# How to derive a Nash equilibrium pure strategy in a linear Cournot Model [duplicate]

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.

## 1 Answer

• 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*}