Whither inflation?

(Note: This post uses mathjax to display equations and has several graphs. I've noticed that the blog gets picked up here and there and mangled along the way. If you can't read it or see the graphs, come back to the original .)

The news reports from Jackson Hole are very interesting. Fed officials are grappling with a tough question: what will happen to inflation? Why is there so little inflation now? How will a rate rise affect inflation? How can we trust models of the latter that are so wrong on the former?

Well, why don't we turn to the most utterly standard model for the answers to this question -- the sticky-price intertemporal substitution model. (It's often called "new-Keynesian" but I'm trying to avoid that word since its operation and predictions turn out to be diametrically opposed to anything "Keyneisan," as we'll see.)

Here is the model's answer:

Response of inflation (red) and output (black) to a permanent rise in interest rates (blue). 

The blue line supposes a step function rise in nominal interest rates. The red line plots the response of inflation and the black line plots output.  The solid lines plot the answer to the standard question, what if the Fed suddenly and unexpectedly raises rates? But the Fed is not suddenly and unexpectedly doing anything, so the dashed lines plot answers to the much more relevant question: what if the Fed tells us long in advance that the rate rise is coming?

According to this standard model, the answer is clear: Inflation rises throughout the episode, smoothly joining the higher nominal interest rate. Output declines.

The model: \begin{equation} x_{t} =E_{t}x_{t+1}-\sigma(i_{t}-E_{t}\pi_{t+1}) \label{one} \end{equation} \begin{equation} \pi_{t} =\beta E_{t}\pi_{t+1}+\kappa x_{t} \label{two} \end{equation} where \(x\) denotes the output gap, \(i\) is the nominal interest rate, and \(\pi\) is inflation. The solution  is \begin{equation} \pi_{t+1}=\frac{\kappa\sigma}{\lambda_{1}-\lambda_{2}}E_{t+1}\left[ i_{t}+\sum _{j=1}^{\infty}\lambda_{1}^{-j}i_{t-j}+\sum_{j=1}^{\infty}\lambda_{2} ^{j}E_{t+1}i_{t+j}\right] \label{three} \end{equation} \begin{equation*} x_{t+1}=\frac{\sigma}{\lambda_{1}-\lambda_{2}}E_{t+1}\left[ (1-\beta\lambda_1^{-1}) \sum _{j=0}^{\infty}\lambda_{1}^{-j}i_{t-j}+(1-\beta \lambda_2^{-1}) \sum_{j=1}^{\infty}\lambda_{2}^{j}E_{t+1}i_{t+j}\right] \end{equation*} where \[ \lambda_{1} =\frac{1}{2} \left( 1+\beta+\kappa\sigma +\sqrt{\left( 1+\beta+\kappa\sigma\right)^{2}-4\beta}\right) > 1 \] \[ \lambda_{2} =\frac{1}{2}\left( 1+\beta+\kappa\sigma -\sqrt{\left( 1+\beta+\kappa\sigma\right)^{2}-4\beta}\right) < 1. \] I use \(\beta = 0.97, \ \kappa = 0.2, \ \sigma = 0.3 \) to make the plot. As you see from \((\ref{three}\)), inflation is a two-sided geometrically-weighted moving average of the nominal interest rate, with positive weights. So the basic picture is not sensitive to parameter values.

The expected and unexpected lines are the same once the announcement is made. This standard model embodies exactly zero of the rational expectations idea that unexpected policy moves matter more than expected policy moves. (That's not an endorsement, it's a fact about the model.)

The Neo-Fisherian hypothesis and sticky prices

A bit of context. In some earlier blog posts (start here) I explored the "neo-Fisherian" idea that perhaps raising interest rates raises inflation. The idea is simple. The nominal interest rate is the real rate plus expected inflation, \[ i_t = r_t + E_t \pi_{t+1} \] In the long run, real rates are independent of monetary policy. This "Fisher relation" is a steady state of any model -- higher interest rates correspond to higher inflation.

However, is it a stable steady state, or unstable? If the nominal interest rate is stuck, say, at zero, do tiny bits of inflation spiral away from the Fisher equation? Or do blips in inflation melt away and converge steadily towards the interest rate? I'll call the latter the "long-run" Fisherian view. Even if that is true, perhaps an interest rate rise temporarily lowers inflation, and then inflation catches up in the long run. That's the "short-run" Fisherian question.

One might suspect that the new-Fisherian idea is true for flexible prices, but that sticky prices lead to a failure of either the short-run or long-run neo-Fisherian hypothesis. The graph shows that this supposition is absolutely false. The most utterly standard modern model of sticky prices generates a short-run and long-run neo-Fisherian response. And reduces output along the way.

Multiple equilibria and other issues 

Obviously, it's not that easy. There are about a hundred objections. The most obvious: this model with a fixed interest rate target has multiple equilibria. On the date of the announcement of the policy change, inflation and output can jump.

Inflation response to an interest rate rise: multiple equilibria

The picture shows some of the possibilities when people learn rates will rise three periods ahead of the actual rise. The solid red line is the response I showed above. The dashed red lines show what happens if there is an additional "sunspot" jump in inflation, which can happen in these models.

Math: You can add an arbitrary \(\lambda_{1}^{-t}\delta_\tau \) to the impulse-response function given by (\(\ref{three}\)), where \(\tau\) is the time of the announcement (\(\tau=-3\) in the graph), and it still obeys equations \( ( \ref{one})-(\ref{two})\). These are impulse response functions and sunspots must be unexepected. So the only issue is the jump on announcement. Response functions are thereafter unique.

A huge amount of academic effort is expended on pruning these equilibria (me too), which I won't talk about here. The bottom two lines show that it is possible to get a temporarily lower inflation response out of the model, if you can get a negative "sunspot" to coincide with the policy announcement.

But I think the plot says we're mostly wasting our time on this issue. The alternative equilibria have the biggest effect on inflation when the policy is announced, not when the policy actually happens. But we do not see big changes in inflation when the Fed makes announcements.  The Fed is not at all worried about inflation past that is slowly cooling as the day of the rise approaches, as these equilibria show. It's worried about inflation or deflation future in response to the actual rate rise.

The graph suggests to me that most of the "sensible" equilibria are pretty near the solid line.

The graph also shows that all the multiple equilibria are stable, and thus neo-Fisherian. At best we can have a short-run discussion. In the long run, a rate rise raises inflation in any equilibrium of this model.

Yeah, there's lots more here -- what about Taylor rules, stochastic exits from the zero bound, off-equilibrium threats, QE, better Phillips curves with lagged inflation terms, habits in the IS curve, credit constraints, investment and capital, learning dynamics, fiscal policy, and so on and so on. This is a blog post, so we'll stop here. The paper to follow will deal with some of this.

And the point is made. The basic simplest model makes a sharp and surprising prediction. Maybe that prediction is wrong because one or another epicycle matters. But I don't think much current discussion recognizes that this is the starting point, and you need patches to recover the opposite sign, not the other way around.

Data and models

I started with the observation that it would be nice if the model we use to analyze the rate rise gave a vaguely plausible description of recent reality.

The graph shows the Federal Funds rate (green), the 10 year bond rate (red) and core CPI inflation (blue).

The conventional way of reading this graph is that inflation is unstable, and so needs the Fed to actively adjust rates. Inflation is like a broom held upside down, with inflation on the top and the funds rate on the bottom. When inflation declines a bit, the Fed drives the funds rate down to push inflation back up, just as you would follow a falling broom. When inflation rises a bit, the Fed similarly quickly raises the funds rate.

That view represents the conventional doctrine, that an interest rate peg is unstable, and will lead quickly to either hyperinflation (Milton Friedman's famous 1968 analysis) or to a deflationary "spiral" or "vortex."

And this instability view predicts what will happen should the Fed deliberately raise rates. Raising rates is like deliberately moving the bottom of the broom. The top moves the other way, lowering inflation. When inflation is low enough, the Fed then quickly lowers rates to stop the broom from tipping off.

But in 2008, interest rates hit zero. The broom handle could not move. The conventional view predicted that the broom will topple. Traditional Keynesians warned that a deflationary "spiral" or "vortex" would break out. Traditional monetarists looked at QE, and warned hyperinflation would break out.

(I added the 10 year rate as an indicator of expected inflation, and to emphasize how little effect QE had. $3 trillion dollars of bond purchases later, good luck seeing anything but a steady downward trend in 10 year rates.)

The amazing thing about the last 7 years in the US and Europe -- and 20 in Japan -- is that nothing happened! After the recession ended, inflation continued its gently downward trend.

This is monetary economics Michelson–Morley moment. We set off what were supposed to be atomic bombs -- reserves rose from $50 billion to $3,000 billion, the crucial stabilizer of interest rate movements was stuck, and nothing happened.  

Oh sure, you can try to patch it up. Maybe we discover after the fact that wages are eternally sticky, even for 7 to 20 years while half the population changes jobs, so, sorry, that deflation vortex we predicted can't happen after all. Maybe the Fed is so wise it neatly steered the economy between the Great Deflationary Vortex on one side with just enough of the Hyperinflationary Quantitative Easing on the other to produce quiet. Maybe the great Fiscal Stimulus really did have a multipler of 6 or so (needed to be self-financing, as some claimed) and just offset the Deflationary Vortex.

But when the seas are so quiet, and the tiller has been locked at 0 for seven years, it's awfully hard to take seriously the Captain's stories of great typhoons, vortices, and hyperwhales narrowly avoided by great skill and daring.

Occam's razor says, let us take the facts seriously: An interest peg is stable after all.  The classic theories that predict instability of an interest rate peg -- and consequently that higher rates will lead to lower inflation -- are just wrong, at least in our circumstances (important qualifier follows).

But if those classic theories failed dramatically, what can take their place? Fortunately, I started this post with just one such theory. The utterly standard sticky-price model, sitting in Mike Woodford's and Jordi Gali's textbooks, predicts exactly what happened: inflation is stable under a peg, and thus raising interest rates to a new peg will raise inflation.

The difference between traditional Keynesian or Monetarist models and this modern sticky-price model is deep and essential. In this model, people are forward-looking. In the standard unstable traditional-Keynesian or Monetarist model, people look backward. When written in equations, the traditional "IS" curve (\(\ref{one}\)) does not have \(E_t x_{t+1} \) or \(E_t\pi_{t+1}\) in it, and the "Phillips curve" (\(\ref{two}\)) has past inflation in it,
not expected future inflation.

Forward looking people generates stability, and backward looking people generates instability. If you drove a car by looking in the rear-view mirror, the car may indeed regularly veer off the road, unless the Fed sitting next to you yells about things to come and stabilizes the car. But when people drive looking through the front windshield, cars are quite stable, reverting to the middle of the road when the wind buffets them to one side or the other.

The response function is also consistent with the experience of a few countries such as Sweden which did raise rates and swiftly abandoned the effort. Those rises didn't do much either way to inflation, but they did lower output. Just as the graph says.

What to do? A robust approach

I will not follow the standard economists' approach -- here's my bright new idea, the government should follow my advice tomorrow. Is this right? Maybe. Maybe not. I'm working on it, and hoping by that and this blog post to encourage others to do so as well.

But if you're running the Fed, you don't have the luxury of waiting for research. You have to face an uncomfortable fact, which the news out of Jackson hole says they're facing: They don't really know what will happen or how the economy works. Nor does anyone else. They know that their own forecasts and models have been wrong 7 years in a row -- as has everyone elses', except a few bloggers with remarkably spotty memories -- so pinpoint structural forecasts of what will happen by raising rates made by those same models and logic are darn suspect.

A robust policy decision should integrate over possibilities. So as far as I'll go is that this is a decent possibility, and should add to the caution over raising rates. Raising rates if there is a fire -- actual inflation -- might be sensible. Raising rates because of inflation forecasts from models that have been wrong seven years in a row seems a bit diceyer.

Of course, there is a bit of divergence in goals as well. The Fed wants more inflation, so might take this model as more reason to tighten. And if this model is right, the Fed will produce the inflation which it desires and can then congratulate itself for foreseeing!

I like zero.  Zero rates are pretty darn good. Zero inflation is pretty darn good too. We get the Friedman-optimal quantity of money. And more. Financial stability: With no interest cost, people and businesses hold a lot of money, and don’t conjure complex but fragile cash-management schemes. Three trillion dollars of reserves are three trillion dollars of narrow banking. Taxes: You don’t pay taxes on inflationary gains and taxes erode less of the return on investments.  We don't suffer sticky-price distortions from the economy.  Yeah, growth is too slow, but monetary policy has nothing to do with long-run growth.

So, face it, the outcomes we desire from monetary policy are just about perfect. We don't really know how this happened, but we should savor it while it lasts.

This last point might be the main one. The model I showed above is utterly standard, as is the main result. "New-Keynesian" papers about the "zero bound" have been analyzing this state for nearly 20 years. The result that inflation is stable around the steady state is at least 20 years old.  All the effort, however, has been about how to escape the zero bound. But why? If a very low interest peg is stable, and achieves the optimum quantity of money, why not leave it alone? OK, there's this multiple equilibrium technicality, but that hardly seems reason to go back to "normal."

The only real concern is that some hidden force might be building up to upend this delightful state of affairs. That's behind most calls for raising rates. But clearly, nobody knows with any certainty what that force might be or how to adjust policy levers to head it off.

One warning. In the above model, the interest rate peg is stable only so long as fiscal policy is solvent. Technically, I assume that fiscal surpluses are enough to pay off government debt at whatever inflation or deflation occurs.  Historically, pegs have fallen apart many times, and always when the government did not have the fiscal resources or fiscal desire to support them. The statement "an interest rate peg is stable" needs this huge asterisk.