I am sorry but that's like the opposite of economic logic. Ceritus paribus, more competition means lower prices.
Regarding the actual topic, I've long maintained that a single payer system in America would be much more expensive than a single payer system in Europe because of different priorities of the American consumers (for instance, old people demographic in America, the pro-life nature where a big part of our insurance expenses goes towards saving premature newborn babies). The only way a single payer in America (or in any country) could be cheaper is by offering a different product to the consumers.
It doesn't hold under empirical study of firms of any industry. Competition can be offset by jacking up prices or otherwise hiding the price. Prices are never set by the market, so everyone pays a ransom for the good or service in question, especially in healthcare.
And don't forget that an efficient market needs price transparency. Everybody who has tried asking "How much does this procedure cost" beforehand knows that's far from being there in the US. How can consumers factor in prices in their decision when they simply don't have the ability to compare suppliers (plus people usually want the best and not cheapest healthcare). There is simply no incentive for healthcare provider to lower their cost. Would that bring more patients?
Monopsonies have been well-studied and lower prices without any competition on the buyer's side.
Wait, what? So why is infant mortality so abysmal then?
http://www.cbsnews.com/news/u-s-infant-mortality-rate-worse-...