The existence of private insurance strongly correlates with higher health care costs [View all]
http://theconversation.com/creating-a-better-health-system-lessons-from-norway-and-sweden-30366
Private insurance, however, is an expensive way to fund health care. When we look at the relationship between private insurance and a nations total health-care costs, we find a strong positive relationship: the more a country relies on private insurance the more it pays for health care, without any extra benefit.
Its not just the bureaucratic cost of private insurance; its also the distortions it introduces, for when there is a private insurer in the market, able to pay premium prices for priority access to care, prices throughout the health system rise, both for the public insurer and for those who pay for health care from their own pockets.
The graph below shows the relationship between countries health-care costs (as a proportion of GDP) and reliance on private insurance. These are all countries with only minor differences in their health outcomes. (For the statisticians, R Squared = 0.66.)

At one end of the scale, the United States stands out. At the other end are three countries Sweden, Norway and Iceland, where private insurance is either absent or plays a minuscule role in funding health care. And these countries contain their total health-care costs (as does Australia for now), to around 9% of GDP. This is in spite of the fact that Sweden, the largest of these countries, has a significantly older population than Australia.