We saved money by bringing in...
...from Capita. Our Academy support came in house and we're saving 10's of thousands. It is actually saving public sector redundancies here.
I never have managed to see how it was cheaper to pay another company that has to make profit on top of do the same job. Especially as when there is an organisational change, (as there inevitably is in public sector as governments change the rules) they charge a fortune to change the contract further down the line.
Public sector needs to get to grips with its management problems. That's the only way it will actually save public money. Outsourcing will just cost more and more.
No one has yet convinced me otherwise.