<ul>
<li>firm $i$'s output: $q_i$</li>
<li>total output: $Q = q_1 + \ldots + q_n$</li>
<li>everyone but firm $i$'s output (output of rest): $R_i = Q - q_i$</li>
<li>marginal cost: $c$</li>
<li>inverse demand function: $p(Q) = a - Q$</li>
<li><p>profit for firm $i$:</p>

<p>$$Profit = Total Revenue - Total Cost$$
$$\Pi_i(q_i, R_i) = p(q_i + R_i)q_i - cq_i$$</p></li>
<li><p>At profit maximum, the derivative of the profit equation equals zero (a.k.a. first order condition). So let's take the derivative of the above with respect to $q_i$ holding $R_i$ constant and set it equal to zero:</p></li>
</ul>

<p>$$p'(R_i + q_i)q_i + p(R_i + q_i) - c = 0$$
$$p'(Q)q_i + p(Q) - c = 0$$</p>

<p>Note the above is true for all firms since they are all simultaneously profit-maximizing in the Cornout model.</p>

<p>Now substitute $p(Q) = a - Q$ and $p'(Q) = -1$.</p>

<p>$$-q_i + a - Q - c = 0$$</p>

<p>Let's get all our $Q$s on one side of the equation:</p>

<p>$$q_i + Q = a - c$$</p>

<p>Note we have $n$ equations of that form. We can add them all together to solve for $Q$:</p>

<p>$$
\begin{align*}
(q_1 + \ldots + q_n) + nQ &amp;= n(a - c)\\
Q + nQ &amp;= n(a - c) \\
(n + 1)Q &amp;= n(a - c) \\
Q &amp;= {n \over n + 1}(a - c)
\end{align*}
$$</p>

<p>The best response for any individual firm is therefore:</p>

<p>$$
\begin{align*}
q_i &amp;= a - c - Q \\
&amp;= a - c - {n \over n + 1}(a - c)\\
&amp;= \frac{a - c}{n + 1}
\end{align*}
$$</p>

CC BY-SA 3.0
2017-12-12T02:05:35.823Z
2017-12-12T02:05:35.823Z
fny
<p>Suppose there are $N$ firms each with the same positive marginal cost $c$. How would I go about finding a pure strategy Nash Equilibrium for the firms? Suppose the Inverse Demand curve is defined: $p=a-Q$ with $Q$ being the market output. </p>

<nash-equilibrium>
How to derive a Nash equilibrium pure strategy in a linear Cournot Model
Lola1984
# How to derive a Nash equilibrium pure strategy in a linear Cournot Model

(Asked by: username_0 on 2017-12-12)

Suppose there are $N$ firms each with the same positive marginal cost $c$. How would I go about finding a pure strategy Nash Equilibrium for the firms? Suppose the Inverse Demand curve is defined: $p=a-Q$ with $Q$ being the market output.

**Tags:** nash-equilibrium

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## 1 answers

### Answer by username_1 on 2017-12-12. Score: 0

* firm $i$'s output: $q\_i$
* total output: $Q = q\_1 + \ldots + q\_n$
* everyone but firm $i$'s output (output of rest): $R\_i = Q - q\_i$
* marginal cost: $c$
* inverse demand function: $p(Q) = a - Q$
* profit for firm $i$:

  $$Profit = Total Revenue - Total Cost$$
  $$\Pi\_i(q\_i, R\_i) = p(q\_i + R\_i)q\_i - cq\_i$$
* At profit maximum, the derivative of the profit equation equals zero (a.k.a. first order condition). So let's take the derivative of the above with respect to $q\_i$ holding $R\_i$ constant and set it equal to zero:

$$p'(R\_i + q\_i)q\_i + p(R\_i + q\_i) - c = 0$$
$$p'(Q)q\_i + p(Q) - c = 0$$

Note the above is true for all firms since they are all simultaneously profit-maximizing in the Cornout model.

Now substitute $p(Q) = a - Q$ and $p'(Q) = -1$.

$$-q\_i + a - Q - c = 0$$

Let's get all our $Q$s on one side of the equation:

$$q\_i + Q = a - c$$

Note we have $n$ equations of that form. We can add them all together to solve for $Q$:

$$
\begin{align\*}
(q\_1 + \ldots + q\_n) + nQ &= n(a - c)\\
Q + nQ &= n(a - c) \\
(n + 1)Q &= n(a - c) \\
Q &= {n \over n + 1}(a - c)
\end{align\*}
$$

The best response for any individual firm is therefore:

$$
\begin{align\*}
q\_i &= a - c - Q \\
&= a - c - {n \over n + 1}(a - c)\\
&= \frac{a - c}{n + 1}
\end{align\*}
$$


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