Competition is what drives down costs in a market. If costs are going out of control, perhaps there is something in the current system that is preventing competition? Lack of price transparency?
But again, market-based solutions are just not appropriate for essential health care. You would pay ANYTHING to not die - there's no price elasticity of demand.
Every other developed country has figured out that this is simply immoral.
Cost goes down with increased productivity. Competition sometimes brings that, sometimes it doesn't.
Economics can be complex!