And more on whether price setting is microfounded in RBC models. For macroeconomists.
Why do central banks like using the New Keynesian (NK) model? Stephen Williamson says: “I work for one of these institutions, and I have a hard time answering that question, so it's not clear why Simon wants David [Levine] to answer it. Simon posed the question, so I think he should answer it.” The answer is very simple: the model helps these banks do their job of setting an appropriate interest rate. (I suspect because the answer is very simple this is really a setup for another post Stephen wants to write, but as I always find what Stephen writes interesting I have no problem with that.)
What is a NK model? It is a RBC model plus a microfounded model of price setting, and a nominal interest rate set by the central bank. Every NK model has its inner RBC model. You could reasonably say that these NK models were designed to help tell the central bank what interest rate to set. In the simplest case, this involves setting a nominal rate that achieves, or moves towards, the level of real interest rates that is assumed to occur in the inner RBC model: the natural real rate. These models do not tell us how and why the central bank can set the nominal short rate, and those are interesting questions which occasionally might be important. As Stephen points out, NK models tell us very little about money. Most of the time, however, I think interest rate setters can get by without worrying about these how and why questions.
Why not just use the restricted RBC version of the NK model? Because the central bank sets a nominal rate, so it needs an estimate of what expected inflation is. It could get that from surveys, but it also wants to know how expected inflation will change if it changes its nominal rate. I think a central banker might also add that they are supposed to be achieving an inflation target, so having a model that examines the response of inflation to the rest of the economy and nominal interest rate changes seems like an important thing to do.
The reason why I expect people like David Levine to at least acknowledge the question I have just answered is also simple. David Levine claimed that Keynesian economics is nonsense, and had been shown to be nonsense since the New Classical revolution. With views like that, I would at least expect some acknowledgement that central banks appear to think differently. For him, like Stephen, that must be a puzzle. He may not be able to answer that puzzle, but it is good practice to note the puzzles that your worldview throws up.
Stephen also seems to miss my point about the lack of any microfounded model of price setting in the RBC model. The key variable is the real interest rate, and as he points out the difference between perfect competition and monopolistic competition is not critical here. In a monetary economy the real interest rate is set by both price setters in the goods market and the central bank. The RBC model contains neither. To say that the RBC model assumes that agents set the appropriate market clearing prices describes an outcome, but not the mechanism by which it is achieved.
That may be fine - a perfectly acceptable simplification - if when we do think how price setters and the central bank interact, that is the outcome we generally converge towards. NK models suggest that most of the time that is true. This in turn means that the microfoundations of price setting in RBC models applied to a monetary economy rest on NK foundations. The RBC model assumes the real interest rate clears the goods market, and the NK model shows us why in a monetary economy that can happen (and occasionally why it does not).