Here's a place where the free market doesn't apply for the free market requires rational actors; there's nothing rational about getting hurt, passing out, and waking up in a hospital with a bill.
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Why would the free market not apply? What can be more rational then planning properly for emergencies that involve you living or dying? If anything it's irrational to hope that some government run "free" emergency care will help you out.
Because it's focused primarily not on the best price, but on the best speed; decisions are measured in how fast your emergency can be turned into a non-emergency, and spares no time for you, or the person who will not be footing your bill but calling for you, to decide the best price for the best value. The hospital which is closest always takes precedence over the hospital which provides the best service for the best price; it's not a free market problem once an emergency is involved, because you cannot act rationally while on the verge of dying.
Which is why the problem is primarily solved before there is an issue; the free market applies to everything except this one moment, where you and your saviors are paired by non-market values. If there is a man who will say "No, I'd rather spend another 20 minutes in this ambulance than spend an extra bit of cash at this place", assuming he can still speak and think clearly, I'd like to meet him for I'm positive he's an interesting fellow.
Though I do agree, I am tired of hearing about emergency services justifying taxation and central law.