Cournot Competition game with 3 Firms

Question

Three firms are in Cournot competition. The inverse demand curve is denoted p(q) where p is the price if a total of q units are produced. Assumptions are: p(0)>0 and p'(q)<0 and p''(q) $\le 0$ whenever p(q) >0. The firms have identical and continuous cost functions, c($q_i$) >0 for all $q_i$>0. Assume c'($q_i$)$\ge 0$ , c''($q_i$) $\ge$ 0 for all $q_i$ $\ge 0$ and p(0) > c'(0). We consider pure strategy Nash equilibria. For interior quantities , derive firm i's first order condition. Do you agree that all firms must supply the same quantity in any equilibrium? Do you agree that there is a unique equilibrium? Prove and explain your answers.

What I did is that

First I agree that all the 3 firms will supply the same quantity in the equilibrium and this equilibrium is unique. The Profit Maximization Problem for firm 1 is:

max p($q_1$ + $q_2*$ + $q_3*$)$q_1$ -c($q_1$)

and the FOC is :

p'($q_1$ + $q_2*$ + $q_3*$)$q_1$ +p($q_1$ + $q_2*$ + $q_3*$) -c'($q_1$) = 0

If i had the inverse demand function and the cost function, I could have derive the best response function. But now the only that I can say is that $BR_1$($q_2$*, $q_3*$). Similarly for the other two firms : $BR_2$($q_1*$, $q_3*$) and $BR_2(q_1* , q_2*$). But I stuck to this point and I cannot show how this quantities are equal. Any help will be appreciated. Thank you.

Answer 1

Firm 1's profit maximization problem is : \begin{eqnarray*} \max_{q_1} & \ p(q_1 + q_2 + q_3)q_1 - c(q_1) \end{eqnarray*} Firm 1's response to firm 2's and 3's quantity choices will satisfy the following: \begin{eqnarray*} p(q_1 + q_2 + q_3) + p'(q_1 + q_2 + q_3)q_1 - c'(q_1) = 0 \end{eqnarray*} Likewise, we'll get the implicit best response functions for the other firms as well. \begin{eqnarray*} p(q_1 + q_2 + q_3) + p'(q_1 + q_2 + q_3)q_i - c'(q_i) = 0 \end{eqnarray*} for each $i \in \{1,2 ,3\}$.

There is a symmetric equilibrium of the above game if there exist $q^*$ such that \begin{eqnarray*} p(3q^*) + p'(3q^*)q^* - c'(q^*) = 0 \end{eqnarray*} To show that there exist one, consider the following function \begin{eqnarray*} g(q) = p(3q) + p'(3q)q - c'(q)\end{eqnarray*} Observe that $g(0) = p(0) - c'(0) > 0$. Also, given that $p(q)$ is decreasing and concave, there exist $\overline{q} > 0$ such that $p(\overline{q}) \leq 0$. Consequently, $g(\overline{q}) \leq 0$. Therefore, by intermediate value theorem, there exist $q^* \in [0,\overline{q}] $ such that $g(q^*) = 0$ i.e. \begin{eqnarray*} p(3q^*) + p'(3q^*)q^* - c'(q^*) = 0 \end{eqnarray*} Also, $g'(q) = 4p'(3q) + 3p''(3q)q - c''(q) < 0$. Since $g(q)$ is decreasing in $q$, $q^*$ is unique. So there is a unique symmetric equilibrium. To show that it is the only equilibrium, we'll show that there is no equilibrium where at least two of the firms choose different quantities. To do so, let us subtract the implicit best response function of firm 1 from that of firm 2, \begin{eqnarray*} p'(q_1 + q_2 + q_3)(q_2 - q_1) = c'(q_2) - c'(q_1) \end{eqnarray*} Notice that the above equality only holds for $q_2 = q_1$. This is because

• when $q_2 > q_1$, $p'(q_1 + q_2 + q_3)(q_2 - q_1) < 0$ and $c'(q_2) - c'(q_1) > 0$,
• when $q_2 < q_1$, $p'(q_1 + q_2 + q_3)(q_2 - q_1) > 0$ and $c'(q_2) - c'(q_1) < 0$.

Therefore, there is no equilibrium satisfying $q_2 \neq q_1$. Likewise, there is none satisfying $q_2 \neq q_3$ or $q_1 \neq q_3$.