## 30 August 2015

### 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")