In a private system, all these considerations are micro-economic, as insurees might leave before the insurer gets to benefit from it. So in that case you get much more short term cost reductions: not accepting new insurees because of "pre-excisting conditions", denying coverage, etc. If you have a public system, the cost reductions are more macro-economic: (long term) disease prevention, cost-effective use of drugs, steering of policy and resources at a system level, top down co-ordination of improvement of health care (instead of doctors improving health care only if it improves or does not decrease their income, insurers improving the system if it improves or does not decrease their income, etc). In my opinion, this will lead to higher societal wealth, compared to a private system.

Buying "health" is not the same as buying a car. Indeed, the "need" to buy "health" is fundamentally different from the "need" to buy a car. You can't therefore leave it all to an open market.
There are the matters of rationality assumptions in normal markets (uncertainty - assymetric information/moral hazard), externalities (i.e. contagious diseases), the need for (partly) social insurance (everybody should be able to consume at least ‘sufficient’ care, hence it should not be entirely related to ability to pay), equity and ethical considerations (i.e. solidarity between rich/poor, healthy/sick, old/young), presence of risk-bearing third parties (consumers are not payers).

Mind you, the public system is not without flaws. By far. As economists call it: "government failure" as opposed to "market failure". For example, government often has to rely on information from those who are regulated and who have interest in distorting this information (agency problem). Politicians often are not accountable for long-term effects of policy measures and there are limited incentives for civil servants for technical efficiency (bureaucracy or ‘red tape’) (motivation problem)



Thus ends my lecture on health economics. You can see me do this for three hours on end somewhere in August is you want.