I Don't understand Market Monetarist Logic

So the typical market monetarist view on business cycles is that low NGDP causes low RGDP. Let $p$ indicate whether or not NGDP is lower than normal and $q$ indicate whether or not RGDP is lower than normal. The market monetarist contention can be represented as such:

$$ p \rightarrow q $$

If a central bank successfully targets inflation, then NGDP should track RGDP (because NGDP growth is always equal to RGDP growth plus the inflation target). This looks like

$$ q \rightarrow p $$

These two statements don't seem to make sense when paired with each other... According to market monetarists, low NGDP caused the great recession, but, because of the inflation targeting regime in the US, low RGDP causes low NGDP... Do market monetarists think that the great recession caused itself? Their logic seems to imply either that recessions come from something like multiple equilibria when central banks target inflation or that they just can't happen because in inflation targeting regimes, NGDP doesn't fall unless RGDP does and RGDP doesn't fall unless NGDP does, so neither ever fall. If they think that inflation targeting produces multiple equilibria, then why don't they say so? If the multiple equilibria logic is correct, then they shouldn't be strictly advocating and NGDP target; they should be telling everyone to switch to any target that doesn't make NGDP depend of RGDP...

This is all extremely confusing. 

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.

 

Good Arguments For Deficits

Just a few quick thoughts on reasons that governments should run deficits. I might go more in depth on each of these later on (I've been planning to write a post on #1 and #3 for a while).

1. Seigniorage Revenue:

The growth of real money demand and the price level over time mean that there is a consistent stream of revenue flowing to the government that show that deficits are optimal for ensuring the stability of government debt in the long run.

2.  Government Spending Smoothing:

Tax revenues from distortionary taxes (i.e. nearly every form of revenue that a government can get) are quite volatile, so government spending should be smoothed so as to not exacerbate business cycles.

3. Deficits Do Not Distort; Taxes Do: 

In circumstances when the monetary authority can't stimulate the economy during a recession, it makes sense to increase government spending. The benefit from this new spending would be reduced since if it were funded with distortionary taxes; deficits do not distort.

Dynamics of Government Debt

I hope I don't make Nick Rowe [1], Scott Sumner, and their fellow Monetarists too angry by assuming that central banks can only monetize government debt, but I think this analysis is still relevant since central banks usually refrain from trading assets other than government bonds.

Anyway, on to the post. Imagine a world in which the fiscal authority never issues any debt. In this world, monetary policy would be equivalent to fiscal policy. Every deficit is funded by seigniorage, so either the central bank gets to target some nominal variable or the fiscal authority gets to set the inflation rate. To see how this works, consider eliminating government bonds (and other assets, should they be present) from the governments budget constraint. This gives

$$ (1) \: M_t + P_t \tau_t = M_{t-1} $$

where $ M_t $ is the money supply, $ P_t $ is the price level, and $ \tau_t $ is the treasury's surplus. Assuming the money demand function simplest money demand function possible, $ M_t = L(P_t) = P_t $ and expressing the constraint in real terms gives

$$ (2) \: \pi_t = -\tau_t $$ ($ \pi_t $ is the rate of inflation)

This world has the unfortunate problem of either being ultra-FTPL (fiscal authority determines the inflation rate) or just plain weird (I don't know what else to call a world where the central bank chooses the fiscal authority's surplus/deficit). Aside from the obvious difficulty of Sargent and Wallace's [2] game of chicken, the problem of a serious conflict of interests arises. What if the optimal fiscal policy is austerity, but the optimal monetary policy involves a high rate of inflation and vice versa? [3] Proposition 1: If that situation can arise, then a non-zero level of government debt is optimal. 

Essentially, government debt allows the monetary and fiscal authority to have contradicting policies at any given point in time. So long as there is government debt, the central bank can always control inflation (I think, but I need to look into the FTPL under an exogenous inflation rate or a money growth rule) and the treasury can always control the surplus.

Let's assume the the level of government debt must be positive [4]. Given this constraint, the central bank can at most monetize 100% of current government debt, essentially imposing a maximum rate of inflation that the central bank can achieve. Proposition 2: The ideal level of government debt is whatever is required for the central bank to achieve its nominal target at any point in time. [5] So, if government debt levels are not sufficiently high (or government debt is not growing quickly enough), then the central bank won't be able to attain its goals.

For complete monetary freedom in my model. government debt needs no upper limit, but having infinitely large government debt is not optimal for obvious fiscal reasons. Ideally, the real value of government debt should not be so high debt servicing costs on the part of the fiscal authority demand constant high primary surpluses. Proposition 3: In order to minimize the burden of high real debt levels, nominal debt should grow at a rate consistent with the central bank's nominal target.

Combining Propositions 2 & 3, we get Proposition 4: The level of government debt should always be high enough for the monetary authority to achieve its nominal target and should grow at the minimum rate required for said nominal target to be achieved.

[1] Nick Rowe, 2015. "Why can't central banks monetise something else?" 

Worthwhile Canadian Initiative.

[2] Thomas J. Sargent & Neil Wallace, 1981. "Some unpleasant monetarist arithmetic,"
Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall.

[3] Of course, the optimality of the policies is pretty unnecessary, the problem still arises if the the central bank and the treasury want to pursue opposite policies. Maybe the fiscal authority is being stupid, and the monetary authority chooses to offset its actions, for example.

[4] This assumption is theoretically weak, but given that I don't know of many governments that are net creditors, I think it is acceptable for my current purposes.

[5] David Andolfatto sort of touched on this in his post "Understanding Lowflation"