What do we see? The USA is one of the biggest innovators in medicine and basically decimates the medical innovations coming from Europe (and yes, I know that there is some government involvement in the USA medicine investigation, but the comparation is still valid).
this is the worst example you could choose.
medicine in the US isn't private (as in unregulated free market) at all, especially not research. even the actual services are run by HMOs (so-called "private" insurance), and only for non-military, non-poor, non-public-sector-worker, adult-but-not-retired, and by (state) governments for everyone else.
What would have happened if you left those resources to the private sector?
...and that's very speculative. what would the behaviour of human beings be.