There are some procedures where shopping around is conceivably an option.
Once you're in desperate need (say, when it's bad enough that you're calling an ambulance), or unconscious, it's not realistic to suggest price-shopping.
Markets of the mostly-free sort work well enough for commodity products, like the kind of peanut butter you're buying a jar of every few weeks (though even that's regulated, from things to food safety to nutrition labeling); for everything else, government's a fabulous tool to make sure people aren't taken advantage of, whether it's through heavy regulation (e.g. Switzerland, as someone mentioned) or offering the service itself (e.g. UK).
The vast majority of health care is a commodity product. Unless you have some super rare condition, whatever your problem happens to be it's extremely likely to receive more or less the same treatment from a large number of potential providers.
Once you're in desperate need (say, when it's bad enough that you're calling an ambulance), or unconscious, it's not realistic to suggest price-shopping.
Markets of the mostly-free sort work well enough for commodity products, like the kind of peanut butter you're buying a jar of every few weeks (though even that's regulated, from things to food safety to nutrition labeling); for everything else, government's a fabulous tool to make sure people aren't taken advantage of, whether it's through heavy regulation (e.g. Switzerland, as someone mentioned) or offering the service itself (e.g. UK).