Finding the cost-minimising production allocation in a Cournot merger with symmetric costs

Question

Suppose I have 3 firms in a market. They have an identical, convex cost function, $C(q) = 20q + q^2$ = $C_1 = C_2 = C_3$, and each firm produces in their own factory.

Market demand is linear, $P=200-Q$ , where $Q = q_1 + q_2 + q_3$

Now, suppose that Firm 2 and Firm 3 merge, but costs don't change. The new Firm (call it Firm $M$), has the same cost function as above.

How would Firm $M$ decide whether to produce at 1 factory entirely and close the inactive one down, or split production between two factories? Specifically, which choice would lead to lower costs?

Intuitively, since the cost function is convex, it would be beneficial to split production between two factories. I don't know how to identify Firm M's cost function when it produces in both factories.

Answer 1

You want to model the decision problem of firm $M$. In case they produce a total quantity of $q_M$, they can split this between factories $1$ and $2$. Firm $M$ can minimize the cost of producing $q_M$ units by solving the problem $$ \min_{q_1,q_2} C_1(q_1) + C_2(q_2) = \min_{q_1,q_2} 20q_1 + q_1^2 + 20q_2 + q_2^2 $$ subject to the constraint $$ q_M = q_1 + q_2. $$ (And the usual non-negativity constraints, but those will not matter here.)

The solution to the above problem yields $C_M(q_M)$. If the solution is such that $q_1,q_2$ are both positive, then no factory should be shuttered.