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.