Apropos John Cochrane's writings on what most people don't know about New Keynesian models, consider this working paper by Alvarez and Lippi. It's not a typical New Keynesian model, as there's money in the model, and menu costs (of a particular form) rather than Calvo pricing. There's money in the utility function, and the money supply is changed through lump sum transfers by the government.
Alvarez and Lippi look at a traditional (in the monetary economics literature) type of monetary experiment. There is a permanent, level increase in the money stock. The model exhibits the kinds of money nonneutralities we would expect to see in a sticky price model. There is a temporary increase in output, prices initially increase less than in proportion to the money supply increase, and money is neutral in the long run. But there's no liquidity effect in the traditional sense. In the experiment that Alvarez and Lippi study, there is no response of the nominal interest rate to the money injection.
However, note that the experiment they are looking at is about fiscal policy, not monetary policy. Money is handed out by way of lump sum transfers - it's not an asset swap by the central bank. I think the key point is that - contrary to what is typically asserted - it is not innocuous to leave the asset quantities out of a New Keynesian model. This avoids addressing important questions, particularly those pertaining to the relationship between monetary and fiscal policy, that are at the heart of the matter.
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