Pigou taxes are often recommended to improve allocative efficiency and, therefore, social welfare by internalizing negative externalities to their sources (and their customers). Economists have spilled a lot of ink trying to specify what an "optimal" Pigou tax would be (see, e.g., here). Harvard's Greg Mankiw has even started a "Pigou Club" to advocate for Pigovian taxes (see here). Pigvoian taxes have even been adopted as a rule of international law in the form of the Polluter Pays Principle (see here).
Haven't any of these people read Coase (I mean read him carefully)? One of his explicit aims in "The Problem of Social Cost" (1960) was to correct an important mistake in Pigou's theory of externalities. Reconceiving externalities (convincingly) as joint- or social-cost problems, Coase argued that the socially efficient level of a Pigovian externality is hardly likely to be zero, as Pigou presumed. In virtually all cases, internalizing the first units of pollution, for example, are likely to be ruinously, if not infinitely, expensive, because of transaction costs (among other costs). The costs would surely exceed any conceivable social benefits. Therefore, those costs should be left where they fall; they should not be redistributed (by taxes or theoretically equivalent quantity-based regulations, see here) to the producer and its market.
This is not to say that Coase believes all costs of Pigovian externalities should always be left where they fall. He did not set out to reverse Pigou's policy recommendations for achieving allocative efficiency in the face of market failure, but to correct it at the margins. According to Coase, externalized costs should be reallocated to the producer up to that point where the costs of internalizing the next unit would exceed the social benefits (a standard MC = MB equation). Thus, an "optimal" Coase tax would imply not the Polluter Pays Principle but the "Polluter Pays for Inefficient Externalities Principle."
Coase's argument is both intuitive and obvious. So, it is all the more surprising that economists have not really taken it on board. They keep modeling optimal Pigou taxes, when they really should be modeling optimal Coase taxes. Admittedly, modeling an optimal Coase tax would be a more difficult enterprise because of the need to estimate transaction costs in order to determine where the marginal costs of redistributing joint or social costs would equal the marginal benefits. But that problem should be surmountable, at least in theory, and when modeling "optimal" taxes (or "optimal" anything) we are always living in a purely theoretical world in any case.
I do hope to be able to develop a model of an optimal Coase tax sometime in the next couple of years, in cooperation with one or another economist friend of mine (whose technical chops are better than my own). It will be interesting to see just how such a model differs from the various models of optimal Pigou taxes. In the meantime, I call on Greg Mankiw to rename his club after Coase, instead of Pigou.