## 14 February 2016

### 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.

1. Interesting John. I'm glad you took a look at this.

I think you've got a typo in this bit:

"then the future price level will a higher than the current one"

"a" should be "be?"

1. I left Stephen a link. You might check out his response.

2. I fixed the error. Thanks. When I first wrote the post, I considered tweeting it to Stephen, but my nerves got the better of me. I'll check out his response now.

3. I replied to Stephen on his post.

4. I saw. Interesting. He responded. I've found that Stephen is pretty good about responding to comments.

2. By "Rt is the gross nominal interest rate"

do you mean

Rt = 1 + nominal interest rate?

Example: If nominal interest rate is 5%, Rt = 1.05.

3. This comment has been removed by the author.

Now that I think I understand what you are doing, I will comment further.

The term Mt is commonly called "money supply" as you labeled it. I usually have a problem with that label when it is related to Pt*Yt as you used it. There is nothing wrong with your work; it just seems to me that Pt*Yt is nothing more than a measurement of economic activity within a period. "True money supply" would be something quite different.

To get to "true money supply", we would need to have a clear consensus upon how money is created. We don't need to worry about what money is as much as we need to decide how it is created. (Of course, this is all just my opinion).

I believe that money is created when banks create a loan. Banks write a loan obligation (signed by the borrower) and then credit the borrower's bank account with an increase in the deposit.

We then can see how much money is available to the economy by adding all the bank deposits. A measurement before the loan will find less money; a measurement after the loan will find more money.

We can bring in the central bank when we begin to talk about reserve requirements. The private banks will have reserve requirements; the central bank will have NO reserve requirement.

I am going to try to follow you blog more carefully. I particularly appreciate the care you take to label all of your math terms. Thanks for the effort.

1. In this case, I was not trying to have an effective description of the money supply at all. I know that CIA models are completely unrealistic, but I use them because they are tractable and allow me to easily explain what causes inflation.

Regarding the creation of money, I think you'd probably be interested in reading about financial frictions in DSGE models to see how mainstream economics deals with banks as financial intermediaries.

I'm glad you've taken an interest in my blog. Thanks for commenting.

5. Your math here gave me inspiration. I thought I could extend the concept into an example that provided a contrast in the inflation response between consumers and suppliers. The result is a blog post found at