You have an infinite horizon model of inflation. Your model tells you what happens to the time-path of the price level P(t) if the central bank changes monetary policy.
Please do something:
Convert your infinite horizon model into a finite horizon model. Suppose the price level at time T, when the world ends, is pinned down at some fixed number P(T). Figure out what happens to the time-path of the price level P(t) if the central bank changes monetary policy, holding P(T) fixed.
Now figure out the results of a change in monetary policy in your finite horizon model, in the limit as T goes to infinity.
Do you get the same results as in your original infinite horizon model? If not, I think there is something seriously wrong with your infinite horizon model.
Take an example:
Suppose your infinite horizon model says that the real interest rate is [exogenous and] constant, the Fisher equation always holds exactly (nominal rate = real rate + expected inflation), rational expectations, the central bank sets a nominal interest rate, and if the central bank unexpectedly raises the nominal interest rate by 1% above the real rate at time t0, and holds it there forever, the price level will start rising at 1% per year. So if the price level is initially 100 before the central bank changes policy, and would have stayed at 100 forever, it instead rises to 101 next year, and so on.
A finite horizon version of your model will give different results. Suppose that P(T) is pinned down at 100, and let T=70 years. If the central bank unexpectedly raises the nominal interest rate by 1% above the real rate, as before, the price level immediately drops to 50, and only then starts rising at 1% per year, so it doubles to 100 over the next 70 years.
If T=140, the price level immediately drops to 25.
If T=210, the price level immediately drops to 12.5.
And so on.
In the limit, as T approaches infinity, the results for the time-path P(t) in the finite horizon model differ by a larger and larger amount from the results in the original infinite horizon model.
That is not good. Your infinite horizon model has a problem.
A standard monetarist model, where the central bank sets the money supply, with a standard Cagan-style money demand function (M/P a negative function of expected inflation), would not have that problem. Proof is left as an exercise for the reader.
[This is my version of what I think Narayana Kocherlakota has recently been saying about the Neo-Fisherian model. My version differs in details, but I think it's the same in spirit.]