# 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 '16 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 '16 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 '16 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 '16 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 '16 at 11:31

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.