The diminishing effect of monetary expansion (reference post)

I need a reference post for the diminishing effect of monetary expansion in the US (and the negative effect in Japan). The motivation of the model is here. The following graphs represent fits to the US (and Japanese) CPI (core for the US and so-called 'core-core' for Japan). In both cases we use the currency component of the monetary base (referred to as the 'base'). Here is the model for the US:

We can see the reduced sensitivity to monetary expansion if we look at the derivative of the price level with respect to the monetary base:

Another way to see the same effect is to look at the exponent of MB in the model equation (log P ~ β log MB), shown in this plot, color coded for larger values of β (red) and smaller values (blue):

In the inset graph, I plotted P  vs MB corresponding to the earliest (1960, red) and latest (2014, blue) points on the graph. The slope is lower for the blue line, which corresponds to the right side of the second graph above and higher for the red line which corresponds to the left side.

And finally, here is the plot of the fit to the price level for Japan:

The key driver in the diminishing effect of monetary expansion is the size of the monetary base relative to the size of the economy (measured by NGDP) -- when the base gets large relative to the economy (and the economy gets large), then the same percentage increase in the monetary base leads to a smaller increase in the price level with the larger monetary base. This overall effect can be seen in this graph of the price level vesus the monetary base for several countries:

The diminishing effect of monetary expansion is seen as the curve bends downward as you move to the right. An additional point is that just because a country finds itself on the right side of the graph above, doesn't mean it is doomed to remain there. The US actually had a very large monetary base relative to the size of its economy during the period just before and after the Great Depression. A bout of accelerating inflation during/after WWII caused NGDP to grow much faster than the base, leading to an overall reduction in the ratio of MB/NGDP.

Identifying monetary regime change?

I was looking at a post from a few weeks ago and noticed this diagram of a local parameter value for the information transfer index κ fit to the price level (blue dots) and how it fits with letting κ ~ log MB/log NGDP (the black line):

But notice how constant the value is before 1980? Did we switch over from a constant  κ model to a varying κ =  κ(NGDP, MB) with the Volcker Fed? Here's the kappa values fit to that model:

And the resulting price level graph:

It does improve the fit before 1980. The interesting thing is that a constant κ model leads to rising interest rates for κ<0.9, which was definitely true before 1980. However, lower values of κ lead to rising rates as well.

Moar Monte Carlo

Here's the results of this post with 100 paths and an AR(6) model:

There doesn't seem to be a huge difference from the previous results (i.e. there's still a persistent difference from the mean path in the 1960s and the 1990s). An AR(6) model uses the past 6 observations, which in this case is the past 6 months. That should be sufficient to achieve persistent shocks like that experienced in the US in 2008. Therefore, something seems to be missing and the first best candidate is NGDP shocks correlated with changes in the monetary base.

The monetary base as a sand pile

I will get to the title eventually, but the first part is some fun numerical differential equation solving. In particular, I'll be looking at the equation

$$
\text{(1) } \frac{dNGDP}{dMB} = \frac{\log NGDP/c_{0}}{\log MB/c_{0}} \; \frac{NGDP}{MB} + NGDP\cdot n(\log MB)
$$

where NGDP = NGDP(MB) is a function of the monetary base. Everything except the last term is motivated in this post. This equation models aggregate demand (NGDP) transferring information in the market that is captured/recorded by the money supply (the currency component of the monetary base). The ratio of logarithms (referred to as the information transfer index $\kappa$ in the link) accounts for the fact that NGDP is measured in the same units that are defined by the monetary base (dollars in the US) which creates the "unit of account" effect. This effect allows e.g. persistent deflation as seen in Japan, or low inflation as seen in the US.

An approximate solution to this differential equation assuming $\kappa$ is slowly changing gives us a pretty good fit to the price level and I use it to establish the value of $c_{0} = 0.48$.

The last term is an exogenous noise term (NGDP shocks) I've added to account for the fact that NGDP is not entirely determined by the monetary base -- it experiences exogenous shocks.

You may have noticed that this is all written with the monetary base (MB) as an independent variable, not time. This will lend itself to the sand pile analogy at the end, but mathematically $\log MB$ is a linear function of time so the only effect is a scale factor (linear transform) relating time and the (log of the) monetary base. More on this later.

So let's numerically solve the differential equation (1)!

Wait, what? Of course this is wrong; this first solution neglects the "shocks". Basically, this is what NGDP would be in the absence of exogenous (systematically negative) factors. We'll use this counterfactual to estimate the shocks -- shown in blue here:

The gray line is a random AR(2) process with parameters estimated from the "empirical" shocks (i.e. the difference between the blue and black line in the first graph). We'll take several  (30) of these gray paths and see the effect on NGDP (I'm basically doing a Monte Carlo simulation) -- that's in the graph on the right below. The graph on the left is the result with the "empirical" shocks (effectively showing that the procedure checks out):

Let's average the results on the right (shown with two-sigma errors in the shaded region):

This isn't perfect, but it's pretty good for such a simple model. I've never seen an economic model that tries to model NGDP over such a long stretch of time (correct me in the comments). The model is systematically low in the 1960s, but otherwise the empirical data is within the error.

Now there are two immediate issues/opportunities for future blog posts:

1. Is the AR(2) process correct? It seems to look about right except for a couple of large corrections (see the shocks graph above). We can look at this more closely by doing a larger number of Monte Carlo paths and see if systematic deviations hold up. We know if we get the shocks right, we get the empirical NGDP (the graph on the left in the pair above).
2. As it was modeled, the shocks are uncorrelated with changes in the monetary base. This seems like a bad assumption (remember the financial crisis?) ... Again, this can be seen if we do a larger number of Monte Carlo paths -- do systematic deviations occur near large changes in the monetary base? I'd like to produce a random sequence that is correlated with changes in the monetary base as well. This would lend credence to the theory that the central bank can cause a financial crisis that has larger effects than would come from monetary policy alone.

Now for the sand pile analogy! Sorry about droning on about that stuff before getting to the good bit.

The idea is that instead of NGDP shocks occurring at random moments in time, they occur at random points after the accumulation of money in the economy. Think of a pile of sand with a stream of sand falling on top of it (say, in an hourglass, keeping with the relationship with time ... the picture below is from Wikimedia Commons). The stream is the addition of money to the base, the total amount of sand is the total monetary base, and the height of the sand pile is NGDP [1]. As sand accumulates, there are random moments when small avalanches occur, causing the height of the sand pile to drop -- these are analogous to exogenous NGDP shocks (they are caused by gravity), but -- and this is the insight in this analogy -- are inevitable as you add more sand to the pile. As the central bank adds money to the economy, a recession is inevitable (the central bank can offset the impact by increasing the flow of money/sand). A question I have: are these avalanches related to changes in the flow rate? This is essentially issue #2 above. Certainly a slower flow rate will reduce NGDP growth, but does slowing the flow rate cause an avalanche/recession?

I'll leave these issues for future blog posts.

[1] An additional benefit of this picture is that it eliminates the linear relationship $NGDP \sim MB$ (the quantity theory of money). However, the exponent is wrong. This analogy implies that $NGDP \sim MB^{1/3}$ when in fact it is more like $NGDP \sim MB^{2}$.

The ISLM model (again)

The economist John Hicks wrote out Keynes' prose as an economic model that came to be known as the IS-LM model. I already derived this model before in a way that followed the way it is introduced in macroeconomics classes (as an IS and LM market). This derivation will acheive the same result, but approached fundamentally as an information transfer market system. The basis for the IS-LM model is that there are two markets: the real economy (IS) and the money market (LM) that couple to each other through the interest rate. As an information transfer system, we'll take a more direct route. We will posit that aggregate investment (demand) is a source of information that sends signals into the market that are detected by interest rate changes. The aggregate investment supply (the money supply) receives this information. See this post for a more detailed description of how information moves around the economy.

Let's start with the market $r : I \rightarrow M$, with interest rate $r$, nominal investment $I$ and money supply $M$ so that

$$
\text{(1) } r = \frac{dI}{dM} = \frac{1}{\kappa} \; \frac{I}{M}
$$

from the basic information transfer model. Looking at constant information source $I = I_{0}$, we have

$$
r =  \frac{1}{\kappa} \; \frac{I_{0}}{\langle M\rangle}
$$

where $\langle M \rangle$ is the expected level of the money supply.  Solving the differential equation (1), we obtain

$$
\Delta I \equiv I-I_{ref} = \frac{I_{0}}{\kappa}\log \frac{\langle M\rangle}{M_{ref}}
$$

where $ref$ refers to reference values of the variables $I$ and $M$. We can combine the previous two equations into a single function that defines the IS curve:

$$
\text{(2) }\log r = \log \frac{I_{0}}{\kappa M_{ref}} - \kappa \frac{\Delta I}{I_{0}}
$$

We can also look at constant $M = M_{0}$ so that we have, solving the differential equation (1) again:

$$
r =  \frac{1}{\kappa} \; \frac{\langle I\rangle}{M_{0}}
$$

$$
\Delta M \equiv M-M_{ref} = \kappa M_{0} \log \frac{\langle I\rangle}{I_{ref}}
$$

where we can eliminate $\langle I\rangle$ to produce (after some re-arranging)

$$
\text{(3) } \log r = - \log \frac{\kappa M_{0}}{I_{ref}} + \frac{\Delta M}{\kappa M_{0}}
$$

Equations (2) and (3) represent how the market adjusts the interest rate to changes in the money supply given fixed investment demand and to changes in the investment demand given fixed money supply, respectively.

One more piece -- we need to relate output $Y$ to investment $I$ and money $M$. I'll add a simple market $p : N \rightarrow I$ where $N$ is NGDP (aggregate demand, sending information to aggregate investment) so that

$$
\frac{dN}{dI} = \frac{1}{\eta} \; \frac{N}{I}
$$

which we solve without holding either constant, resulting in:

$$
N \sim I^{1/\eta}
$$

It turns out empirically (see below), $\eta \simeq 1$ so that $\log N$ is proportional to $\log I$. If we hold the price level constant (we are looking at short run effects [1]), then we can say that $Y = N/P$ is proportional to $I$. This is the first part. To get the relationship with $M$, we'll match first order shifts of the IS and LM curves. If we have a small change in $r = r_{0} + \delta r$, then we have for the IS curve:

$$
\log r_{0} + \frac{\delta r}{r_{0}} + \cdots = \log \frac{I_{0}}{\kappa M_{ref}} - \kappa \frac{\Delta I}{I_{0}}
$$

So that

$$
\frac{\delta r}{r_{0}} \simeq - \kappa\frac{\Delta I}{I_{0}}
$$

Similarly for the LM curve

$$
\frac{\delta r}{r_{0}} \simeq \frac{\Delta M}{\kappa M_{0}}
$$

So that we have (after taking the absolute value as a postitive shift of the LM curve causes the interest rate to fall, but a positive shift in the IS curve causes the interest rate to rise)

$$
\Delta I \simeq \frac{I_{0}}{\kappa^{2} M_{0}} \Delta M
$$

Which means $\Delta Y \sim \Delta I \sim \Delta M$ are all proportional to each other and we can scale the IS and LM curves such that they become functions of $Y$. This allows us to plot them on the same graph, like the ISLM model. On the left, we have the relationship between $I$ and $Y$ and on the right, we have the IS-LM model graph (IS curve in blue, LM curve in red):

Empirically, it works best if we take $r \rightarrow r^{c}$ where $c \simeq 1/3$ [2] ($M$ is taken to be the monetary base):

And as promised, here is that plot of investment vs NGDP:

[1] This is effectively where economists' complaint about the IS-LM model not incorporating inflation (by not differentiating real and nominal interest rates) comes in.

[2] This is a fudge that I have yet to figure out. The empirical results for the 10-year and 3-month interest rates are a pretty good motivation. It fits the data. Additionally, it is basically a re-labeling of the interest rates. There is no a priori reason that the "information price" $p_{i}$ that goes into Equation (1) and the real world price represented by the interest rate $r$ need to be related by $r = p_{i}$; any bijective relationship is possible. We encounter this all the time e.g. decibels give us a more intuitive linear feeling of loudness than power. The market treats the cube root of the interest rate as a linear measure of information.

UPDATE 5/22/2015

Fixed the fudge factor. See here. Basically introduced another information equilibrium relationship between $r$ and the price of money $p$ so that the market at the top of the page becomes:

$$
(r \rightarrow p) : I \rightarrow M
$$

And therefore $r \sim p^{1/c}$

The Fisher effect

A quick post; here's more evidence we can label the difference between the model result (green) and the data (LOESS smoothed, black dotted) as the Fisher effect (the effect of expected inflation on interest rates). In the previous post, the 10-year rate is decomposed into its various components giving this graph:

I tentatively labeled this discrepancy the Fisher effect because it occurs during the inflation (stagflation) scare of the late 70s and is reversed after the Volcker disinflation. The Fisher effect is supposedly a "long run" effect on interest rates that causes rates to rise with an increase in the money supply (expected inflation raises interest rates). Monetary expansion in the short run lowers nominal interest rates due to the so-called liquidity effect (or, what we should just call supply and demand because adding money to the economy should lower the rental cost of money).

What is this additional evidence that this really is the Fisher effect? Well I ran the model for the 3-month (short run rate) and discovered the strong Fisher effect in the 70s-80s apparent in the 10-year rate largely vanishes in the 3-month rate (within the data/model error):

The Fisher effect should disappear on shorter time scales as expected future inflation should have a larger effect on long term treasuries than short term -- exactly as seen here. Now this isn't conclusive evidence, just evidence that whatever the gray-shaded deviation is in the 10-year data it has properties consistent with the Fisher effect: greater impact on long term rates and occurring during periods of high/changing expected inflation.

The effects that move interest rates

Scott Sumner laid out the primary forces acting on interest rates (from a monetarist perspective) in a convenient piece of a paragraph:

The short run effect of monetary policy is called the liquidity effect. A change in the money supply causes interest rates to move in the opposite direction, in the short run. The long run effects of monetary policy are called the income effect, the price level effect and the expected inflation (Fisher) effect. These three effects all cause interest rates to move in the same direction as the money supply.

How do these components look in the information transfer model?

We'll call the liquidity effect the effect on interest rates of taking MB → MB + dMB holding NGDP constant

We'll call the income/inflation effect (price level effect) the effect on interest rates of taking MB → MB + dMB accounting for inflationary effects on NGDP and subtracting out the liquidity effect (I will also show this without subtracting out the liquidity effect and call it the combined monetary effects).

Both of these effects act simultaneously if your temporal resolution is quarterly (as we will consider here); the liquidity effect would likely happen on an even shorter time scale -- almost instantly in the interest rate market. This is effectively assuming "the long run" is a quarterly time scale.

There is an additional piece of the model that affects interest rates that I'll call exogenous NGDP effects (or shocks, essentially the difference between the effect of MB → MB + dMB on NGDP and the actual NGDP in the next period).

We leave out the Fisher effect (expected inflation) as the information transfer model does not contain "expectations" as such. The difference between this "expected" inflation and the inflation in the income/inflation effect is that the latter is essentially derived from a quantity theory of money so that locally log NGDP ~ α log MB. In some sense the Fisher effect represents a failure of the interest rate market: it is an incorrect assessment of future inflation that causes interest rates to rise above (or fall below) the value realized a quarter later.

Here are the model baselines:

And we'll jump right to the take-away graph:

We can see that the combined effect of the different components (green) fits nicely to the actual data (black dotted line, LOESS smoothed). The liquidity effect (blue) appears to dominate the total monetary effects (dark purple) after the 1980s as the income/inflation effect (price level effect, light purple) fades away. The major deviations of the data from the model are here tentatively attributed to the Fisher effect (shaded gray).

The allocation to the Fisher effect makes some intuitive sense; fears of inflation plagued the 1970s (recall the WIN campaign). After inflation died down, there was a subsequent correction towards the "expectation-less" effect of monetary policy. However, other than the late 1970s and early 80s, the "Fisher effect" appears to be of negligible importance.

Let's look at the income and liquidity effects as a function of the change in the monetary base:

In the left graph above, we see the liquidity effect (blue) moves in the opposite direction as the change in the base while the income/inflation effect (purple) moves in the same direction -- exactly as described by Sumner in the quote above.

If we look at the combined monetary effects in the right graph above, another pattern emerges. The period before the early 1980s (blue) has a sizable contribution from the income/inflation effect, while data afterwards (purple) shows the liquidity effect dominating interest rates. This is broadly in line with the general path of interest rates over the post-war period: rising before the 1980s and falling afterwards.

UPDATE 3/19/2014: Here's a diagram version of the interest rate model used here:

The informaton transfer version of Nick Rowe's version of Keynes' General Theory

Basing this on Nick Rowe's post on Keynes' General Theory (Chapter 3). It appears to be set up as an information transfer market

$$
\frac{W}{P} : Y \rightarrow L
$$

with real wages $W/P$, real output $Y$ and employed labor force $L$ so that

$$
\frac{W}{P} = \frac{dY}{dL} = \frac{1}{\kappa} \; \frac{Y}{L}
$$

from the basic information transfer model. Looking at constant  $Y = Y_{0}$, we have

$$
\frac{W}{P} =  \frac{1}{\kappa} \; \frac{Y_{0}}{\langle L\rangle}
$$

where $\langle L\rangle$ is the expected value of the labor supply. This is Rowe's second equation ("classical labor demand"), written as $W/P = MPL(L)$ and subsequently $W/P = 1/L$. Ours is technically $W/P = c/L$ where $c$ is a constant.  Solving the differential equation, we obtain

$$
\Delta Y \equiv Y-Y_{ref} = \frac{Y_{0}}{\kappa}\log \frac{\langle L\rangle}{L_{ref}}
$$

where $ref$ refers to reference values of the variables $L$ and $Y$. This is analogous to Rowe's first equation ("classical production function"), written as $Y = f(L)$ and subsequently $Y = \log L$. Ours is technically $Y = c_{1} \log L + c_{2}$ where the $c_{i}$'s are constants.

Rowe's third equation $W/P = MRS(L, Y)$ follows from looking at constant $L = L_{0}$ so that we have (solving the differential equation again):

$$
\frac{W}{P} =  \frac{1}{\kappa} \; \frac{\langle Y\rangle}{L_{0}}
$$
$$
\Delta L \equiv L-L_{ref} = \kappa L_{0} \log \frac{\langle Y\rangle}{Y_{ref}}
$$

where we can eliminate $\langle Y\rangle$ to produce (after some re-arranging)

$$
\log \frac{W}{P} = \log \frac{Y_{ref}}{\kappa L_{0}} + \frac{\Delta L}{\kappa L_{0}}
$$

Rowe's form is $W/P = Y/(1-L)$, but I'm not entirely sure what the $1$ is supposed to mean (full employment?); however we basically obtain this if we take the log of both sides

$$
\log \frac{W}{P} = \log Y + \log \frac{1}{1-L}
$$

and expand around $L = L_{ref}$ so that

$$
\log \frac{W}{P} = \log Y + \log \frac{1}{1-L_{ref}} + \frac{L - L_{ref}}{1-L_{ref}} + \cdots
$$
$$
\log \frac{W}{P} \simeq \log Y + \log c + c \Delta L
$$

Of course, $W/P : Y \rightarrow L$ is a terrible model (Keynes' disagreed with at least part of it) ... here's this model compared to data:

A much more successful market would be $P : W \rightarrow L$:

Where the equations that define the supply and demand curves are (respectively):


$$
\log P = - \log \frac{\kappa L_{0}}{W_{ref}} + \frac{\Delta L}{\kappa L_{0}}
$$

$$
\log P = \log \frac{W_{0}}{\kappa L_{ref}} - \frac{\kappa \Delta W}{W_{0}}
$$

You would plot them as functions of $\Delta L$ or $\Delta W$ so that $P \sim \exp(1-\Delta W)$ or $P \sim \exp(\Delta L-1)$ and they look like this:

Nick Rowe's model of the money stock

Suppose, just suppose, that the central bank does target an exogenously fixed rate of interest, and ignore the Wicksellian indeterminacy this creates, and ignore the fact that this is incompatible with targeting inflation or anything vaguely sensible. That interest rate target is the central bank's supply function, and a change in that interest rate target supply function will cause a change in the stock of money. 

Will it be true that the actual stock of money will always be equal to and determined by the quantity of money demanded at that target rate of interest? ... "no".

That's from Nick Rowe's post today. This is basically true if you set up an inverted version of the information transfer model. Previously, I had taken the monetary base as input (exogenous, in economic language), so that: 

rs = r(NGDP, MB)

rl = r(NGDP, M0)

P = P(NGDP, M0)

(That's not a typo, the function r is the same for the long term, say 10-year, rate rl and short run, say 3-month rate rs.) NGDP is exogenous, and as it is set up, P, rs and rl are endogenous.

Rowe sets up a system with an endogenous monetary base, effectively inverting the first two equations above:

MB = M(NGDP, rs)

M0 = M(NGDP, rl)

P = P(NGDP, M0)

with the same function M. Allowing non-ideal information transfer, the actual monetary base should be less than the quantity demanded (the solution to the equations) at the target rate of interest, answering the question at the end of the quotation above. What happens if we insert the effective Fed funds rate (this is approximately equal to the Fed target rate) for rs and the empirical 10-year rate for rl? Here is the Fed funds rate:

Here is the resulting monetary base (including reserves); the model is the solid line, while the actual value of the base is the dashed line:

Here is the resulting currency component of the monetary base (excluding reserves); the model is the solid line, while the actual value of the base is the dashed line:

It is in this last graph that the actual currency component is smaller than the quantity demanded. However, if we were to fit the model parameters to the short term interest rate, then the previous graph would show that the base including reserves demanded was always less greater than the actual monetary base. In fact, here it is (using the model fit to the short term interest rate to set the parameters):

For fun, here are the plots of the price level fit to these endogenous monetary bases (using the original fit to the long run interest rate):

It works reasonably well for the price level, but does show that an interest rate inflation target is nigh impossible due to the fluctuations.

Update 4/5/2014: the "less than" crossed out above should have been "greater than". H/T Tom Brown.

Unsolved problems in information transfer macroeconomics [update: solved]

Once the theory was invented, [Einstein] didn’t have a monopoly on it; it was out there for anyone to understand and move forward with. ... Your theory should have a life of its own; it should be a machine that I (or anyone) could use to make predictions.

--Sean Carroll (The alternative science respectability checklist)

To this end, I'd like to put forward a (living) list unsolved problems information transfer macro that maybe people out there can help with or solve themselves:

1. In the interest rate market r:NGDP→MB, the interest rate r itself is not the "price" ... it's actually r^c where c is a constant (≈ 2.8). This is a bijective function on the domain [0, ∞), so from a mathematical standpoint there really isn't a meaningful difference. However it's an additional parameter that may be derivable from e.g. the byzantine system of determining a bond's price. It doesn't appear at first glance to be related to the term of the bond (my initial guess).
2. In the labor market P:NGDP→L, there is a significant component that isn't explained that is approximately linear with time if you take the ratio P/(NGDP/L). Including this ratio to incorporate this unknown effect is critical to understanding the unemployment rate. Originally, I thought it might be related to nominal wage flexibility and that does seem to explain a piece of it. I also considered inequality might be involved. I haven't nailed this down yet.

Update 26 November 2016

Both of these have been solved. In 1), the interest rate model should be seen as a pair of information equilibrium relationships (where the interest rate is in information equilibrium with the "price of money"):

p:NGDP→MB

r→p

See here (draft paper) or the actual paper.

In 2), the labor market ends up relying a bit on the Solow model, and the difference mentioned above is due to capital (and is different for different countries that have different capital-labor mixes). Here is the description of the "quantity theory of labor and capital".

How money transfers information

In this post I showed how the information flows in a simple market for apples, but here I'm going to show what I hinted at in the earlier post: money is a tool to transfer information from the demand to the supply.

Let's start with a simple system of $d$ buyers (green) and two suppliers of gray bundles of something. Each sale adds $+\log_{2} d$ bits of information [1] about the size of the market $d$ and the distribution of demand to a supplier. Or at least it would if the supplier had some knowledge of the size of $d$ in the first place. If there was only one supplier, that supplier could use the total amount of goods sold $s$ as an estimate since the total amount of information would have to be less than or equal to the information coming from the buyers (the source), i.e. (dropping the base of the $\log$'s):

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

You can't get more information from a message than the message contains! However, there is more than one supplier, so each supplier only sees a fraction of the total supply and a fraction of the total demand. If this were all there is to it, then while each transaction could transfer $\log d$ bits of information, the supplier would have no idea. Enter money; now with each sale a supplier acquires a few inherently worthless tokens:

How does this help? Well, because these tokens are available to everyone (either set up by some government or based on custom), the supplier has some idea of how many are out there (let's call it $m$):

Now each sale is accompanied by $n_{m}$ tokens of money, so that each token transfers $+ \log m$ bits of information from the demand to the supply. This monetary system could potentially work well enough so that we can say the information captured by the supply is equal to the demand, thus equation (1) becomes:

$$
\text{(2) }I_{s} = n_{s} \log s = n_{m} \log m = n_{d} \log d = I_{d}
$$

We call this ideal information transfer when we use the equal sign. If we take $n_{s} = S/dS$ where $dS$ is the smallest/infinitesimal unit [2] of supply and likewise $n_{d} = D/dD$ for demand and assume $D, S \gg 1$ (a very large market), we can write:

$$
\frac{S}{dS} \log s = \frac{D}{dD} \log d
$$

$$
\frac{dD}{dS} = \frac{\log d}{\log s} \frac{D}{S}
$$

$$
\text{(3) } \frac{dD}{dS} = \frac{1}{\kappa} \frac{D}{S}
$$

where we've defined the information transfer index $\kappa \equiv \log s/\log d$. The left hand side of equation (3) can be identified with the price $p$ because it is proportional to $dM/dS$, or the rate of change of money for the smallest increase in quantity supplied.

But wait, there's more! See, money can be exchanged for all kinds of goods and services:

This means that the total demand of all goods and services ($AD$, or aggregate demand) is related to the total amount of money ($M$) so that, again assuming ideal information transfer:

$$
\text{(4) } P = \frac{dAD}{dM} = \frac{1}{\kappa} \frac{AD}{M}
$$

where $P$ is an overall measure of the price of all goods and services ($AD$); it's called the price level. The rate of change of the price level over time is inflation. Now for the totally awesome part: we can solve equation (4) for aggregate demand in terms of the money supply. It's a differential equation that can be solved by integration. We re-arrange equation 4 like this:

$$
\text{(5) }\frac{dAD}{AD} = \frac{1}{\kappa} \frac{dM}{M}
$$

and integrate

$$
\int_{AD_{0}}^{AD} \frac{dAD'}{AD'} = \frac{1}{\kappa} \int_{M_{0}}^{M} \frac{dM'}{M'}
$$

$$
\log \frac{AD}{AD_{0}} = \frac{1}{\kappa} \log  \frac{M}{M_{0}}
$$

$$
\frac{AD}{AD_{0}} = \left( \frac{M}{M_{0}}\right)^{1/\kappa}
$$

Using equation (4) again, we have

$$
\text{(6) } P = \frac{1}{\kappa} \frac{AD_{0}}{M_{0}} \left( \frac{M}{M_{0}}\right)^{1/\kappa -1}
$$

If $\kappa = 1/2$, then $P \sim M$, and the price level rises with the money supply, i.e. the quantity theory of money. Awesome, huh? Except the quantity theory doesn't really work that well ($\kappa \sim 0.6$ works better for the US and other countries, except Japan where $\kappa \sim 1$ is a better model). But we left out a big piece: aggregate demand (e.g. NGDP) is measured in the same units as the money supply. And to top it off the money supply is adjusted by the central bank based on economic conditions! This is the picture of the macroeconomy:

What does it mean? It means that $\kappa = \log m/\log d$, or the amount of information transferred from the demand to the supply relative to the amount of information transferred by money, is changing! If we assume this happens somewhat slowly [3], we can transform equation (6) into:

$$
\text{(7) } P = \alpha \frac{1}{\kappa (MB, NGDP)} \left( \frac{MB}{MB_{0}}\right)^{1/\kappa (MB, NGDP) -1}
$$

$$
\kappa (MB, NGDP) = \frac{\log MB/c_{0}}{\log NGDP/c_{0}}
$$

where we've replaced M with the monetary base, AD with NGDP, grouped a bunch of constants as $\alpha$, and introduced a new constant $c_{0}$ since the units of money are arbitrary. We can fit this model to the price level (it works best, see the lower right graph, when the monetary base is actually just the currency component of the monetary base):

Pretty cool, huh? Using this model (the information transfer model or ITM), we seem to get less inflation from a given increase in the money supply as the money supply and the economy get bigger. In fact, it can even go the other way -- in Japan an increase in the money supply (blue) decreases the price level (brown):

The rest of this blog is devoted to exploring this model and the concept of information transfer in economic, even commenting on current events based on these ideas. Have a look around!

Footnotes

[1] You can think of it as a sale discovering the ID number of a buyer. If the ID number is a member of the set {1, 2, 3, 4, ... , d}, then you need $\log_{2} d$ bits to represent the ID number. Thus, a sale transfers $\log_{2} d$ bits from the buyer (demand) to the supplier (supply).  I will drop the subscript 2 for the binary log in the rest of the post.

[2] We are looking at infinitesimal units to see how supply and demand change with respect to each other. for those unfamiliar with this concept, it forms the basis of calculus. And as a physicist, I am given to frequent abuses of notation.

[3] Technically, integrating equation (5) as we did assumes $\kappa$ is independent of $AD$ and $M$. However, if we assume $\kappa$ doesn't change rapidly with $M$ or $NGDP$, then the integration can proceed as shown, but is only an approximation.

What's up with NGDP?

Scott Sumner has a post up where he makes the (very interesting from a philosophical standpoint) claim that the recession was caused by the Fed deciding to keep NGDP growth at a constant 4-4.5% per year from 2009-2016, hence likely raising rates in 2015. Problematic causality aside (which arises from causality-violating effects of expectations), I wanted to address a related question:

Why is NGDP growth what it is? This is partially related to Noah Smith's critique of macro where what you call a fluctuation is in some sense a free parameter.

Some people might look at recent growth level and see that a linear extrapolation from the early 2000s, with a bubble along the way that pops in 2008:

But what about earlier growth? Well, maybe there is a longer term trend towards lower growth ... here is a quadratic fit and you can see the tech boom and the so-called housing boom:

Monetary economists (like Sumner) tend to believe the rate of NGDP growth is for the most part what the monetary authority wants it to be. This could make sense if you look at the first graph, but it seems strange if you look at the second. Has the Fed become more pessimistic over time about the growth prospects of the US? Also, looking at the first graph, it seems interesting that the post-2008 growth rate matches up with a brief period before the "housing boom". Is the Fed targeting a growth rate that happened for a couple of years in the early 2000s (and nowhere else in the history of NGDP)?

There's an additional way to look at the data -- each of these booms over the past seem to involve decreasing growth rates shown as a set of log-linear fits:

However! Not only do the boom periods seem to have decreasing growth rates over time, so do the lulls between the booms! This could imply that our current path might undergo a correction towards an even slower rate of growth at some point in the future (shown in red). This is a picture of an underlying sustainable rate of NGDP growth that is slowing down, and any monetary policy that attempts to target earlier rates is doomed to become "too loose" resulting in an eventual downward correction. This is also the picture of the secular stagnation discussed by Larry Summers and Paul Krugman (among others).

This picture also poses serious problems for market monetarist theories -- the Fed can't pick NGDP growth rates unless it picks this sustainable rate.

This view also poses serious problems for macroeconomics in general -- why would nominal economic growth peter out over time in a country with increasing population? Will it take off again? Will it go to zero? Less than zero? If the population is increasing, you'd expect non-zero real growth. This means that you'd have to have deflation in order to reconcile zero NGDP growth with positive RGDP growth.

The information transfer model predicts Japan-style lost decades (persistent liquidity traps) in this scenario. The only way I've seen so far to escape is a bout of hyperinflation, however Japan has lived with this scenario for 20 years so maybe it'll be all right? In the "information trap" scenario, monetary policy ceases to be a useful method of macroeconomic stabilization and inflation becomes almost meaningless. It's a liberal utopia! Macro stabilization through fiscal policy! No one takes inflation hawks seriously, money illusion goes away, and no more fighting to keep means-tested benefits in pace with inflation.

Of course, this could all just be a lack of informative macro data coupled with the common human cognitive bias towards believing you exist in a meaningful time. There are only a few events of falling NGDP in the well-documented post-war period. Maybe it will all go away and the long run really is a log linear function. 

Interest rates in the UK

A quick one; here are the model results for the UK short (3-month) and long (10-year) interest rates (based on this version of the information transfer model using the same fit parameters for the short and long rate, with the long rate model fit as an upper bound):

 

The light blue line includes central bank reserves, while the dark blue line is the currency component (notes and coins).

Apples, bananas and the information transfer model of supply and demand

I was asked by Nick Rowe to explain where the information comes in:

"If I trade an apple for a banana, there is supply and demand, but where does information come into it?"

My initial reply on the post was less clear than I'd like, so I'm going to try and do better here. First we'll start with a single good (apples).

Let's say there are d potential apple buyers, labelled 1, 2 ... $d$. Selling one apple to #42 uncovers $\log_{2} d$ bits of information (the number of bits required to describe the ID number). Selling a second apple to #42 uncovers another $\log_{2} d$ bits of information for a total of

$$
\log_{2} d + \log_{2} d = 2 \log_{2} d
$$
bits of information. Selling a third apple to #1005 uncovers another $\log_{2} d$ bits of information, and so on, until we've sold $n_{d}$ apples and uncovered $n_{d} \log_{2} d$ bits.

This uncovered information is transferred from the buyers (the demand) who ostensibly know how much they'd like to buy (at a given price) to the sellers (the supply) who only have some vague idea of the size of the apple market after they start to sell some apples (or do a little market research).

The optimal way to register this information would be to keep a log of each apple and each buyer's ID number. In that case, the information captured by the supply would be

$$
\text{(1) }I_{d} =  n_{d} \log_{2} d
$$

However, the suppliers don't actually know the size of their market d so they actually capture

$$
\text{(2) } I_{s} = n_{s} \log_{2} s \leq n_{d} \log_{2} d
$$

bits of information where $s$ could be e.g. the sellers' estimate of $d$ [1]. We have $I_{s} \leq I_{d}$. In an ideal world, this would be equality. If only there were something that existed to help gauge the size of the market for a product allowing a seller to capture all the information ... (hint: it's called money).

Which reminds me, we haven't actually discussed what the buyer's are buying these apples with yet. We'll start with Nick's suggestion of using a banana to buy an apple. In that case the informaton the sellers collect (by acquiring a banana from the apple buyer) is

$$
n_{b} \log_{2} b = n_{s} \log_{2} s
$$

where $b$ is the number of potential banana buyers. We can use this to determine the "exchange rate" (the price of a banana in terms of apples). If we take the smallest unit of bananas to be $dB$ so that $B/dB = n_{b}$, then

$$
B \log_{2} b = S \frac{dB}{dS} \log_{2} s
$$

where $n_{s} = S/dS$ the number of apples supplied to the apple buyers. If we assumed the market for apples bananas is about the same size as the market for apples, we could say that $\log_{2} b \sim \log_{2} s \sim \log_{2} d$, so that:

$$
B \sim S \frac{dB}{dS} \leq n_{d}
$$

where $dB/dS$ (where we let $dS$ and $dB$ become infinitesimal) is the "exchange rate" (price) of apples to bananas. What if the apple buyer gave the apple seller something else instead of bananas? Well, starting with equation (2) you'd have

$$
n_{s} \log_{2} s \leq n_{d} \log_{2} d
$$
$$
\frac{S}{dS} \log_{2} s \leq \frac{D}{dD} \log_{2} d
$$
$$
\frac{dD}{dS} \leq \frac{\log_{2} d}{\log_{2} s} \frac{D}{S}
$$

We'll call the left hand side the price $P$ (like the exchange rate above) and define $\log_{2} s/\log_{2} d \equiv \kappa$ (the information transfer index). Leaving us with:

$$
\text{(3) } P = \frac{dD}{dS} \leq \frac{1}{\kappa} \; \frac{D}{S}
$$

where we can call $S$ the supply and $D$ the demand.

If we take the thing exchanged to be dollars, then we could potentially take $\log_{2} s \rightarrow \log_{2} m$ where $m$ is the size of the money supply (monetary base). This would allow you to get a really good estimate of the potential market size d by looking at the price (or at least changes in the price) while knowing the size of the money supply.

In the information transfer model, I have typically assumed equality in equation (3) and taken $\kappa$ to be an unknown constant I fit to the data. One notable exception is the money market where I took $d = NGDP$ and $s = MB$ and used it to describe the price level. In many of the posts on this blog, I've used the shorthand notation $P:D \rightarrow S$ for a model that transfers information from the demand $D$ to the supply $S$ with price $P$.

PS Some notes:

1. Equation (1) assumes transactions are maximally uninformative, or equivalently, all microstates with $n_{d}$ apples sold are equally likely. One way of thinking about that is that it makes the fewest assumptions about potential microfoundations.

2. Equation (3) is the simplest possible relationship between supply and demand that maintains homogeneity of degree zero (related to the long run neutrality of money).

3. Per Nick Rowe's original post, if the network of exchanges collapsed, we'd see a fall in the total amount of information being exchanged so that if we looked at the market $P:AD \rightarrow AS$ for aggregate supply and aggregate demand, we'd see a fall in $AD$ and/or $P$.

Update: Forgot to add a very important note ...

4. Equation (3) above can be solved to recover supply and demand curves and Marshall's diagrams.

[1] Footnote added in update 3/14/2014. The a priori estimate information about the size of $d$ would actually have to come from somewhere else. Being strict about it, all the seller would know is that $s$ is at most the size of the total number of apples he or she has sold and at least the number of different people sold to. In this way, $s \rightarrow d$, eventually but only if the market for apples was dominated by a monopoly. Competitive sellers really can't get at the information about the size of $d$ without some sort of tool to measure the size of the market -- that's what money does. It allows a seller to gauge the size of the market in order to calibrate the information they receive (they don't know if $I_{s} = $ 10 bits or 100 bits) so they can use that information to supply the market. If the price suddenly goes up, the amount of information being received suddenly increases ($I_{s}$ goes from say 10 bits per apple to 20 bits per apple), telling the supplier that demand ($d$) has increased (or supply from all the suppliers has fallen).