Friday, August 30, 2013

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.

Sunday, August 25, 2013

The interest rate in the information transfer model

This post is a first step towards integrating the information transfer versions of the IS-LM model and the quantity theory. We'll begin with one of the basic equations from the information transfer framework:

$$ P = \frac{1}{\kappa}\frac{Q^d}{Q^s} $$

In the LM market, we have aggregate demand represented by $NGDP$ as an information source sending information to the money supply (the monetary base, $MB$) with the interest rate as the price (the detector that detects the signal transmitted from the demand to the supply). We'll write this price $P \rightarrow r$ as

$$ c \log r = \log \frac{1}{\kappa}\frac{NGDP}{MB} $$

With $c$ being an arbitrary constant. If I fit this equation to the Effective Fed Funds rate, I get a very good fit (model is blue, data is green):


The fit parameters are $1/\kappa = 39.1$ and an overall normalization of $c = 0.279$ (assuming of course that the fed funds rate is divided by 100 to change from a percentage into a dimensionless number).

An interesting way to visualize this data is to plot the interest rate versus real GDP (aka real output, denoted $Y$ in the ISLM model). With the information transfer model providing both the interest rate (above) and RGDP derived from $NGDP$ and the price level in the quantity theory, we can observe "LM curves" in the data where increasing $Y$ traces out an upward sloping curve with the interest rate.


It appears as though the LM curves are "reset" by the Fed lowering interest rates to "heat" the economy (by increasing the monetary base via the equation above). There was a period of relative constant interest rates (dashed red line) from the mid-90s to the early 2000s (the late 90s "tech boom") where the economy grew with limited intervention from the Fed. That last statement would probably send Scott Sumner up the wall. The Fed is always intervening, but in this case by limited intervention I mean keeping the monetary base growing roughly at the same rate as NGDP. This keeps interest rates constant via the equation above.

In 2008 we see the data bump up against the zero lower bound. The LM market stops sending information detectable by the interest rate. Here we would obtain the "flat" LM curve of the liquidity trap. This is different from the constant interest rate of the late 1990s, which is the LM market equilibrium moving along at roughly a constant interest rate as $Y$ increases until 2001.

It appears there is a qualitative change in the properties of the LM market that begin in the early 1990s -- interestingly the first recession with Alan Greenspan as Fed chair. For some reason, no LM curve appears after the recession ends. Could raising interest rates in the late 1990s have helped us avoid the Great Recession later on by leaving interest rates high enough to avoid the zero lower bound?

Should proper macroeconomic stabilization produce a picture that looks (heuristically) like this:


And do the 90s look like a massive failure in retrospect?

The next step (next post maybe?) is to see if using the real interest rate (related via the Fisher equation to the nominal rate) shows any significant difference in qualitative behavior.

The messy stuff is supposed to be an equation

Let me apologize to those of you who view this blog in various readers (I just viewed this blog in Feedly to see how it renders) -- the equations come out as a mess like this:

$$
\log r = \log \frac{Y^0}{\kappa_{LM} LM_{ref}} - \kappa_{LM}\frac{\Delta Y}{Y^0}
$$

I use mathjax to render LaTeX, which works great when rendered in a common browser (I highly recommend it). However, it doesn't make it through to some RSS readers.

Friday, August 23, 2013

Visualizing the diminishing marginal utility of monetary expansion

Building off of this post, I'd like to show how you might visualize how adding to the monetary base can cause your economy to expand ... up to a point. Base money is behaving both as the number of "bits" used to describe the economy (i.e. receive the information transferred from the aggregate demand) as well as a determination of the meaning of the unit "bit" (i.e. how many bits the aggregate demand consists of). In analogous economic terminology, money is the medium of exchange (the monetary base determines how many dollars are available) and the unit of account (the definition of a dollar unit).

In the model adding bits will capture more of the information being transmitted from the aggregate demand and allow the economy to grow. Simultaneously, the value of these bits is decreasing since their supply is increasing. Let's imagine each of these bits as boxes and the economy as a stack of boxes, like this:


As you add boxes to a stack, two things happen. One, the number of boxes in the stack gets larger (the monetary base grows). Two, the size of the boxes shrink (the value of each dollar decreases). Depending on the size of these competing effects, the result can vary. In the information transfer model, a series of these stacks at a fixed aggregate demand looks like this:


You can imagine this falling to zero eventually:


Note that at fixed aggregate demand, the height of the boxes represents NGDP for a fixed RGDP, a measure effectively equivalent to the price level. The price level (height of the stack) grows with the size of the base initially, but then starts to shrink as the size of the boxes gets smaller. Thus, we recover the picture from this post:


This is the mental picture I have in my mind when thinking about an information trap in the information transfer model.

What is the stance of monetary policy?

What is the stance of monetary policy? I mean that not in the sense of what is the current stance of monetary policy, but what does it mean for monetary policy to have a stance?

Scott Sumner has a list of indicators here:
http://www.themoneyillusion.com/?p=23083

He is disagreeing with Paul Krugman's assertion that interest rates are a good indicator. I think the unwritten assumption here is whether monetary policy is effective and that determines if the idea of a "stance" makes sense. A fighting stance in an arena means you mean business. A fighting stance at a dinner party is something different. In Krugman's view, rates are great: they drop when the Fed loosens and rise when the Fed tightens and are stuck near zero when the Fed is ineffective (they can't drop, so the Fed can't loosen, but it could tighten).

Sumner says monetary policy is always effective and dominates the economy therefore tight policy must (tautologically) be associated with poor economic performance (low NGDP growth) and vice versa for loose policy. This characterization is a tad uncharitable, but I don't think it is wrong.

Effectively the theoretical assumptions define the indicator in this case. Sumner's assertion of NGDP as a better indicator contains no information beyond a re-assertion of his priors.

Friday, August 16, 2013

The liquidity trap and information transfer

In the discussions mentioned in the last post, I became interested in seeing why the information transfer model didn't seem to do so well during the Great Depression. Of course, we need a starting point to seeing what doing well means. I'd like to start with economists' favorite first order approximation: $P \sim MB$ (with $P$ the price level being the dashed line and $MB$ the monetary base being the solid lines, the darker one of which is divided by 300).


This of course doesn't work so well and led to e.g. Milton Friedman proposing M2 (see the last post) as the "real money supply". Let's zoom in on two pieces: pre-WWII and post-WWII:


Notice however that you could probably change the constant of proportionality (in this case 300) to be different before and after WWII and it might work better. That gave me an idea; what if we fit the ITM model parameters to pre-war and post war data? It works remarkably well in the specific domains (but terribly outside them):


And the analogous zoom-in:


Pretty amazing, no? So I decided to do the fit for 10 year intervals and look at the parameter values. Here is the supply reference constant ($\sim 700 \text{ G\$}$ for post war fits) with the dashed lines being the pre-war (red) and post-war (blue) reference constants (the red and blue solid lines represent the value for the fit parameter when fit to the entire pre-war data and post-war data, respectively):


The other free parameter (I kept $\gamma$ fixed) $Q^d_{ref}/Q^s_{ref}$ shows the exact same behavior. You can see what I did next already in the graph. The gray line is a typical curve arising in phase transitions (like at the critical temperature in an Ising model). What happens if you use this curve as the reference constant? This:


You get a model that works well pre-WWII and post-WWII but not at the "phase transition". Since the price level equation is analogous to an isentropic process (reversible adiabatic process) one can imagine this as the economy moving along one adiabat for a time, going through an irreversible process (WWII or maybe Bretton-Woods as shown on the graph), and moving over to another adiabat, like for example, here. Basically, the constant in $P V^\gamma = \text{ constant}$ changed.

I'm going to move into $\sigma \kappa$ space (normalized monetary base and information transfer index), but first, here is the graph of $\kappa$:


Recall that $\kappa = 1/2$ is the pure quantity theory of money. And here is the path in $\sigma \kappa$ space:


I had mentioned previously that the ITM bascially had the monetarist view of the Great Depression. That was incorrect and based on the fact that I thought $\kappa$ was small. If this phase transition view is correct, then we can see that the Great Depression and the Great Recession have much in common -- along with Japan's lost decade -- every one of those times occurs on the other side of the ridge line where monetary policy is orthogonal to aggregate demand. Our path over the past century appears to have brought us over the ridge into the region of ineffective monetary policy twice, and even in the information transfer model, the Great Recession, the Great Depression and Japan's lost decade represent the same forces at work. I will call this an information trap in the future, analogous to the liquidity trap.

The implications are not pleasant. Do we have to redefine money? During the depression countries left the gold standard, an option not available now. Maybe we abandon physical currency for digital currency? Do we simply muddle along like Japan has for decades? Does there have to be a situation where the government takes the commanding heights again, like in WWII? If any of these are true it means we are in for a long languishing.

Thursday, August 15, 2013

The MANetarist

There has been an interesting argument happening in the economics blogosphere about whether Friedman was a Keynesian or not. Participants included Glasner, Krugman and Sumner. I'm not really going to weigh in on the debate ... ah, who am I kidding. Friedman is to Keynesians what a manarchist is to anarchists. I'd like to coin the term manetarist for economists who apply economic theory in order to reinforce conservative priors. In this way, Friedman tried to re-purpose Keynesian economics (and the quantity theory of money) in order to have the government be the root cause of the Great Depression and advocate limited government macroeconomic stabilization. [As an aside, I think it is telling that neither Friedman nor his defender Sumner have real mathematical models.]

There is one thing I thought was particularly egregious. Krugman's 1998 paper on liquidity traps has a graph in it on page 155; he says the "basic facts [in the graph] underlie two influential views of the Depression". Instead of showing the graph starting in 1929, I will create its analog starting in 2008. The graph plots the Monetary Base and the M2 Aggregate as an index referenced to 2008 vs time:


In the original graph starting in 1929, M2 falls, only to return to 100 in 1939, but is essentially similar to the graph above.

Krugman's "two influential views" of the depression-era data mentioned above are both monetarist views (probably since he is writing in 1998 and bringing back the idea of a liquidity trap). I believe this graph underlies two more disparate views: Keynes and Friedman's approach to economics.

Keynes' analysis said: Look at the graph: MB rose which ISLM (not actually Keynes, but Hick's shorthand version of Keynes' arguments) says should generate a boost in aggregate output and lower the interest rate, but the interest rate is low, therefore liquidity trap. Or, more concisely, here is information that doesn't work with prior model, so modify model with a new effect.

Friedman's analysis said: Look at the graph: MB rose, but monetarism, so notice M2 didn't rise enough. Or, less concisely, here is information that doesn't work with prior model so use different information that confirms prior model. 

Now, of course, Friedman later abandoned M2 because he really didn't have a good reason to use it in the first place except to hold on to his priors. If you normalize to any given year except during the Great Depression or the Great Recession, M2 and MB roughly track each other. To me, that is strong evidence that these periods represent a new effect, and not just that you should use one measure or the other. [As an aside, M2 includes money created by fractional reserve banking and it seems to me that the lending involved should be closely tied to interest rates, hence involve an interest rate model.]

If you get down to it, Keynes was about new ideas while Friedman was about old ideology.


Thursday, August 8, 2013

Deriving the IS-LM model from information theory

I would like to use this derivation to illustrate a point: the information transfer framework is more general than the specific application to a quantity theory of money that has made up the bulk of the blog posts over the past month or so. The framework allows you to build supply-and-demand based models in a rigorous way. I will use it here to build the IS-LM model.

The IS-LM model attempts to explain the macroeconomy as the interaction between two markets: the Investment-Savings (goods) market and the Liquidity-Money Supply (money) market. The former effectively models the demand for goods with the interest rate functioning as the price (with what I can only guess is "aggregate supply" acting as the supply). The latter effectively models the demand for money with the interest rate functioning as the price (with the money supply acting as the supply). In the most basic version of the model, there is no real distinction made between the nominal and real interest rate.

Economists might find my "acting as the supply" language funny. I am only using it because in the information transfer framework, we have to know where the information source is transferring information: "the supply" is the destination. In our case, we are looking at two markets with a single constant information source (the aggregate demand) transferring information to the money supply (in the LM market) and the aggregate supply (in the IS market) via the interest rate (a single information transfer detector). The equation that governs this process is given by Equations (8a,b) in this post:
$$
\text{(8a) }P=
\frac{1}{\kappa }\frac{Q_0^d}{\left\langle Q^s\right\rangle }
$$
$$
\text{(8b) }
\Delta Q^d=\frac{Q_0^d}{\kappa }\log \left(\frac{\left\langle Q^s\right\rangle }{Q_{\text{ref}}^s}\right)
$$

However, each market employs these equations differently. The IS market is a fairly straightforward application. The price $P$ is replaced with the interest rate $r$, and the constant information source $Q^{d}_{0}$ becomes the equilibrium aggregate demand/output $Y^0$ (although we will also take it to be $Y^0 \rightarrow Y^0 + \Delta G$ in order to show the effects of a a boost in government spending, which shifts the IS curve outward). The expected aggregate supply is put in the place of $\langle Q^s \rangle$ is the variable used to trace out the IS curve. It can be eliminated to give a relationship between the interest rate and the change in $Y$ ($\Delta Y$ put in the place of $\Delta Q^d$). Thus we obtain
$$
\log r = \log \frac{Y^0}{\kappa_{IS} IS_{ref}} - \kappa_{IS}\frac{\Delta Y}{Y^0}
$$

The LM market employs an equilibrium condition in addition to Equations (8a,b), setting $\Delta Q^s = \Delta Q^d$ via the money supply $\Delta Q^s = \Delta M$ (this selects a point on the money demand curve). The constant information source $Q^{d}_{0}$ is still the equilibrium aggregate demand/output $Y^0$, but in the LM market we look at the curve traced out by the equilibrium point for shifts in the money demand curve (changing the "constant" information source, $Y^0 \rightarrow Y^0 + \Delta Y$). These two pieces of information allow us to write down the LM market equation:
$$
\log r = \log \frac{Y^0 + \Delta Y}{\kappa_{LM} LM_{ref}} - \kappa_{LM}\frac{\Delta M}{Y^0 + \Delta Y}
$$
Plotting both of these equations we obtain the IS-LM diagram which behaves as it should for monetary and fiscal expansion:


In both cases, $\kappa_{xx}$ and $XX_{ref}$ are constants that can be used to fit the model to data (I basically set them all to 1 because all I want to show here is behavior). The interest rate and output are in arbitrary units (effectively set by the constants).

As an aside, there is an interesting effect in the model. It basically breaks down if $r = 0$ (in the thermodynamic analogy, it is like trying to describe a zero pressure system -- it doesn't have any particles in it). As it approaches zero, the LM curve (and the IS curve) flatten out, producing the liquidity trap effect in the IS-LM model as popularized by Paul Krugman. Here is the graph for a close approach to zero:


This is not to say the zero lower bound problem is "correct" anymore than the IS-LM model is "correct". The results here only say that the IS-LM model is a perfectly acceptable model in the information transfer framework, which serves more to validate the framework (since IS-LM is an accepted part of economic theory ... economists may disagree whether it describes economic reality, but they agree that it e.g. belongs in economic textbooks).

What use is couching the IS-LM model in information theory? In my personal opinion, this is far more rigorous than how the model appears in economics. It is also possible information theory could help give a new source of intuition. To that end, let me describe the IS-LM model  in the language of information theory:
Aggregate demand acts as a constant information source sending a signal detected by the interest rate to both the aggregate supply and the money supply. Changes in aggregate demand are registered as changes in the information source in the LM market, but are registered in the response of the aggregate supply in the IS market [1]. Aggregate supply shifts to bring equilibrium to the IS market (the supply reads the information change), but M is set by the central bank and so does not automatically adjust. This creates a disequilibrium situation in which $I_{AD} = I_{AS}$ but $I_{AD} \neq I_{M}$; in order to restore equilibrium, either AD must return to its previous level or M must adjust (adjusting the interest rate) [2]. This defines what a recession is in the IS-LM model: a failure of the central bank to receive information (in information theory, we must have $I_{M} \leq I_{AD}$, i.e. the central bank cannot receive more information than is being transferred). A shift in output (e.g. by increasing government spending) is registered as a change in the information source in both the IS market and LM market so we can maintain $I_{AD} = I_{AS}$ and $I_{AD} = I_{M}$ by letting the interest rate adjust to the new equilibrium (e.g. crowding out).
[1] This difference is due to a modeling choice in order to represent empirically observed behavior.
[2] In a more complicated model there may be other possibilities.



Tuesday, August 6, 2013

What is a supply and demand diagram, anyway?



Here is the process for how you construct a supply and demand diagram in the information transfer model. You start with a 3D surface (think $x,y,z \rightarrow Q^d, Q^s, P$) defined by 
$$ P = \frac{1}{\kappa}\frac{Q^d}{Q^s} $$
and perform two projections into the $(Q^d , P)$ and $(Q^s , P)$ planes. You keep the line that "varies". For example, in the $(Q^s , P)$ plane, the supply curve doesn't vary with the quantity supplied because it is the set of points on the surface where $Q^s = \text{const}$ that includes the equilibrium point where the curves cross. This means the red supply curve appears as a vertical line, but the demand curve falls with increasing quantity supplied. The opposite goes for the projection in the $(Q^d , P)$ plane: here the demand curves are constant versus demand while the supply curves vary with the quantity demanded.

The dashed line represents a shift in the demand curve (it appears as a vertical line shifted to the right in the $(Q^d , P)$ plane).

Not all supply and demand diagrams look like the basic one. For example, sometimes you project both curves into the same plane (e.g. here). However, I think this is a good way to get a handle on the various ones that appear out there.

Monday, August 5, 2013

On Krugman's Models and Mechanisms

This is a thinking out loud post -- I am still a novice when it comes to how to translate results in the information transfer model (ITM) into the language of economics. If any economists are out there reading this, I'd appreciate some guidance.

Paul Krugman wrote up a blog post where he talks about the AD-AS model picture. I was initially drawn in by his comment
"... deflation could be expansionary if it is perceived as temporary, so that deflation now gives rise to expectations of future inflation."
I found this interesting -- it gives a way to achieve a weird effect in the information transfer model: expansionary contraction of the money supply. However, the ITM version doesn't depend on microfoundations or expectations. It simply happens when the the base is large compared to NGDP.

After musing on that for awhile, I decided to look at supply and demand curves in the information transfer model (since that's what Krugman was talking about). The keen thing about this model is the way it represents these curves. The supply and demand diagrams you frequently see in economics are a projection from a three dimensional space to a two dimensional space. I show what these look like for 1985 and 2010:



Both the red and blue curves show what the supply and demand curves look like at the given value (solid curve) and for shifts of +3% (dashed curves).

The red curves show what I call the supply curve, which is defined to be the equilibrium point at constant values of the monetary base for various values of AD and has more in common with the LM curve in the IS-LM model than the AS curves (SRAS or LRAS) in Krugman's pictures. I am also showing these curves vs NGDP which is different (I will show RGDP like Krugman does below). One interesting thing that is apparent is the MB-NGDP orthogonality line makes the AD curves bend over on themselves when viewed in the 2D space for 2010.

I took 1985 to constitute a "normal" time relative to the orthogonality line. The supply and demand curves for 1985 (graphed vs RGDP) look like this:


Here is the logic of the diagram. A boost of real AD at a constant monetary base should be deflationary as the monetary base is now too small relative to the new economy -- the real value of a dollar has gone up because each dollar now buys the same fraction of a larger economy. Increasing the monetary base without an increase in AD should raise the price level. Each dollar now buys a smaller fraction of the same real economy and real output falls.

Now we look at our current predicament. The supply and demand curves for 2010 (graphed vs RGDP) look like this:


Note that the "SRAS" curve has the same slope identified as showing a liquidity trap by Krugman in his post. However, the AD curves slope downward as opposed to upward. 

Here is the logic of the diagram in this case. A boost of AD at a constant monetary base will generate inflation because the monetary base is now too large relative to the size of the economy -- we are in a Japanese-style lost decade. Increasing the monetary base without an increase in AD will be slightly deflationary and lead to only a small increase in the new equilibrium RGDP (basically, the new money you printed doesn't budge inflation, so becomes real money). Since we are near the orthogonality line, monetary policy doesn't accomplish terribly much.

Where this boost in AD could come from is not specified in the information transfer model. It could be increased government spending. It could even potentially come from a market rally based on what the conventional wisdom says about monetary policy targets. In this second scenario, talk from the Fed about expanding the monetary base could cause people using one traditional model (i.e. not listening to me or people who say monetary policy is ineffective) to expect a boost in AD causing markets to go up producing an actual boost in AD. According to the ITM any concrete steps (concrete steppes?) to enact said policy would lead to some actual deflationary forces. If the AD boost had a larger effect on the price level than the deflationary MB effect, then the Fed could potentially eke out a victory. If the reverse occurred (MB effect overpowering the AD boost), then the deflation would catch up ... looking eerily like Japan's experience since the 1990s.

On a side note, Scott Sumner had a post up August 2nd or 3rd (2013) apparently critical Krugman's post (it simply quoted some of Krugman's post and said something along the lines of economics professors should take econ 101 again), but it has since disappeared. Maybe it was a mistake.

Thursday, August 1, 2013

Econophysics for fun and profit

If you are a physicist planning on revolutionizing economics with your bold new theory, I highly recommend reading Cosma Shalizi's very excellent rant [1] "Why oh why can't we have better econophysics?". For a short version, check out the entry on "scientists" at Noah Smith's Econo-troll bestiary. In [1] Shalizi says:
Let me also complain that there isn't enough physics: the repertoire of ideas taken from physics is very impoverished. Basically, we see random walks, power laws, and spin systems over and over again. These are important ideas, but they're just a small part of theoretical physics! To give an example, Eric Smith and Duncan Foley have a fun paper working through detailed mathematical analogies between the axiomatic versions of utility theory and thermodynamics, leading to a reversible "engine" that runs on credit.
The link is broken and should actually point to this working paper [2] Is utility theory so different from thermodynamics? which has subsequently been published. What follows is kind of an unstructured comment on that paper and borrows liberally from it and other related materials by the authors.

It turns out attempting to find analogies between physics and economics has a long history, including Walras (1909) and Fisher (1926) among others. Eventually, economists got fed up with this. Paul Samuelson has a fit in 1960:
The formal mathematical analogy between classical thermodynamics and mathematical economic systems has now been explored. This does not warrant the commonly met attempt to find more exact analogies of physical magnitudes -- such as entropy or energy -- in the economic realm. Why should there be laws like the first or second laws of thermodynamics holding in the economic realm? Why should ``utility'' be literally identified with entropy, energy, or anything else? Why should a failure to make such a successful identification lead anyone to overlook or deny the mathematical isomorphism that does exist between minimum systems that arise in different disciplines?
Oh snap. Anyway, some of the basic ideas that came out of the thermodynamic analogy seem to be that goods are extensive measures like energy or volume and prices are intensive measures like pressure or temperature. Hey, that's what I found! Prices are like pressure, the quantity supplied is like volume and the quantity demanded is like energy in the information transfer model.

Foley and Smith [2] make a really interesting point about the thermodynamic analogy: economists tend to study what seem to be irreversible processes (people will not make exchanges to undo their utility gains) and physicists tended to study reversible processes (at least when they started coming up with thermodynamic laws). This difference changes the whole approach to problem solving, leaving the fields looking completely different. However, this is the point where I think [2] goes down a rabbit hole the information transfer model avoids. The authors make the mistake that Samuelson derides above: they make a homological association (in order to avoid the word "analogy") of utility with entropy.

The idea of utility maximization is pervasive in economics and inextricably links it with the normative ethical theory with the same concept. "Rational" expectations has economic agents out there maximizing individual utility. Cue Shalizi: "Alas, experimental psychology, and still more experimental economics, amply demonstrate that empirically [the neoclassical framework is] just wrong." If your economic framework has utility maximization as a fundamental theorem in the same way that thermodynamics has a second law, then the framework itself really is just a (likely normative) description of a particular class of states (since utility maximization is not generally true in the real world) and your entire mode of study as a would be econophysicist is to calculate expansions around your theory. This is perverse. It would be like building a particle theory in a quantum field framework, finding the vacuum state and then making up an entirely new theory to study deviations from that vacuum. That is to say the framework should ideally describe the fluctuations around the equilibria. Barring that, the framework should at least allow the kinds of fluctuations you see. If some of your fluctuations are meaningful and actually, whoops, violate fundamental theorems in your framework, then what good are your equilibria? 

My personal feeling is that the normative stuff got economics in trouble in a lot more ways than physicists trying to make analogies with thermodynamics. Homo economicus is an alien. If your models require humans to behave like this in order to be solvable then there is something seriously wrong. (For another interesting take, check out this great article on Nietzsche and Austrian economics -- the successful capitalist as √úbermensch, rather than, as most economics research shows, mostly just lucky or sitting on economic rent. I mean, basic analysis shows that without barriers to entry prices should become the marginal unit cost of production and profit should go to zero, no?)

The information transfer model avoids this morass the same way Shannon made information theory into a field in its own right: not caring about the content of the message. Information content is maximized in a random string! I don't care what the signal being sent from the demand to the supply actually is. This is not to say the information transfer model is right; it's just not normative. For example, diminishing marginal utility sounds pretty dumb if you try and ascribe it to the thermodynamic analogy in the information transfer model (weirdly quoting myself):
Translating [diminishing marginal utility] to the thermodynamic analogy the ridiculousness becomes obvious: it says "when undergoing an isothermal expansion, the pressure an atom is willing to exert falls because of the diminishing marginal utility of extra volume". Diminishing marginal utility for goods is actually a sign choice and is due to choosing demand as the information source rather than destination.
I precede that statement by noting the fact that in the information transfer model, diminishing marginal utility is not even a property of an economic agent (which aren't even defined), but rather an ensemble of economic agents.

I think if you are a physicist with an eye to revolutionize economics, you should keep a few things in mind. Theorems and fundamental laws that always apply don't translate well to the human domain. Be cautious when making normative claims and be aware of normative assumptions that are baked into your model. This is especially dangerous in economics due to its closeness to normative ethics and due to the moral gut feelings humans have about e.g. debt. And be nice: make sure your model reduces to stuff economists already know (for some reason people get angry when you declare their entire grad school education was all for naught).

PS What is it with Smiths and economics? Noah Smith, Eric Smith, Adam Smith. My last name is Smith, too.