Actually, there is- it's when the socialized system decides they're good enough, and stops trying.
There is literally no way a private company could offer something cheaper and better than the government?
Obviously in practice that's not how things work out, so in most cases private companies end up moving faster, leveraging efficiencies, utilising innovation better, and generally being cheaper. The lesson shouldn't be that means the government shouldn't provide services though, but rather that the government should act more like a business and do things businesses do to be competitive.
It's very noticeable that when government does this it works really well. The UK government has brought a lot of it's web and IT services in-house in the past decade, and they're so much better now.
And no... there is no way that a private company could offer something cheaper because both entities function the same way except that a private company also has to make a profit. Governments are not profit-making entities. Take whatever method a private company has to make something cheaper and remove profit and you have something the government can handle.
Industries like healthcare are plagued with government interference. Some regulations are important. Many just serve as a moat around the oligopoly that is the current private health Care system.
You're either being intentionally obtuse or you're intentionally being misleading for some reason. This isn't rocket science. If two things are equal and one has the requirement to also generate profit, there's no way the other can possibly cost more.