Assume prices are perfectly flexible. Assume full information rational expectations. Assume money is neutral, super-neutral, and super-duper-neutral, etc. The Fisher equation holds exactly. And, the central bank sets the nominal rate of interest. And the economy is always in equilibrium.

The model predicts that if the central bank increases the nominal interest rate by 1%, the rate of inflation will instantly increase by 1% too. That follows directly from the Fisher equation, since the real rate will be independent of monetary policy.

That's the model that I have been discussing with George Selgin and amv on Scott Sumner's blog. (amv has a post here).

What's wrong with that model?

It is very, very, very, "fragile". It is "fragile in the limit" (I need to find a better term). We should pay no attention to such models.

All models are false. No model has assumptions that are exactly true. But we don't lose sleep over that fact. It doesn't stop some models from being useful.

Some models are "robust", and others are "fragile". If a large change in the model's assumptions causes only a small change in the model's predictions, a model is "robust". If a small change in the model's assumptions causes a large change in the model's predictions, a model is "fragile". I find these two concepts useful in talking about models, even though I don't know exactly how best to define "large" and "small" in this context.

Other things equal, a robust model is better than a fragile model. It doesn't matter "much" if the assumptions are a "bit" wrong.

But there's a special type of fragility that I think is a death sentence for any model. A model that is "fragile in the limit" is, I think, a totally useless model. Let me explain what I mean by that.

The standard model of a perfectly competitive firm is a model that is not fragile in the limit. In the limit, as the elasticity of demand approaches infinity, so the assumptions of monopolistic competition approach the assumptions of perfect competition, the predictions of monopolistic competition also approach the predictions of perfect competition. Price approaches marginal cost, for example.

Suppose that weren't true. Suppose that the predictions in the limit (as the assumptions approached the model's assumptions) were totally different from the predictions at the limit. That would be a model that is fragile in the limit. And that, to my mind, would be a totally useless model. An infinitesimal change in the assumptions would be enough to change the predictions by a discrete amount. We could never trust such a model.

The model I introduced at the beginning is "fragile in the limit".

To see this, suppose we changed the model to suppose that some fraction f of the prices adjust slowly. And that the people adjusting those prices have non-rational expectations. They expect next period's inflation will be the same as last period's inflation. Suppose initially that f=1, so everybody adjusts prices slowly and has non-rational expectations. If the central bank raises the nominal interest rate by 1% the expected real rate will initially rise by the same 1%, so inflation will fall. And then fall some more the next period, and so on forever.

As we reduce f below 1, so there are some rational agents, who adjust prices instantly, they will anticipate falling inflation, so the average expected real interest rate will rise even more, and inflation will fall even more quickly. In the limit, as f approaches 0, a 1% rise in the nominal interest rate will cause instant explosive deflation.

How that model behaves in the limit, as f approaches 0, is diametrically opposed to how the model behaves at the limit, when f=0.

Let's take a slightly different model. It's exactly like the last model, except that the central bank sets the money supply, instead of the nominal interest rate. It's the "monetarist" version, as opposed to the "Neo-Wicksellian" version.

If we add a standard money demand function to the model, the equilibrium where f=0 will be exactly the same in the two versions of the model. An outside observer, who didn't know whether the central bank was setting the money supply or the nominal interest rate, wouldn't be able to tell the difference. The outside observer cannot tell whether it's the money supply or the nominal interest rate that's exogenous.

But there's a very big difference between the two models.

The monetarist model is robust in the limit. It won't be exactly right, if f is greater than 0, so some prices are sticky, and some people have non-rational expectations. But it will be approximately right. In the limit, as f approaches 0, the predictions of the monetarist version of the model will be identical to the predictions at the limit, when f=0.

In the olden days, we would have said that the Neo-Wicksellian version of the model has an "unstable" equilibrium. But modern monetary theorists don't like to talk about "stability" in that sense. You can't ask whether the economy would tend to return to equilibrium if it were ever away from equilibrium, because it never is away from equilibrium, so the question doesn't make sense.

Another way to look at it is in terms of adaptive learning. If people weren't born knowing the structure of the model, but had to learn it, would the learning process converge to the rational expectations equilibrium? But that sounds too much like adaptive expectations. Which it is.

So, if you don't like talking about stability of equilibrium, and you don't like talking about adaptive learning, maybe we could talk about fragility?