*raises*inflation, and not the other way around?

Most recently, Steve Williamson plays with this idea towards the end of a recent provocative blog post. Most of Steve's post is about the Phillips curve, but he concludes

If the Fed actually wants to increase the inflation rate over the medium term, the short-term nominal interest rate has to go up.Conventional wisdom says no, of course: raising interest rates lowers inflation in the short run and and only raises inflation in a very long run if at all.

So, here's the policy advice for our friends on the FOMC...If there's any tendency for inflation to change over time, it's in a negative direction, as long as the Fed keeps the interest rate on reserves at 0.25%. Forget about forward guidance...So, as long as the interest rate on reserves stays at 0.25%...you're losing by falling short of the 2% inflation target, which apparently you think is important. And you'll keep losing. So, what you should do is Volcker in reverse.. For good measure, do one short, large QE intervention. Then, either simultaneously or shortly after, increase the policy rate. Under current conditions, the overnight nominal rate does not have to go up much to get 2% inflation over the medium term.

The data don't scream such a negative relation. Both the secular trend and the business cycle pattern show a decent positive association of interest rates with inflation, culminating in our current period of inflation slowly drifting down despite the Fed's $3 trillion dollars worth of QE.

To be sure, I left the grand Volcker stabilization out of the picture here, where a sharp spike in interest rates preceded the sudden end of inflation. And to be sure, there is a standard story to explain negative causation with positive correlation. But there are other stories too -- the US embarked on a joint fiscal-monetary stabilization in 1982, then under the shadow of an implicit inflation target gradually lowered inflation and interest rates. Other countries that adopted explicit inflation targets have similar-looking data. And every time George Washington got sicker, his doctors drained more blood.

So much for data, how about theory? Why do we think that higher interest rates produce lower inflation? We are now, in fact, in a new environment, and old theories may not apply any more.

The first standard story was money. In the past, when the Fed wanted to raise rates, it sold bonds, cutting down on the $50 billion of non-interest-paying reserves. The standard story was, with less "money" in the economy and somewhat sticky prices, nominal interest rates would rise temporarily. The less money would eventually mean less inflation, and then and only then would nominal rates decline. In this view, running the Fed was a tricky job, like driving 68 Volkswagen bus in a crosswind, since the steering was connected to the wheels in the wrong direction in the short run.

However, we are likely to stay with huge excess reserves and interest on reserves. When the Fed wants to raise interest rates now, it will simply pay more on reserves and bingo, interest rates rise. We will remain as awash in interest-paying reserves as before. So this 1960s monetary mechanism just won't apply. Is it possible that in the interest-on-reserves world, raising interest rates translates right away into larger inflation?

More recent economic thinking has (rightly, I think) left the money vs. bonds distinction in the dust. The "Paleo-Keyneisian" (credit to Paul Krugman for inventing this nice word) models in policy circles state that the Fed raises rates, this lowers "demand," and through the Phillips curve, lower demand means less inflation. No money in sight here, but yes a negative effect. The first half of Steve's blog post tearing apart the Phillips curve at least should question one's utter confidence in that mechanism.

Paleo-Keynesian models aren't really economics though. What do new-Keynesian (DSGE) models say? Interestingly, new-Keynesian models can quite easily produce a positive effect of interest rates on inflation. Here are two examples (The models I use here are discussed in more depth in "Determinacy and Identification with Taylor Rules" and "The New-Keynesian Liquidity Trap")

Lets' start with the absolutely simplest New-Keynesian model, a Fisher equation and a Taylor rule,

\[ i_t = E_t \pi_{t+1} \] \[ i_t = \phi_{\pi} \pi_t + v_t \]

The standard solution ( \(\phi_{\pi} \gt 1 \) and choosing the nonexplosive equilibrium) is

\[ \pi_t = -E_t \sum_{j=0}^{\infty} \phi^{-(j+1)} v_{t+j}. \]

So, suppose \(v_t\)=0 for \(t \lt T\) and imagine an unexpected permanent tightening to \(v_t=v \) for \(t \ge T\). Interest rates and inflation are zero (deviations from trend) until T, and then

\[ \pi_t =i_t = -\frac{1}{\phi_{\pi}-1} v \]

Both inflation and the interest rate jump down together. Wait, you say, I thought this was a tightening, why are interest rates going down? It is a tightening -- v is positive. The Fed deviates from its Taylor rule, so interest rates are higher than they would be for this inflation rate. But an observer sees interest rates and inflation move together, both going down. Conversely, if the Fed were to "loosen" by deviating from its Taylor rule in a lower direction, then we would see inflation and interest rates move immediately and positively together. I'm not sure news papers would call this "tighter interest rates!"

A better way to think of this experiment is, what if the Fed adopted a higher inflation target? Rewrite the Taylor rule as

\[ i_t = \phi_{\pi} \left(\pi_t -\pi^*_t \right) \]

You see this is the same, with \(v_t = -\phi_{\pi}\pi^*_t \). So, if the Fed suddenly (and credibly!) raises its inflation target from \(\pi^*_t=0\) to \(\pi^*_t=\pi^* \gt 0 \) at \(t=T\), inflation and interest rates jump from zero to

\[ \pi_t =i_t = \frac{\phi_{\pi}}{\phi_{\pi}-1}\pi^*. \]

The higher inflation target gives instantly higher inflation -- and must come with a sudden

*rise*in the Fed's interest rate target!

Blog readers will know I'm not much of a fan of the standard New-Keynesian equilibrium selection devices. But since this "model" is only an Fisher equation, obviously it's going to be even easier to see a positive connection between interest rates and inflation in other equilibria of this model. For example, take \(\phi_{\pi}=0\) (as we must at the zero bound anyway) and choose the equilibrium that has zero fiscal effects, i.e. no unexpected inflation at time T.

\[ i_t = E_t \pi_{t+1} \] \[ i_t = v_t \] Now, a sudden unexpected rise from \(i_t=0\) to \(i_t=v\) for \(t \ge T\) gives us \(\pi_T\)=0 (no unexpected inflation) but then \(\pi_t=v\) for \(t=T+1,T+2,....\). In words, the Fed raises rates at \(T\), there is a one-period pause and then inflation rises to match the higher interest rate after this one-period pause.

"But what about price-stickiness?" I hear you protesting, and rightly. The whole story about a temporary effect in the wrong direction hinges on price stickiness and Phillips curves. I happen to have a paper and program handy with explicit solutions so let's look. The model is the standard continuous time New-Keynesian model,

\[ \frac{dx_{t}}{dt} =i_{t}-\pi _{t} \] \[ \frac{d\pi _{t}}{dt} =\rho \pi _{t}-\kappa x_{t}. \]

Now, suppose the Fed raises the interest rate from zero to a constant i starting at time T. This is a simple matrix differential equation with solution

\begin{equation*} \left[ \begin{array}{c} \kappa x_{t} \\ \pi _{t} \end{array} \right] =\left[ \begin{array}{c} \rho \\ 1 \end{array} \right] i+\left[ \begin{array}{c} \lambda ^{p} \\ 1 \end{array} \right] e^{\lambda ^{m}\left( t-T\right) }z_{T} \end{equation*}

where

\begin{eqnarray*} \lambda ^{p} &=&\frac{1}{2}\left( \rho +\sqrt{\rho ^{2}+4\kappa }\right) \geq 0 \\ \lambda ^{m} &=&\frac{1}{2}\left( \rho -\sqrt{\rho ^{2}+4\kappa }\right) \leq 0. \end{eqnarray*}

There are multiple solutions, as usual, indexed by \(z_T\), equivalently by what inflation does at time T. The inflation target or Taylor rule selects these, but rather than get in to that, let's just look at the possibilities:

Here I graphed an interest rate rise from 0 to 5% (blue dash) and the possible equilibrium values for inflation (red). (I used \(\kappa=1\, \ \rho=1\) ).

As you can see, it's perfectly possible, despite the price-stickiness of the new-Keynesian Phillips curve, to see the super-neutral result, inflation rises instantly. The equilibrium I liked in "New-Keynesian Liquidity Trap" with no instantaneous response produces a gradual rise in inflation. The only way to get a big decline in inflation is to imagine that by a second "equilibrium selection policy" the Fed insists on a quick jump down in inflation.

Obviously this is not the last word. But, it's interesting how easy it is to get positive inflation out of an interest rate rise in this simple new-Keynesian model with price stickiness.

So, to sum up, the world is different. Lessons learned in the past do not necessarily apply to the interest on ample excess reserves world to which we are (I hope!) headed. The mechanisms that prescribe a negative response of inflation to interest rate increases are a lot more tenuous than you might have thought. Given the downward drift in inflation, it's an idea that's worth playing with.

I don't "believe" it yet (I hate that word -- there are models and evidence, not "beliefs" -- but this is the web, and it's easy for the fire-breathing bloggers of the left to jump on this sort of playfulness and write "my God, that moron Cochrane 'believes' monetary policy signs are wrong" -- so one has to clarify this sort of thing.) We need to explore the question in a much wider variety of models. But it is certainly a fascinating question. What

*is*the connection between interest rates and inflation in the interest-on reserves world? If one wants to raise inflation, is Steve right that

*raising*rates does the trick?

By the way, none of this is an endorsement of the idea that more inflation is a good thing. If interest rates stay low, and we trend to zero inflation or even slight deflation, why wouldn't we just welcome the Friedman rule -- inflation policy has attained perfection, on to other things? Technically, welfare calculations come after understanding policy in these models, and "believing" that all our seemingly endless doldrums can all be fixed with a little monetary magic like taxing reserves is another proposition that needs a lot more support. More likely, if you don't like the long-term economy, go fix "supply" and growth where the problems are. Nobody's Phillips curve gives a big output gap with steady inflation.

History: I last thought about this question here, in response to a John Taylor Op-Ed also suggesting that

*raising*rates might be stimulative. This sign is an old question. The last time it came up was around the stabilization of 1980-1982. A school suggested money was "superneutral." They were wrong, I think, in the short run, at the time. I wrote my thesis showing there is a short run effect of money on interest rates, in the expected direction, which tells you a bit about how long monetary controversies go on. But both interest rates and unemployment did come down much faster than the Paleo-Keynesians of the time thought possible. It's definitely time to rethink it.

There is lots more good stuff in Steve's post. Like causality and Japan:

.. There used to be a worry (maybe still is) of "turning into Japan." I think what people meant when they said that, is that low inflation, or deflation, was a causal factor in Japan's poor average economic performance over the last 20 years. In fact, I think that "turning into Japan" means getting into a state where the central bank sees poor real economic performance as something it can cure with low nominal interest rates. Low nominal interest rates ultimately produce low inflation, and as long as economic stagnation persists (for reasons that have nothing to do with monetary policy), the central bank persists in keeping nominal interest rates low, and inflation continues to be low. Thus, we associate stagnation with low inflation, or deflation.and the value of forward guidance without commitment

You've [Fed] pretty much blown that, by moving from "extended period" language, to calendar dates, to thresholds, and then effectively back to extended periods. That's cheap talk, and everyone sees it that wayAnd the whole Phillips curve thing is good stuff too. But we're here to talk about the possible negative sign.

(Thanks to Frank Diebold for showing me how to get MathJax to work in blogger. )

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