# Cournot game with 2 firms

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?

• 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? Mar 29, 2016 at 17:27
• In my opinion, the question is unnecessarily specific right now. Let me show what I mean and you can reverse if you want to. Mar 29, 2016 at 17:31
• 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). Mar 29, 2016 at 17:39
• 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! Mar 29, 2016 at 17:40
• @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. Mar 30, 2016 at 11:31

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.