Some prediction fun (ECB edition)

Neo-Fisherite Edward Lambert makes a prediction of lower inflation as the ECB lowers interest rates via the "Fisher effect" (inflation is low because nominal interest rates are low and expected to stay low), which is similar to what I called the Fisher effect where expected inflation influences nominal interest rates -- however, I think the effect was largely confined to the 1970s and 1980s and to long term interest rates.

I'll jump in and make a prediction of lower inflation in the Euro zone (I was saying we're in for lower inflation in the EU since last November ) as well, but via the diminishing effect of monetary expansion (aka the "information trap"). Here is another post on the idea in relation to the neo-Fisherite rebellion. And this is another one.

However, due to the short time scale the EU has been in its current configuration, it's quite a bit more uncertain where the EU is in terms of the information transfer model. Check out the wild behavior of the monetary base and the currency component (the Euro was introduced in 2000 as a non-physical currency and was introduced as a physical currency in 2002, but other currencies continued to be used for awhile afterwards):

It appears only the post-crisis (2009 and beyond) Euro data has an EU and a Eurozone area in a stable configuration, so I just ran the models for that. Here are the CPI and the long term (10 year) and short term (interbank) interest rates:

And here are the resulting IS curves for the long term interest rate (analogous to this analysis, each line is taken at the start of each year, with the red one being 2014):

You can see the IS curves and the "zero bound" sliding backwards from 2009 (curves 2005 through 2014 are shown), a sign of being in a liquidity trap. But they are still downward sloping, not yet vertical.

But I still want to jump in and make a prediction that we'll get disinflation in the Eurozone as they go forward with their policies.

Reconciling expectation and information

There was a brief back and forth between Scott Sumner and me in comments on his post on whether money is tighter or looser given a drop in interest rates. Sumner said (the edited version):

The fed funds change is different. Whereas a change in IOR affected the demand for base money, a change in the few funds rate is an effect of a change in the supply of base money. That’s easier money in the short run, ceteris paribus. But whether it is actually easier money depends on how the action impacts the expected future path of monetary policy.

The short run effect is the liquidity effect, which I describe in more detail at this link. I commented that the long run effect was the same result as I gave in this post, except the "depends" clause was different -- in the information transfer (IT) model result the depending is measured by the size of the base relative to the size of the economy (e.g. NGDP). Sumner disagreed with it initially, but then agreed with my contention that if the base is small, the future path of monetary policy is likely expansionary (i.e. looser money, leading to higher interest rates).  I'd like to say more about that with this post.

But first, I have to correct some sloppy language on my part. I have a tendency to say "the size of the base relative to NGDP" or write "MB/NGDP" when I actually mean a) not MB but M0, the currency component of the base (i.e. without reserves) and b) not the ratio but the ratio of the logarithms log M0/log NGDP (aka the IT index, denoted by the Greek letter kappa κ elsewhere in this blog). You can see what I mean when I plot the two ratios here:

We can see that κ goes from a smaller value in the 1960s to a larger value today, whereas the simple ratio goes down and then up (it is normalized to 2011). The IT index κ is more relevant in the model because the price level is determined by the base via log P ~ (1/κ - 1) log M0 which means that P ~ constant for κ = 1. Another way to look at the IT index is that it is the fraction of a unit of source information that can be transferred with the unit of account, i.e. 1/logM0 NGDP = log M0/log NGDP (using my poor man's subscripting technique to represent the logarithm in base M0) analogous to Sumner's share of the economy that can be bought with one dollar (value of money = 1/NGDP).

Now how do we reconcile the expectation of future base growth ("loose money") with the size of the base M0? Well, if we plot the numerator (blue) and denominator (red) of the IT index and assume a uniform probability distribution (maximum ignorance) using the denominator (log NGDP, i.e. where κ = 1 if log M0 = log NGDP) as the upper bound and the quantity theory of money (κ = 1/2) [1] as the lower bound (dashed gray) for the possible values of log M0, we can see that if the IT index is below κ = 3/4, more states in log M0 space exist above than below the current value of the base in 1970. See the relative length of the green line segments the diagram below:

We can put this in Sumner's language in the quote above. In 1970, there are more places for log M0 to be above its current value than below, meaning a larger base and therefore looser money is expected. That means interest rates will tend to rise as an effect of base expansion. In 2005, a larger or smaller base are closer to being equally likely (i.e. that base expansion is less likely than it was before in the 1970s), meaning that interest rates will tend to stay constant or fall as an effect of base expansion.

The uniform distribution over the values of κ ∈ [1/2, 1], of course, is an oversimplified picture. The actual probability distribution over the possible expected values would likely involve more understanding of human behavior (in response to Mike Freimuth's comments here) -- the distribution is probably peaked at the value of the (log of) the base and goes to zero at (the log of) NGDP as well as zero (log 0 = -∞), but its exact form would depend on how humans discount rare events (see e.g. prospect theory).

The actual cross-over point, based on the behavior of IS curves (i.e. the effect of expansionary policy on interest rates), seems to be in the late 1970s or early 1980s. This is where the IS curves turn over and loose money means lower interest rates.

[1] If we have log P ~ (1/κ - 1) log M0 then κ = 1/2 means that  (1/(1/2) - 1) log M0 = (2 - 1) log M0 = log M0 = log P, i.e. P ~ M0 (the quantity theory of money) and the rate of growth of the price level is equal to the rate of growth of the base. [Added in update 6/14/2014]

Is this what Noah Smith is referring to?

Noah Smith wrote a piece for Bloomberg View:

[Shinzo Abe] went the opposite direction of Europe, and -- unlike the U.S. -- he gave every indication that the shift toward monetarism was permanent. The result: Japan has escaped deflation.

It took me awhile to track down the deflation victory point (it's not yet in the data from FRED):

See it? It's the one point at the end above the 1-sigma model error band from April 2014. It is actually quite remarkable (a 2-sigma jump), and for the sake of the people of Japan I hope it holds up. I mentioned in an earlier post that this rise over the past few months could be due to the impact of fiscal policy. Before someone says I scaled the axes to make the jump look small, note that the US CPI traversed the entire range in the same time period (2% inflation would take you from 0.9 in 1994 to 1.1 in 2014).

A jump in CPI due to monetary policy should affect interest rates, but nothing remarkable seems to be happening. Interest rates are low given the model result (again 1-sigma error band shown):

Krugman, Keynes and the liquidity trap

There was some lively discussion in the comments on this post about the liquidity trap. I realized 1) I hadn't worked out the details of what happens in the information transfer model to the level of detail of that post and 2) I found the liquidity trap model from Krugman's 1998 Brookings paper to be essentially mute on the reason I wrote the post in the first place: I wanted to find out if the ECB raising rates in 2011 or keeping them above zero after 2008 had anything to say about the efficacy of monetary policy or monetary offset.

To that end, I thought I'd try to reproduce the IS curve with the information transfer model (ITM). The results were pretty neat, and they follow in a long line of IT model results where everyone (monetarists, Keynesians) is correct but only in at certain times.

I started from this post and looked at the effect of monetary expansion (red arrows in the graph on the left below) on the interest rate (red lines). The model also incorporates the impact of monetary expanion on the price level -- it's a highly nonlinear system. The "information trap" criterion ∂P/∂MB = 0 is represented as a dotted line. At this line is where monetary policy has no impact on the price level or nominal output -- effectively the liquidity trap Krugman describes in his 1998 paper. I sampled points given by the rainbow dots in the graph below on the right (and the colors correspond in the rest of this post as well).

At each of the points in the graph on the right above, I evaluated the interest rate and nominal output (NGDP) for various values of monetary expansion (and contraction) and used those values to trace out an IS curve (interest rate vs output). One difference from Krugman is that I used nominal output instead of real output. I did run the numbers for real output, and the results were basically the same, but the graphs didn't look as nice. Maybe I'll put them in a follow up.

Here was the result:

The black points on the rainbow curves represent the starting value and the curves trace out monetary expansion and contraction. The gray points represent data from the US. Note that the curves are not required to follow the data exactly; one way to think of the curves is as the effect of steering the economy by central bank near the given point, but things like population growth steadily moves the economy to the right. However, when the points do follow one of the curves closely, that gives an indication of how much power monetary policy has over the economy. In the graphic below, I show each curve in a normalized window (+/- 30% of the center value) that gives a better view of the changing IS curves over time:

In the graph above, we can clearly see 1) the monetarist view (higher interest rates are a sign monetary policy has been loose) as the purple and violet curves in the beginning, 2) the Keynesian view of a downward sloping IS curve (greens) where lowering interest rates boosts the economy and 3) the liquidity trap (orange and red), where the IS curves become almost vertical -- output becomes almost independent of interest rates and raising or lowering interest rates has no impact. Since this effect enters via the impact of monetary policy on the price level, it means that monetary policy is ineffective in the AD/AS framework and there is no monetary offset.

This answers two questions. First, the non-zero ECB interest rates (and raising or lowering them) probably had no monetary impact (they might have had a fiscal impact on the member states). This is how I interpreted the liquidity trap in the earlier post. Second, the economy doesn't need to be at the zero lower bound (ZLB) to experience a liquidity trap -- but it does have to have "low" interest rates close to the information trap criterion line in the graphs at the top of this post. This is the connection with Keynes -- the original liquidity trap he describes does not have to occur at the ZLB.

But what actually happens at the ZLB? We can use the last two red/orange graphs in the series if I show them with a linear scale:

It looks like pushing a piece of uncooked spaghetti against the countertop with the spaghetti bending away. Pushing down towards the ZLB produces a negative impact on NGDP i.e. "deflationary monetary expansion". Notice that there is actually a peak output for the curve on the left -- I imagine this is Krugman's point 2 in Figure 2 in his paper. In fact, here is one of the IS curves derived here (blue) alongside a schematic of the "traditional" IS curve (dashed) from e.g. Krugman's 1998 paper (along with the labels from that paper):

This a possible resolution of the discussion I was having with Mark Sadowski in comments on the earlier post.

Comments from Free Radical

Tom Brown and Mike Freimuth discussed the information transfer model on the latter's blog Free Radical. Mike apparently is a PhD candidate at the same school I went to for my PhD. Small world. Anyway, Mike brings up a couple of points that I thought I'd address here.

It seems like he is just saying that the price level is correlated with the size of the money base.

Actually it gives an explicit functional form for the price level:

$$
P(M) = \alpha \frac{1}{\kappa} \left( \frac{M}{M_{0}} \right)^{1/\kappa - 1}
$$

Where $\alpha$, $M_{0}$ and $\kappa$ are free parameters that can be used to fit the empirical data. Since constant $\kappa$ doesn't work very well,  I tried assuming that $\kappa$ could vary since it is actually based on the potentially changing information content of 1 unit of NGDP vs 1 unit of currency (it is proportional to the ratio of the Hartley information for the demand "states" and supply "states"). This gives us:

$$
P(M, N) = \alpha \frac{\log N/c_{0}}{\log M/c_{0}} \left( \frac{M}{M_{0}} \right)^{\frac{\log N/c_{0}}{\log M/c_{0}} - 1}
$$

A different motivation of this equation, entirely from long run neutrality of money, is presented at the link here. The model is the simplest model of economics consistent with long run neutrality (homogeneity of degree zero of supply and demand functions).

The model is also a model of supply and demand in general and is capable of constructing many traditional diagram-based micro and macro economic models.

... it just seems like it boils down to MV=PY to me.

It is, but with a specific model for $V$ and $Y$, namely $V = \kappa P$ and $PY \sim M^{1/\kappa}$.

Putting aside the fact that I’m not exactly sure what he means by the information-carrying capability, one must wonder why this wouldn’t be constantly in question. If the theory is that it works because nobody questions it then I find that theory unconvincing.

The theory doesn't explain why money has value (information carrying capacity, i.e. the capability to be used to mediate information transfer). It is agnostic on that. What it explains are the dynamics of money in terms of the size of the economy and the quantity of money. Given a value of money at one time, it gives the value at another (via the price level). Actually, the model can give the price level from 1980-2014 given the data before 1980 -- and you can extrapolate our current low inflation environment starting in the 1970s. See here:

http://informationtransfereconomics.blogspot.com/2014/05/out-of-sample-predictions-with.html

An analogy: Thermodynamics doesn't explain what atoms are (it was actually worked out without knowing what atoms are) -- it explains the dynamics of huge ensembles of atoms. It doesn't even work out e.g. what the volume is for an ideal gas at a given pressure, but rather figures out the relationship between pressure and volume (e.g. $P_{1} V_{1} = P_{2} V_{2}$). Something like backing theory or some other consideration gives money its value just like quantum mechanics gives the theory behind atoms.

Regarding the "questioning", this was probably unwarranted theorizing on my part. There was something that changed in the US and UK (maybe more countries but I don't have data) that corresponded to the end if WWII that made the coefficients in the equation above change. I think it was Bretton-Woods, but the theory also allows for other major changes in currency to reset the coefficients of the model. In terms of the thermodynamics, we have a law like $P V^{\gamma} = \text{ constant}$; certain things can happen (like a chemical reaction) that causes the "constant" to change.

Answering some technical questions

An anonymous commenter has been working to reproduce the results on this blog and had some questions that I thought would be beneficial for others trying to do the same thing.

1) I'm a little stuck on how you're getting the blue & red vectors. I grok the general concept but are those vectors representing changes in P or NGDP? My first attempt was just taking the gradient of the P surface but that doesn't seem to line up...

The arrows represent changes in NGDP and MB. It is kind of like the gradient, except that an increase in the price level P increases NGDP = P*Y (also I didn't add $\delta MB$ and $\delta NGDP$ and divide by the step -- I just multiplied by a factor $\alpha$ to look at relative fractional increases). The red arrows are defined by

$$
M \rightarrow \alpha M \text{ and } N \rightarrow N \frac{P(N,\alpha M)}{P(N, M)}
$$

i.e. an increase in the base changes the price level, affecting NGDP. The blue arrows are given by

$$
M \rightarrow M \text{ and } N \rightarrow \alpha N
$$

i.e. an increase in NGDP doesn't change the base. Here is the Mathematica code:

In the picture above, I cut out some of the pieces that make the other lines in the graph to simplify the presentation. The vector "yearlist" is a list of years where the arrows are being evaluated. There is also an overall scale factor to make the pictures look nice. It is a 20% increase (1.2) and a scale factor of 2.

As a side note, I look at the gradients in more detail in this post:

http://informationtransfereconomics.blogspot.com/2013/10/the-1970s.html

2) I don't know if this is just the vagaries of the Matlab solver(s), but I'm not getting the exact fits that you are for the parameters. A lot of the time I don't seem to get proper fits at all and it's extremely dependent on starting values.. is this your experience as well?

Yes, the parameter values are fairly sensitive to initial conditions. The data is noisy, so the objective function over the parameter space $p$ is noisy, and you can get stuck in a lot of local minima. I set the problem up as a minimization problem over the parameters $\min_{p} f(p, t) = |P(M(t), N(t), p) - CPI(t)|$ over a grid in the time variable (using monthly or quarterly values depending on the resolution of the data, so for e.g. a year of data, I'm solving 12 or 4 separate optimizations ... over the 50 years, that is 600 optimizations) using this method which is a little more robust. The Mathematica code is in the picture below:

3) Do you have a data source for countries other than the US? I'm struggling to find good data series for places like Japan etc. and have had it with navigating the often byzantine central bank websites of these countries

Most of the GDP and CPI data is from FRED. Monetary base data is also available there for Argentina and Switzerland (which I've used ... and apparently Russia, China, Poland and South Africa, which I haven't). For the monetary data, I frequently do have to go to the central bank websites.

For Japan, the long term time series for the monetary base is here:

https://www.boj.or.jp/en/statistics/boj/other/mb/index.htm/

For Canada, you can find monetary base data here:

http://homes.chass.utoronto.ca/~floyd/macro.html

I link to 300 years of UK data in this post:

http://informationtransfereconomics.blogspot.com/2013/11/the-long-run-in-uk.html

For the EU, I sometimes go here (but only if I'm desperate and up for a maddening experience):

http://epp.eurostat.ec.europa.eu/portal/page/portal/eurostat/home/

4) When you put multiple countries on one chart, how are you normalizing P? I assume with a P0, but is that fitted value.. or do you say normalize all your CPIs before feeding them in..?

I use the fitted coefficient P0 (in the pic above, I refer to it as ΔUS in the code and α on the diagram). I also normalize all CPIs to 1 instead of 100 (out of habit) before feeding them in. The fits have a strong tendency to make P = P0 at M = M0; it has worked that way in every fit, except for Switzerland (because it is normalized to 100 in 2010 in the data from FRED). The normalization of the price level is completely arbitrary, so there was an adjustment to the Swiss data to make 1982-1984 = 100 like the US data (and then normalizing to 1).

5) I've been trying to read up on information theory and the original paper - curious as to if there's any way of combining multiple sources, detectors etc. into one model - my math/intuition is not developed enough to work out how that might be done but it seems like an interesting avenue to explore...

I have looked at multiple (well, two) interacting markets with the same price "detector" and same information source (aggregate demand):

http://informationtransfereconomics.blogspot.com/2013/08/deriving-is-lm-model-from-information.html

http://informationtransfereconomics.blogspot.com/2013/08/scott-sumners-model-part-2_30.html

The main "information transfer model" is a combination of three markets: P:NGDP→M0 (price level, with endogenous NGDP and M0), P:NGDP→L (labor market, with exogenous NGDP and endogenous L) and r:NGDP→M (interest rate market, with exogenous NGDP and exogenous M = M0 or MB for long and short interest rates, respectively).

I started down the path of combining several markets, but have only gotten to the point of Walras' law (which involves an assumption about how markets interact):

http://informationtransfereconomics.blogspot.com/2013/09/walras-law.html

Thanks for the questions; I hope this answers them!

How does a liquidity trap work?

I've been getting in arguments with market monetarists (in particular Scott Sumner and Mark Sadowski) about the liquidity trap lately. See e.g. comments here [1] and here [2]. I don't think they understand the liquidity trap model. Maybe they just discount the model assumptions that disagree with their assumptions and are therefore effectively talking about a different model. It's possible it's my fault because I don't understand the liquidity trap argument or the monetarist argument (or both). So I decided to make some nice pictures and put forward what I think the Keynesian and the monetarist views of the liquidity trap are.

Here is the basic picture, borrowing from Figure 1 (on page 9) Figure 2 on page 13 (aka p149) [H/T Mark Sadowski] of Paul Krugman's 1998 Brookings paper (we start before the recession hits that causes the liquidity trap; Krugman starts from an economy in a liquidity trap). The dashed curve represents the Hicksian IS curve in the pre-liquidity trap picture with the black point representing the equilibrium interest rate and output (the money market equilibria at a given output are given by the gray vertical line). Suddenly a shock hits, moving the output to the left and now the set of money market equilibria are given by the red line (which would select the red point on the dashed IS curve). However, the shock also drives the IS curve to the blue curve. The equilibrium solution Z is now below the zero lower bound (ZLB), and the region where the IS curve is < 0 is shaded in blue.

Note that it is the fact that the equilibrium point Z is below the ZLB that indicates the liquidity trap, not the position of the current state of the economy in the light blue region (the red or black dots). I think this is a major source of misunderstanding between the two sides. It doesn't matter if a country has positive interest rates ... it can still be in a liquidity trap.

The first question we need to ask is: Where does the red dot go when the IS curve shifts?

The Keynesian answer: nowhere, necessarily. The IS and money markets have become disconnected and their intersection no longer determines the equilibrium. The central bank may try to move the red dot to the ZLB (zero interest rate policy, ZIRP) straight down along the red line from point #1 to point #2.

The monetarist answer? I'm not sure. The red dot may stay where it is (#1), but some evidence indicates that market monetarists think the red dot moves to the zero lower bound at point #2 or that the money curve shifts to the left, moving the equilibrium to point #3 (see figure on the left above). The latter is more consistent with the market monetarist belief that the central bank causes the recession through monetary policy.

Scott Sumner and Mark Sadowski point out that the ECB has positive interest rates, which means -- to them at least -- the EU isn't even at the ZLB. This may be evidence that the point stays at point #1 when the shock hits (like the Keynesian answer), or it may be evidence that the money curve shifts even farther to the left and the ECB is at equilibrium #4 in the figure on the right above (or even all the way over to point #5 that keeps the interest rate constant).

Only points #1, #2 and #3 could be characterized as an economy in a liquidity trap (the last one, only marginally). However, only points #1 and #2 are consistent with the modern description of the liquidity trap. Krugman states that moving to #2 doesn't hurt, and if the fall in the IS curve is slow, the equilibrium point will follow the curve down until it hits point #2. My interpretation is that Krugman believes point #1 is the EU and point #2 is the US.

The second question we need to ask is: Where does the red dot go when the central bank raises interest rates in a liquidity trap? We'll say the economy starts at point #2, which is Krugman's point #3 in Figure 1  2 in his 1998 paper (see the figure below), showing an economy in a liquidity trap.

The Keynesian answer: the economy moves to point #6 in the diagram above, or possibly the point next to it. The equilibrium point is free to move up or down (at least, down to the ZLB). To leading order, the IS and LM markets are disconnected, the central bank has no traction and the effect of interest rate policy has no impact on output.  Raising rates does send a contractionary signal, which can cause deflation/disinflation through expectations (e.g. the ECB raising rates in 2011) reducing output. Raising rates also will cause e.g. budget constrained EU member nations to reduce government spending and put the savings toward increased debt service. The central bank could send an expansionary signal but this is hard to do (it is hard to promise the inflation will stick around). Effectively, the central bank cannot move the red line (output) left or right.

The monetarist answer: the economy moves to point #7. The economy is no longer in a liquidity trap or at the ZLB. By assumption, the central bank can generally offset changes in output through monetary policy (moving the red line to the left or right), effectively taking the blue curve to be the dashed blue curve in the ZLB region (blue shaded region). This means that monetary policy can offset fiscal policy through its effect on output in the AD/AS model. This is the diagram monetarists say describes the ECB raising rates in 2011. (Note that point #7 here is Krugman's point #1 in Figure 1 2 in his 1998 paper, which shows an economy that's not in a liquidity trap.)

The third question we need to ask is: How do we get back to pre-recession output?

The Keynesian answer: fiscal stimulus is the only thing that will move us from point #2 to point #8 in the diagram above. In terms of monetary policy, we're stuck at point #2.

The monetarist answer: if the central bank targets the output level given by point #8 (moving the red line to the right), we can move along the dashed blue curve to point #8. Monetary policy effectively selects the level of output ... by assumption.

The fourth and final question we need to ask is: Which answers are right?

This is the Keynesian/Hicksian/Krugmanian liquidity trap model, so I'd defer to those answers for how a liquidity trap works. But I think I've captured where different model assumptions enter. Because macro data is uninformative, the argument between both sides tends to take the form of "obviously the (Keynesian/monetarist) model assumptions are correct" and "such and such random piece of (monetarist/Keynesian) model confirming data show that the (Keynesian/monetarist) model is wrong". Neither side cedes that model assumptions on the other side could be correct, nor do many of the blog posts out there look at all the data together. The two sides are talking past each other. They have different models and assumptions and each side doesn't believe the other one.

Now I believe the evidence is strongly in favor of the liquidity trap model, at least for the US, Japan and the EU. The information transfer model allows both explanations to exist, but selects which one based on the empirical data. It favors the liquidity trap view in the case of Japan, the US and the EU. It favors the monetarist view for countries like Canada, Australia and Sweden. In order to do this, it looks at all of the data (price level, NGDP, monetary base, and interest rates) together.

Now maybe the information transfer model is incorrect, but at least it doesn't assume the result.

Money: the unit of information and the medium of information exchange

We're all familiar with "bits" as units of information and "bits" as the medium of information exchange. We intuitively understand how a 1 GB SD card is a medium for exchanging 1 GB, or 8 billion bits of information. Why, then, is the medium of exchange and unit of account functions of money such a hotly debated topic on the econ blogs? For example, these posts by Sumner and Rowe have over 100 comments each as of last check

First, I don't think I have to convince you that money is a medium for exchanging information. By giving you 5 dollars, I am sending you the information "you need to give me 5 dollars worth of goods and services and we're square" or as Narayana Kocherlakota would say "remember to give me 5 dollars worth of goods and services later" in his paper Money is Memory [2]. The money in your bank account is essentially a record of the work you've done. Actually, one of the oldest ways of storing information in the world was likely a form of economic transaction. Accounting is just a very specific form of recording history.

When money is just the medium of exchange, for most practical purposes it really doesn't matter if someone gives you some MZM, M1, M2, MB or even short term treasury bonds (these days); we seem to understand that intuitively. Where I think this gets confusing is when we think of money as defining the unit of information. This messes with our minds in two ways: 1) when money defines the unit of information, it is dependent on how much medium of exchange is out there and 2) what do we mean by money (M1, MZM, M2, etc)?

Let's attack the first question first. I think it's fairly easy for us to see how money can be a varying unit of measurement. A pound sterling doesn't mean the same thing in 1700s as it does today -- that is we have to go back and covert the weird units British people used in the 1700s called "pounds sterling" into the rather practical units British people use today call "pounds sterling". That is to say, we adjust for inflation. But that adjustment requires knowing a lot about the value of goods and services that were available in the 1700s and today.

The information we're familiar with doesn't work this way. A bit today is a bit in a billion years. We don't have to adjust for infor-flation because there are gazillions of GB out there on the internet. The basic unit of information, the bit = $\ln 2$ "nats", or the information in a flip of a fair coin, is as static as mathematics. We can exchange 8 billion bits and the result is the same the same way we exchange 8 bits [1].

In the transfer of economic information, the total amount of money does define the unit of information. To be precise, both the total amount of money and the size of the economy come together to define the unit of information we refer to colloquially as "value". And actually, it's "nominal value", since we'd say a pound of bacon is worth a dollar (money units) in the 1970s and 5 dollars (money units) today.

With constant units of information, changing units is pretty easy. We have

$$
\log_{2} 2 = \ln 2/\ln 2 = 1 \; \text{bit}
$$

or 1 bit = (ln 2 nat)/(ln 2 nat/bit). To deal with variable units of information we need some tools. The base of the logarithm determines the unit of information and we have the relationships

$$
\log_{b} x = \frac{\log_{c} x}{\log_{c} b} = \frac{\log_{e} x}{\log_{e} b} = \frac{\ln x}{\ln b}
$$

If the base of the logarithm $b = 2$ we're measuring information in bits. If it's $b = 8$, we're measuring in bytes. We'd like to measure information in terms of money. That means, if $M$ is the amount of money out there and $NGDP$ defines the size of the economy, the unit of information defined by money is

$$
\log_{M} NGDP = \frac{\ln NGDP}{\ln M}
$$

I call this the information transfer index  here (or actually its inverse$1/\kappa$), but really it just defines the units of information (i.e. "value") and is connected to the price level (which is how you adjust for inflation). I've referred to the changing value of $\kappa$ as the "unit of account effect"; we could call the basic mechanism in the quantity theory where approximately $P \sim M$ the "medium of exchange effect" analogously.

As for my second question above: which monetary aggregate? Let's leave that up to observational evidence. It seems $M$ is the currency component of the monetary base (physical currency including printed bills and coins).

[1] At least if are well below the Bekenstein bound. When we start transferring information on the order of the number of bits defined by the causal horizon of the exchange, we'll need to take general relativity into account.

One interesting side note: if we measure information in terms of a fraction of the information in the causal horizon defined by the Hubble volume instead of bits, the Bekenstein bound would become:

$$
I \leq -\frac{RM}{2 \log \ell_{p}H_{0}}
$$

where the constants are the Planck length and the (inverse) Hubble time. This makes the information time dependent in these "natural" units since the Hubble length c/H0 is based on the age of the universe.

[2] Actually, a better way to describe Money as Memory is to say that "I have 5 dollars because I did something worth 5 dollars in the past that the community owes me for, and I'm discharging the community's obligation because you are giving me 5 dollars worth of goods or services in return".

Update (6/8/2014): Reworded a couple sentences above, added a few lines for clarity and added the second paragraph to the footnote.

Update (6/14/2014): Added the second footnote.

Seattle's new minimum wage and information theory

I mentioned yesterday I was working on a post about the minimum wage increase in my hometown of Seattle to 15 dollars per hour [1]. I worked up an analysis with the information transfer model and it turns out it is largely consistent with Scott Sumner's rather even-handed assessment of the possibilities of the impacts. In particular, Sumner points out that the effect of a minimum wage increase is largely model-dependent (although he doesn't use those words exactly).

One take-away from this analysis, though, is that the "economics 101" view that an increase in the minimum wage leads to fewer jobs involves unstated assumptions, and, as it is usually presented, wrong. The easy way to see that (from my perspective) is that a number p > 7.5 and a number p > 15 carry the same amount of information, so that inside the market itself, there really is no difference between them. Just like moving the decimal place to the right two places on all prices (a scale factor), shifting all minimum wage workers hourly wage by 7.5 (constant shift) has no impact on its own. The impact lies entirely in the interaction of the labor market with the markets for goods and services.

Let's begin at the beginning with an ideal system with a labor supply S, money supply M (it doesn't matter which aggregate at this point) and labor demand D. We can follow this post and write down a relationship between the information transferred to the supply from the demand that is mediated by money

$$
\text{(1) } I_{s} = n_{s} \log s = n_{m} \log m = n_{d} \log d = I_{d}
$$

We'll ignore the demand side of this (it isn't necessary here, basically we can say that demand and money supply are the same thing in this analysis) and just write down the differential equation resulting from the second equals sign (taking $n_{s} = S/dS$ and $n_{m} = M/dM$)

$$
\text{(2) } \frac{dS}{dM} = a \; \frac{S}{M}
$$

where $a = \log s/\log m$ is a constant. The left hand side of equation (2) is the exchange rate of labor supply for money basically the price of labor (i.e. wages). This differential equation can be solved holding each of $S$ and $M$ constant, resulting in supply and demand curves pictured in the graph below. This result has a mimimum wage of zero. But before we get to the graph, let's apply a finite minimum wage by adding a constant $c$ to the left hand side (or subtract it from the right hand side), leaving you with the equation

$$
\text{(3) } \frac{dS}{dM} = a \; \frac{S}{M} - c
$$

If you solve this differential equation you get the thick blue curve in the graph below:

We've recovered the "economics 101" picture: the price of labor goes up, the number of jobs goes down. However, there is something that's been left out. See, the "demand curve" (black) is the solution to equation (2), not equation (3). In the graph below, I've added in the actual minimum wage price (= 1.5 here) as well as a dotted gray line:

Our supply and demand curves for a given market should be the solution to the same differential equation. The result of using (3) for the demand curve as well gives the picture below:

The equilibrium price has gone up (from 1 to 2.5 = 1 + 1.5), but the equilibrium quantity demanded is the same as for the zero minimum wage solution. The implied assumption in the "economics 101" analysis must be that the supply curve shifts to the right (or the demand curve shifts to the right):

This assumption can follow from budget constraints, supply shocks or even expectations, but it is importatnt to realize it is model dependent.

Now let's look at non-ideal information transfer. In that case, the supply and demand curves (via Gronwall's inequality) form a bound on the equilibrium price and quantity demanded/supplied. Let's also assume that every state (point in quantity-price space) is equally likely, i.e. maximum ignorance. We can use that to see where the averge wage would fall over the purple triangle of allowed states (you'll need to integrate $e^{-x}$ to $\infty$). In the graph below, the area above the purple line and below it are equal:

One thing to note is that in this picture, the most likely wage is anchored at the minimum wage (in this case = 0), while the average is a little bit higher (it's about = 0.17).

If we perform the shift in equation (3), the whole diagram basically shifts upward. This view makes sense intuitively -- if we assume the minimum wage labor market is small enough to be considered independent of aggregate demand, then when apply a minimum wage increase, the most likely wage is at the new minimum wage and the average wage is slightly above it.

The key take-away here is that the fall in employment with a minimum wage increase is not "economics 101", but rather a model dependent result. "Economics 101" should say that an arbitrary shift in the price of good in a single market does not matter.

[1] I try to avoid "dollar signs" because they mess with the mathjax and I haven't figured out a way to fix it.

On travel again = light blogging

I'm on travel for work again (LA). It will probably mean light blogging and lack of response to comments and emails.

I'm working on a post about the minimum wage as my adopted hometown of Seattle is poised to phase in 15 dollars per hour. It is not obvious that the simple supply and demand argument follows. The information detecting capacity of p > 0 and p > 15 are pretty much identical, which means that the primary effect would have to come in through some other market. That means traditional economics is making implicit assumptions in treating this problem.