(a) Health insurance is the collective provision of all health care.
(b) Health insurance is the sharing of extreme risk in health care spending.
In my view, (a) represents what most people think of as good health insurance. For example, I have a friend who says her health insurance is great because she can get new eyeglasses every year for everyone in her family for a co-payment of only $10.
We have never observed (b). (b) would mean something where you only make a claim when your expenses are going to run into the tens of thousands of dollars. Claims would be rare and large, as in fire insurance. Premiums would be low, as in fire insurance.
Since we never have observed (b), we do not know whether it is something that could be provided by the market or would have to be provided by government. I am willing to concede that it may be the latter. However, what most people mean by universal health coverage is (a), which has some pretty obvious incentive problems. [Update: see Regina Herzlinger on universal coverage.][Further update: see Michael Cannon's counter-argument to Herzlinger.]
When people cut back on health care, what happens? The evidence (from the famous Rand experiment) is that they cut back on "necessary" care, as defined by doctors (who are not exactly disinterested on the topic of how much people should spend on medical services). The result on health outcomes of these cutbacks is not visible.
Peter Orszag (now Obama's had of Office of Management and Budget), Jason Furman (an Obama adviser), and Shannon Brownlee (author of the book Overtreated) are among those who acknowledge that much of our health care spending (a standard estimate is about 30 percent) goes for medical procedures that are not cost-effective. [update: Michael Cannon emails that the claim is even stronger--that 30 percent of medical spending creates no value at all. Actually, I am not exactly sure what these folks would claim if pinned down. But Michael may be right.]
The foregoing are among those on the left who believe that government-run health care can save money. But if you believe that government can save significant money on health care, then by the same token you have to believe either:
(i) our current mix of medical services is optimal, and we could deliver it much more cheaply;
(ii) our current mix of medical services is far from optimal, with too many cost-ineffective services provided.
Most people who can do arithmetic realize that (i) is not the answer. The standard bogeymen of drug industry profits and insurance company overhead are not big enough to make a major difference. Moreover, the evidence for (ii) is broad and overwhelming.
The bottom line is that what we think of as health insurance is not going to survive if we are going to get control of health care costs. Either health insurance is going to become very intrusive about our choices of medical services (the top-down, government option, under the guise of "health care quality"), or we are going to see much higher deductibles and co-payments (the bottom-up option).
For an example of health insurance that tries to get at the insurance aspect, see Health-status insurance, by John Cochrane.