Market monetarism falls in the liquidity trap

We've seen that the choices of a constant (outside the market) or floating (inside the market) information sources and destinations has a considerable impact on the resulting model. Having a market aggregate demand and a non-market monetary base with the price level detecting the signal from the demand to the base can lead to hyperinflation (or runaway interest rates). Choosing a market aggregate demand and a market base leads to very powerful monetary policy if you don't include the effect of the base modifying the unit of account, but leads to "liquidity trap"-like conditions if you do. In the market shorthand, all of these markets are represented by P:NGDP→MB where P is the price level.

There are three main cases

• Market source, set destination: This includes the hyperinflation case P:NGDP→MB and generally describes a supply curve. 
• Market source and destination: This includes the quantity theory of money and the MS market in the LS/MS model (for labor supply/money supply). Both of these are also P:NGDP→MB as in the previous case but the monetary authority conducts its operations in a market (the MB "floats"). This combination generally describes an equilibrium path for a market (both supply and demand reacting to price signals). 
• Set source, market destination: This includes the LS market (P:NGDP→LS) in the LS/MS model and both markets in the IS/LM model (r:NGDP→AS aggregate supply/goods market and r:NGDP→MB the money market, respectively with the interest rate r as the price). This generally describes a demand curve. 

There is an additional modification mentioned in the introduction. In the special case of P:NGDP→MB, the monetary base describes both the amount of money in the supply as well as the units all three pieces are measured in (the price level in the US is a ratio of two prices in dollars at different times). In economics this is known as two of the functions of money: the medium of exchange and the unit of account, respectively. We incorporate this by allowing the information transfer index κ, which measures the number of symbols used to describe the signal in the information source and destination (demand d and supply s, respectively, and sorry about the s-d source-demand and d-s destination-supply pairings), to vary with the size of the monetary base and the aggregate demand. In particular,

$$
\text{(1) } \kappa = \frac{K_0 \log \sigma^s}{K_0 \log \sigma^d}= \frac{\log MB/c_0}{\log NGDP/c_0}
$$

The constant $K_0$ determines the unit of information ($=1/\log 2$ for bits and $= 1$ for nats) and $c_0$ is a dimensionful constant that is fit to the data. Ever since this post (refer to it for the rationale), I've taken $c_0 = \gamma Q^s_{ref} = \gamma MB_{ref}$ where $\gamma ∼ 0.016$ is a universal constant across countries. There is one additional assumption: κ is slowly varying which allows us to pull it out of the integral. The logarithmic dependence on MB and NGDP as well as the empirical fact that the value only changes from ∼0.6 to ∼0.8 over 60 years justifies this assumption.

This modification allows this excellent fit to the price level (and inflation rate and RGDP growth, as well as describing Japan). I did not include it in the LS/MS model above because I didn't think Scott Sumner would appreciate it (because it leads to the information trap, analogous to the liquidity trap popularized by Paul Krugman, where monetary policy loses effectiveness). I even included it in a fit to the interest rate, which led to a marginal improvement.

Putting all of these pieces together, I'd like to present a "theorem" ...

Information Trap Theorem Given the market $P : NGDP\rightarrow MB$ with a floating information source and destination and the functional form $\kappa(NGDP,MB)$ given by $\text{(1)}$ above, with $0 \lt \kappa \lt 1$ for $MB = MB_0$, then for some value $MB = MB_*$ with $MB_0 \lt MB_*$ we have

$$
\frac{\partial P}{\partial MB} \bigg|_{MB = MB_*} \leq 0
$$

Proof Given the function
$$
P(NGDP,MB) = \frac{\log NGDP/c_0}{\log MB/c_0} \left( \frac{MB}{Q^s_{ref}} \right)^{\frac{\log NGDP/c_0}{\log MB/c_0}−1}
$$
then the derivative $\partial P/\partial MB$ is initially positive for $\kappa \ll 1$ and goes through zero at the solution of the transcendental equation
$$
MB^0 = c_0 \exp \left(−\frac{\kappa (MB^0)+ \log \frac{c_0}{Q^s_{ref}}}{\kappa (MB^0)} \right)
$$
such that if $MB_* \gt MB^0$ then
$$
\frac{\partial P}{\partial MB} \bigg|_{MB = MB_*} \leq 0 \;\;\; \square
$$

What does this mean in terms of economics? Well, here is a less mathematical version:

Information Trap Theorem Given a market determination of the monetary base via a price mechanism with the monetary base acting as the medium of exchange and the unit of account, then for a given level of aggregate demand there is some level of the monetary base where increases in base money will not increase the price level.

This is actually fairly general and follows from information theory (assuming a market is transferring information). It says that any open market monetary policy target (with floating aggregate demand and floating monetary base) -- be it interest rate, price level, inflation rate, NGDP level, or NGDP growth rate -- will suffer from a situation where monetary policy becomes ineffective (resulting in "liquidity trap"-like conditions). The key requirements are the market determination of the base and money acting as the medium of exchange and the unit of account. Monetary aggregate targets (like the Friedman rule for the constant growth of M2) and other formulas with constant growth in the money supply won't suffer from an information trap, but will lead to accelerating inflation if they don't include a market mechanism.

This means that the market monetarists will fall in the information trap just like any other type of open market monetary policy (besides, say, a constant rate of growth in the base which unfortunately leads to accelerating inflation). Or using Scott Sumner's metaphor, the steering wheel doesn't work when you need it when you use NGDP targeting either.





Hyperinflation

If you let the unit of account enter into the equations in the information transfer model (i.e. let $\kappa \rightarrow \kappa (NGDP, MB)$), you get the information trap. In that situation, the price level has only a weak response to changes in the monetary base. This allows an excellent fit for the model to the empirical price level (data is in green, model fit is blue):

If we look at a given year (say 1985) and look at the model response to an increase in the monetary base (at fixed $NGDP$) we have a price level that rises for a bit and subsequently falls. In terms of the value of a dollar ($1/P$), we get a picture with a fall in the value of a dollar followed by a flattening out (actually a rise) in the value:

This is the information trap (diminishing marginal utility of increasing the monetary base). The dashed line shows the traditional economic view (a long run quantity theory of money) that $P \sim MB$. In that case, increasing the base decreases the value of a dollar and leads to inflation ... and eventual hyperinflation.

The question I want to explore is How do we get hyperinflation in the information transfer model? Hyperinflation was declared to be a political phenomenon by Matthew Yglesias, and there isn't complete agreement on the mechanisms behind it. The government's use of seigniorage as a source of revenue for a failing economy is a common theme, as well as the so-called inflation tax. This all points to the use of a model.

I will take an agnostic approach and only make the assumption that in the market $P : NGDP \rightarrow MB$, the monetary base represents a constant information destination that isn't determined by signals from the aggregate demand (as opposed to a floating information source, i.e. a market determined one). One way of viewing this is that the government or central bank begins issuing base money without regard to any market mechanism while the aggregate demand ($NGDP$) does respond to the signal from the detector of the information transfer ($P$). Performing the integral here results in the equations:

$$ \text{(1a) }P= \frac{1}{\kappa }\frac{\left\langle Q^d\right\rangle }{Q_0^s}$$
$$ \text{(1b) }\Delta Q^s = \kappa Q_0^s \log \left(\frac{\left\langle Q^d\right\rangle}{Q_{\text{ref}}^d}\right)$$

These equations actually define a supply curve in the information transfer model (as opposed to a demand curve). If we eliminate $\langle Q^d \rangle$ and replace the supply and demand with $MB$ and $NGDP$, respectively, we obtain:

$$
P = \frac{NGDP_{\text{ref}}}{\kappa MB_0} \exp \frac{\Delta MB}{\kappa MB_0}
$$

If we include the function of the monetary base $MB$ as the unit of account, then $\kappa = \kappa (NGDP, MB)$, we arrive at

$$
P = \frac{\log NGDP/c_0}{\log MB/c_0} \frac{NGDP_{\text{ref}}}{MB_0} \exp \left( \frac{\log NGDP/c_0}{\log MB/c_0} \frac{MB-MB_0}{MB_0} \right)
$$

If we add this function starting at the given year (1985, fitting to the price level in 1985 and borrowing $c_0$ from the price level fit at the top of this post), we happily (well, in the academic sense) get a rapidly decreasing value of the dollar (red curve):

The price level associated with this curve if we fit to the rate of growth versus time starting in 1985 is shown in this graph (again, red):

The price level in this case basically assumes that the monetary base increases at a constant rate (specifically, the average rate from 1983-1987). This path of the price level results in accelerating inflation:

This is an interesting result! It says that a constant rate of increase in the monetary base results in accelerating inflation (as opposed to the traditional economic view where a constant rate of increase of the base, ceteris paribus, is supposed to lead to constant inflation).

Ah, but this is just accelerating inflation. I wanted to achieve hyperinflation. We can do that if we increase the rate of increase of the monetary base (say, to 30% per year), which results in a faster increase in the price level:

We achieve hyperinflation (>50% annual inflation rate) in no time:

The key choice appears to be selecting a constant information destination in the market $P : NGDP \rightarrow MB$. This choice represents a scenario where the aggregate demand responds to price signals, but the monetary base is instead set by the monetary authority without regard to price signals. Aggregate demand "floats" inside the market, while the monetary base doesn't float and is set outside the market.

The price level fit and information trap at the top of this post occur for both a floating information source and floating destination. In that case, information transmission and reception occur in a market (floating = market): the aggregate demand shifts to respond to the price level and the central bank adjusts the monetary base through open market operations that take into account the price level (in the US there is a de facto inflation target on the order of 2%).

In my next post, I plan to take these observations further, summarize the past several market constructions (IS-LM and LS-MS) and outline a "theorem" where any use of a market to set monetary policy (including interest rate, inflation or even NGDP growth or level targeting) can result in an information trap.

In the singular interest of accounting for the unit of account in the interest rate

This is an update of this post where I've taken $\kappa = \kappa (NGDP, MB)$ such that

$$ \text{(1) } P = \frac{\log NGDP/c_0}{\log MB/c_0}\frac{NGDP}{MB} $$

The fit results in a marginal improvement (in the sense that deviations are more Gaussian) over the original (data is dashed green, light blue is the old fit and dark blue is the new fit):

Interestingly, the modified equation $\text(1)$ seems to do well (well, better than the original calculation) as a price ceiling where:

$$ \text{(2) } P \leq \frac{\log NGDP/c_0}{\log MB/c_0}\frac{NGDP}{MB} $$
which is relevant to the case of non-ideal information transfer (data in green, model in gray):

The Fisher equation and information transfer

David Glasner talks about the Fisher equation breaking down:

 "In other words, if r + dP/dt < 0, where r is the real rate of interest and dP/dt is the expected rate of inflation, then r < -dP/dt. But since i, the nominal rate of interest, cannot be less than zero, the Fisher equation does not hold, and must be replaced by the Fisher inequality i > r + dP/dt." 

The Fisher equation represents a "minimal coupling" between the price level and the interest rate, which would be implemented in the information transfer framework ($i : NGDP \rightarrow MB$) as:

$$ \text{(1) } i = r + \frac{dP}{dt} = \frac{1}{\kappa}\frac{NGDP}{MB} $$

Glasner refers to it as a pathology (that applies to our current circumstances in the US since 2008):

"Perhaps one way to think about the nature of the pathology is that the Fisher equation has been replaced by the Fisher inequality, a world in which changes in inflation expectations are reflected in changes in real interest rates instead of changes in nominal rates ... "

And I'd agree that it is a pathology. In particular, it seems to be a specific pathology where the market defined by Eq. (1) breaks down (in the thermodynamic analogy, zero price means a zero pressure system -- no atoms, no pressure). The market no longer transfers information that is detected by the interest rate.

The artificial separation of expectations

Imagine you are buying a car. What might you consider?

• Price of the car (financing, quantity available)
• Dimensions (number of seats, cargo capacity, urban vs rural use)
• Amenities (leather seats, A/C)
• Gas mileage (weighted by cost of gas)
• Maintenance costs
• Resale value
• Traffic congestion
• Policy changes (registration fees, congestion pricing, tolls)

Why separate these pieces of information into sources of "current value" and sources of "expected value"? It is true this seems more relevant for an options contract:

• Market value
• Strike price

However, it is important to realize in this case the strike price anchors the future expected value (measured as a difference from the strike price). The two measures are completely intertwined -- the market value as a piece of information has no meaning without the strike price. The discounted expected value is basically the measure that takes into account both these pieces of information. The success of the Black-Scholes model lies in its ability to account for the supply and demand price of the stock simultaneously with the expected value of the derivative.

There are behavioral factors (like money illusion) that show there are differences between "expected value" and "current value" and people seem to be given to hyperbolic discounting. But this is an even more serious problem for rational expectations where the deviation from the model price is assumed to be an unbiased random fluctuation. This seems all the more reason to just lump expected value into the other reasons people buy (or don't) at the current price.

The distinction between sources of expected value (based on market psychology and economic models) and sources of current value (based on the invisible hand and the price on an open market) is artificial. They both represent pieces of information used to determine if an economic agent should buy at the current price or not. We already assume people randomly buy at the "incorrect" price for reasons we don't model (I will buy a pack of post-its from my local grocery store even though they are cheaper at an office supply store or online due to spatial, temporal and accounting convenience). That price I paid enters into the "going rate" of post-its. It communicates information to the supply as well as any other purchase.

The real interest rate in the information transfer model

I mentioned doing this with real rates. Well, no big surprises, it works just as well (this fit was done partially by eye using the values from the nominal rates as a starting point):

The useless power of expectations

Scott Sumner has another post up heavy with expectations. He rhetorically asks for the step in his argument at which a reader disembarks. Well, Step 1 is supply and demand with a shift in the supply curve resulting in a drop in price. I'll buy that one. Step 2 goes too far:

2. Same thing, but assume the company merely announces the big gold discovery, but it is credible. Now gold prices would plunge on the announcement.

How far do they plunge? In Step 1 it was a series of market transactions causing the price to take a random walk with a downward drift of unknown duration. We can't know the final equilibrium price (EMH). If people knew the final equilibrium price (they expected it) the price it would head directly there. But it doesn't.

There are some cases I would not have a problem with the "expectations" argument. If company ABC announces it will buy company XYZ at 10 dollars per share, then the price of XYZ's stock fairly rapidly becomes 9.94 dollars per share. However this is not really a functioning market; it is similar to government price setting on a smaller scale. And the mining company is not even announcing a huge gold discovery at $600 per ounce. It is announcing it will sell some its stock of company XYZ. Where does the price go? Who knows?

Now go a step further imagine that people are "wrong" about the economics. The dominant paradigm includes the idea that a huge gold discovery will ruin the economy because it is ripe with moral hazard -- too much money makes people lazy. Widespread deflation will result. This expectation will push down the price of gold even farther than than the equilibrium price due to supply and demand. Now what happens?

• The price drifts back to the equilibrium price in the case of no moral hazard. [Did the power of expectations vanish?]
• The price stays at a level expected by moral hazard. [Did the law of supply and demand vanish?]

These two cases are particularly relevant to the case of Japan. Is the current price level where it is because the BOJ is expected to take back all the base money and thus supply and demand is powerless? Or is the current price level where it is because of supply and demand and thus expectations are powerless?

The information transfer model says the latter. Or more precisely, the information transfer model doesn't make the distinction between prices that come to equilibrium because of market forces or expectations. Expectations are part of the market forces (we make no assumptions about the information being transferred from the demand to the supply; it includes "Kevin thinks $5 is too high" and "Kyoko thinks the price will rise to 1800 ¥ given the current trend of exchange rates before falling to 200 ¥") and their effect on equilibrium prices is indistinguishable from the invisible hand. This resolves the tension between the two bullets above. There never was a price expected by moral hazard that differed from the equilibrium price.

I refute it thus

I posted a comment on Sumner's post about the difference between money as a medium of exchange and money as a unit of account. His response was: As far as QE [is concerned], it’s well understood why there is little inflation. We don’t need new models.

I'd like to see this model. I assume by the tone of the blog that it's not Krugman's liquidity trap model [pdf] and it has something to do with expectations. In his short course on money, Sumner has segment about expectations, which are apparently awesome:

Unfortunately the role of expectations makes monetary economics much more complex, potentially introducing an “indeterminacy problem,” or what might better be called “solution multiplicity.” A number of different future paths for the money supply can be associated with any given price level. Alternatively, there are many different price levels (including infinity) that are consistent with any current money supply.

Indeterminacy problem. So your model describes what actually happens equally as well as an infinite number of things that didn't happen. Sounds like a success to me.

Before we got to this happy point where expectations can describe whatever we want, Sumner says that it's hard to say if the lack of inflation is actually more IOR or expectations. We now have one tool that does whatever we want and another tool that seems like a realistic mechanism until you see that a 0.25% IOR should probably have a hard time hold back a doubling of the monetary base. I could see that 0.25% IOR holding back on the order of 1% inflation. But Sumner is actually unclear on the extent it is expectations versus IOR:

It’s not clear the extent to which the recent low inflation in the US reflects this factor [a government injects large quantities of money into the economy, but tells the public that it will be removed as soon as there is any sign of inflation], and how much reflects the 2008 decision to pay interest on reserves.

Apparently the same expectations mechanism is operating in Japan.

This occurred in Japan between the early 2000s, when lots of base money was injected, and 2006, when the monetary base was reduced by 20% in order to prevent inflation from occurring.

Of course this 20% decrease occurs in the middle of a 350% increase. Up 300%, drop 20%, then up another 150%. The 20% drop was obviously what ruined it. It was apparently so amazingly bad that Japan has failed to have any net inflation for the past 20 years. Imagine if they paid IOR from 2001 to 2008.

The liquidity trap seems to have some explanatory power here, but 20 years still seems like a long time to deleverage. However, the information trap should remain in effect until the ratio of NGDP to MB drops to a low enough level. In that link I show that the information transfer model recovers the lack of inflation during the Great Depression. In the graphs below, the information transfer model recovers the lack of inflation in the US and Japan:

In my opinion, this is far superior to "monetary policy rules except for and because of expectations". Maybe the answer should be that the lack of inflation from QE is well understood ... with an information transfer model. No expectations required.

Scott Sumner's Model (Part 2)

After the dramatic failure that was the "nominal hourly wage model" in the previous post, I decided to try and build Sumner's model (which he asked for over a year ago) from the more vague instruction in his recent post on a third way:

... I greatly prefer a third approach; labor and money.

Following the information transfer framework approach to the IS-LM model, and reading into Sumner's use of Okun's law, I wrote down two markets (in my new notation for describing an information transfer market, $\text{detector }: \text{ source } \rightarrow \text{ destination}$):

$$
P : NGDP \rightarrow LS
$$
$$
P : NGDP \rightarrow MB
$$

The price level ($P$, using FRED CPI less food, energy) is detecting the signal from the aggregate demand ($NGDP$) to both the labor supply $LS$ (where I use FRED's all employees measure PAYEMS) and the money supply (or monetary base $MB$). I will now build the two markets: the labor market and the money market.

The Labor Market

Like the IS-LM model, I took the labor market to have aggregate demand ($NGDP$) as a constant information source, hence (using Equations $\text{(8a,b)}$ from here):

$$
\text{(1) } P = \frac{1}{\kappa_L}\frac{NGDP_0}{\langle LS \rangle}
$$
$$
\text{(2) }\Delta NGDP = \frac{NGDP_0}{\kappa_L} \log \left( \frac{\langle LS \rangle}{L_0} \right)
$$

where $NGDP_0$ is the constant information source and $\Delta NGDP$ measures deviations from the equilibrium value. $L_0$ and $\kappa_L$ are arbitrary constants that will be used to fit to empirical data. Values of the expected labor supply $\langle LS \rangle$ move you along a labor supply (LS) curve. To that end, we can re-write these equations as an explicit curve in $P$, $\Delta NGDP$ space (by eliminating $\langle LS \rangle$):

$$
\text{(3) }\log P = \log \frac{NGDP _0}{\kappa_L L_0} - \kappa_0 \frac{\Delta NGDP}{NGDP _0}
$$

This equation is analogous to the IS curve here. This allows us to draw a labor supply (LS) curve versus real output (RGDP) $Y = NGDP/P$ (with the price level normalized to 1995 dollars and output shown in 1995 dollars):

The LS curve slopes slightly downward like the AD curve in the AD-AS model and the IS curve in the IS-LM model. You can think of it as the diminishing marginal utility of labor (at fixed money supply, which we will treat later). We can also show the model fit (blue) to the price level (green) via Equation $\text{(1)}$:

The constants are $\kappa_L = 0.041$ and $L_0 = 112 \text{ million people}$ (PAYEMS is given in units of thousands of people). Additionally, we can derive a version of Okun's law starting from Equation $\text{(1)}$:

$$
P = \frac{1}{\kappa_L}\frac{NGDP}{LS}
$$
$$
LS = \frac{1}{\kappa_L}\frac{NGDP}{P} = \frac{1}{\kappa_L} RGDP
$$
Taking the time derivative (and then dividing the equation by $LS = RGDP/\kappa$):
$$
\frac{d}{dt} LS(t) = \frac{d}{dt} \frac{RGDP(t)}{\kappa_L}
$$
$$
\frac{1}{LS} \frac{d}{dt} LS =\frac{\kappa_L}{RGDP}  \frac{d}{dt} \frac{RGDP}{\kappa_L}
$$
So that finally
$$
\frac{d}{dt} \log LS(t) = \frac{d}{dt} \log RGDP(t)
$$
Which allows us to plot this rather remarkable fit (at least for economics):

Or in the way Okun's law is usually presented (as a plot of a set of points with one axis being change in the labor supply and the other being the change in output):

This defines the labor market (LS market); it already delivers some empirical success describing the price level and Okun's law. To build a (simplified) complete economy however, we must include the effect of the money supply (I will use the monetary base here) on the price level. Changes in monetary policy that affect the price level will cause effects in the labor market since both use the price level as the measure that detects a signal from the aggregate demand.

The Money Market

Unlike the labor market (and the money market in the IS-LM model) I will build the model using a "floating information source". This assumption is where the purported power of monetary policy (in the monetarist view) enters into the model as will be made clear later. In that case the solution to the differential equation Eq. $\text{(5)}$ from here no longer has the form Eq. $\text{(2)}$ above, but instead:

$$
NGDP \sim MB^{1/\kappa}
$$

Here we use $\kappa$ without a subscript since it is different free parameter from $\kappa_L$. Starting from the price equation for the money market

$$
P = \frac{1}{\kappa}\frac{NGDP}{MB}
$$

and substituting the equation above it, we can write the price level in terms of the monetary base or NGDP:

$$
\text{(4a) } P = \alpha \frac{1}{\kappa} \left( \frac{NGDP}{N_0} \right)^{1 - \kappa}
$$

$$
\text{(4b) } P = \alpha \frac{1}{\kappa} \left( \frac{MB}{M_0} \right)^{1/\kappa-1}
$$

where $\alpha$ is an arbitrary overall normalization (the magnitude of the "price level" is arbitrary). Basically, we have a quantity theory of money with an equation that can be rewritten as the equation of exchange. These fits do a bit worse than the information transfer model where $\kappa$ becomes a function of the base (which is the successful version of the quantity theory of money I put together a couple of months ago) but are reasonable over some fraction of the time domain:

Note that in the picture using the monetary base, the recent QE appears to have been irrelevant (this is fixed in the information transfer quantity theory and the effect I refer to as an information trap). However, during normal times we can plot a money supply (MS) curve (vs real output $Y = NGDP/P$) using Equation $\text{(4a)}$ above (on a graph with the LS curve in blue, again choosing 1995 as the reference year):

The resulting LS-MS model (for labor supply, money supply)  is similar to the "Keynesian cross" AD-AS model with the MS curve analogous to the "45 degree" $AD = Y$ equilibrium line which does not shift (you can only move along it, as it represents a fixed relationship between the price level, NGDP and MB given by equations $\text{(4a,b)}$). Shifts in the LS curve (increases/decreases in aggregate demand) cause the price level to rise or fall, respectively (if it is supported by a change in the monetary base). An increase is shown with the dashed curve. However, increasing the monetary base causes the equilibrium point on the MS curve to move up and to the right, causing a shift in the information source in the LS market  ($NGDP_0$ in Equation $\text{(3)}$), causing the LS curve to shift up from the solid blue curve to the dashed blue curve. 

Discussion

In Sumner's version of this LS-MS model, he concentrates on expectations of what monetary policy will be in the future (large shifts that the market believes will vanish via e.g. "tapering" or inflation targeting won't cause the LS curve to move). That implies that the "transmission mechanism" from the MS market to the LS market is only approximately $P_{LS} \approx P_{MS}$ instead of the $P_{LS}= P_{MS} = P$ in the set up of the model at the top of this post.

Note that this model includes monetary offset (you can't move along the MS curve unless the central bank lets you, and in Sumner's expectations language, you can't expect to move along the MS curve unless the you expect the central bank to let you). Fiscal stimulus that directly employed people (e.g. the WPA) might break model and therefore break the relationship between these markets. In that case, the price level will not be what detects a signal from the aggregate demand to the labor supply ($P : NGDP \rightarrow LS$), but instead it will be set by government fiat.

Additionally note that these models assume ideal information transfer i.e. the information in the AD is equal the information detected at the LS and MS markets ($I_{AD} = I_{LS} = I_{MS}$). Deviations from the ideal (likely!) will result in $I_{AD} \geq I_{LS}$ and $I_{AD} \geq I_{MS}$. In those cases solutions usually fall at prices below those indicated by the ideal price.

And finally (note) the empirical success of the LS-MS model is comparable to the IS-LM model. The liquidity trap seems like a robust empirical result of the latter (it is an explicit breaking of the "$P$ transfers information" assumption when the interest rate is at the zero lower bound) while the former seems to break down and requires additional assumptions beyond a simple price mechanism like "expectations" [1]. I'd count that as a strike against LS-MS. It does do a good job describing the price level and Okun's law. While my modified quantity theory is the best (of course), the final outcome of this exercise is that both models are of similar power. The LS-MS model includes a strong prior (monetary policy roolz) that has some empirical evidence, but it is not overwhelming enough to discount e.g.  fiscal policy or the liquidity trap. When you use it, you are assuming monetary policy is powerful.

[1] I put this is scare quotes on purpose. These expectations are like superluminal signals in physics. It is important to point out that in this model (and the IS-LM model) the price level is detecting all signals from the AD to the markets. It is inconsistent to say that some of the information transferred from the AD to the MB is detected by the price level, but some (i.e. superluminal expectations) isn't. The information being transferred necessarily includes expectations. Therefore the more likely solution (IMHO) isn't that some additional channel of communication from the demand to the supply is disrupting the normal channel but that the normal channel you have is breaking down (e.g. the zero lower bound mechanism in IS-LM).

Scott Sumner's Model (Part 1)

I've been working on trying to build different economic models in the information transfer framework. I have had some success with the quantity theory of money (here, here and here) and the IS-LM model (here and here). The "holy grail" as it were is Scott Sumner's model. Not because it is the best, but because it doesn't exist!

I started to believe it would fall out quickly when I saw the following graph on a flight from LA back to Seattle (from this post from a couple days ago):

At least that's my version of the graph from FRED data. It plots the unemployment rate and the ratio of hourly nominal wages to NGDP. I saw that and thought (in the information transfer framework) is the unemployment rate a "price" detecting a signal from the aggregate demand to nominal hourly wages? ($r:NGDP \rightarrow NHW$) In the information transfer framework we'd write the "price equation" like this:

$$ r = \frac{1}{\kappa}\frac{NGDP}{NHW} \text{ ???} $$

Unfortunately, my initial idea crashed and burned when I realized after I got a chance to plot it myself that the correlation in the graph is a trick of normalization and selective windowing. Here is a version over a longer period:

Apparently the model in Scott Sumner's head has more variables than the version he writes down. You can see that there is an approximate overall $1/\text{year}$ bias. However this graph was useful in the sense that it helped me write down the real thing, which is the subject of the next post.