Don't Be Fooled By Annual Inflation

If you look at Fred right now, you'll see that the CPI is currently 1.3% higher than it was a year ago. This figure can mislead people into think that inflation is a lot closer to target than it really is. To understand this, you need to look at the actual graph for the CPI over the last 12 months or so.

As you can see, the current CPI is really barely any higher than it was last month and is a long way away from the level of CPI inflation consistent with the Fed's 2% annual PCE inflation target. In fact, the CPI is just under 0.03% higher than it was a month ago.

If CPI were to grow at 2% each year, this would require 0.16% inflation each month, which is a whole lot higher than the current 0.03%. Basically, even though the current price level is 1.3% higher than it was a year ago, the price level has not been growing even that fast for quite a while. If you want to see whether or not inflation will be on target, you should look at the compounded annual rate of change, and not percentage change from a year ago.

Update:

The new PCE numbers are in and, guess what, inflation is indeed chronically below target; at least the y/y rate is closer to the compounded annual rate of change, though.

What 'Off-the-Shelf' Monetary Models Actually Say about Neo-Fisherism

Stephen Williamson wrote this in a blog post today:

Standard off-the-shelf monetary models essentially all exhibit a neo-Fisherian effect. That's nothing special. The Fisher effect is important. Typically increases in nominal interest rates lead to increases in inflation.

Testing this claim should be pretty simple, all we need is to get some "off-the-shelf" monetary models and see what happens when a central bank switches from one interest rate peg to another. I guess the only difficulty here would be that most standard monetary models exhibit indeterminacy when there's an interest rate peg, but I guess that doesn't seem to phase Williamson.

Let's start with the most basic monetary model I can think of: Cash-In-Advance. I won't bother with much derivation here since I've already talked in detail about CIA models in previous posts, so here is a simple CIA monetary model (assuming that the nominal interest rate is always greater than the interest rate that money pays):
$$(1)\: M_t = P_t y $$
$$(2)\: R_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t} $$
where $M_t$ is the money supply, $P_t$ is the current price level, $y$ is the constant level of output, $R_t$ is the gross nominal interest rate, and $\beta$ is the representative agent's discount rate.

It's clear from this that, by pegging the nominal interest rate, the central bank can peg the ratio of the future price level to the current price level to whatever it wants, but here's where the indeterminacy comes in. The actual current price level is not determined under a pure interest rate peg; only the expected rate of inflation. If the central bank raises the nominal interest rate, then the future price level will be higher than the current one, but what happens to the current price level is unclear.

If, instead of pegging the interest rate, the central bank decides to set the money supply to the desired level each period, the current price level is actually determined, but the relationship between the current price level and the nominal interest rate is still unclear. If, for instance, the central bank temporarily lowers the money supply in the current period then returns it back to it's previous level in the following period, the nominal interest rate will go up, but the current price level will fall. Any permanent changes in the money supply impact the price level without changing the nominal interest rate at all and any temporary increases in the money supply result in a higher price level and a lower nominal interest rate.

Of course, if the nominal interest rate is at the zero lower bound, the basic CIA model predicts that the money supply no longer determines the price level, so the effect of a change in the nominal interest rate on the price level is even more unclear. The only reliable way to exit the zero lower bound (without using fiscal policy) in a CIA model is to shrink the current money supply until the cash in advance constraint once again binds, which at the very least means that the current price level doesn't increase. Effectively, even when escaping the zero lower bound, increases in the nominal interest rate mean a lower current price level.

It's interesting that even this exceedingly simple monetary model fails to fully support the neo-Fisherian hypothesis. Granted, it comes closer than most models: a higher nominal interest rate given the current price level does mean a higher future price level, but this can be easily dealt with if we add some kind of nominal rigidity to the model. Furthermore, CIA models don't exhibit interest elasticity in money demand, which, if present, would mean that permanent increases in the money supply would result in a lower nominal interest rate and a higher price level. Perhaps Stephen spoke too soon.

Demystifying Neo Fisherism

Misunderstanding of monetary economics abounds in the econoblogosphere. Since I'd like to think I know a decent bit about this issue, I think I might try and clarify some things with a pretty simple model.

There exists a household with the utility function $U = E_0 \sum^\infty_{t=0} \beta^t \left(u(c^1_t) + u(c^2_t)\right)$ where $0 < \beta < 1$ is the household's discount factor, $E_t$ is the rational expectations operator given information known in period $t$, $c^1_t$ is a consumption good that can be purchased using cash only, and $c^2_t$ is a good that can be purchased using cash or credit. The household uses government bonds and money carried from the last period as well as a constant endowment to purchase government bonds, money, and both consumption goods and to pay a lump sum tax levied by the government. The household's budget constraint is
$$ (1.1)\: M_{t-1} + B_{t-1} + P_t y = M_t + Q_t B_t + P_t \tau_t + P_t (c^1_t + c^2_t)$$
where $M_t$ is the money supply that will be carried into the next period, $B_t$ is the stock of government bonds that will be carried into the next period, $Q_t$ is the price of government bonds maturing in period $t+1$, $P_t$ is the price of both consumption goods, $y$ is the endowment, and $\tau_t$ is the real lump sum tax. $c^1_t$ must be paid for in cash, so the household faces a cash in advance constraint where it must hold at least enough money to cover $P_t c^1_t$.
$$ (1.2)\: M_t \geq P_t c^1_t $$

I assume that the government sets $B_t = 0\: \forall t$, so the government's budget constraint, given zero government bonds, is
$$ (1.3)\: M_t + P_t \tau_t = M_{t-1} $$
The government sets the lump sum tax so that $M_t = \mu_t M_{t-1}$ where $\mu_t$ is an exogenous policy parameter set by the central bank.

 The household maximizes $U$ subject to $1.1$ and $1.2$ which gives the following maximization problem
$$ (2)\: \mathcal{L} = U + \lambda_t \left(M_{t-1} + B_{t-1} + P_t y - M_t - Q_t B_t - P_t \tau_t - P_t (c^1_t + c^2_t)\right) + \gamma_t \left(M_t - P_t c^1_t\right)$$
which  yields
$$(2.1)\:\frac{\partial \mathcal{L}}{\partial c^1_t} = \beta^t u'(c^1_t) - \lambda_t P_t - \gamma_t P_t = 0$$
$$ (2.2)\: \frac{\partial \mathcal{L}}{\partial c^2_t}= \beta^t u'(c^2_t) - \lambda_t P_t = 0 $$
$$ (2.3)\: \frac{\partial \mathcal{L}}{\partial B_t} = -\lambda_t Q_t + E_t \lambda_{t+1} = 0 $$
$$(2.4)\:\frac{\partial \mathcal{L}}{\partial M_t}=-\lambda_t + E_t \lambda_{t+1} +\gamma_t=0$$

$2.1-4$ and $1.3$ can be combined to form an equilibrium for $P_t$, $c^1_t$, $c^2_t$, $M_t$, and $Q_t$:
$$ (3.1)\: u'(c^1_t) = u'(c^2_t) (2  - Q_t) $$
$$ (3.2)\: M_t = P_t c^1_t $$
$$ (3.3)\: y = c^1_t + c^2_t $$
$$ (3.4)\: u'(c^2_t) = \beta u'(c^2_t) \frac{1}{Q_t}E_t\frac{P_t}{P_{t+1}} $$
$$ (3.5)\: M_t = \mu_t M_{t-1} $$

With the equilibrium, it is possible to get a bit of an answer to the questions that Neo-Fisherians raise. Firstly, the long run inflation rate is equal to the growth rate of the money supply and the euler equation shows that, in the long run, the inflation rate is a constant different from the nominal interest rate. This means that, were the central bank to choose a low path for $\mu_t$, both inflation and the nominal interest rate would be lower. Of course, that's completely standard, it's just a lot more sensible to have a model where it's clear that this is a long run tightening of monetary policy. (Point Neo Fisherians)

This means that a disinflation, i.e. a reduction in the path of $\mu_t$, is consistent with a low nominal interest rate in the long run. In the short run, a higher value of $\mu_t$ can either take the form of higher inflation or lower interest rates. This is because a higher value of $Q_t$ (the inverse of the nominal interest rate) induces the household to shift demand from the credit good to the cash good because the nominal interest rate represents a cost to holding cash (and therefore buying the cash good) which can almost be considered a "shadow price" for the cash good. When the "shadow price" falls, as happens when the nominal interest rate falls, $c^1_t$ goes up which, given equation $3.2$, puts downward pressure on the price level. Because of this effect, increases in $\mu_t$ in the short run result in lower interest rates. The effect is exacerbated if the money supply is assumed to be auto-regressive. (Point everyone else)

The real problem with Neo Fisherism, as John Taylor points out in the post that Cochrane links to, is that the money supply is not modeled. High interest rates mean that the future price level is high relative to the current price level, but does that mean that the current price level has fallen to produce this, or that the future price level has increased? Adding the money supply solves this entirely. Interest rates can be high because the future money supply has been raised relative to today or because the current money supply has been reduced; only now the central bank has complete control over it.

The addition of the cash and credit goods to the basic cash in advance framework helps to illustrate that some (pseudo) non-neutrality of money can cause low interest rates and high expected inflation to coincide, something that doesn't happen in New Keynesian models unless the Taylor Rule has extremely persistent shocks. Also key here is that the interest rates are indicative of expected inflation, not current inflation and any apparent relationship with current inflation is either coincidence -- because the money supply auto-regresses, e.g. -- or a result of temporary money non-neutrality.

Also, the idea that forcing interest rate to be low actually causes high inflation is completely wrong; it's all about the money supply, and high inflation only happens if the money supply is growing quickly. Deliberately setting a low nominal interest rate must eventually result in low money growth (unless you are in a liquidity trap. See here), so it's pointless to suggest such a policy in the hopes of deliberately causing higher inflation. The endgame is to stop thinking about monetary policy in terms of interest rates at all and switch to thinking about movements in the money supply.

Monetary Policy Effectiveness In Liquidity Traps

As I've argued here, conventional money demand models suggest that the price level becomes indeterminate at the zero lower bound and monetary expansion can not do anything to change inflation. In a recent conversation with Scott Sumner, Scott pointed to Paul Krugman's 1998 paper about this issue. Krugman suggests in his paper that only current monetary expansions are useless, but commitments to larger money supplies in the future (or, as Scott would probably like me to say, commitments that the current monetary expansion will be permanent) can both alleviate the liquidity trap and raise the current price level.

So, in line with Krugman's model, let's assume that there is a representative household that maximizes the utility function

$$(1)\: U = \sum^\infty_{t=0}\beta^t\left(u(c_t)\right) $$

where $\beta$ is the household's discount factor and $u(c_t)$ is the utility that the household gains from its consumption, $c_t$, in period $t$. The household is endowed without output $y$ every period and participates in an asset market where it trades one period government bonds and government money. The household's budget constraint is

$$(2)\: M_{t-1} + (1 + i_{t-1}) B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t $$

where $M_t$ is the money supply, $B_t$ is the household's holding of government bonds, $i_t$ is the nominal interest rate that government bonds pay, $P_t$ is the price level, and $T_t$ is the lump sum tax from the government. The household also faces a cash-in-advance constraint; it must finance its consumption with government cash. This constraint takes the form

$$(3)\: M_t \geq P_t c_t $$

Notice the fact that this is an inequality constraint. The household can hold as much money as it wants, but must at minimum have enough cash on hand to pay for its consumption. The household maximizes $1$ subject to $2$ and $3$ which yields the following first order conditions:

$$(4a)\: M_t = P_t y\: \mbox{if}\: i_t > 0$$ 

$$(4b)\: M_t \geq P_t y\: \mbox{if}\: i_t = 0$$

$$(5)\: 1 + i_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t}$$

If, like Krugman did, we assume hat next period's price level is constant, we can draw a nice diagram with $4a$, $4b$, and $5$:

The solid blue line is the curve from $5$, the dotted blue line marks the zero lower bound, and the black lines represent the money supply. Normally, the central bank is in complete control of the price level and can move it around by moving the money supply around. But, because the cash-in-advance constraint does not bind at the zero lower bound, increases in the money supply at the zero lower bound will not be immediately spent by the household. This means that, given a constant future price level, the central bank can only push the price level up until it hits the zero lower bound. After that, no amount of current monetary expansion can increase the current price level.

Of course, all that was exactly in line with Krugman. Here's where it gets interesting, though. Krugman assumes in his paper that the central bank has control of the future price level the entire time and can easily increase the future money supply to end the liquidity trap. If we drop the assumption that the cash-in-advance constraint must bind in the next period, can the monetary expansion, regardless of permanence be effective? In order to escape the liquidity trap, the central bank needs to make the household expect that the price level next period will be higher than the price level this period (this would shift the solid blue curve in the graph to the right). 

I'm having a lot of trouble wrapping my head around it, but I think that everything hinges on expectations. The cash-in-advance constraint will only bind in the next period if the price level two periods ahead is expected to be higher than the price level next period and so on, ad infinitum. This means that the central bank can only exit the liquidity trap if the household subjectively expects inflation to be greater than the rate of time preference (the inverse of the discount factor subtracted by one) in the future. This is independent of the path of the money supply; not only is there no equilibrium for the price level in the static analysis at the zero lower bound, there is no equilibrium for the entire path of the price level once the zero lower bound has been reached.

The alternative is to reduce the current money supply until the cash-in-advance constraint binds once again; basically to cause a bunch of deflation now instead of in the future. The problem with this is that prices are sticky and a massive monetary contraction would cause a recession.

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:

The Trouble With The Zero Lower Bound

Most of the time, it seems that the monetarist view of inflation is pretty much correct. Inflation roughly tracks the monetary base and velocity is pretty stable and almost directly follows short term interest rates. Unfortunately, there is this thing called the zero lower bound that seems to throw monetarism off.

The US has been at the zero lower bound twice in the last 150 years, and both times monetary expansion has seemed to have an irrelevant - even a negative - impact on inflation.

Here's 1934-1945:

And here's 2009-2015:

Most monetarists seem to have trouble coping with the irrelevance of the monetary base at the zero lower bound, even though it does seem to be part of a lot of basic monetary models. Take the most simple of money demand functions - cash-in-advance. It is easy to figure out that as long as there is a cost to the household incurred by holding money, the cash-in-advance constraint will bind, but whenever there isn't a cost, the constraint ceases to bind. This effectively means that, rather than being stuck at unity, the velocity of money is indeterminate; increases in the money supply will no longer have any effect on the price level.

Money-in-the-utility-function models have similar properties in the sense that velocity also becomes indeterminate. MIUF models are slightly strange though because money demand itself actually goes to infinity when the zero lower bound binds. But, MIUF is a pretty bad assumption anyway, so it's fine to ignore this.

An easy modification to CIA models that, when calibrated properly, might be able to make them match the data pretty well is the addition of a non-cash good to the economy. The income-velocity of money will now fluctuate with the nominal interest rate while the effects above will still be present.

I digress, the key idea of this kind of rambling post is that the zero lower bound seems to do strange things to monetary policy which precludes central banks from being omnipotent as some would suggest...

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")

Some Fun With Money Demand

I was doing some thinking about augmenting my liquidity trap post with an interest elastic money demand function of the form:

$$ (1) \: \log M_t + i_t = \log P_t $$

Solving for inflation in this model results in

$$ (2) \: \pi_t = \Delta \log M_t + \Delta E_t \pi_{t+1} $$

(The nominal interest rate term changes to expected inflation because the real interest rate is assumed to be constant). So far, everything looks unremarkable, but solving forward yields interesting results.

$$ (3) \: \pi_t = \Delta \log M_t - E_t \sum^{\infty}_{j=0} \Delta \log M_{t + 1 + j} - E_{t-1} \sum^{\infty}_{j=0} \pi_{t + j} $$

So, what does this tell us? Well, a couple things. Namely,

1.  Inflation is dependent on three things: the growth rate of the money supply this period, the sum of all expected money growth, and the sum of all expected inflation.

2. In order for an increase in the money supply to cause inflation it must not be accompanied by a reduction in expected money growth or an increase in expected inflation.

Knowing these two things allows us to come to the conclusion that increases in the money supply that are expected to be reversed in the future will not be inflationary.

This post kind of lacks a conclusion, but I'm going to make up for it by writing a post that evaluates quantitative easing using the findings from this post and the 'How to Escape a Liquidity Trap' post.

The Nominal Interest Rate and Inflation Determination

The dynamics of inflation in relation to the nominal interest rate are generally assumed to be governed by the liquidity effect. That is, increases in the money supply temporarily decrease the nominal interest rate because of money demand and some form of nominal rigidity. In a perfectly friction-less world, increases in the money supply (particularly increases in the growth or future path of the money supply) cause the nominal interest rate to increase instantly, rather than after the economy returns to its natural level (some economists would say equilibrium, but defining equilibrium as simply a solution to a model makes more sense to me). Of course, explaining this whole thing with math is a whole lot more descriptive, so here I go:

I'm going to assume households have a linear utility function and derive utility from consumption, so $ u(c_t) = \ln c_t $ where $ c_t $ is consumption. This implies the "consumption Euler equation" that is used to determine inflation given the nominal interest rate.

$$ (1) \:  i_t = \left(\frac {E_t c_{t+1}}{c_t}\right)\left(\frac {1 + E_t \pi_{t+1}}{\beta} \right) - 1 $$

$ i_t $ is the nominal interest rate, $ \pi_t $ is the rate of inflation, and $ \beta $ is the constant discount factor. Assume consumption grows at a constant gross rate "$g$",

$$ (2) \: c_t  = g c_{t-1} $$

and the interest rate equation becomes

$$ (1a) \: i_t = g \left(\frac {1 + E_t \pi_{t+1}}{\beta} \right) - 1 $$

Now, if the central bank targets inflation so that it evolves according to

$$ (3) \: \pi_t = \bar \pi + \rho (\pi_{t-1} - \bar \pi) + \epsilon_t^\pi $$

where $ \bar \pi $ is the "trend" rate of inflation, $0 < \rho < 1$ is the "shock stickiness" parameter, and $ \epsilon_t^\pi $ is white noise, and $ E_t \epsilon_{t+1}^\pi = 0 $, then expected inflation, $ E_t \pi_{t+1}$, is defined by

$$ (3a) \: E_t \pi_{1+1} = \bar \pi + \rho (\pi_t - \bar \pi) $$

To fill in the model, the final interest rate equation becomes

$$ (4) \: i_t = g \left( \frac {\bar \pi + \rho (\pi_t - \bar \pi)}{\beta}\right) - 1 $$

This shows essentially what the "Neo-Fisherian" assertion is. The nominal interest rate and inflation rise with each other. In fact,

$$ (5) \: \frac {d i_t}{d \pi_t} = \frac {g \rho}{\beta} $$

So, if the rate of inflation increases by 1%, then the nominal interest rate will increase by $ \frac {g \rho}{\beta} $%.

Of course no monetarist will be happy until I use the money supply as a determinate of the price level, so assume a cash in advance constraint:

$$ (6)\: M_t = P_t c_t $$

where $ M_t $ is the money supply and $ P_t $ is the price level. Since consumption still grows according to (2), the interest rate equation is redefined to

$$ (7)\: i_t = g \left( \frac {E_t P_{t+1}}{P_t\beta} \right) $$

$ M_t $ grows at gross rate "$ m_t $", so its law of motion is

$$ (8)\: M_t = m_t M_{t-1} $$

and, given (6), 

$$ (9)\: E_t P_{t+1} = \frac{m_{t+1} M_t}{g c_t} $$

Integrating all this back into (1) gives

$$ (10)\: i_t = \frac{m_{t+1}}{\beta}-1 $$

If the central bank permanently increases $ m_t $, which is equivalent to $ \pi_t +1 $, by 1%, the nominal interest rate will increase by $\frac{1}{\beta}$% ($ \frac {d i_t}{d m_t} = \frac {1}{\beta} $).

Basically, absent nominal rigidity, the nominal interest rate and inflation have a positive, even causal, relationship not afforded to them by conventional wisdom. Of course, this is really driven by the way that the money supply interacts with the nominal interest rate. "Neo-Fisherism" is really an incomplete hypothesis because of this. More focus should be given to the effects of open market operations as non-nominal-rigidity ways of explaining the liquidity effect.

Rational Partisan Theory

Recently, I have been intrigued by a fragment of the macroeconomics literature known as rational partisan theory. It looks into the effects of changes in the expectations of government policy that occur during elections. Usually, there are two "parties": one that wants high inflation and another that wants low inflation. Expectations of future policy are contingent upon the probability of each party being in power after the next election.

The simplest way to analyse this is to use the Fisher Equation and assume the central bank follows a rule and targets an inflation rate (both of which are unknown to the agents in the model).

First, the Fisher Equation:

$$ i_t = E_t \pi_{t+1} + \rho $$

where $i_t$ is the nominal interest rate, $\pi$ is the rate of inflation, and $\rho$ is the real interest rate. 

Second, the policy rule:

$$ i_t = \phi (\pi_t - \bar\pi) + \bar\pi + \rho $$

where $\bar\pi$ is the central bank's inflation target.

Here's where the "Partisan" part comes in. Each period, there is an $L_t$ percent chance that the left wing (higher inflation) party will be elected and an $R_t$ percent chance that the right wing (low inflation) party will be elected. So, by definition, $L_t + R_t = 1$. Suppose the right wing party wants the inflation rate to be $\pi^R$ and the left wing party wants the inflation rate to be $\pi^L$.

Since the agents don't know the central bank's policy rule or it's inflation target, there best guess for next period's inflation is:

$$ E_t \pi_{t+1} = L_t\pi^L + R_t\pi^R $$

If there is always a 50% chance that the left or right wing party will win the election, and $\pi^R = 0.01, \pi^L = 0.02$, expected inflation will always be $0.015$. 

Assume that the probability that the right wing party will win the election tomorrow evolves like this:

$$ (R_t - 0.5) = \theta (R_{t-1} - 0.5) + \epsilon_t $$

to close the model.