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.
That's what it is focused on now precisely because it isn't a free market. If it was then companies providing those emergency services to you would care about doing a good job just like any other servics or they go out of business losing their invested capital. Unless it's total charity all decisions are still paired by market forces. The problem is with the current system a lot of those forces are illogical and invisible to the consumers of those services and therefore are devoid of any feedback regarding their success or failure at solving the problems. Hospital selection isn't dictated by the consumer but by "laws" so no adjustments can be made and everything costs more as a result.