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"Debasing the coinage was a form of “monetary policy” uniformly condemned by Medieval economists."

Sweet, sweet Peter Olivi wrote on this. Mirabile dictu.

Everything here I agree with and is my account of the reason for the target as well.

Do I read correctly that you are saying a higher target implies a greater quantity of or possibility for positive shocks?

If we are expecting a stagnation, "Fed" would target lower rates? I don't know what to do with "stagflation" in this model.

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Yes, if we conceive of a central bank choosing an optimal rate of inflation based on its view/estimate of the size and frequency of shocks, when that view changed toward expecting larger or more frequent shocks (or toward greater stickiness in the downwardly sticky absolute prices) it would raise its target rate.

"Stagflation" in this model is stickiness in REAL relative prices, possibly through expectations/contractual inflation adjustments. In this situation, a higher target rate would not help.

"Secular stagnation," is a very different animal from "Stagflation" if that is what you had in mind. Secular stagnation, as I understand it is the view that the central bank is unable to achieve its target rate (or perhaps what ought to be its target rate to achieve full employment) with conventional policies like control of ST interest rates and that requires larger fiscal deficits than would be called for from the government acting like a microeconomic income maximizer.

But your question deserves more thought than I have given it. Thanks!

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