Purely private goods suffer from neither of these problems. Examples include, food, clothing, dwellings, books, newspapers etc. In fact most goods consumers typically purchase are almost purely private goods. Clearly some publicly provided goods are not public goods, just as some pure public goods aren't publicly provided.
Non-excludability and non-rivalness together make it difficult for private markets to produce public goods in a manner that is both profitable and allocatively efficient. While some technical changes have removed some of the non-excludability problem for some public goods there are still numerous public goods problems that remain.
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