Veteran's Affairs, the postal service, social security benefit enrollment and payment processing, the irs, and public education is generally more efficient than commercial offerings at multiple points in their historical existence. Many of them are currently hampered by underfunding though, so I don't know how efficient they are now (being unable to invest enough to maintain themselves would force them to make long-term inefficient tradeoffs).

> How many are the examples when the federal government was more efficient (as in effect per unit money) than a commercial company?

The real question I think is whether there are examples where larger companies gain efficiencies, and if the most efficient way to do something is at a national or international scale, do you want to depend on a massive private companies (likely heavily subsidized) to do it?

That's not handing a public function to the private sector for the sake of finding efficiencies or to fund emergent innovation through competition, that's handing risk-free concessions to a cronies.

> How many are the examples when the federal government was more efficient (as in effect per unit money) than a commercial company?

Here is the thing: commercial companies have incentive to stay efficient as long as there is viable competition from a public agency, which provides a baseline in terms of cost and quality.

As soon as public agencies are out of the picture, then there is little incentive for private companies in staying lean and efficient: Why deliver what you promised when you can go over-budget? The threat that an expensive project will be left unfinished gives you leverage to ask and approve excess costs. Why compete with other companies when you can collude with them and laugh together on your way to the bank?

If the government is a monopsony for a good like “subways” or “highways”. It’s advisable to have an in-house alternative to the commercial sector. Commercial firms are obligated by incentive to increase profits over time - whereas in-house teams are obligated to increase organizational size/pay over time.

Competition between the two should lead to an equilibrium between organizational bloat and profit taking.

Read up on the design and construction of the pentagon and los alamos national laboratory. The relationship between the US federal government and private sector changed tremendously over the 20th century.

Amtrak is a disaster, both inefficient and undercapitalized and dangerous. The U.S. Congress nationalized passenger rail in 1971, gave Amtrak the extraordinarily Northeast Corridor backbone, and yet America has easily the worst passenger rail in the developed world. Amtrak's cost per passenger mile is 3x or 4x higher than flying with a worse safety record.

Something like the https://en.wikipedia.org/wiki/Works_Progress_Administration

I'm thinking health care and education.