Outsourcing is a monument to the failure of education systems. It turns out all that useless algebra and esoteric calculus was incredibly useful after all.
Regardless, outsourcing isn't about saving money, it never has been. Outsourcing is about accounting and that's what confuses people. Outsourcing increases costs, by a significant amount, but it lets you book the expenses in temporary categories as opposed to indefinite categories like HR.
There are about 5,357 different ways to do it, but the end result is that you're amortizing HR and reducing headcount (often through attrition, but sometimes redundancies) to free up funds to distribute to other budget lines. It appears as if you're doing more with less, and contemporary accounting supports that notion, but you're just kicking the can.
Outsourcing let's you, legitimately, say you are shrinking the organizations footprint (aka: smaller government) and reducing expenses, which is true. But in reality you've just broken expenses apart and thrown the pieces ahead. The trouble is that everybody is doing that and what would otherwise be little bumps you could absorb by virtue of sheer scale start to pile up. You're building your own ramp and sooner or later you're going to launch off the end and fall a very long way.
In a commercial setting you can get by with screwy books because you've got all sorts of options and failure is one of them. You can cut losses and go home. You really can't do that with a government though. Outsourcing government processes is a fashion, not a goal. In the end it's a terrible idea.