Hope is that when big entities like the state or insurances have to pay the big bill, they have far superior means to enforce lower prices. That's not everything that needs to be done, but it is still useful.
The really big gap between the U.S. and the rest of the developed world on medical costs isn't the cost of individual procedures (once you've taken into account what insurers actually pay), but the much larger amount the U.S. spends on last-6-months-of-life "heroic care". If that were brought more in line with international norms, the overall cost-per-person numbers would be much better.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1361028/
If I get table 4 right, people who manage to live past 85 years will have 1/3 of their health care spending after their 85th birthday (not entirely accurate, but a reasonable simplification).