Experience and economics support the view that the best way to control costs without sacrificing quality is to give consumers more power to act as a check on erratic pricing, deteriorating quality and excess care. Competition and consumer choice are the most powerful cost-control mechanisms ever devised. Our plan includes both, and that’s why we are confident that our targets are achievable.In many areas, it is true that competition and consumer choice act to hold prices down. But health care is not like other segments of the economy. (For the reason why, see here.) If you look around the world, you see evidence that Ryan's argument is exactly wrong. Among rich nations, the more socialistic the health care system, the cheaper it is. The cheapest among rich nations is Britain's entirely socialistic system. Mostly socialistic systems like the French or the Swedish are next. Mostly private systems like those in Switzerland and the US are the most expensive.
Within the US, government-run Medicare is cheaper than private insurance.
Ryan is living in an Ayn Rand novel, not on planet earth. His model makes some theoretical sense but we already know, using the whole world as our experiment, that in practice he is completely wrong.