First, we don't have a pure free-market health care system, we actually have some form of a bastardized socialized health care system. Our government pays approximately 50% of healthcare receipts.
Second, there are numerous economic models on how a single-payer system would work, and where the money would come from. Some models have a gross levy on income taxes, some on businesses, and some have it as a valued-added tax. So it is not entirely clear where the money would come from because there are competing models.
As to whether we would pay more? Perhaps, but it's hard to see that. Right now we spend approximately 2x over other top nations in the world. We could continue to spend that amount, and if we did would probably provide superior care (under a single-payer system) than those other nations once our single-payer system reached maturity. We would probably see some savings though, because there are other studies that estimate that 1 out of every 3 health care dollars go towards funding the massive bureaucracy. It is a fact that if we have a single payer that is responsible for collecting bills, paying receipts, instituting one set of billing codes, one agency for managing disputes, etc, that we would save money. We would also probably save money because our government could negotiate the cheapest drug rates (because they would represents all 300 M Americans.)
So I do not think there is a clear-cut answer to your question above regarding whether we would save money (or pay more) under socialized medicine.
Try reading, if you get a change, a book called "Critical Condition: How Health Care in America Became Big Business--and Bad Medicine" by Donald L. Barlett, James B. Steele (ISBN: 0385504543). It gets into these topics.
