In cases of both Medicare and Defense budgets, the government is essentially overpaying private business (healthcare providers and bomb makers) to provide public services. There's a huge lobby that no doubt pays everyone off to keep the status quo. Having all that government money artificially inflates both the costs of bombs and health, because they can essentially charge what they want. I'm not sure how you get private enterprise to avoid screwing the government, without more bureaucracy and regulation (more government). As Smerf says audits, but I think that's another layer that could be easily manipulated too.
Look, my stance on government run institutions is this: I can live with inefficiency because there's no incentive for the government to screw its employees or to inhibit its service to make more and more money. (A government institution would never have an idea like planned obsolescence or patenting biological seeds or withholding technology for the iphone 2020) I know that its a bit rosy, but generally i think there's some truth there.