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I am tagging this as multivariable calculus because it potentially involves taking partial derivatives.

I am working on some mathematical treatment for Cournot duopoly models (not homework, just preparing for a test tomorrow). A simple two player Cournot model is of the form $p=1000-q_1-q_2$ where $p$ stands for price, and $q_i$ is the quantity produced by firm $i$. The payoffs of each firm are given by $pq$ minus some marginal cost times $q$, say $-10q$. In this case, to find best response for firm $i$, I would take its payoff function $\pi_i=(1000-q_i-q_j)q_i-10q_i$ and take its derivative with respect to $q_i$, getting $1000-2q_i-q_j-10$. I would then set it equal to $0$, solving for $q_i$, and getting $q_i=\frac{990-q_j}{2}$, thus getting an expression maximizing $\pi_i$ in terms of $q_j$. Doing the same thing with $q_j$, I get $q_j=\frac{990-q_i}{2}$ since the function is symmetric. Then I set equations equal to each other/substitute one into the other, and get the Nash equilibrium (profit maximization for both), easy.

The issue I am having is dealing with $n$ firms. Assume a>c>0, b>0. Suppose we have total quantity of output $Q=\sum_{i=1}^{n} q_i$ and price be given as $p=a-bQ$. Then let profit of firm $i$ be defined as $\pi_i=p(Q)q_i-cq_i=(a-bQ)q_i-cq_i$. Defining $Q_{-i}$ to denote the sum of quantities of all firms but of firm $i$, we can rewrite the profit function of $i$ as $\pi_i=(a-bq_i-bQ_{-i})q_i-cq_i$ (observe that $q_i+Q_{-i}=Q$). From here on, I can again take a partial derivative with respect to $q_i$, and find the payoff maximization function to be $\pi_i'=a-2bq_i-bQ_{-i}-c$. Rewriting for $q_i$, I get $q_i=\frac{a-bQ_{-i}-c}{2b}$. So, for an arbitrary firm $j$, the maximization function in terms of $Q_{-j}$ will be given by $q_j=\frac{a-bQ_{-j}-c}{2b}$.

The problem I have here is that I can't anymore substitute as easily as I did in the first example with only two firms. The book gives me a hint: Summing the best-response functions over the different players will help. I do not know what to make of it. Could you explain how to find an intersection given $n$ such functions in the form presented above?

EDIT: I might be onto something here, feedback appreciated!

So I decided to follow the books advice and see what comes of it. Summing both sides of $q_i=\frac{a-bQ_{-i}-c}{2b}$ over all the players I get $q_1+\dots +q_n=Q=\frac{na}{2b}-\frac{nc}{2b}-\frac{(n-1)(q_1+\dots + q_n)}{2} \to 2Q=\frac{na}{b}-\frac{nc}{b}-(n-1)(q_1+\dots + q_n)$=$\frac{na}{b}-\frac{nc}{b}-(n-1)(Q) \to (n+1)Q=n\frac{(a-c)}{b} \to Q=\frac{n}{n+1} \frac{(a-c)}{b}$. Thus $q_i=\frac{n}{n+1} \frac{(a-c)}{b} - Q_{-i}$. Hopefully I am at least on the right track, though I can't see where this leads me yet.

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I think you get it. – GWu Apr 19 '11 at 4:29
up vote 3 down vote accepted

As you pointed out, $q_i+Q_{-i}=Q$. Summing over all $i$ yields $$Q+\sum_i Q_{-i}=nQ.$$ Then sum over all $j$ for the best-response functions: \begin{eqnarray} Q&=&\sum_{j}q_j=\frac{n(a-c)}{2b}-\frac 1 2 \sum_{j}Q_{-j}=\frac{n(a-c)}{2b}-\frac 1 2(n-1)Q\\ \Rightarrow Q&=&\frac{n(a-c)}{(n+1)b}. \end{eqnarray} From here, you can solve for $Q$, and then get each $q_i$ because \begin{eqnarray} q_i&=&\frac{a-bQ-c}{2b}+\frac{q_i}2\\ \Rightarrow q_i&=&\frac{a-bQ-c}{b}=\frac{a-c}b-Q\\ &=&\frac{a-c}b-\frac{n(a-c)}{(n+1)b}=\frac{a-c}{b(n+1)}. \end{eqnarray}

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@GWu: Thanks! I just did that, and edited my post -- it brought me to something. I am not sure if that helps me much though because just as when I started, I have $q_i$ in terms of fractions and $-Q_i$. How do I get a Nash equilibrium from here? I could have presumably gotten it from $q_i=\frac{a-bQ_{-i}-c}{2b}$ too, then... – gametheorist Apr 19 '11 at 4:30
I just edited mine too since I guessed you didn't figure out this. – GWu Apr 19 '11 at 4:32
@GWu: I see how you got to that equation since adding $$\frac{q_i}2$$ to the rest of the equation results into $$\frac{a-bQ_{-i}-c}{2b}$$. Now, if I understand correctly this is better because I can plug in any $q_j$ into the equation since the equation is now dependent on total quantity and not $Q_{-j}$? – gametheorist Apr 19 '11 at 4:39
And plugging in everyone's best response into that one function I get $$q_i=\frac{a-b(n\frac{(a-bQ-c)}{2b}+\frac{Q}{2})-c}{2b}+\frac{q_i}2$$, am I correct? – gametheorist Apr 19 '11 at 4:44
Not that complicated. You have already found what $Q$ is. I wrote down the complete solution. – GWu Apr 19 '11 at 4:46

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