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.