Public Good arguments are often used by proponents of government intervention in the economy to justify their arguments. They use it to justify government intervention in education, health care, utilities, the Internet, etc. But these arguments do not hold up. To know why, we must first discuss what exactly a public good is.
A public good is a good/service that is non-rival (one’s consumption doesn’t prevent someone else from consuming it) and non-excludable (one cannot prevent another from enjoying the resource). Fireworks are the textbook example of a public good. If a person shoots off fireworks, he cannot reasonably prevent people from watching them. Likewise, each person’s ability to enjoy the firework show doesn’t inhibit anyone else’s ability to enjoy the show. National defense is another example.
With public goods, one tends to get the free rider problem. People can benefit from the good/service without paying (think, for example, a person who watches the fireworks show from his porch rather than buying a ticket to the field). This can cause too few of the public good to be produced even if there is ample demand. Government can step up to provide the good (and, in many cases, this is a legitimate function of government. Police and national defense come readily to mind).
So, why don’t other things, like medical care or education fit this bill? Simply because they are not public goods. Medical care is both rival and excludable (one person’s “consumption” of a doctor’s time prevents another from consuming it) and excludable (a doctor can turn away patients). Education is similar, although it is just excludable.
In conclusion, there may be other arguments for medical care and education to be supplied by government, but public good analysis isn’t one of them.