So here are my thoughts on neo-Fisher. I'll write down remarks, then read Williamson, then see how my remarks might change. I think that's the honest way to go.

Remember when Krugman had to repeat "expansionary fiscal policy is expansionary" a thousand times during the austerity debates? I feel like I'm in that boat. "Contractionary monetary policy is contractionary..."

1. In any macro model, the following equation holds: i = r + Ep, where i is the nominal interest rate, r is the real interest rate, and Ep is the expected inflation rate. This equation may be motivated in several ways. I like to think of it as a no-arbitrage condition between real and nominal bonds. Others like to use it as the definition of the ex-ante real interest rate. Notice that this relationship specifies a positive d(Ep)/dr.

2. In many macro models, we write down a Taylor rule: i = d*p + e. In this equation, i and p remain the nominal interest rate and inflation rate; d is a parameter and e is an exogenous shock. So the interest can rise for two reasons: either because inflation rose, or because of an exogenous shift towards tighter monetary policy. Just saying "let's raise the interest rate and see what happens" commits the fallacy of reasoning from a price change. Notice, however, that this equation specifies a positive dp/dr. (Technically, it specifies a positive dr/dp, but neo-Fisherites like to pretend causality goes the other way.)

3. Monetary policy is not just "one damn interest rate after another," but a whole generation of economists thinks it is, and it's probably Woodford's fault (God bless him).

4. In the time-series data generated by a model, like in the time-series data in the real world, inflation and the nominal rate of interest are positively correlated. That does not mean that an exogenous shift towards tighter monetary policy (higher interest rates conditional on inflation) will increase the inflation rate; indeed, contractionary monetary policy is contractionary, i.e. it reduces the inflation rate.

5. Expansionary monetary policy might be so expansionary as to increase expected inflation, which increases current inflation, which in turn might cause the nominal interest rate to rise after all is said and done.

6. A one-time, permanent shift towards more expansionary monetary policy (say, a once-and-for-all doubling of the money growth rate) may well increase the nominal interest rate over the longer term. That does not mean that an exogenous shift towards tighter money (a higher interest rate conditional on inflation) will lead inflation to rise.

7. I am not convinced that Garcia-Schmidt and Woodford's paper, which relies on learning dynamics, is a good counterargument to the neo-Fisherians.

This post is already too long, so I may need to cut it into parts.