What 'Off-the-Shelf' Monetary Models Actually Say about Neo-Fisherism

Stephen Williamson wrote this in a blog post today:

Standard off-the-shelf monetary models essentially all exhibit a neo-Fisherian effect. That's nothing special. The Fisher effect is important. Typically increases in nominal interest rates lead to increases in inflation.

Testing this claim should be pretty simple, all we need is to get some "off-the-shelf" monetary models and see what happens when a central bank switches from one interest rate peg to another. I guess the only difficulty here would be that most standard monetary models exhibit indeterminacy when there's an interest rate peg, but I guess that doesn't seem to phase Williamson.

Let's start with the most basic monetary model I can think of: Cash-In-Advance. I won't bother with much derivation here since I've already talked in detail about CIA models in previous posts, so here is a simple CIA monetary model (assuming that the nominal interest rate is always greater than the interest rate that money pays):
$$(1)\: M_t = P_t y $$
$$(2)\: R_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t} $$
where $M_t$ is the money supply, $P_t$ is the current price level, $y$ is the constant level of output, $R_t$ is the gross nominal interest rate, and $\beta$ is the representative agent's discount rate.

It's clear from this that, by pegging the nominal interest rate, the central bank can peg the ratio of the future price level to the current price level to whatever it wants, but here's where the indeterminacy comes in. The actual current price level is not determined under a pure interest rate peg; only the expected rate of inflation. If the central bank raises the nominal interest rate, then the future price level will be higher than the current one, but what happens to the current price level is unclear.

If, instead of pegging the interest rate, the central bank decides to set the money supply to the desired level each period, the current price level is actually determined, but the relationship between the current price level and the nominal interest rate is still unclear. If, for instance, the central bank temporarily lowers the money supply in the current period then returns it back to it's previous level in the following period, the nominal interest rate will go up, but the current price level will fall. Any permanent changes in the money supply impact the price level without changing the nominal interest rate at all and any temporary increases in the money supply result in a higher price level and a lower nominal interest rate.

Of course, if the nominal interest rate is at the zero lower bound, the basic CIA model predicts that the money supply no longer determines the price level, so the effect of a change in the nominal interest rate on the price level is even more unclear. The only reliable way to exit the zero lower bound (without using fiscal policy) in a CIA model is to shrink the current money supply until the cash in advance constraint once again binds, which at the very least means that the current price level doesn't increase. Effectively, even when escaping the zero lower bound, increases in the nominal interest rate mean a lower current price level.

It's interesting that even this exceedingly simple monetary model fails to fully support the neo-Fisherian hypothesis. Granted, it comes closer than most models: a higher nominal interest rate given the current price level does mean a higher future price level, but this can be easily dealt with if we add some kind of nominal rigidity to the model. Furthermore, CIA models don't exhibit interest elasticity in money demand, which, if present, would mean that permanent increases in the money supply would result in a lower nominal interest rate and a higher price level. Perhaps Stephen spoke too soon.