I would prob say the hospital knows the optimal price already. It might not be 100% applicable to the US but here is an example in Ireland, if I go to the hospital without any referral or by not being brought in an ambulance which would make it free, I have to pay. The funny thing is that they ask you beforehand if you have insurance and substantially increase the price of an overnight stay, etc if you do, to which point insurers are suggesting customers to withhold such a fact and lower the bill.