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A public good is a good that it is difficult to charge for, perhaps for political reasons, perhaps because it is physically difficult to charge for it. Lots of people benefit, whether they paid for it or not. As a result, people are disinclined to pay for it, so absent state intervention, the good tends to be underprovided. A lighthouse is a public good, since you cannot stop people from taking advantage of it regardless of whether they have paid for it. Economists say it is “non excludable” – those who provide it have difficulty excluding those who will not pay. They also say it is “non rivalrous”. One person using a lighthouse does not diminish someone else's use of the lighthouse. If a good is non excludable, but rivalrous, they call it a “commons”, leading to the “the tragedy of the commons” – that stuff that cannot be owned and defended tends to be overused, misused and abused. The fish in the ocean are a commons, and so tend to be overfished.
An “externality” is similar to a public good. We tend to call positive externalities “public goods” and call public bads “externalities”. Air pollution is an “externality” because the pollution tends to float across property lines, thus winds up polluting someone external to the decision that led to the pollution. We call air pollution an externality, but call cleaning up air pollution a public good.
A public good, a commons, or an externality is in a sense a transaction cost problem. If everyone affected could just get together and agree on something, there would be no problem, but of course a committee of seventeen people could not agree to adjourn a meeting held in a burning building.
Externality, public good, commons, or high transaction costs, are merely different ways of looking at much the same problem.
Genuinely anarchist societies are apt to under provide public goods. On the other hand governments tend to provide the wrong kind of public goods, providing what serves their purposes rather than the supposed purpose of the public good, Further, when a government gets in the business of providing a some supposed public good, it unavoidably creates a lobby, which results in the public good being overprovided, thus for example ever lengthening copyright, ever more expansive patents for ever more trivial “inventions”, and, of course, the infamous military-industrial complex, such as Haliburton.
The lobby helps the bureaucracy expand, and the bureaucracy returns the favor, helping the lobby at the expense of the public and at the expense of the supposed purpose of the bureaucracy.
Every time there is a crisis somewhere that arguably threatens Americans, the Pentagon sends some US troops, and no one seems to notice that they pour concrete a lot more than they confront bad guys. They build an overseas base that continues indefinitely, providing permanent careers to a multitude of officers long after the crisis has been forgotten, and the base continues to stir up resentment, and attract trouble, in countries that few Americans have ever heard of. Americans now support about a thousand overseas bases, most of them originally built to contain long forgotten aggressive acts by the Soviet empire, and though the empire fell sixteen years ago, the bases are still slowly growing as I write.
The holocaust of the Jews was a public good. Germans (and perhaps Frenchmen and Poles for that matter) experienced the removal of Jews as a pleasant thing that is nonexcludible and nonrival in consumption, making it a public good. Taking beliefs and preferences as given, Aztec human sacrifice also has public good properties, as they thought that they were keeping the world from ending by appeasing the gods. With private goods, individual bad decisions are eventually corrected by the example of individual wise decisions. Collective bad decisions are not. Thus decisions about public goods are usually bad decisions. We should always deliberately under provide “public goods”, because of the high probability of error.
The most important public goods, such as war, are also public bads, thus if it is plausible that if we were less able to produce public goods efficiently, we would be considerably better off. Uncontroversial public goods, such as lighthouses, are not that big a deal, and there usually is some workaround. For example most of the benefit provided by a lighthouse was captured by a one or two ports, each with one or two major docks. The owners of those docks did not find it very difficult to overcome such a public good problem. Since there are only a few of them, it is a transaction cost problem, and not an intolerably large one.
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