Should all of our economy be privately owned?

Answer 1

There are some things that should not be done on a for-profit basis. Bridge and road construction and maintenance. Public education. Public utilities. Police and Fire departments. Probably a few other things.
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Answer 2

There are certain things that have to be done by the government, because a market does not exist for their supply otherwise. For instance, the miliatry and courts would not exist in a market because there is no individual demand to supply an adaquate amount. Also, there are cases where a market would exist, but a compeditive one can not. For instance with airports or a subway system. There is a market for a subway, but you cant have many competeing subways in a city, so you would have a monoply. In this case, either the government has to own it, or if it is privately owned, it has to have a regulatory scheme placed on it. This is how a lot of utilities work. They are privately owned, but government regulates them heavily.