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Medicare's Mythical Administrative Cost Savings
...Rather, private insurers have costs that Medicare doesn't have within the agency. Private insurers bill. Medicare does too, but the IRS has its own budget--hell, its own courts--which don't show up on Medicare's balance sheet. Private insurers negotiate with suppliers. Medicare does too, but most of the negotiation takes place between lobbyists and Congressmen who again, do not show up on Medicare's balance sheet. The Federal government has all sorts of these little items which relieve government agencies of reporting certain costs. But the costs remain.

My guess would be that these explicit costs are still lower than Medicare's. But then there are implicit costs to government fiat that markets don't have. As Tyler Cowen points out, taxation has deadweight losses, and Medicare is a tax on employment, which is something we are particularly anxious not to suppress right now.

The final point is that while people commonly think of administrative costs as "wasted", in fact, they are an important part of the market system....

...Third, we're still driving quite a bit of product innovation. Our messy, organic, wasteful, unfair, irrational system allows experimentation, and they cherry pick the best results. If we stopped doing this, their system would stop looking so good.