The Trouble With Taylor Rules

Proponents of tight monetary policy like to show variants of the following chart

as evidence that the Federal reserve has kept interest rate too low since the financial crisis and that policy leading up to the financial crisis was too weak. To many this argument is compelling; as inflation is currently near target and output isn't too far from potential, the Federal Funds Rate should be higher, shouldn't it?

No. This analysis is flawed on multiple levels, most importantly its failure to acknowledge the nonneutrality of money. Of course no one who has shown this chart thinks that money is neutral, but they are nevertheless making a massive mistake. In reality, the interest rate implied by the Taylor Rule is affected by the actual level of interest rates. Say, for example, inflation is 2%, the output gap is zero, and the nominal interest rate is 3%. This implies an interest rate of roughly 4% in most specifications of the Taylor Rule, so the Federal Reserve raises interest rates by 1%. But now both output and inflation are lower, so the Taylor Rule suggests a lower interest rate. Thus, had the Fed raised interest rates back in 2011 when the Taylor Rule said it should have, since money is not neutral, inflation and output would have subsequently been lower, implying a lower rate suggested by the Taylor Rule.

Following a Neo-Wicksellian framework, any time inflation and/or output are above target, the interest rate either is currently or is expected to be below the natural rate. Given that both output and inflation are currently close to target, it is clear that the Wicksellian natural rate is about zero, or at least that the expected difference between all future interest rates and all future natural rates is about zero. Now is not the time to raise rates, then, as some Taylor Rules suggest.

Taylor Rules have the obvious problem of not taking the natural rate into account, or at least assuming that it is constant over time. This leads people to have mistaken expectations that interest rates should be at a certain level at full employment, not realizing that the level of interest rates determines whether or not we are at full employment; money is not neutral.

Inflation ≠ Expected Inflation

Neo-Fisherian arguments seem to rest on the idea that expected inflation is somehow related to current inflation - almost to a point of equivalence. A typical argument would be: look at the fisher relation $i_t = r + E_t \pi_{t+1}$. Notice that the nominal interest rate, $i_t$, and the expected inflation rate, $E_t \pi_{t+1}$, are related. Increasing the nominal interest rate must therefore cause the rate of inflation to increase. Before you accuse me of debating a straw man, read this from John Cochrane's recent post:

If you parachute down from Mars and all you remember from economics is the Fisher equation, this looks utterly sensible. Expected inflation = nominal interest rate - real interest rate. So, if you peg the nominal interest rate, inflation shocks will slowly melt away. Most inflation shocks are individual prices that go up or down, and then it takes some time for the overall price level to work itself out.

The problem with this argument is that the current rate of inflation is never modeled; the central bank can choose expected inflation, but there is no reason that the actual rate of inflation must change in response to higher expected inflation. There are a couple of ways around this problem. In the interest of keeping the model as simple as possible, you could assume that the central bank sets the nominal interest rate in response to the current inflation (i.e. a Taylor Rule) or, in the interest of coming of with a more structural model, you could try and come up with a variable that actually does cause current inflation (e.g. the money supply).

The Taylor Rule approach is the way that most economists have gone in the last twenty years or so. Positive deviations of the nominal interest rate from the level implied by the Taylor Rule result in lower rates of inflation. This is itself enough to prove that, as long as a central bank follows a Taylor Rule, a Neo-Fisherian analysis is wrong. There are still some valid contentions that a Neo-Fisherian might make though: a.) central banks set interest rates by discretion, not by adherence to a Taylor Rule b.) Taylor Rules don't actually produce a unique equilibrium value for the initial rate of inflation or the initial price level. In order to deal with contention a, it is clear that a more structural model of inflation is necessary since interest rates clearly do not cause inflation. Contention b is a bit more complicated. In order to make sense of it, it is helpful to look at the coefficient on inflation in the Taylor Rule. If that coefficient is less than one, then any initial rate of inflation will converge to the central bank's inflation target; there are multiple equilibria. Alternatively, the coefficient can be greater than one which will cause the rate of inflation in the future to explode unless the initial rate of inflation is equal to the target rate. The only reason that this calibration works is because economists have chosen to rule out explosive solutions which may make sense for real variables, but does not make sense for nominal variables like inflation.

Contentions a and b leave two options for revision to the conventional approach: come up with a more structural model of inflation or come up with a model that determines a unique equilibrium for "passive" Taylor Rules (i.e. Taylor Rules where the coefficient on inflation is less than one). For some reason, the price determination literature failed to go down the first route and instead chose to come up with 'the fiscal theory of the price level'. Basically, the fiscal authority can threaten to disobey its budget constraint unless the initial inflation rate does not jump to the correct level. I don't really understand how this is all that much better than the trick with active Taylor Rules though. After all, both involve threats to either cause a hyperinflation or not pay off debt at some point that force the initial inflation rate to be on target. 

Because of this, it seems obvious to me that the structural path should be taken. There should be a way to determine a unique equilibrium rate of inflation without requiring that fiscal or monetary policy be intentionally unstable. Of course, a cursory analysis using something like the money supply is easy. Current inflation is caused by current money growth and expected inflation is caused by expected money growth, so interest rates and inflation will go up in response to an increase in the growth rate of the money supply that is expected to be persistent. The debate should end there.

P.S. I don't necessarily mean to say that applies to the current situation; the zero lower bound is special both in theory and in practice.

P.P.S. For the more visually oriented, here's a graph that illustrates the explosive behavior of inflation under a Taylor Rule:

Notes on Taylor Rules, Inflation, and Neo-Fisherism

I've been working on writing a paper outlining my views on three topics in monetary economics (the three things in the title). Click here for the pdf of what I have written so far. Here is the text if you don't want to download the pdf:

UPDATE: The pdf link should update automatically to changes, but I won't change the text in the blog post. Just download a copy of the pdf every time you want the most up to date version.

Introduction:

There has been quite a bit of discussion about the relationship between the nominal interest rate and the rate of inflation recently among economists. To my knowledge, the problem began when Cochrane (2007) challenged the idea that the inflation rate could be determined with simply a Taylor Rule and a Fisher relation in combination with a commitment to active monetary policy and implicitly passive fiscal policy (see Leeper (1991) for example). Cochrane's key insight was that, in these models, the central bank is essentially committing to cause inflation to explode by increasing the nominal interest rate (effectively the expected inflation rate) more than one for one with current inflation. Because economists had ruled out explosive solutions, the only other equilibrium – one in which the inflation rate jumps immediately at period zero to the central bank's target – was considered. As Cochrane noted, there is not necessarily any reason to rule out explosions in nominal variables as they have no impact on the real economy in the models in question.

\par In the years since then, the failure of zero interest rate policies to generate inflation became of interest. Pretty soon, a similar yet entirely different debate came into existence. A few of economists (to my knowledge, Williamson and Cochrane) had the novel idea that the nominal interest rate had a causal relationship with the rate of inflation. This notion had long existed in the literature and is even a property of just about every macroeconomic model; the problem, in fact, was not the notion that high inflation and high interest rates happened at the same time. Rather, it was the idea that central banks could deliberately cause inflation by setting the nominal interest rate at a higher level. The consensus that active monetary policy was required for inflation stabilization and that positive deviations from the target interest rate implied by a Taylor Rule would result in lower inflation was in direct opposition to these "Neo-Fisherian" claims, so a debate that pulled in a slew of other economists ensued.

The difficulty in this case is that both sides are right in their own way. The consensus was correct that, so long as the central bank uses a Taylor Rule to target inflation, positive deviations from that target would result in a lower inflation rate. The Neo-Fisherian view is correct in the sense that if the central bank does not follow a rule and deliberately loosens monetary policy, the inflation rate and the nominal interest rate will increase. The issue with both views is that the underlying assumptions are either not understood or not made clear by there proponents. Economists putting forward the conventional wisdom don't make it clear that the Taylor Rule is the sole cause of inflation dynamics in their model and Neo-Fisherians fail to put forward that the result that they purport is highly dependent on how the money supply (or in some cases fiscal policy) acts when the nominal interest rate is increased. Each model relies heavily on a set of implausible assumptions about the way central banks behave. It is clear that central banks don't behave in the way implied by the consensus models and it is equally clear the the Neo-Fisherian result only occurs when monetary policy has taken a permanently more accommodative stance; even though this assumption is not put forward by its proponents.

If, as I suggest, the "Neo-Fisherian problem" and the "Taylor Rule problem" are all about assumptions, then their respective solutions are simple: just add some microfoundations. When it comes to arguments about monetary policy, the necessary microfoundation is painfully obvious. These models all need money in order for their implications to be understood. Interest elastic money demand functions solve Cochrane (2007)'s critique as they prohibit real explosions of the money supply – something that would happen if the nominal interest rate expanded or collapsed infinitely and money demand functions in general can determine when high interest rates mean tight monetary policy and when high interest rates mean loose monetary policy without appealing to dynamics implied by implicit monetary policy rules and without simply assuming that all high interest rates are do to loose money. 

Model:

We will begin by adding a simple ad-hoc money demand function to a two equation frictionless New Keynesian model and looking into the implications of the simple addition for monetary modeling. As usual, there is a Fisher equation relating the nominal interest rate to expected inflation and a Taylor Rule relating current inflation to the nominal interest rate.

$$i_t = \rho + E_t \pi_{t+1}$$

$$i_t = \rho + \phi \pi_t$$

$i_t$ is the nominal interest rate, $\pi_t$ is the inflation rate, $E_t$ is the period $t$ rational expectations operator, $\phi$ is the "inflation reaction parameter" on the Taylor Rule, and $\rho$ is the constant real interest rate. The sole addition that we will add to this basic model is a simple money demand function which sets real money demand equal to 

$$m_t - p_t = y - \eta i_t$$

where $m_t$ is the nominal money supple, $p_t$ is the log price level ($\pi_t = \Delta p_t$), $y$ is the (constant) level of output, and $\eta$ is the interest-elasticity of the money supply.

\par With the addition of the money demand function, so long as $\left|\eta\right| > 0$, Cochrane's problem with ruling out nominally explosive equilibria disappears. Now, a real variable depends on the nominal interest rate and prevents hyper inflations that are not caused by excessive money growth. In fact, adding money demand changes nothing about the dynamics of the model; following a Taylor Rule still gives the conventional wisdom about monetary policy without having to deal with the difficult problem of ruling out nominal explosions.

\par The interesting thing about this model is that it can replicate the Neo-Fisherian result easily. Consider a deterministic economy where the central bank permanently increases the growth rate of the money supply, $m^g_t = \Delta m_t$, from $m^g_0$ to $m^g_1 > m^g_0$.

If you have any feedback or suggestions before I continue to write, feel free to comment. I intend to continue by expanding my analysis to a more full fledged New Keynesian model with different types of money demand ranging from Money-In-The-Utility-Function to Cash-In-Advance and explain the mixed signal problems of using interest rates as an indicator of the stance of monetary policy. (I also plan to refer to more of the relevant literature than just "Determinacy and Identification with Taylor Rules")

New Keynesian Central Bankers Are Stupid

Imagine that there is a central bank that targets the inflation rate successfully every period because there are no real shocks in the economy. In this model, inflation looks like this:

$$ \pi_t = \pi^* $$

where $ \pi_t $ and $ \pi^* $ are the inflation rate and the inflation target, respectively. The nominal interest rate in this model will always be $ \pi^* $ higher than the constant (no real shocks) real interest rate, $ \rho $ and can be written as 

$$ i_t = \rho + \pi^* $$

Now suppose that the central bank is not omniscient and occasionally misses its target either on accident or because of some unforeseen shock. The inflation rate is now

$$ \pi_t = \pi^*+ \epsilon_t $$

where $ \epsilon_t $ is the central bank's error every period. Since the inflation rate is not serially correlated, the nominal interest rate remains equal to $ \pi^* \: \forall t $. Let's add some real shocks into this economy, so the real interest rate fluctuates over time, adjusts slowly, and is equal to $ r_t $.

$$ r_t = (1 - \rho^r)\rho  + \rho^r r_{t-1} + \nu_t $$

The nominal interest rate now moves around with the real shocks:

$$ i_t = r_t + \pi^* $$

For some unknown reason, the central bank decides to adopt a floating inflation target, $ \bar\pi_t $, and sets it so that it becomes a weighted average of $ \pi_{t - 1} $ and $ \pi^* $.

$$ \bar\pi_t = (1 - \rho^\pi)\pi^* + \rho^\pi \pi_{t-1} $$

The central bank still occasionally misses its target, so $ \pi_t $ is not always equal to $ \bar\pi_t $ and is instead

$$ \pi_t = (1-\rho^\pi)\pi^* + \rho^\pi \pi_{t-1} + \epsilon_t  $$

The nominal interest rate is related to the current inflation rate now because of the auto-regressive process that the rate of inflation fallows and can now be expressed as

$$ i_t = r_t + (1 - \rho^\pi)\pi^* + \rho^\pi \pi_t $$

By sheer assumption, let's say that $ \epsilon_t $ and $ \nu_t $ are negatively correlated. What is this model now? Well, it's New Keynesian, isn't it.

Think about it: There is a central banker that knows it could keep the real interest rate constant (or equal to its natural rate) by pegging the inflation rate, but instead he or she chooses to make it serially correlated by following a Taylor Rule. The real interest rate falls when inflation is above "target" and the nominal interest rate and the rate of inflation are positively correlated. Really, all you need to do to make the dynamics exactly like that of a New Keynesian model is to add a variable $ x_t $, call it the "output gap" and say that $ \dot x_t = r_t - \rho $.

Of course the real aspects of this "model" are really irrelevant (and pretty weak as assumptions go, replacing nominal rigidity with "shocks are negatively correlated" is pretty bad, the AR part of the real interest rate can make sense if the capital stock takes time to adjust, for example). What's really important is that NK central bankers are stupid. They know that they should be targeting a constant rate of inflation, but they abandon that for the sake of Taylor Rules and avoiding the money demand function.

I obviously don't think that central bankers can simply choose to their nominal target be achieved every period, but they should at least refrain from "endogenizing" the money supply in favor of a tool that can mean different things at different times depending on your assumptions.