (Here, you just pasted the whole question and answer, without mentioning where *exactly* you had troubles).
FooBar
CC BY-SA 3.0
2016-03-29T17:40:14.060Z
I see what you mean now and I agree with you that its definitely less specific and can be applied more easily to other situations that are similar. Thank you!
Astag
CC BY-SA 3.0
2016-03-29T17:40:26.440Z
Note that you can use dollar signs for latex-like ("mathjax") notation.
FooBar
CC BY-SA 3.0
2016-03-30T11:31:01.857Z
@FooBar since you put a lot of work into this I think you could answer "Yes, the approach is correct." so that the OP can accept an answer and the question is resolved.
Giskard
<industrial-organisation><oligopoly>
Cournot game with 2 firms
Astag
# Cournot game with 2 firms

(Asked by: username_0 on 2016-03-29)

I have given

* 2 firms in a market with constant marginal costs and no fixed costs
* market demand has $D(p)$
* The ﬁrms play a **Cournot** game

I'm supposed to Calculate the equilibrium quantity for each ﬁrm and the market equilibrium quantity and price. Also I have to find the implied proﬁt for each ﬁrm.

My approach:

* Isolate the profit functions as $\pi\_i = F(q\_A, q\_B)$ for $i \in \{A, B\}$
* Then I compute marginal costs using that MC = the derivative of total costs w.r.t. Q
* Then use that MR = MC for each firm, and solve for $q\_A, q\_B$
* Then using this I was able to find the equilibrium quantity for each firm by substituting one into the other
* Finally I subbed these quantities into $D(p)$ to get the equilibrium price

Among my results, I get that $q\_A = q\_B$. I'm not sure if I've solved this correctly but it seems to make sense to me that the firms have the same equilibrium quantities due to it being a Cournot game.

Was the approach correct?

**Tags:** industrial-organisation, oligopoly

> **Comments:**
> **username_0**: I did have a look and I believe that I did try the problem sufficiently myself and that others could learn from it as well. Am I missing something?  
> **username_1**: In my opinion, the question is unnecessarily specific right now. Let me show what I mean and you can reverse if you want to.  
> **username_1**: Generally, we don't do homework corrections here, so don't expect others to redo everything with your numbers. If you have a specific issue, point that out. (Here, you just pasted the whole question and answer, without mentioning where *exactly* you had troubles).  
> **username_0**: I see what you mean now and I agree with you that its definitely less specific and can be applied more easily to other situations that are similar. Thank you!  
> **username_1**: Note that you can use dollar signs for latex-like ("mathjax") notation.  
> **username_2**: @username_1 since you put a lot of work into this I think you could answer "Yes, the approach is correct." so that the OP can accept an answer and the question is resolved.  

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## 1 answers

### Answer by username_3 on 2016-04-02. Score: 1 | Accepted answer

The approach is correct, but it is not true that firms have the same equilibrium production quantities because we are dealing with Cournot competition - it is because the firms have the same profit functions. You can see for yourself with an example that if the two firms had different cost functions, they would not necessarily produce the same outputs as each other.

The step of "substituting one into the other" implicitly uses the concept of Nash equilibrium, i.e. each firm's production quantity is a best response to the other's. Given that $A$ produces some amount $q\_A$ and $B$ knows it, $B$ would make the amount $q\_B^\*$ is profit-maximizing for $B$. Then we fix $q\_B^\*$ and find what $A$ would make as a profit-maximizing response - call it $q\_A^\*$. If I then fix $q\_A^\*$ and find $B$'s profit-maximizing response is the same $q\_B^\*$, $(q\_A^\*,q\_B^\*)$ is a Nash equilibrium: each firm's action is optimal, given the other's. This line of reasoning can be implemented by a solving a system of two equations, given by the profit-maximizing condition $MR=MC$ for each firm that you mentioned.


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