Is there really an "efficient equilibrium *price*" with externalities? In the "Intro to Micro" approach to externalities, I've always felt uneasy as a teacher with the idea that there is something like an "efficient equilibrium" with a single "efficient price".

To keep things simple, I will focus on external costs, but everything applies symmetrically to external benefits.

The graphs you often see are similar to those in the Wikipedia article on externalities. These graphs use the term "ideal equilibrium" instead of "efficient equilibrium", but the rest is standard (I include the captions from Wikipedia):


 "Demand curve with external costs; if social costs are not accounted for price is too low to cover all costs and hence quantity produced is unnecessarily high (because the producers of the good and their customers are essentially underpaying the total, real factors of production.)" (https://en.wikipedia.org/wiki/Externality)

I have of course no problem with recognizing and explaining that there is a unique efficient quantity. 

My main issue is with the idea that there is a unique "ideal price" above the actual equilibrium price, and secondarily with the fact that this price is dubbed an "equilibrium" price (however "ideal" or "efficient"). 

At first sight, it seems ok to explain that, in the presence of an external cost, the good is under-priced because the actual equilibrium price only takes private costs into account (as opposed to social cost). 

However, this only makes sense if we are talking about the price for consumers. For producers, a higher price would mean a higher quantity supplied. So in the presence of an external cost, the good is actually overpriced for producers, and underpriced for consumers only. 

This is, in fact, the reason why a Pigouvian tax is a good instrument to solve the external cost problem: Because it increases the price for consumers and decreases the price for producers. Under a Pigouvian tax, the efficient quantity is indeed the equilibrium quantity. However, we now have two equilibrium prices (these are real equilibrium prices this time), one for consumers --- higher than the old equilibrium price --- and one for producers --- lower than the old equilibrium price.

This issue becomes even more salient if, after discussing external costs in the context of the above graph, you attempt to make sense of external benefit in the context of the symmetric graph (https://en.wikipedia.org/wiki/File:Positive_externality.svg) which again suggests that the good is underpriced at the actual equilibrium (whereas it is really underpriced for producers, and overpriced for consumers).

My questions are:


Am I missing something? Is there really a sense in which the price $P_S$ in the above graph is the "correct efficient price" across the board, both for producers and consumers?
Presentations of externalities that rely on graphs like the one above are typically agnostic about the "source" of the externality (whether it is a consumption or production externality) and only care about its sign (i.e., whether it is an external benefit or an external cost). I guess if we assumed that the external costs came from a consumption externality, then $P_S$ would indeed an efficient price (because it would reduce consumption to the efficient quantity, which is all we would care about). But then again, isn't it a stretch to call $P_S$ an "equilibrium" price when, clearly, imposing $P_S$ across the board for both consumers and producers would induce a shortage?
If you were teaching externalities out of a textbook using similar graphs and terminology, how would you try to make sense of all this for your students? Depart from the textbook and explain you don't agree with the idea of an "efficient equilibrium" with a single "efficient price"? Or try to work around the issue without opposing the textbook?
The blue lines are labelled as being costs, not prices. The line labelled "Private Cost" represents quantity supplied for a particular cost to the consumer. The line labelled "Social Cost" represents the quantity supplied for a particular cost to society. While, for the former, the cost is equal to the price received by the supplier, for the latter the cost is equal to the price received by the supplier plus the externality. $P_S$ therefore represents the price paid by the consumer, not the price received by the supplier.

In the case of steel smelting causing air pollution, the free market price represents the equilibrium if steel consumers are required to compensate steel suppliers for the cost that suppliers incur in smelting steel. The ideal price represents the equilibrium if steel consumers are required to compensate steel suppliers for the cost the suppliers incur, and also compensate the general population for the cost that they incur through pollution.