Private Sector Solutions
One school of thought states that the role of government is to establish the "rules of the game" and to enforce/uphold people's property rights. Ronald Coase of The University of Chicago School of Law has done a lot of work based on the importance that property rights are clearly defined and upheld. Some of his work can be used as a base for potential private sector solutions to "internalizing" an externality.
The Coase Theorem
Under the Coase Theorem, we say that the efficient outcome will be reached when property rights are well defined (and well enforced!) and when the bargaining between parties to address the externality is costless. By costless bargaining we mean that negotiating costs are zero or minimal. For example, bringing a law suit against another party has high bargaining costs.
To demonstrate the Coase Theorem, imagine that you are one of many fishermen on a large lake outside of Marsland's capital of Phobus. The local government has given the property rights of the lake to the fishermen to fish. Now suppose there is also a textile mill on the banks of the lake that dumps its waste into the lake harming some of the catchable fish that can be sold at market. In fact, five units of waste from the textile plant detrimentally impacts $200 worth of catchable fish. Further, assume that it costs the plant $95 to dispose each unit of waste in a different way than dumping in the lake, and that the textile mill could not make a profit if it had to pay the $95 to dump elsewhere. They would be willing to pay up to $90/unit to dump their waste, at which point they would break even. The fishermen receive a constant marginal benefit of $50 from the sale of fish.
Because the property rights have been given to the fisherman, they have the right to demand that the textile mill end their practice of dumping in the lake (this is enforced by Marsland's government). However, by not being allowed to dump in the lake, the mill loses its incentive to be in business because it can no longer operate profitably. But, as the Coase Theorem suggests, there may be room for compromise. To stay in business, the textile mill should be willing and able to pay an amount up to $90/unit to the fishermen to allow them to dump in the lake. The fishermen should be willing and able to accept an amount more than $50/unit to compensate for lost fish. This negotiation will internalize the loss of efficiency caused by the negative externality that was created by the textile's dumping.
Another important element of the Coase Theorem is that it does not matter which party has the property rights for the efficient outcome to be reached. Let's now assume that the textile mill has been assigned the property rights and that the mill's marginal benefit is $35 per unit produced. Just as when the property rights were flipped, there is room for negotiation. In this scenario, the fisherman should be willing and able to pay the textile mill an amount up to $60 to get the mill to reduce its dumping. The mill will be willing and able to accept an amount greater than $35.
All externality problems stem from not having property rights well defined or enforced (or both!)
Is no dumping the efficient outcome? No! Remember that the textile mill is also a part of society; the mill is likely involved in the production of the fisherman's cloths and fishing nets. Remember that the efficient market operates where marginal social benefit is equal to marginal social cost (or as close as possible without mb < mc). In our scenario, we can say that the marginal revenue is the same as marginal benefit.
Consider this table where the textile mill has the property rights;
Based on the cost and benefit structures given in the table above, we find that the socially optimal level of dumping is three units. We know this because this is where the mb is closest to marginal cost without mc > mb.