## Thursday, September 18, 2014

### The liquidity effect and the inflation/income effect

This is an update to this post where I've instead set the change in NGDP based on the increase due to the change in the monetary base (the NGDP shift is described here, the methodology for these shifts in monetary policy matches up with this post). I show the results for both an increase and decrease in the monetary base (I assume M0 follows MB up to the reserve requirement and the shift is 5% of M0):

There is a disinflationary dip (inflationary spike) at the onset of the policy change, followed by inflation (disinflation) in each case. In the 1993 and 2005 cases, nominal interest rates follow what you'd expect from an decrease (increase) in the base: rates rise (fall). What is interesting is that in the 1970s the income/inflation effect (which raises NGDP and tends to raise interest rates) is offset by the liquidity effect (which lowers interest rates). Essentially, in the equation

log r = c log (NGDP/M) + b

NGDP and M go up (or down) by approximately the same amount, leaving NGDP/M unchanged. When inflation is high, the income/inflation effect cancels the liquidity effect; when inflation is low, the liquidity effect dominates.

In Williamson's post, he gives a model where the Fisher effect causes nominal rate increases to produce inflation. I have used the term "Fisher effect" in different ways on this blog. In this post, I use it to describe the direct correlation of inflation an nominal interest rates. In this post, I attributed the deviation from the model to the "Fisher effect" -- i.e. a higher interest rate than the information transfer model predicted due to some un-modeled force (like expectations). However, the information transfer model seems to confine this "Fisher effect" to long term interest rates and to the 1970s. Williamson's approach models the Fisher effect (expectations of higher inflation). However these two models can be consistent if inflation expectations are viewed in the light of this post -- agents expect future inflation if the monetary base is small relative to the size of the economy [1].

Overall, the information transfer model takes what closer to an orthodox view -- expansionary monetary policy lowers interest rates and creates a spike in inflation, while contractionary policy raises interest rates and creates a deflationary dip. The dip in inflation is followed by a steadily higher than expected inflation after the onset of the new policy -- this effect is much smaller, but lasts longer (the integrated result is that both of these effects cancel by the time the shock of the change in monetary policy wears off). The information transfer model does indicate that the effect of monetary policy on inflation steadily diminishes from 1973 to 2005, which isn't part of orthodoxy -- except of course, this is how the model describes the liquidity trap.

Essentially, the experiment Williamson describes doesn't have an exact analogy in the information transfer model. The central bank can't choose an interest rate to hold policy at indefinitely (too high or too low would eventually lead to a boom or bust, respectively). What I'm trying to do here (and here and here) is show that the information transfer model can manifest the effects described by Williamson (under certain circumstances) in order to make contact with "real economics" as practiced by real economists.

[I'm not sure if I'm happy with this post -- it seems pretty disjoint. Consider it some "thinking out loud" ... ]

[1] This does seem to be backwards in the sense that if the monetary base is large, people seem to fear inflation is right around the corner. However an expectation that presumed ignorance of the future direction of policy would see that there are more possible states of the economy with a smaller monetary base than a larger one as the base approaches NGDP.

## Wednesday, September 17, 2014

### The path of policy is strongly dependent on the path of policy

The redundant title was intended as something of a joke. I was hoping to try and do an apples to apples comparison to Stephen Williamson's last graph at this post:

However, it is impossible to construct an economy that maintains a constant nominal interest rate in the information transfer model, partially for reasons given here (constant inflation requires ever increasing NGDP), and partially for reasons given here (the interest rates need to follow a specific trend to follow the path of NGDP-M0).

Therefore I looked at the difference between the (smoothed) model result and the (smoothed) model counterfactual result. This means that the results below indicate the difference between the given variable and the expected variable path (i.e. a positive nominal interest rate means that the interest rate would be higher than the trend path). The results also strongly depend on how the nominal interest rate increase happens (in particular because inflation is a derivative, so sudden jumps have strange effects) [1]. Additionally, the inflation rate depends on the size of the economy and the monetary base, so I chose three years for the onset of the nominal interest rate increase: 1973, 1993 and 2005.

I tried three different approaches to how the nominal rate changed, and they all end up with slightly different results. The first assumes a sudden jump in the nominal interest rate that goes from being slightly below the NGDP-M0 trend to being slightly above.

The second assumes that there is a smooth (but still narrow) rise in the nominal rate with a long decay back to the trend

The third assumes that there is a smooth rise and shorter decay back to the trend

(Also note that the scales aren't all the same.)

The end result is that inflation always seems to increase a bit with a nominal rate increase (the increase gets smaller and smaller over time, both in terms of the onset year -- 1973 to 2005 -- and in terms of inflation eventually returning to the trend set by NGDP-M0), broadly in line with Williamson's graph. The path inflation takes to get there varies from a sudden jump to disinflation (of varying magnitude) followed by increased inflation depending on the precise path.

It appears that nominal rate increases (which involve a reduction in the monetary base -- i.e. contractionary policy) tend to be inflationary (relative to a case where there was no rate increase) in the medium to long run because they move the price level down (P decreases if M0 decreases, ceteris paribus), so inflation must increase in order to return to the price level back to its expected long run path. It's a bit like how digging a hole in front of a mountain can make the average grade steeper on the far side of your hole. Here is a graph of the price level (with red being the counterfactual contractionary policy):

That dip and the return to the trend is responsible for a nominal interest rate increase leading to inflation.

[1] I also tried to make this as pure of a nominal interest rate increase as possible by making the change in the path of NGDP-M0 perpendicular to the lines of constant interest rate illustrated e.g. here. This differs somewhat from the methodology I used here, where the NGDP increase is dependent on the M0 increase.

## Tuesday, September 16, 2014

### Another analogy for monetary policy and recessions

The traditional mental model seems to see contractionary monetary policy as adding friction or letting up on the gas pedal in a car, for example, Noah Smith: "When the economy is doing well, raise interest rates to slow things down ..."

This is very different to the information transfer model view; I'd previous likened contractionary monetary policy to piling snow on a mountain until an avalanche occurs.

I'd like to add another mechanical analogy: contractionary monetary policy is like stalling an aircraft to lose altitude. Aircraft can become difficult to control at their stall angle/speed and you can end up losing much more altitude that you intended. Another interesting extension of this analogy is that at "high speed" (i.e. high inflation) a stall is less likely than at "low speed" (low inflation). In a sense, the economies of the US, EU and Japan all seem to be flying near their stall speeds -- contractionary policy could induce a recession.

### Distilling Fisher relationship data

I mentioned I was going to say some more about Stephen Williamson's piece on the EU and the Fisher relationship (where higher interest rates are associated with higher inflation and lower rates with lower inflation). The plan of action is to remove the empirical noise from the Fisher relationship Williamson presents by finding the underlying trends in the data in the fluctuations (based on smoothing the model inputs).

Here is the model (blue) and the data (green) -- removing the 500 € notes -- for the price level (CPI less food and energy):

Here are the inflation rates (year over year for the data in green, instantaneous derivative of the price level for the model in blue):

Both the inflation based on CPI and CPI less food and energy are shown. And here are the interest rates (long rate is red, short rate is blue):

So now finally we are equipped to reproduce Williamson's second to last graph:

You can see how much of the scatterplot (green points) is a deviation from the model (blue line). The noise dominates the model; it would be hard to associate any particular movement of interest rates or inflation with this relationship (only the long run trend over several years). Due to some of the data not going back to 1997, only the larger green dots are from the same period as the model result. However those points cover much of the same range as the full data set shown with the smaller green dots (although I couldn't find data on the short term interest rate that goes all the way back to 1994 like Williamson's data -- so it misses the highest interest rate part).

We do recover the Fisher relationship where inflation and interest rates are directly correlated, though. I'm not sure this is a causal relationship -- at least in the way a neo-Fisherite would see it. The neo-Fisherite view is that the central bank could e.g. raise rates and produce inflation (or keeping rates low leads to deflation). However, in the information transfer model, high inflation means that the monetary base is small with respect to the size of the economy (NGDP) which in turn means expansionary policy increases interest rates increase through the income-inflation effect. Low inflation means that the base is large compared to NGDP and the liquidity effect dominates interest rates (expansionary policy reduces rates). In this scenario, any attempt to raise interest rates (reducing M0 or MB growth) to try and increase inflation would throw the economy off the long run NGDP-M0 path likely leading to recession.

## Saturday, September 13, 2014

### What is inflation?

Stephen Williamson asks: "How do macroeconomists think about inflation?" and answers with a very long (and very interesting -- I plan on doing a post dedicated to it later) history of thought, ending with an analysis of the situation in the EU. According to Williamson, it's not obviously wage-price spirals (e.g. the Phillips curve) and its not directly linked to the money supply -- empirically, these seem to lack stable relationships.

Simon Wren-Lewis says: "The idea that we can take one variable, or one equation, and distill from that the future price level is a fantasy. What is surprising is that this fantasy has been, and still remains, so attractive for some economists."

Wren-Lewis cites Frances Coppola, who says: "Empirically, it is abundantly clear that there is no clear relationship between the quantity of monetary base and the price level."

But Nick Rowe disagrees, he believes that printing more money than you would have otherwise leads to higher inflation that otherwise would have been.

Scott Sumner thinks inflation is irrelevant. He says: "When I say inflation is a meaningless concept I’m suggesting that the concept is not well defined, despite the BLS’s attempts to do so." He also says: "There is no such thing as a “true rate of inflation,” but there’s also no reason to assume that inflation has not averaged 2% in recent decades. It’s just as reasonable as any other number the BLS might pull out of the air." Sumner references the hedonic adjustments made by the BLS (adjusting for the quality of goods) -- which of course makes it seem very arbitrary. I got into a discussion with Sumner and commenter Dustin about it awhile back.

Overall, there seem to be a lot of ideas of what inflation is or isn't without any rigorous definitions. I was reading Romer's Advanced Economics (also on my flight home last night) where he says economists don't even necessarily know if inflation is good or bad (Nick Rowe has a concise list of the mechanisms that make it bad: "Economists would instead talk about shoe leather costs, menu costs, relative price distortions, difficulties of indexing taxes, confused accountants, etc").

In short, there seems to be a lot of confusion.

In the information transfer model, however, the price level is pretty well defined:

$$P = \frac{dNGDP^{*}}{dM0}$$

The price level is the increase in NGDP given an infinitesimal increase in currency M0. These days you get about twice as much NGDP for the same amount of printed currency as you did in the 1980s (in the US) [1]. The inflation rate is just the time rate of change of this (the derivative of the log gives the fractional rate):

$$i = \frac{d}{dt} \log P = \frac{d}{dt} \log \frac{dNGDP^{*}}{dM0}$$

Now you may have noticed that I slipped an asterisk on NGDP -- that's because it's not really measured NGDP, but rather the theoretical shock-free NGDP I've shown here. Here are the two compared (the blue curve is the shock-free NGDP and the red curve shows the empirical data that includes shocks):

Now the price level model in terms of the information transfer index $\kappa (NGDP, M0)$ is an approximation, so the model isn't perfect, but it is pretty good (taking the derivative of the blue line above gives the points and the inflation data is the green line):

So inflation is the time rate of change at which the "friction-free" "ideal" $NGDP^{*}$ goes up for a given expansion in currency $M0$. In terms of equations we have [2]:

$$NGDP^{*} \simeq \beta \left( \frac{M0}{M_{0}} \right)^{1/\kappa (NGDP, M0)}$$

$$P = \frac{dNGDP^{*}}{dM0} \simeq \alpha \frac{1}{\kappa (NGDP, M0)} \left( \frac{M0}{M_{0}} \right)^{1/\kappa (NGDP, M0) - 1}$$

Why don't we include the shocks in $NGDP^{*}$? Here's one plausible story. Maybe that derivative is supposed to be a partial derivative -- the price level is the change in NGDP that is due to a rise in the supply of currency only, independent of other factors. If the shocks $\sigma$ are independent of currency M0 (or only weakly dependent, say on $\log M0$), and $NGDP(M0, \sigma) = NGDP^{*}(M0) + \sigma$ then we have:

$$P = \frac{\partial}{\partial M0} NGDP(M0, \sigma) = \frac{\partial}{\partial M0} NGDP(M0, \sigma = 0)$$

$$P \simeq \frac{\partial}{\partial M0} NGDP^{*}(M0) = \frac{dNGDP^{*}}{dM0}$$

Now how does the BLS measure CPI if it doesn't know about $NGDP^{*}$ -- I'm pretty sure they've never heard of the information transfer model. It is likely helpful that the individual prices the BLS measures won't be experiencing the same shocks at any given time, so on average a price for a typical good will give insight into $NGDP^{*}$. However, since NGDP (without the asterisk) measures all goods produced, it will be affected by the distribution of prices for a given good. BLS also drops the highly fluctuating food and energy from "core" inflation. There is no "core" NGDP. Both of these effects mean that the BLS CPI statistics may in fact be measuring $NGDP^{*}$ when they measure inflation.

PS I wrote this while I have a pretty serious cold, and am feeling a bit loopy; please excuse me if this doesn't make any sense.

Footnotes:

[1] Just because the price level is going up, it doesn't mean that the relative expansion of NGDP is getting bigger for a given expansion in M0 -- for that to happen the price level would have to go up at the same rate as NGDP -- ie. RGDP growth is zero.

[2] In the function $\kappa (NGDP, M0)$ we have the actual realized NGDP because that is what a dollar buys a fraction of.

### The bitcoin wall

There have been a couple of references to bitcoin on this blog (including a couple comments), but I was reading this on my flight back home last night and something clicked when I got to this:
Most digital currencies incorporate a pre-determined path towards a fixed eventual supply.

Since I've been looking at the path of NGDP-M0 in the past few posts (e.g. here), I realized that a bitcoin economy (where a bitcoin defines the unit of account) would reach some point along the NGDP-M0 path, stop and fluctuate around that point.

The Fed came close to experimenting with this kind of trajectory just before the Great Recession -- currency in circulation started to reach a plateau. Let's take the bitcoin counterfactual where that plateau continued indefinitely (i.e bitcoin existed since the 1960s or so and reached fixed supply in 2008):

Although I guess it is possible to delete bitcoins, I also altered the time series so the bitcoin M0 never decreases. Now, what would happen to the path of NGDP-M0? Well, you hit a wall:

I'm not sure what would constrain the size of the shocks -- you'd have massive recessions and large booms (somewhat like the pre-Fed US, except even during those times the gold supply would increase). I would imagine that the series of shocks would eventually cause the bitcoin economy to collapse as people abandoned it for some kind of fiat currency.

Of course, if the bitcoin path were defined by the theoretical curve then it wouldn't be doomed. Maybe someone should make info-coin that follows the theoretical path of NGDP vs M0? It is unfortunate that the name bitcoin already used "bit".

But fiat currency already seems to follow the path, so why introduce something new that does the same thing?

## Wednesday, September 10, 2014

### Suggestive graph juxtaposition

Metastability?

Speculative, but something to think about ...