Top ten successes of the information transfer model

(1) Explaining supply and demand: If you set up a description of an information transfer process moving information through a channel from a source to a destination and make some identifications (the demand as the information source, the supply as the information destination and the price as a detector of a signal sent from the demand to the supply), you basically get out fundamental economic supply and demand logic including supply and demand diagrams (another way to visualize diagrams is here). I've used the notation Price:Demand→Supply to describe these models of a market.

(2) Modeling the price level: This is basically done with a quantity theory of money, using the information transfer framework with a market P:NGDP→MB (the price level detects signals from the aggregate demand or NGDP to the monetary base). The major results for the US are here. The fit to RGDP growth is rather remarkable. I also fit the data for the entire period from 1929 to 2013 using three monetary policy regimes with great success (market-based systems before and after WWII, with a pegged interest rate model in between). More on monetary regimes is here including a theorem that increasing the monetary base will eventually lead to an information trap (basically a liquidity trap). One of the key pieces of the model that allows such success in fitting the data is the way the behavior of money as a unit of account is incorporated in the model via a varying information transfer index. In a sense, the monetary base is the number base you count the GDP in much like binary is the number base you count data in, but in the latter case the base is fixed at two. It also works for the EU and Japan.

(3) Modeling interest rates: The best result was the same fit the data from 1960-2013 also fit 1929-2013. Overall, the simple model (derived from the IS-LM model) does a good job of describing the US, Japan and the EU.

(4) Modeling Japan's lost decade: Japan's sluggish economy and immunity to inflation is a major problem in economics. The information transfer model describes the price level and interest rates and the major conclusion is that Japan is in an information trap, a condition where monetary policy has little influence on the price level.

(5) Deriving the quantity theory of money (and explaining deviations): The basic equation in the information transfer model for the market P:NGDP→MB looks like the equation of exchange. You can also derive the major result of the quantity theory from the model (specifically that for high inflation, the rate of monetary base growth equals the rate of inflation or r = i). But it is even better than that. The information transfer model explains the deviations at low inflation from the basic r = i picture.

(6) Deriving the IS-LM model: The supply and demand framework allows a straightforward derivation of the curves in an IS-LM diagram.

(7) Okun's law: A relatively straightforward application of the model building capability of the information transfer model allows us to build a market P:NGDP→LS (price level detecting a signal from the aggregate demand to the labor supply) that recovers Okun's law.

(8) Walras' law: This follows from the supply and demand model via some algebra.

(9) Explaining how the EU can be in a liquidity trap but not at the zero lower bound: Keynes' original work allowed a liquidity trap to occur at any interest rate. Later economists argued that it could only occur at zero interest rates (as they can't be lowered). Now Paul Krugman argues that the EU is in a liquidity trap even though EU rates are not actually zero (as Scott Sumner points out); Krugman's explanation is that they are close enough. The information transfer model shows that the liquidity trap rate is actually a function of GDP and the monetary base. For example, it is about 2% for the EU, 1% for Japan and 0.1% for the US.

(10) Explaining the history of economic thought in the US since the early 1900s: Since the path of GDP and the monetary base give us interest rates and the price level since the early 1900s, we can use the fact that the model appears to be a quantity theory of money and an IS-LM model at different times to understand the currents in the history of economic theory. In particular, we expect to get quantity theories before the depression (check) displaced by interest rate and liquidity trap theories in the 1930s (check), a resurgence of quantity theories in the 1960s-70s (check) and a return to interest rate and liquidity trap theories today (check).

Ch-ch-ch-changes.

I was re-reading this post (and the one Noah Smith links to in it) about what we mean by economic cycles, (or even how to extract the effects of policy changes; this seems like a bad way to do it) and thought I should update my take on it.

The procedure I came up (for what is essentially a quantity theory of money) is to extrapolate where the economy would be if the monetary base increased from time T1 to time T2 (which generally increases the price level) but GDP remained the same when adjusted for inflation. Call this point GDP0(T2). I then looked at the remainder when I took the difference GDP(T2) - GDP0(T2). Basically this is the distance the actual GDP is from the GDP you expected to arrive at given monetary policy. Graphically, the process is described here where I referred to that remainder as a "nominal shock". Here is a graph based on NGDP data (dark blue is a LOESS smoothing and red indicates recessions):

I also went back to this fit to all the data since 1929 (using GNP instead of GDP) and extracted these nominal shocks (the colors correspond to the different monetary policy regimes described in the link):

Six points

Borrowing a theme from my other blog Spittle-Flecked Ire (which I've been ignoring), I'd like to say these statements:

1. David Beckworth looks [at] a bunch of natural experiments on the efficacy of monetary policy at the zero bound ... [1] 
2. These three quasi-natural experiments indicate that there is much monetary policy can do at the ZLB. [2]

are based on exactly six data points across three countries. That is neither "a bunch" nor do they "indicate" anything. Here are the two data points for Japan:

It seems like confirmation bias. I still say RGDP growth will continue its post-2000 average for Japan,  its post-2009 average for the US, and its post-2009 average for the EU. These predictions are based on hundreds of data points (in fact, all available data).

For completeness, here are the four other points:

Analyzing the EU with the information transfer model

Today I analyzed the EU with the information transfer model and produced the analogous graphs from this post (the price level), this post (interest rates) and this post (interest rates, the liquidity trap and the information trap). The monetary base data came from here and the GDP, GDP deflator and interest rate data came from FRED. Altogether the story is similar to the US and Japan. The EU is in an information/liquidity trap. The interesting result is that this tests the hypothesis that the zero lower bound is not a necessary requirement to be in the trap. Scott Sumner frequently points out that the ECB is not at the zero lower bound (rates are at ~1%) as an argument against Paul Krugman and the liquidity trap. However, this analysis shows that the "liquidity trap" interest rate is almost 2% in the EU (in Japan it is 1% and in the US it is 0.1%).

First I will start with the price level fit (model in blue, GDP deflator data in green):

For reference, here is the information transfer index (quite high since the Euro's inception):

For fun, here is the 3D price level surface (again, model in blue, deflator data in green):

Now we come to the interest rate models. I used the discount rate, which roughly tracks the 3-month rate. There are probably some issues -- the ECB (and markets) seem to treat Greek debt differently than German debt so there is no overall "EU" rate -- but the fit does a decent job anyway (discount rate in green, model calculation in blue):

The money-graph behind the bold claim at the top of this post is last. The constant interest rate contours in (MB, GDP) space are shown as red lines, the information/liquidity trap criterion is shown as a black line and the path followed by the empirical data is shown in blue. The key takeaway is that the "liquidity trap" rate (given by the location of the black line given the current level of GDP) is almost 2%:

Addendum

I'd like to point out that the first graph is a non-trivial result; the Euro monetary base does not have a immediately obvious relationship with the price level:

Exit through the hyperinflation

Update 5/14/2015 

I updated the graphs at this link. It doesn't change any of the conclusions, so I'll leave this post unchanged.

In the last post I mentioned that I would look at accelerating inflation as a potential way to exit from a liquidity/information trap. First let me start with interest rates. The information framework describes the effective fed funds rate very well from the 1960s onward. In order to look at even earlier times, I looked at the 3-month interest rate instead (which is approximately equal to the Fed funds rate, and is shown in green) and added a few points I managed to get from a Fed paper on the Depression for the years 1929-1931 (green dots). Even without re-fitting the data to the entire 1929-2013 period (fit only to 1960-2013), I got a remarkable fit:

The model result is shown in blue. There is no "information trap" in the interest rate market model r:NGDP→MB, (NGDP standing in for aggregate demand, and MB being the monetary base) just a zero lower bound problem (they can be related in the sense that they happen at the same time).  Therefore the same parameters can be used for interest rates across the phase transition between the Depression and Post-WWII periods. However, if we look at the interest rate data we see the first sign of how a central bank can exit a liquidity/information trap and usher in that phase transition. In the highlighted region in the graph (World War II and afterwards), you can see the Fed pegged interest rates. That is key to leaving a path given by a market-based monetary policy and induces hyperinflation (or just accelerating inflation). In the information transfer framework, that is incorporated going from a floating, market-adjusting information source i.e. aggregate demand and destination i.e. money supply to a floating, market-adjusting information source and a constant, market-ignoring information destination.

This all means we need to use different models to form the complete story. We have P:NGDP→MB with market MB before WWII, statist MB during WWII and a return to market MB after WWII. In the graph below, I fit the models to the data before (red) and after (blue) WWII (from this post). The new result is adding the piece in the middle using the accelerating inflation model (purple):

This is a much better description than the original phase transition model. Since the 3D price level diagram includes what are basically three distinct models, they don't represent a single surface. I've shown them as patches in this figure (with the empirical data shown as a blue line):

One additional interesting result is that it is possible this non-market/statist/constant information destination (money supply/central bank) continued until the 1980s, at least in terms of the price level. One thing to recall is that the Federal Reserve Act was amended in the 1970s to include inflation as part of a dual mandate. With that legislation the market P:NGDP→MB goes from having a non-market information destination to a market destination. Why do I include this possibility? Well, the accelerating inflation model works well all the way up to the 1970s if you let it:

Although in that case the fit no longer works during WWII.

What does this exit via accelerating inflation picture mean for the US today? Well, it appears as though the Fed could peg interest rates a given time period (say, 10 years), markets be damned, and we would leave the liquidity/information trap. This rate doesn't even have to be zero as far as the model goes -- it could be 3%. That would be illegal (because of the dual mandate) so Congress would likely have to amend the Federal Reserve Act. This is politically impossible right now, so we're probably doomed to a Japan-like lost decade or two.

Interest rates and monetary policy or: Never reason from a price change

Scott Sumner has a great post up today. Great, in the sense that he lays out with some clarity how he sees interest rates (I will call these rules S1 - S4):

1. Moves toward easier money usually lower short term rates. The effect on long term rates is unpredictable.
2. Moves toward tighter money usually raise short term rates. The effect on long term rates is unpredictable.
3. Extremely easy money policies (hyperinflation) almost always raise interest rates.
4. Vice versa. [Which I have taken to mean "Extremely tight money policies (disinflation) almost always lower interest rates".]

But his immediate reaction (as well as Yglesias's) to the immediate reaction to the Fed's guidance that it won't start "tapering" violated his maxim to never reason from a price change.

So what does the information transfer model say about monetary policy and interest rates? I will make a list like Sumner did, but the interesting piece comes in when you recognize that markets don't know about the information transfer model. Hence the market reaction to easier money is always lower rates, but the long run depends on whether you are in a high information transfer (IT) index regime or not:

1. If the IT index is high, increases in the monetary base lowers interest rates (increasing MB causes NGDP to stay the same or drop slightly, decreasing NGDP/MB ~ r). Markets will also shift to lower rates as MB is increasing (perception of easier money).
2. If the IT index is high, decreases in the monetary base raises interest rates (decreasing MB causes NGDP to stay the same or increase slightly, increasing NGDP/MB ~ r). Markets will also shift to higher rates as MB is decreasing (perception of tighter money).
3. If the IT index is low, increases in the monetary base will raise interest rates (increasing MB causes NGDP to increase more, increasing  NGDP/MB ~ r). Markets will initially shift to lower rates as MB is increasing (perception of easier money) but the equilibrium will drift to higher rates over the long run.
4. If the IT index is low, decreases in the monetary base will lower interest rates (decreasing MB causes NGDP to drop, decreasing NGDP/MB ~ r). Markets will initially shift to higher rates as MB is decreasing (perception of tighter money) but the equilibrium will drift to lower rates over the long run.

High and low IT index is dependent on how close the economy is to the information trap criterion, and some of the reasoning can be understood from these two posts. I will call these rules I1 - I4. Let's review Sumner's rules in the light of the information rules:

1. Moves toward easier money usually lower short term rates. The effect on long term rates is unpredictable. This is because this case includes both I1 and I3. Markets like what they think is easier money, but the long run depends on whether the information transfer index is high or low. 
2. Moves toward tighter money usually raise short term rates. The effect on long term rates is unpredictable. This is because this case includes both I2 and I4. Markets don't like what they think is tighter money, but the long run depends on whether the information transfer index is high or low.
3. Extremely easy money policies (hyperinflation) almost always raise interest rates. This is rule I3. Sumner is basically recalling the 1970s here when the information transfer index was low. However this does not apply during the 1930s or since 2008. Sumner rationalizes this by saying money during the 1930s and 2010s is tight (see next rule) despite the massive increases in the monetary base (the increase is expected to vanish ... eventually).
4. Extremely tight money policies (disinflation) almost always lower interest rates. This is rule I4. Sumner believes this is the situation in the 1930s, the 1980s and now (his views are consistent with the information transfer model if the IT index is always low). Rule I4 only applies if the IT index is low, which works in the 1980s, but is incorrect in the 1930s and since 2008 since the IT index is high. We do not have "tight" money that is leading to low interest rates, we are in a liquidity trap which happens at low interest rates and monetary policy is ineffective: money is neither tight nor loose.

Another way to see this is by eras:

• The 1930s: Sumner says S4 explains. I say I1 explains. (We disagree on whether the Fed could intervene.)
• The 1970s: Sumner says S3 explains. I say I3 explains. (We agree inflation was a monetary phenomenon.)
• The 1980s: Sumner says S4 explains. I say I4 explains. (We agree that monetary policy brought inflation under control.)
• The 2010s: Sumner says S4 explains. I say I1 explains. (We disagree.)

I think this is also a good place to place to compare and contrast the information transfer model with two other views of our current situation:

Scott Sumner believes the current situation is like the US in the 1930s and Japan. Money is tight and the Fed could create expectations that allow economic growth but it is failing. Fiscal stimulus will be offset by continued Fed failure (and would be unnecessary if the Fed stopped failing). 

Paul Krugman believes the current situation is like the US in the 1930s and Japan. Monetary policy is ineffective at the zero lower bound (a liquidity trap). The Fed could create expectations that allow economic growth but it is almost impossible for the Fed to do credibly. Fiscal stimulus will boost the economy because the Fed has no traction to offset it. 

The information transfer model shows the current situation is like Japan [1]. Monetary policy is ineffective when the base becomes too large relative to NGDP (an information trap). The Fed cannot generate inflation via expansionary monetary policy nor expectations of expansionary monetary policy unless they abandon targets [2]. Fiscal stimulus could [3] boost the economy because the Fed has no traction to offset it (monetary policy shifts are orthogonal to NGDP shifts). [1]

I hope I was fair to the viewpoints of Sumner and Krugman. Overall the information transfer view is more similar to Krugman, but does not include the ability to leave the liquidity trap via expectations (the central bank credibly promising to be irresponsible). However there is a possibility to leave the trap by generating accelerating inflation (see footnote [2]) by, say, printing money without regard to macroeconomic targets (inflation, NGDP growth) and giving it to people. I guess this is credibly promising to be irresponsible. The key is that the central bank has to stop receiving information from the economy and reacting to it. This may have been what happened in the 1940s. As you can see from the following graph the ratio of NGDP to the MB grew significantly (inflation rates reached 15-20%) which would have significantly decreased the IT index:

I've gone a little off the original subject of interest rates and monetary policy. I think I will devote a future post to this hyperinflation exit picture of the Great Depression and what it could mean for today.

[1] If this phase transition picture is correct, then a) the situation is like the 1930s as well and b) the only likely solution to our current problems is to create a phase transition/redefine money (e.g. leaving the gold standard and WWII in the 1940s, or potentially the accelerating inflation -- see [2] below). The modern equivalent may be switching to electronic money. I am uncertain of these conclusions and monetary reset may be easier than it seems from the only available case in the data. 

[2] Hyperinflation/accelerating inflation will still leave us without a monetary policy that functions with targets. We would just have another recession in the future as the Fed tried to stop inflation. The key question is whether this will leave us at a higher NGDP/MB ratio or not. NGDP grows faster than MB as long as the IT index is below 1 (at IT index = 1 they grow at the exact same rate). Therefore engaging in accelerating inflation can move us to lower IT index. Low inflation targets will move us back towards higher IT index causing the cycle to start anew. 

[3] I haven't worked it out in detail, but it seems that the primary mechanism for fiscal stimulus failing is monetary offset and that is impossible in a information/liquidity trap. In fact, fiscal stimulus may have offset the negative impact of monetary stimulus in an information trap.

Japan, interest rates and the liquidity trap

As promised, here is the version of the figure from the link with data from Japan. Following that diagram, the level curves of the interest rate surface (constant interest rate) are plotted in red and the information trap criterion is plotted as a dashed black curve. The modeled path of the Japanese economy in the space is plotted in dark blue with a few years indicated:

Interestingly, Japan seems to have stumbled into the information/liquidity trap in the 1990s because rates were low and the liquidity trap rate was (relatively) high. The liquidity trap rate reached 1%, whereas the US pushed itself over the dashed line and rates below 1% with quantitative easing while the liquidity trap rate was of order 0.1%.

The liquidity trap and the information trap

I am going to explore the relationship between liquidity traps (LT) and what I called information traps (IT) which manifest similar phenomena. The original meaning of the liquidity trap is a horizontal demand curve for money at some interest rate. In Keynes' work, it was possible for this to happen at any interest rate, however later work purported that this could only happen at the zero lower bound (interest rate r ≈ 0). As there has been no evidence for an economy experiencing a liquidity trap at r > 0 (even during the Depression, rates were near zero), the economic consensus is that the liquidity trap and the zero lower bound problem (ZLB) are one in the same. It turns out Keynes' original formulation is more accurate in the information transfer framework; LTs can happen at any interest rate -- it's just that past experience makes us think LT and ZLB are the same.

In the information transfer framework, the key indication of an information trap is an information transfer index, which determines how much effect the unit of account has in the economy, is closer to 1 than to 0.5 (in the latter case, the information transfer model of the market P:NGDP→MB with the unit of account effect and the quantity theory of money are the same thing). In the liquidity trap (IS/LM) model, the key indicator is the interest rate. In this graph, I've plotted the Fed Funds rate (green, left axis) and the information transfer model of the rate (blue, left axis) alongside the information transfer index (red, right axis):

It is immediately obvious that the relationship is not immediately obvious. The index rises monotonically while the rates rise and fall. I decided to go back to the 3D picture of the price level surface (where the information trap criterion appears as a ridge on the surface) and plot the equivalent interest rate surface. Here is that graph with the interest rate surface in red and the price level surface in white (constant values i.e. level curves of the z-coordinate are shown with lines on the surfaces the units of the z-axis are log interest rate r and log price level P, but the relative magnitudes don't matter):

I've drawn the ridge line (information trap criteria) as a dashed curve on the price level surface. Everything with a larger monetary base than this curve is in an information trap -- the derivative of the price level with respect to the monetary base is zero or negative meaning the central bank can't create inflation with monetary policy. You can see that the projection of the dashed curve onto the red interest rate plane results in a non-trivial relationship. You can see it more clearly if you look at the interest rate plane itself:

The level curves of the red surface (constant interest rate) are plotted in red, two lines of constant information transfer index are plotted in black along with the information trap criterion (dashed). The modeled path of the US economy (from the graph at the top of this post) in the space is plotted in dark blue with a few years indicated. You can see that the relative size of the monetary base that leads to an information trap gets smaller as the economy (NGDP) grows (for a larger NGDP, a larger MB is necessary to be in an information trap but MB/NGDP decreases).

According to the information transfer model, interest rates are always "low" when you're in an information trap, but "low" is a relative term. From the 1960s until the present, you had to have interest rates below 1% to be in an information trap. In the 1970s the rate had to be below ~ 0.01%. Starting in the 1990s, an interest rate below 0.1% became sufficient (the current August 2013 effective Fed funds rate is 0.08%). The information trap rate will reach 1% when NGDP reaches about 22 trillion dollars. 

Note that the validity of the interest rate model may not reach such low interest rates, so at best we can say is that the information trap interest rate has been r ≈ 0 and will not become large for quite some time. Therefore, ZLB ≈ LT ≈ IT for now but it is theoretically possible for LT ≈ IT above the zero lower bound representing a flat money demand curve at non-zero interest rate as Keynes' originally formulated it. 

PS An upcoming post will reproduce this graph in the case of Japan.

Modeling interest rates in Japan

I should have added a piece on the interest rates to this post on the information transfer model of the price level in Japan. Anyway, following this post on modeling the interest rate in the US, we get a fairly good model for Japan (using the estimated data for NGDP):

This uses this measure of the interest rate, but I still haven't found a robust way to choose a particular interest rate to act as the empirical data.

One bank to topple them all

This is the 5th anniversary of the bankruptcy of Lehman Brothers; I'd like to point out that the economic shock that caused the recession occurred precisely in September (at least in the information transfer model).

It was this large shock coupled with the relatively high information transfer index at the time which rendered monetary stimulus ineffective that led to our current predicament. In the graph below, aggregate demand perturbations are in blue while monetary perturbations are red.

The mystery of the Japanese monetary base

I'm going to start with a quote from Scott Sumner:

Just to be clear, in this post I’m treating QE [Quantitative Easing] as temporary, as does [Paul] Krugman. We both believe that permanent QE is highly effective (at least I think we both do.)

Since Krugman and Sumner agree, I will take this view as some kind of consensus view. It does allow you to make some sense of, say, the US monetary base (MB) and the US CPI (less food, energy). You see MB chugging along and the price level following suit until, boom, in 2009 as the Fed conducts some QE. The base shoots up and and the price level just sits there. The graph at the end of this post shows what I mean. Obviously, the economy thinks the QE is temporary and so is experiencing inflation based on the expected base after the QE (which is based on inflation guidance from the Fed and therefore the expected path of the monetary base ex-QE). It seems like a logical argument, doesn't it?

This view is insane.

Let's take a look at the Japanese monetary base to see how insane this view is. In the graph below I plot the Japanese CPI (green) and the Japanese monetary base (dark blue) along with the latter's trend (dashed blue); both are normalized to 2005.

Why is the consensus view insane? Well I've taken three tangents to the CPI at different times. We can see the first one roughly parallel with the MB trend. But the next one is almost horizontal. And the third tangent has a negative slope. To me, it is obvious the direct relationship between the monetary base and inflation is breaking down. The consensus view instead believes either the Bank of Japan will eventually cut the monetary base by about 80% to the location marked with a blue asterisk or what can only be described as hyperinflation will take hold leading to a five-fold increase in the CPI at some undetermined point in the future as the markets realize the trend increases since the 1990s are going to stick around.

There is a third possibility: the Bank of Japan has convinced everyone that it is so good at fighting inflation it can grow the MB at any rate it desires and inflation will stay the same. But if they're so good at controlling inflation then why has inflation gone negative and they've struggled to hit their targets? Yes, Abenomics, but the markets think monetary policy has some effect (remember the consensus view above). Of course markets will react positively to Abenomics.

Now I believe expectations are part of the invisible hand, that is to say, I see no reason to separate them out as different information processed by the market. I have previously showed that Japan's situation can be explained with the information transfer model; I am going to expand on that here and show how the changing relationship between the base and the CPI can still come from a purely monetary theory. In the previous work, I only looked at the period since 1994 because that was the only data that exists of sufficient quality in the FRED database. There was some data from before 1994, but it was in US dollars which does not work because the data needs to be in the same units. Therefore I used exchange rate data to try and figure out what the Japanese GDP was before 1994. In this graph I show the 1994 data (dark red), the GDP in dollars converted to yen (gray) and the estimate of the Japanese GDP since 1970 derived from that (light red):

If I use this extrapolated GDP data in the information transfer model (along with the monetary base) use the fit to the CPI (green) for the post 1994 data, I get a pretty good fit (blue, the "out of sample fit" is dashed blue):

If I do the fit over the entire range (1970-2013), it is even better (I show the exchange rate derived data in gray):

With this graph I show how the changing relationship between the monetary base and the price level can be incorporated in a quantity theory without expectations. The key difference from the quantity theory is the inclusion of the unit of account effect that comes from building the model in information theory.

Addendum: Do we need GDP data?

Actually, this fit was so good I wondered what the result would be if I used the monetary base and the CPI to extract the GDP data over the entire range. This unfortunately didn't work all that well as the numerical method failed to find a solution for the 1980s (literally, 1980-1989, so the results before and after that period are just connected with an interpolated line):

It is quite noisy and the best that can be said about it is that it is generally of the right order of magnitude. Here is the numerical solution (blue) evaluated for the CPI (data in green) with the linear interpolation (the only region it doesn't match) included:

The equation involves GDP inside a logarithm and in the exponent, so numerically this is a difficult situation and I should stick to the fit with the extrapolated data.

Walras' law

Paul Krugman made a reference to Walras' law today which inspired me to try and figure out what the "law" is in the information transfer framework. In this first attempt, I managed to force the incompressible fluid condition (divergence is zero) into something resembling Walras' law and tentatively declared success. Start with a series of markets: 

$$ P_i : Q^{d}_{i} \rightarrow Q^{s}_{i} $$

And take the differential of the demand vector $\mathbf{Q}^{d}$ and set it to zero

$$\text{(1) } \sum_{i} dQ^{d}_{i} = 0 $$

We can rewrite this as

$$ \sum_{i}\sum_{j} \frac{dQ^{d}_{i}}{dQ^{s}_{j}}dQ^{s}_{j} = 0 $$

If we consider that the markets are independent (the goods supplied are not close substitutes so that the demand for $i$ is not directly [1] changed by a change in the supply of $j$), then it is reasonable to assume 

$$\text{(2) }\frac{dQ^{d}_{i}}{dQ^{s}_{j}} = \delta_{ij} \frac{dQ^{d}_{i}}{dQ^{s}_{i}}$$

So that

$$ \sum_{i}\sum_{j} \delta_{ij} \frac{dQ^{d}_{i}}{dQ^{s}_{i}}dQ^{s}_{j} =  \sum_{i} \frac{dQ^{d}_{i}}{dQ^{s}_{i}}dQ^{s}_{i} = 0$$

And using the definition of the price we find

$$ \sum_{i} P_{i} dQ^{s}_{i} = 0 $$

This is one half of Walras' law where the sum of the values of the excess supplies ($dQ^{s}$) is zero. To get the other half (the sum of the values of the excess demand is zero) involves a little more work. This is because in the information transfer model we have the relationship $P_{i} dQ^{s}_{i} = dQ^{s}_{i}$; effectively, the units in Walras' law as written are wrong. Demand has units of value already (price times number supplied). However, if we have floating information sources and destinations, the solution to the differential equation relating the supply and demand is:

$$ \frac{Q^{d}_{i}}{Q^{d}_{i, \text{ref}}} = \left( \frac{Q^{s}_{i}}{Q^{s}_{i, \text{ref}}} \right)^{1/\kappa} $$

Now if $\kappa = 1$ (an assumption about the elasticities of supply and demand), then we can say

$$dQ^{d}_{i} \frac{Q^{d}_{i, \text{ref}}}{Q^{s}_{i, \text{ref}}}  = dQ^{s}_{i} $$

So that finally

$$ \sum_{i} P_{i} dQ^{s}_{i} = \sum_{i} P_{i} \frac{Q^{d}_{i, \text{ref}}}{Q^{s}_{i, \text{ref}}} dQ^{d}_{i} = 0 $$

Which mostly recovers Walras' law. The aspect about the excess supply adding to zero is a direct result of the assumption of a divergenceless demand vector $\text{(1)}$, while getting the demand side involves some additional assumptions about the elasticity of demand. However! If we don't care about getting Walras's law exactly right, we can start from the differential of the supply 

$$ 0 = \sum_{i} dQ^{s}_{i} = \sum_{i}\sum_{j} \frac{dQ^{s}_{i}}{dQ^{d}_{j}}dQ^{d}_{j} $$

Making the close substitutes assumption $\text{(2)}$, we find

$$ \sum_{i} \frac{1}{P_{i}} dQ^{d}_{i} = 0 $$

So that in the information transfer framework, Walras' law is

$$ \sum_{i} \frac{1}{P_{i}} dQ^{d}_{i} = \sum_{i} P_{i} dQ^{s}_{i} = 0 $$

And is related to $\mathbf{Q}^s$ and $\mathbf{Q}^d$ being incompressible vector fields (with can then be related to conservation of mass).

[1] By directly changed, I mean via and explicit dependence $Q^{d}_{i} = f(Q^{s}_{i}, ... Q^{s}_{j} ... )$. We have all the markets interacting with each other via the divergenceless condition, but absent that, the markets would not depend on each other.

An information transfer history of the United States

I mentioned earlier that I would discuss how the historical theoretical framework of economics would look if the information transfer model was correct in a post; this is that post.

From the point of view of information transfer economics what we should see is a pre-Depression vogue for quantity theories of money (since the information transfer index $\kappa$ would be low, on the order of $\kappa \sim 0.5$, which means the quantity theory is a good description) that drifts into interest rate theories as the quantity theories lose their power (an interest rate description is not necessarily affected by the unit of account effect). Then would come the Depression in the 1930s and theories where monetary policy loses any effectiveness as $\kappa \rightarrow 1$. While previous recessions would have been "V"-shaped, appearing mostly monetary in nature since $\kappa \sim 0.5$, this would begin a "U"-shaped depression. After the depression would follow a resurgence of quantity theory as $\kappa \sim 0.5$ (likely with a revisionist bent towards why quantity theory failed in the Depression). Again these would lose power and give way to interest rate-based theories and "U"-shaped recessions starting in the 1990s as $\kappa$ increases. Finally economic theory would culminate in a resurgence of Depression-era theories as $\kappa \rightarrow 1$ in the late 2000s. This path of the information transfer index $\kappa$ looks like this one below (from this post):

So how well does this picture work? Stunningly well.

Fisher published The Purchasing Power of Money in 1911 (containing the equation of exchange) and the "Cambridge school" put forth their quantity theories of money in the 1920s. Keynes published his Tract on Monetary Reform in 1923 (Brad DeLong called it the best monetarist book ever written). In these cases, M1 was the aggregate of choice but when $\kappa \sim 0.5$ M1 is parallel to MB in logarithm (differing by just an overall scale factor). As $\kappa$ increased, Fisher moved on to interest rates in his magnum opus The Theory of Interest in 1930. Keynes published his General Theory in 1936 which contains the guts of the IS/LM interest rate model and showed how it was possible for the economy to be mired in a "liquidity trap". Keynesian theory took over the economic mainstream. However, since the Keynesian view concluded that the money supply doesn't directly affect inflation (it had been developed in an environment where $\kappa$ was large, thus this was an empirically true relationship at the time and had been for decades before). Concentrating on unemployment, the lack of focus on inflation in a low $\kappa$ environment would lead to inflation. The Keynesian theory would remain a relatively accurate description of the economy as long as interest rates remained above zero. Recessions were "V"-shaped after the war.

However, as inflation increases in the information transfer model, the quantity theory becomes more accurate. Friedman and Schwartz published A Monetary History in 1963. They put forth the revisionist monetary theory that it is M2, not MB or M1, that controls the economy (it is more likely M2 dropped in response to the Depression than caused it). As $\kappa$ approaches its lowest value in the late 1960s and 1970s, inflation gets bad enough to cause a paradigm shift. The Philips curve breaks down because the quantity theory explains the price level (since $\kappa \sim 0.5$), not unemployment. Lucas issues his critique in 1976. The Fed even claims to try Friedman's idea to target M2, but it is far too unstable since MB controls the price level (in the information transfer model) and M2 is derived from it by the private banking sector hence contains a "business cycle" component -- which is why it dropped during the Depression. The Fed is given its dual mandate in 1977 to control inflation and unemployment. The Fed decides to cause a serious recession, raising interest rates starting in the late 70s in order to control inflation. It is a success (because $\kappa \sim 0.5$) and monetary policy becomes a prominent component of macroeconomic stabilization.

As the information transfer index continues to increase, the direct relationship between between the price level and the money supply begins to fade again. Plus there were all the old Keynesian successes to understand (or revise)! Rational expectations based theories enter the field. Monetary policy no longer controls the empirically estimated price level, but instead controls the "expected" price level. Keynesianism plus NAIRU or monetarism plus interest rates dominate "saltwater" and "freshwater" economics respectively, essentially interpolations between the $\kappa \sim 1$ and $\kappa \sim 0.5$ information transfer theories (the former expands around $\kappa \sim 1$, while the latter expands around $\kappa \sim 0.5$). This is somewhat successful and we get "the great moderation", but "U"-shaped recessions have returned starting in the 1990s. The Fed doesn't cause recessions with monetary policy as it used to. Japan reaches $\kappa \sim 1$ earlier than other major countries and stagnates with respect to monetary policy. Krugman writes in 1998 about the return of the liquidity trap. In the late 2000s, the US is back near $\kappa \sim 1$ and we get Krugman's "return of depression economics" and the Keynesian liquidity trap. Interest rates are at zero. Large increases in the monetary base do little to the price level. New "market monetarists" (aka "monetarists") like Scott Sumner claim that monetary policy can put the economy back on track if central banks manage expectations correctly and/or change from interest rate targets to higher inflation targets or ideally NGDP targets. (As there is no measure of inflation expectations that differs significantly from empirically measured inflation, I personally don't see how inflation expectations could explain anything empirically measured inflation isn't explaining. Why are expectations different from market forces? [1])

Of course, different market-based targets or expectations management will not work according to the information transfer model. Keynesian stimulus might work. Accelerating inflation is a possibility. However, what appears to have accomplished the exit from $\kappa \sim 1$ was exiting the gold standard. This caused a phase transition, redefining the monetary base.

The following graph illustrates the preceding account of economic history; it shows the monetary base (dark blue), M2 (light blue and dashed), and the price level (green) over the past several decades. The prominence of the quantity theory of money happens right when these three lines become parallel (gray region, $\kappa \sim 0.5$). On either side of it, the direct relationship between the measures of the money supply becomes murky since $\kappa \sim 1$. We can see the "V"-shaped recessions happening at low $\kappa$ and $U$-shaped for high. We can see the problem that quantity theories had with MB increasing during the Depression and the revisionist view that M2 is responsible. We can see where inflation expectations come into vogue as the price level falls away from its direct dependence on the MB as $\kappa \rightarrow 1$. Most of all, we can see (as red bars) the three major macroeconomic events of this and the last century and how they occur when $\kappa \sim 1$ (the Great Depression of the 1930s), $\kappa \sim 0.5$ (the Great Inflation of the 1970s) and when $\kappa \sim 1$ again (the Great Recession of the late 2000s).

Compare the shifting paradigms in economic theory with these fits (red, blue) to the price level (gray dashed curve) in the information transfer framework that operates over all values of $\kappa$:

[1] There is a glib answer (proponents of expectations based theories are attempting to make up for the fact they are not using an information transfer model) and a less glib answer (proponents of expectations based theories are attempting to fix monetarism and allow it to explain difference between the times when quantity theories work like the 1970s and when they don't work like the 1930s and 2000s).

The unit of account effect and the number base

Here and here I refer to the "unit of account effect" and I think the properties of the logarithm may help illuminate what this means. The unit of account effect follows from the prescription:

$$
\kappa \rightarrow \kappa (NGDP, MB) \equiv  \frac{\log MB/c_0}{\log NGDP/c_0}
$$

that derives from the definition of the information transfer index $\kappa$ in the information transfer model. The effect itself arises because $\kappa \rightarrow 1$ meaning that $\partial P /\partial MB \sim 0$. Using the properties of the logarithm under a change of number base, we can rewrite this as:

$$
1/\kappa =  \log_{MB/c_0} NGDP/c_0
$$

where $\log_{b} x$ means the logarithm of $x$ in base $b$. This means the unit of account property of money is analogous to the number base. (There is a happy coincidence of number base and monetary base terminology as well.) And we can easily see that as $MB \rightarrow NGDP$ we have $\kappa \rightarrow 1$. Basically, $x$ has one digit in base $x$.

Using values for 2013 (and $c_0 \simeq 11$), the (inverse) information transfer index is given by

$$
\log_{MB/c_0} NGDP/c_0 \simeq \log_{283} 1443
$$

Is the idea that the macroeconomy is communicating information via the price level in base 283 right now useful? Probably not in itself [1]; however, the idea the the monetary base represents the base of the number system used by the economy to transfer information is an interesting way to think about the unit of account in economics.

[1] Also not useful: $NGDP \sim 16$ trillion dollars in its "natural base" is $\text{5U}_{283}$.

Business cycles, units of account and the meaning of constant

This is a short one; there are three marginally related points made in the past few posts I'd like to expand on:

• I don't think I've ever made this point explicitly clear: the information transfer framework does not describe "the business cycle" i.e. changes in aggregate demand. It allows you to describe the detector (price) given the information source and destination. So the output of the model calculations are the prices in the markets Price:Demand→Supply. In the examples in the blog, we are talking about the price level (or inflation rate) given the monetary base and the level of aggregate demand (using NGDP as a proxy). Or in another in another case in the IS/LM model the market r:NGDP→MB allows you to calculate the interest rate given the base and AD. One of the successes is that the price level doesn't have to respond to a major increase in the base (and may even shrink in the case of Japan), but again that is calculating the price given the inputs. The causes of the business cycle would involve a model for aggregate demand. Once you have that however, you can use the information transfer model to determine the other variables.
• I don't know if the unit of account effect (i.e. information transfer index dependence on the base) should or should not be included in the interest rate calculations. It would make sense to me historically that the success of interest rate theories (say, Keynesianism) in describing the macroeconomy would occur first in a world without the unit of account effect because interest rates are not impacted by it (or less impacted by it). The post WWII world was consistent with a monetarist view that ignores the unit of account effect, but pure monetarism does not work during the depression or our current recession. The historical sequence of the relative apparent correctness of the theories at the time would be consistent with not knowing about the unit of account effect until the information transfer model was developed: Keynesian during the depression, monetarist during the 60s and 70s, then Keynesian during the current recession. Milton Friedman's apparent success was that he used a different measure of the money supply (M2) to claim monetarism could describe the depression. [I think I may want to devote an entire post to this point later.]
• In this post and others I refer to floating information sources and constant information sources (or destinations). By constant, I don't mean in time, but relative to itself. Temporal changes in a constant supply change the supply explicitly and the demand responds to changes in the supply.Temporal changes in a floating supply change the supply explicitly and the supply and demand both respond to changes in the supply. In pure mathematical terms, constant means the source or destination Q is constant with respect to dQ (its behavior with respect to dt is not specified).