Free Markets Would Halt Capitalism
The only truly market-based solution in the realm of air pollution is to let the people who bear the costs of the pollution set the price at which they're willing to allow it. That is, we would have to privatize the air, so that each breather had the right to a share of clean air, and could sell part of that right to a plant or other facility that wanted to put a certain amount of pollution into it. Under this system, the market truly would determine what level of pollution is acceptable, on the basis of contracts between polluters and pollutees. The government's role would be reduced to courts that enforce breaches of contract and illegal encroachment on individuals' air. For the sake of this thought experiment, we'll make the assumption that such courts would have access to the information needed to determine whose pollution is in whose airspace, and that they would consider measurable levels of pollution to be actionable trespasses that harm the air owner*. But such a truly market-based pollution control system would bring both pollution and pollution-creating activities to a screeching halt.
The trick is that you can't pick and choose whose air to pollute. Air flows and mixes and affects everyone within a certain area. That means that for a plant to emit some pollution, it must buy the right to pollute from everyone in the area. And that means that all it takes is one holdout to bid the polluter up to an astronomical rate, or refuse to sell altogether. This is a perfectly economically rational decision, if the person values clean air more than they value any amount that the polluter is able to pay -- and even if it's irrational, the real world is populated with plenty of irrational people.
The only reasonable solution, if we think that some level of polluting activity ought to be allowable, is to move back toward that dreaded institution, common property. The air in some suitably defined intermixing bubble** is the common property of all of the people who use it. These people can then make a collective, and hopefully democratic, decision about how much pollution ability to sell, overruling any holdouts. The result is effectively a cap-and-trade or emission tax system. Indeed, the air-owning collective could even impose an end-of-pipe solution, drawing up a contract that allows the plant to emit as much as it likes, provided it installs a certain type of scrubber on the smokestack. Such collectivization also reintroduces all of the problems of politicization of environmental management that advocates of market-based solutions are trying to avoid.
*In the real world, both of these assumptions would be highly questionable, dealing a further blow against common-law-based solutions to pollution. Even in a Hayekian paradise, the costs of acquiring the information necessary to pin a particular individual's air quality, and the resulting harms, on a particular emitter, would be monumental. And the innocent-until-proven-guilty standard and considerations of how much harm gives an individual standing would incline the courts to be very permissive toward polluters, particularly when there are any uncertainties. Across-the-board restraints on risk-creating activities are necessary to circumvent these issues.
**The size of this bubble would vary depending on the pollutant. For some, like greenhouse gasses, it would be global. The bigger the bubble, the more opportunties there are for holdouts under the privatized system.