# A Very Simple Neo-Fisherian Model

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Includes: RINF
by: John Cochrane

A sharp colleague recently pushed me to write down a really simple model that can clarify the intuition of how raising interest rates might raise, rather than lower, inflation. Here is an answer.

(This follows the last post on the question, which links to a paper. Warning: This post uses mathjax and has graphs. If you don't see them, come back to the original. I have to hit Shift+Refresh twice to see the math in Safari.)

I'll use the standard intertemporal-substitution relation, that higher real interest rates induce you to postpone consumption:

I'll pair it here with the simplest possible Phillips curve, that inflation is higher when output is higher:

I'll also assume that people know about the interest rate rise ahead of time, so:

Now substitute

So, the solution is:

Inflation is stable. You can solve this backwards to:

Here is a plot of what happens when the Fed raises nominal interest rates, using :

Click to enlarge

When interest rates rise, inflation rises steadily.

Now, intuition. (In economics, intuition describes equations. If you have intuition, but can't quite come up with the equations, you have a hunch, not a result.) During the time of high real interest rates - when the nominal rate has risen, but inflation has not yet caught up - consumption must grow faster.

People consume less ahead of the time of high real interest rates, so they have more savings and earn more interest on those savings. Afterwards, they can consume more. Since more consumption pushes up prices, giving more inflation, inflation must also rise during the period of high consumption growth.

One way to look at this is that consumption and inflation were depressed before the rise, because people knew the rise was going to happen. In that sense, higher interest rates do lower consumption, but rational expectations reverse the arrow of time: higher future interest rates lower consumption and inflation today.

(The case of a surprise rise in interest rates is a bit more subtle. It's possible in that case that jump down unexpectedly at time jumps up. Analyzing that case, like all the other complications, takes a paper not a blog post. The point here was to show a simple model that illustrates the possibility of a neo-Fisherian result, not to argue that the result is general. My skeptical colleague wanted to see how it's even possible.)

I really like that the Phillips curve here is so completely old-fashioned. This is Phillips' Phillips curve, with a permanent inflation-output trade-off. That fact shows squarely where the neo-Fisherian result comes from. The forward-looking intertemporal-substitution IS equation is the central ingredient.

Model 2:

You might object that with this static Phillips curve, there is a permanent inflation-output trade-off. Maybe we're getting the permanent rise in inflation from the permanent rise in output? No, but let's see it. Here's the same model with an accelerationist Phillips curve, with slowly adaptive expectations. Change the Phillips curve to:

or, equivalently:

Substituting out consumption again:

Explicitly:

Simulating this model, with

Click to enlarge

As you can see, we still have a completely positive response. Inflation ends up moving one for one with the rate change. Consumption booms and then slowly reverts to zero. The words are really about the same.

The positive consumption response does not survive with more realistic or better grounded Phillips curves. With the standard forward-looking new Keynesian Phillips curve, inflation looks about the same, but output goes down throughout the episode: you get stagflation.

The absolutely simplest model is, of course, just:

Then, if the Fed raises the nominal interest rate, inflation must follow. But my challenge was to spell out the market forces that push inflation up. I'm less able to tell the corresponding story in very simple terms.