There were lots of good answers after my previous post. Commenters dlr (first) and then Rajat provided my preferred answer, and there were some other good options as well (Friedman, Lucas, etc.) Here I'll explain the special connection between Coase and Krugman.
In 1960, Coase developed a radically new way of thinking about externalities. At the time, Pigou's interwar theory of externalities was very well established, almost unquestioned. When a person or company does something that imposes external costs on others, there is a market failure. The optimal public policy is a remedial tax, equal to the size of the external cost.
Coase's alternative view was the sort of shocking "bolt from the blue" that almost never occurs in a mature science like economics. There was a famous seminar at the University of Chicago, where almost everyone went in convinced Coase was wrong, and he convinced them all, one by one. Coase's basic insight is that external costs, by themselves, are not market failures. The victim would have an incentive to bribe the entity imposing external costs. That bribe has a similar impact to an optimal tax. Thus before Coase, economists thought there was an economic rationale for government regulation of indoor smoke. After Coase, economists recognized that the owner of the property, not the government, should regulate indoor smoke.
But Coase did not stop there. He also showed that when many people are harmed by externalities, there may be "transactions costs" in privately negotiating an agreement. In that case, Pigou's suggestion that a remedial tax is needed might be correct. But the real problem is not externalities, it's transactions costs.
In 1998, Krugman came up with a radically different way of thinking about liquidity traps. During the interwar period, Keynes had argued that monetary policy may become largely ineffective at zero interest rates, as money and bonds become very close substitutes. He recommended government actions such as fiscal stimulus.
Krugman showed that even at zero interest rates, monetary injections should be effective. That's because the liquidity trap is presumably not expected to last forever (an assumption that Krugman himself later questioned) and thus an increase in the money supply should raise prices once the liquidity trap had ended. Krugman showed that in a rational expectations model, the mere expectation of a higher future price level would tend to raise the expected long-term rate of inflation, and reduce real interest rates on long-term bonds.
Thus monetary policy would continue to be effective at zero interest rates. And if real interest rates did not decline, then nominal rates would rise, which would end the liquidity trap---also making monetary policy effective. No need for activist governments engaging in fiscal stimulus
But Krugman didn't stop there. He noted that (conservative) central banks might not be able to convince the public that currency injections are permanent. In that case, future expected inflation would not rise, and the monetary injections would be ineffective. Krugman argued that the real problem was not that cash and bonds are perfect substitutes at zero rates, producing a "liquidity trap", but rather that central banks might not be able to convince the public that they will allow higher inflation in the future, creating what Krugman called an "expectations trap."
If there is an expectations trap, then the original Keynesian policy of fiscal stimulus might make sense, but not for the reason assumed by Keynes. This is similar to Coase's argument that corrective taxes might be called for, but not for the reason originally assumed by Pigou. But Krugman also indicated that something like a higher inflation target ("promising to be irresponsible") was a first best policy, and indeed Krugman recently cited Abe's decision to raise Japan's inflation target to 2% as an example of what he had in mind (although Krugman would probably prefer an even more aggressive target.)
I'll add my own wrinkle here. In a 1999 Economic Inquiry piece I argued that the constraints of the gold standard were always lurking in the background of Keynes's thinking. When I first wrote that paper, I was not aware of Krugman's 1998 paper. But in retrospect, I was claiming that the gold standard created a sort of expectations trap, which prevented central banks from raising the expected rate of inflation.
My previous post was not just intended to be a diverting puzzle. I believe that seeing these sorts of underlying similarities allows us to better understand each theory separately. Indeed being a good economist is largely a matter of seeing common underlying factors behind a lot of seemingly disparate phenomena. This sort of intuition is what puts Coase and Krugman ahead of most economists. Coase had written a paper back in 1937, pointing out how "transactions costs" help to explain why firms are big. If transactions costs did not exist, the principle/agent problem would cause films to contract out almost every single specific task they do to other smaller and more specialized firms--even to individuals. Big firms would be almost completely hollowed out.
Then, 23 years later, Coase recognized that these same transactions costs explain why the private sector may have trouble negotiating solutions to complex externality problems.
I can't disagree with people who pointed to Friedman and Phelp's insight that it's not high inflation that matters, but rather higher than expected inflation. That was a really important breakthrough. But the way Coase and Krugman reframed the longstanding dogma on externalities and liquidity traps seems like more of a radical intellectual shift--not just adding one derivative. And in both cases the original policy suggestion became a sort of special case, a policy that is called for when other options are not available.