Defining Public Good

From Wikipedia:

In economics, a public good is a good that is non-rivalrous. This means: consumption of the good by one individual does not reduce the amount of the good available for consumption by others.[1] For example, if one individual eats a cake, there is no cake left for anyone else; but breathing air or drinking water from a stream does not significantly reduce the amount of air or water available to others.

The term public good is often used to refer to goods that are non-excludable as well as non-rival. This means it is not possible to exclude individuals from the good's consumption. Fresh air may be considered a public good as it is not generally possible to prevent people from breathing it. However, technically speaking such goods should be called pure public goods. These are highly theoretical definitions: in the real world there may be no such thing as an absolutely non-rival or non-excludable good; but economists think that some goods in the real world approximate closely enough for these concepts to be meaningful.

Naive Objection (by David Isenberg): Think about the limits of the above definition! Suppose you have a unit if fresh air, a public good by the above definition, and somebody lights a fire, consuming the oxygen. Suddenly the air is rival. Now is "fresh air" no longer a public good? In other words, perhaps the term, "public good," is defined by an additional characteristic, i.e., the use of that good, by an *agreement* that the good is public.

Naive Objection 2 (by same guy): Above it says in the real world there may not be absolutely non-rival goods, but what about **IDEAS**? Ideas are always non-rival -- if I tell you my idea, there's more of it, so ideas are actually anti-rival or negatively rival, i.e. subject to increasing returns. Ideas are excludable, of course; if I don't tell you mine you may never learn it. But once the secret is out, ideas are the ULTIMATE public good, according to the above definition.

But, as I say above, I (naively) believe that the above definition is incomplete, or otherwise deficient.