Health Care, The US V Spain

Atrios has an interesting anecdote on health care with some interesting statistics.

Spain spends about 7.5% of its GDP on health care. We spend about 13.9%. About 4% of our GDP is spent on a subset of health care called... "health care administration."

So, Spain spends 7.5% of it's GDP on state administered universal health care and the US spends 13.9% of it's GDP on a private health care system that covers fewer people all of the time including 4% of it's GDP on "administrative costs". So what does that do to the argument that a government administered health care system would be too inefficient?