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".