generally all citizens paying into something makes it cost more, not less, as the ready availability of other people's cash saps the desire to cut costs.
This is an oft touted claim of conservatives, but it just isn't true. Socialized health care gives far more bang for the buck than privatized. Take a look at how much Americans pay in healthcare costs pro capita (pre-Obamacare if you like, so you won't have that to blame), and compare that with factors like lifespan and health. Columbia pays far more and gets far less precisely because it's so privatized.
The problem is that with a capitalized system, what matters is maximizing profit, and prices will converge at the highest cost the buyers are is willing to pay. When what you pay for is your life and health, the sky is the limit.
Why do Americans go to Canada to buy prescription drugs? Because the free market does not mean lower costs. Rather the opposite.