Cartel profit maximizing quantity question (Cournot game)

Question

Find the Nash equilibrium of Cournot’s game when there are two firms, the inverse demand function is P(Q) = α – Q when α ≥ Q and 0 otherwise, and the cost function of each firm I is Ci(qi) = qi2. If the firms collude in this situation to create a cartel to maximize their profits, how much would each firm produce?

I'm able to find the Nash equilibrium quantities when there is no collusion - however, I'm stuck on the cartel part.

We want to find joint profit maximizing quantities under collusion here. When the marginal cost to each firm is constant this is easy to do, since $C(Q) = C(q_1) + C(q_2)$. However, in this case, $C(Q) = q_1^2 + q_2^2$.

Thus, our joint profits are $\pi(Q) = Q(\alpha - Q) - (q_1^2 + q_2^2) \neq Q(\alpha - Q) - Q^2$. I'm not sure how to take this first order condition (with respect to $Q = (q_1 + q_2)$) given the non-linearity of the cost functions. Any ideas as to how to solve this would be greatly appreciated (not looking for a direct answer; simply a nudge in the right direction).

Answer 1

Hint

By colluding, presumably the two firms would like to figure out the least costly way to produce in order to maximize profit. Treat the two firms as two production plants of a single "parent company". The total cost of the cartel can then be found by solving \begin{equation} C_\text{cartel}(Q)=\min_{q_1,q_2}\;[C_1(q_1)+C_2(q_2)],\quad\text{s.t.}\quad q_1+q_2=Q. \end{equation} This will give you a the optimal way of distributing production across the two firms, $q_i^*(Q)$, as a function of aggregate output $Q$. Then the cartel's total cost is just the sum of the member firms' costs evaluated at the optimum: $C_\text{cartel}(Q)=\sum_i C_i(q_i^*(Q))$.