Cournot-game problem I'm so stuck with an exercise about Cournot game and was hoping if someone could help me out here. Would appreciate all the help. This is the exercise:
Consider the market for Blue Turtle (a new de-energizer) in which demand $Q$ is related to price $P$ by $P(Q) = Q^{-0.5}$.
In this market there are m identical producers, say firm 1; 2; up to m which can
produce any non-negative quantity of Blue Turtle say $q$ at costs $C = q^2$.
Assuming these firms are playing as in a non cooperative (Cournot) game. Determine the market price for Blue Turtle and the quantities produces by each of these firms? Show that price falls if the number of producers grows.
thanks again.
 A: Since all Cournot exercise are similar, it might still help someone, even with a slightly different problem
Each firm maximizes:
$\begin{align}\max \pi_i &= P(Q)q_i - c(q_i)\\
&= (\sum_{i\neq j} q_j+q_i)^{-\frac12}q_i - q_i^2\end{align}$
So the first order condition for firm $i$ is
$-\frac12Q^{-\frac32}q_i  +Q^{-\frac12} -2q_i = 0 $
This has to hold for all firms in Equilibrium. Hence summing that up over all $i\in I$ gives you:
$\begin{align}
\sum_{i=1}^m\left(-\frac12Q^{-\frac32}q_i  +Q^{-\frac12} -2q_i\right)&=0\\
-\frac12Q^{-\frac32}\sum_iq_i  +mQ^{-\frac12} -2\sum_iq_i&=0\\
-\frac12Q^{-\frac32}Q  +mQ^{-\frac12} -2Q&=0\\
(m-\frac12)Q^{-\frac12}   -2Q&=0\\
\frac{m}2-\frac14   &=Q^\frac32\\
(\frac{m}2-\frac14)^\frac23  &=Q \Leftarrow\text{Equilibrium Quantity}\\
\end{align}$
Now, if I did not make an error here, we can already say that prices decrease if $m$ increases, because $P(Q)$ is decreasing in $Q$ and the equilibrium $Q$ is increasing in $m$.
To continue you'd now have to plug this equilibrium $Q$ back into the FOC
