You often get those kind of problems when playing strategy games, especially 4X (civ-like) games, with developing your cities/bases vs producing military units. The main difference with your "toy problem" is that in games the N isn't fixed, but probabilistic, which makes it much harder.

I often tend to spend most of the time developing my production capacity (as you said, it's the most efficient thing to do with a fixed N) but sometimes I do it too much and I get caught unprepared by an attack...