While I highly recommend reading Landsburg's post itself, the short versions is this. Pigou identified the problem of externalities (i.e., cost of transactions that were not born by the transacting parties) and proposed a straight-forward solution, internalize the cost by imposing a tax equal to the cost of the externality (what is known as a Pigovian tax). Pigou's primary argument wasn't that this was fair (though it certainly seems so) but that, by incorporating the costs into transactions, the market would produce optimum levels of external costs. In terms of pollution, it would mean that we would get neither too little pollution (at the cost of more valuable economic activity) nor too much of it.
Coase proposes a wrinkle to this argument that made many people's head explode. He argued that it wasn't clear who was imposing costs on who. As Landburg sums it up:
If you breathe the pollution from my factory, I’m imposing a cost on you—but at the same time, you’re imposing a cost on me. After all, if you lived somewhere else, you wouldn’t be complaining about the smoke and I wouldn’t be getting punished for it.
While this example might seem a bit bizarre, consider the 19th century problem of railroads running through farms. If farmers plant their crops too close to the tracks, sparks from the old steam engines could (and did) set the fields on fire. Since the externality originates with the railroad, Pigou would argue that they should bear the costs of ant fire damage. But according to Coase, the farmer's choice of planting close to the tracks is as much of a factor as the railroad's choice to run trains down them. Furthermore, if the railroad is held accountable for the cost of fire, farmers have no incentive to avoid planting close to the tracks.
Here is where Coase's ideas become really interesting. Suppose that it would be cheaper for have farmers not plant within 50 feet of the tracks than it would be to have the railroad compensate the farmers for fire losses. This might well be the case since fires destroy entire fields and not planting 50 feet of crops along the track would not reduce output by much. If this was the case, it would be more efficient to impose the cost of the externality on the farmer rather than on the railroad. Conversely, imposing the cost on the railroad would not be the optimum solution as farmers would have no incentive to keep their crops back and more fires would result.
Of course, in a perfect world where parties could costlessly negotiate with one another, it wouldn't matter what the rule was. If railroads were charged with paying damages for every crop fire and it was much cheaper to have farmers not plant near the tracks, the railroad could simply pay farmers to keep their crops back. The railroad would have to pay out less money and the farmers would be compensated for the loss in production. Such bargaining over externalities is known as Coasian Bargaining and the Coase Theorem argues that, if transaction costs for such bargaining are sufficiently low, then it will not matter what rules the government establishes. The parties involved will work out the most efficient solution among themselves.
Unfortunately, the real world is fraught with transaction costs that would make it hard for railroads to do this with large numbers of farmers. Therefore, the liability rules established by the government will matter and they can produce more or less efficient results depending on the specifics of the situation. Sometimes, indeed very often, a Pigovian tax or similar solution is the best way to go, other times not. In the example above, it would be more efficient to have the farmers not plant close to the tracks and the best rule may be that farmers are not allowed to plant with so many feet of the tracks.
As Landsburg points out, the real message Coase has for us is to move away from an obsession with identifying who is at fault and focus more on finding an efficient solution to the problem. Though the theory can not say what the best solution would be in any given situation, it does suggest that having the parties negotiate is a good approach in general. Landsburg notes that Coases's work led to the development of the field of law and economics.
On a tangential note, Abraham Lincoln gained prominence as a lawyer when he successfully argued a case for the Rock Island Railroad before the Supreme Court that dealt with externalities. The railroad had a bridge across the Mississippi that was struck by a riverboat. The boat owner sued for damages, claiming the bridge was a hazard to navigation. This was 1857, railroads were new and this bridge was the first one over the Mississippi. At stake was the right of railroads to build bridges over navigable rivers without being liable for every boat that bumped into them. Lincoln's victory established the liability rule (i..e, that the boat operator's are liable) that allowed railroads to span rivers (not a trivial matter for subsequent US economic development). [And, yes, Honest Abe was a corporate lawyer to the extent that he counted railroads like the Illinois Central among his biggest clients.]
Note: Lansburg also provides a link to Coases's 1960 paper "The Problem of Social Costs."
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