You're assuming that there exists some finite $X to develop some invention, as if you simply pour enough money into a problem, a solution will be found. This i

You're assuming that there exists some finite $X to develop some invention, as if you simply pour enough money into a problem, a solution will be found. This is faulty logic. There is no direct correlation between money spent to develop, and outcome of desired invention/solution. Name one disease that needs to be treated/cured, and how much more money needs to be spent on R&D to find the solution. Nobody can do that. Ergo, your math does not apply. When you let free-riding happen, (e.g. ban patents and allow copying of inventions, not to be confused with copying of brand), you let grass-roots innovation happen. More people get to understand the problem domain, more cheaply than any monopoly can manage.