Rivalry and excludability define what goods the government should provide or manage. The government should provide pure public goods, and regulate natural resources and natural monopolies.
Excludability is the ability of producers to detect and prevent uncompensating consumption of their products. Rivalry is the inability of multiple consumers to consume the same good.
multiple consumers cannot consume the same good
multiple consumers can consume the same good
consumption can be detected and prevented by producer
Efficiently produced and allocated by markets
High fixed costs, low marginal costs => inefficient competition
consumption cannot be detected or prevented
Tragedy of the commons, negative externalities => overconsumption
Free riders, positive externalities => underproduction
A public good is a non-rival non-excludable good that benefits almost everyone in a polity. Because public goods are not excludable, they get under-produced. The pricing system cannot force consumers to reveal their demand for purely non-excludable goods, and so cannot force producers to meet that demand. Thus government should produce non-excludable goods that aren't supplied by nature — namely, pure public goods. (Impure public goods are those that are partly excludable — those for which producers can capture some but not all of the benefits that the goods provide. Examples are education, technology development, landscaping, broadcasting, and the arts. Impure public goods need not be produced by government.)
A natural resource is any rival non-excludable resource. Because natural resources are not excludable but still rival, they get over-consumed. Thus government should police the use of natural resources, preferably by taxing resource use by the amount that the use costs the rest of the polity.
A natural monopoly is any non-rival excludable good that benefits almost everyone in a polity. Because natural monopoly goods have high fixed costs and vanishing marginal costs, they cannot be produced efficiently through market competition. Thus government should regulate the provision of natural monopoly goods.
A private good is any rival excludable good. Markets are able to manage their production and allocate their consumption more efficiently than government can.
The above is standard textbook analysis. See e.g.:
- Ch. 11 Public Goods and Common Resources in Principles of Economics by Greg Mankiw (Harvard economist fired as Chairman of White House Council of Economics for defending outsourcing).;
- The Free Rider Problem in the Stanford Encyclopedia of Philosophy;
- Public Goods and Externalities in the Concise Encyclopedia of Economics by Tyler Cowan (libertarian economist at GMU, co-blogger at Marginal Revolution).