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It's an understatement to say his theories led to carbon trading. He wrote two of the most important articles in modern economics, ones that are more directly relevant to public policy than nearly anything else that came out in the last century. His ideas have been tremendously influential not just in economics, but in law and political science.

The Nature of the Firm: http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0335.1937.... (1937) (introduced the theory of transaction costs).

The Federal Communications Commission: http://old.ccer.edu.cn/download/7874-1.pdf (1959) (introduced an economic theory for allocation scarce resources using market mechanisms).

The latter paper looks at the basic models of allocating spectrum (at that time a classical rival resource) and demonstrates how market trading of spectrum rights would lead to optimal allocations, assuming no transaction costs. The former paper looks into transaction costs in the context of asking why firms bother to hire permanent employees instead of contracting everything out, which would in theory lead to more efficient allocations of labor.

Anyone interested in public policy should definitely read these. Unusually among economists, Coase had a gift for clear prose and preferred to build arguments from basic principles, rather than relying on jargon or trying to bludgeon his readers with mathematics (something he contemptuously referred to as 'blackboard economics'). The patience and rigor of his arguments won him great popularity among lawyers, and his theories are central to the 'law and economics' movement.

I am sad to learn of his passing but it was a life well lived.

The article does a poor job of describing the Coase Theorem. This New Yorker article [1] does a much better job. I like the example they give, which comes from Coase's paper:

    "To illustrate this, Coase used the example of a cattle
    farmer’s herd straying onto the cultivated fields of a
    neighboring grain farmer.

    If the law said the cattle farmer was legally responsible, 
    then mostly likely he would end up paying for the cost of a new fence. 
    If putting up a fence was prohibitively expensive, he might end up paying the 
    grain farmer for the occasional damage to his crops. But what if the cattle 
    farmer wasn’t legally responsible, perhaps because the property lines were 
    unclear? In these circumstances, Coase said, the grain farmer would likely end 
    up paying for a fence. Or, if the grain farmer believed that constructing a 
    fence would be more costly than the occasional damage done by straying 
    cattle, he would reluctantly accept the odd incursion.

    The assignment of property rights would determine who pays. But from a 
    broader economic point of view, Coase argued, the outcome would 
    essentially be the same, regardless."
This doesn't hold up in real life with negative externalities and transaction costs. The article cites an example of pollution, but pollution can affect thousands of people, and there's no way to get all of them together to reach an efficient outcome without transaction costs.

[1] http://www.newyorker.com/online/blogs/johncassidy/2013/09/ro...

The Volokh Conspiracy had some good posts today about how the Coase theorem doesn't ignore externalities and transaction costs: it demonstrates that externalities are a distraction (since they aren't, on their own, enough to explain why a laissez faire isn't perfectly efficient). Transaction costs combined with some arbitrary set of property rights are sufficient on their own to entirely capture the problems people use "externalities" to explain, but much more amenable to reasoning about and fixing.

To come back to the pollution example, it doesn't matter if the factory has the right to pollute or the people have the right to clean air: what matters is that transaction costs prevent them from working out their competing interests without intervention.

http://www.volokh.com/2013/09/02/coase-externalities/

http://www.volokh.com/2013/09/03/coase/

'Transaction costs combined with some arbitrary set of property rights'

Surely this means assuming the externality is covered by property rights. In which case it's no longer really an externality. In practical terms assuming adequate property rights is assuming half the problem away.

I'm not sure I know what you mean by an externality being "covered" by property rights. An externality arises when the social cost or benefit is different from the private cost of an action. I assume you are talking about having the actor internalize the costs of the externality by expanding the scope of his property rights. However, I think the parent post was just referring to a result of the Coase theorem. Namely that, no matter the initial allocation of property rights, externalities can be traded away, as long as there are no transaction costs. In practice, negotiating away externalities is questionable for other reasons too, like contract credibility. E.g, if I were to pay someone in front of me on a plane not to recline their seat, they might just pocket the cash and recline anyway.
"Namely that, no matter the initial allocation of property rights"

Under the assumption that property rights can and have been allocated and can be and are enforced.

This is more a point about how the Coase theorem gets interpreted rather than about a flaw in it.

Better summary: pay extortionists, but only if transaction costs are low!
Coase's book The Nature of the Firm, which I read for some reason for pleasure while I was in grad school for a completely unrelated subject, stands out to me as one of the most elegant, lucidly-thought and well-presented books I've ever encountered. I'm not particularly a fan of the market-worshiping policy aims in service of which Coase's work was (ab?)used, but he's well worth reading as an exemplar for anyone whose job is to turn mental constructs into plain text.
If he was also responsible for sulfur emissions cap and trade, that would be a more substantive achievement. Carbon trading hasn't been similarly successful at having an environmental impact, AFAIK.
The headline is horrible—it's kind of like saying "Scientist's Theories Led to Atomic Bomb" for an Einstein obit—but Ronald Coase was one of the most creative economists of the 20th century and deserves to be widely known. For a great introduction to Coase's Theorem and its implications, I recommend Law's Order by David Friedman (available in a nearly illegible online edition at http://www.daviddfriedman.com/laws_order/, but I recommend the print version). It will challenge many of the things you think you know about externalities.