Blowing the anti-neo-Fisherite model out of the water

Scott Sumner, to put it lightly, doesn't agree with Noah Smith. And neither does Nick Rowe. The claim is that Canada's inflation target has worked out just fine blows the neo-Fisherite model out of the water. But let's look at where Canada is on my favorite graph of the price level vs the currency component of the monetary base:

Canada is where the US is in the mid-1990s -- still on the highly sloped part of the curve where monetary policy is effective (also note that Sumner quotes Milton Friedman from 1998). The neo-Fisherite piece of the curve is on the right hand side -- where the US was during the depression, where Japan is now and where the US and EU are moving.

Canada shouldn't have a problem hitting its inflation targets yet. Let's check back in 20 years.

Sumner then moves on to say the Fisher effect in interest rates is confusing neo-Fisherites. However, this also doesn't demonstrate anything -- the direction of interest rate changes is well described by the information transfer model (in the diagram below). The Fisher effect only strongly impacted long term interest rates during the 1970s and 1980s. Here's the model diagram (note that the price level vs the monetary base curve is effectively the curve at the top of this post):

Equilibrium in a two-good market

Nick Rowe put up a reference post about capital and interest. I thought I'd follow along with the information transfer model. Rowe initially sets up two markets: apples and bananas. If you'd like to start from the beginning with this, here's my original post on information theory in a two good market.

If we solve the equations (8a,b) and (9a,b) in this post for the price in terms of a change in quantity supplied $\Delta Q^{s}$ or demanded $\Delta Q^{d}$, we have

$$
P = p_{0} e^{\Delta Q^{s}/c_{s}} = p_{0} e^{- \Delta Q^{d}/c_{d}}
$$

These equations define supply and demand curves. We'll assume the markets for apples and bananas are independent in the sense that the supply and demand curves for one market don't depend explicitly on the other. This means that if we plot the supply and demand curves for e.g. bananas in the 3D space $(\Delta A^{(s,d)}, \Delta B^{(s,d)}, P)$, they are constant surfaces in the $\Delta A$ (apples) direction, and every slice at constant $\Delta A$ is the same supply and demand curves for bananas. For visual people, pictures are best (bananas on left, apples on right):

The same goes for the apples, but it's constant along the $\Delta B$ direction (on right, above). If I add the supply surfaces prices (times the quantity supplied) for both goods, I get the Production Possibilities (PPS) surface (in relative terms)

$$
PPS = a_{0} \Delta A^{s}  e^{\Delta A^{s}/a_{s}} + b_{0} \Delta B^{s} e^{\Delta B^{s}/b_{s}}
$$

If I do the same for the demands, I get the Indifference Surface (IS):

$$
IS = a_{0} \Delta A^{d}  e^{-\Delta A^{d}/a_{d}} + b_{0} \Delta B^{d}  e^{-\Delta B^{d}/b_{d}}
$$

I plot the PPS in orange and the IS in blue here:

These two surfaces are tangent at the black dot. The locations where $PPS = 0$ represent the production possibilities frontier (PPF), dotted orange curve. This is the line where the total cost to supply the given amount of apples and bananas at particular location on the curve is the same as the cost to supply the amount in equilibrium (i.e. at $\Delta A = \Delta B = 0$ recall we are looking at $\Delta Q^{s}$, not the absolute level $Q^{s}$). Likewise, $IS = 0$ represents the indifference curve (IC). One way to think about these lines is that the represent lines of constant aggregate supply (PPF) and constant aggregate demand (IC) with a changing mix of apples and bananas produced/consumed. In fact, the vertical axis is basically aggregate demand (supply) relative to equilibrium.  By "aggregate" here, we mean just apples and bananas in this two-good economy.

The last piece required to reproduce Rowe's graph is the "budget constraint". This is basically the equation:

$$
a_{0} \Delta A + b_{0} \Delta B = 0
$$

It's the amount you can change quantity of apples or bananas without spending more or less money (note that $a_{0}$ is the equilibrium price of apples). Here are the two curves and the budget constraint, plotted on the same (2D) graph (essentially the slice through zero on the z-axis = relative aggregate demand = relative aggregate supply):

That's all for now. I'll get to the rest of Rowe's post later.

PS Here's another view of those tangent surfaces:

Good ad hoc vs bad ad hoc

This goes with this post and Tom Brown's comment asking what I meant by bad ad hoc models. Plastic, like expectations in economics, can be molded into many form factors.

The effect of expectations in economics (third addendum)

Here's a point I left off that didn't fit well in the post from yesterday. In the paper [pdf] Mark Sadowski links to, the mechanism for how expectations enter into the model is illustrated (as an extreme case) in their Figure 1. This is ad hoc in the worst way. The model does not actually add any information aside from the original observation that "calling" a recession has an impact -- a classic example of (the original meaning of) begging the question. If I have an empirical effect and I assume a mathematical model that gives me that effect, what have I learned?

This is an example of what I called the useless power of expectations. I should have been more explicit by calling it the useless power of theoretical models of expectations. I can achieve whatever I'd like by assuming some piece of a model that accomplishes that effect.

For another example, Scott Sumner put forward a verbal model of expectations. Let gold sell at a price p. Sumner first asked what happens if a mining company discovers a huge amount of gold and brings it to market. Well, as supply and demand goes, the price of gold falls. He even provides a just-so story mechanism (the "hot potato effect") whereby individuals unload gold because now there is more gold than individuals want to hold in equilibrium. Eventually the price falls over these transactions until everyone wants to hold as much gold as they have at the new price p'.

Sumner then asks what happens if the company just announces the gold discovery. He says gold prices plunge. I'd agree, but I ask: how far? The worst ad hoc theoretical model of expectations would be to say they fall to p'. But if it's not p', what is it? And why does this new p'' differ from p' derived from the "hot potato effect"? It is possible people will use the heuristic of the relative size of discovery versus the amount of gold previously in the market (e.g. a discovery of 10% the size of the gold market should cause the price to fall about 10%, ceteris paribus). Maybe that heuristic will anchor the expected value at p''. Even then, the distribution of prices achieved [1] in the market grinding down to p' via the "hot potato effect" in the first case will in general differ from the distribution of the expected price p''. Since those probability distributions differ, there is information loss in Hayek's information aggregating function of the price mechanism [2] (i.e. the KL divergence of the two distributions).

This is why rational expectations is one of the only ways forward-looking expectations [3] have been incorporated in economics in a tractable way -- it assumes away the inconsistency between not only the prices p'  and p'', ascribing it to random error, but also the distributions. But that means rational expectations is precisely the assumption that the price falls to p' -- the worst ad hoc theoretical model.

[1] In different possible worlds, the price p' is different -- that's the distribution.

[2] In the information transfer model, we'd say that the information loss due to expectations means that p'' is initially lower than p', eventually drifting back towards p' (ceteris paribus).

[3] Backward-looking expectations (like adaptive expectations or the simple martingale of this post) don't present the same issues as forward-looking expectations. Generally, when these are dramatically wrong it's because of an unforeseen shock.

Back to the abstract

A frame from the probably-never-to-see-the-light-of-day video introduction to information transfer economics. Additionally, this is probably not an accurate depiction of Fielitz and Borchardt. Graphics borrowed from here.

I saw that Fielitz and Borchardt, whose pre-print inspired me to develop the information transfer model of economics, have added a new reference to economics (in bold) in an update (v3):

Information theory provides shortcuts which allow to deal with complex systems. The basic idea one uses for this purpose is the maximum entropy principle developed by Jaynes. However, an extensions of this maximum entropy principle to systems far from thermal equilibrium or even to non-physical systems is problematic because it requires an adequate choice of constraints. In this paper we apply the information theory in an even more abstract way and propose an information transfer model of natural processes which does not require any choice of adequate constraints. It is, therefore, directly applicable to systems far from thermal equilibrium and to non-physical systems/processes (e.g. biological processes and economical processes). We demonstrate the validity and the applicability of the information transfer concept by three well understood physical processes. As an interesting astronomical application we will show that the information transfer concept allows to rationalize and to quantify the K effect.

http://arxiv.org/abs/0905.0610

I wonder if they saw my blog? I have about 1000 pageviews from Germany ...

UPDATE! I forgot this blog's first birthday! I just noticed the date on the first post (linked up at the top) was April 24, 2013. As a little celebration, here are the top three posts since the blog's inception (I'm actually really proud of all three of these):

1. How money transfers information

http://informationtransfereconomics.blogspot.com/2014/03/how-money-transfers-information.html

2. Entropy and microfoundations

http://informationtransfereconomics.blogspot.com/2014/02/ii-entropy-and-microfoundations.html

3. The link between the monetary base and interest rates

http://informationtransfereconomics.blogspot.com/2014/02/the-link-between-monetary-base-and.html

The effect of expectations in economics (second addendum)

Just a little graph for thought: I went and looked at the distributions of the difference between the empirical and theoretical short term (3-month) interest rate curves from this post when separated into positive and negative differences. I'll suggestively put it up next to the KL divergence graph from this post ...

Hmm ...

The effect of expectations in economics (addendum)

I mentioned in the previous post that I would show and updated diagram:

In thinking about this while writing this post, maybe the theoretical price [interest rate] should be fit to the empirical data (as is done here) instead of being fit to an upper bound of the empirical data as is done above [in the post]. This solution would represent the gray "least informative prior" solution running through the data and the blue more accurate expectations would rise above the theoretical curve and the inaccurate expectations would still fall below the theoretical curve. I will update with this version in a follow-up post.

Here is the new graph:

In this presentation, the theoretical model curve (dark gray) follows the peak of the least informative prior (gray histogram). The incorrect expectations (red histogram) match up with negative deviations and the more accurate expectations (blue histogram) match up with the positive deviations.

For completeness, here is the fit to the short term (3-month) interest rate (the graph above falls in the gray box):

The effect of expectations in economics

Beef. Wikimedia Commons (link).

I started out a post a few days ago with saying I had a beef with use of expectations in economics that I thought I'd flesh out a bit in light of some comments from Tom Brown and Mark Sadowski. I didn't mean to imply that expectations didn't have an effect on the direction of prices. The beef was with the use of expectations in theoretical models in economics. The particular case I called out was inflation expectations -- a theory including measured expectations was no more predictive of inflation than a simple martingale that used past inflation to predict future inflation. In fact, the expectations model was systematically inaccurate. 

The point of my post was to show that systematic inaccuracy leads to information loss which translates into a negative impact on the economy. However I didn't emphasize enough that it doesn't matter the sign of the inaccuracy -- even over-optimism should lead to information loss and a negative impact on the economy. The systematic high inflation expectations above could lead to sub-par NGDP growth relative to the fundamentals. In fact, ever since inflation reached a peak in the late 1970s in many industrialized countries, people have expected higher inflation than actually occurred and NGDP growth has been lower (I'm not saying this is the correct reason for this; there are many different explanations).

Mark Sadowski linked [pdf] in his comment to a report that showed an impact of expectations of uninformed vs informed consumers based on a recession being "called". Essentially when a recession is called, even if it is just a "red zero" where it could have easily been negative because of random chance (not fundamentals), it creates a disproportionate negative impact on consumer spending by "uninformed consumers". In the information transfer model we'd explain this by saying some subset of the population updated their probabilities of expected growth rate incorrectly, leading to information loss, creating a negative impact on the economy. This is just another example of being incorrectly pessimistic about the fundamentals leading to negative impact.

The interesting cut of the beef, and the one that diverges from mainstream economic thought, is where inaccurate optimism about growth also leads to lower growth. This has particular relevance to monetarist theories purporting that more effective central bank communication could lead to a greater impact of policies such as quantitative easing (even allowing less monetary expansion to achieve the same effect, see e.g. here). Is the sub-par growth in the post-financial crisis period due to expectation of recovery? Maybe. Maybe it's just secular stagnation.

This idea that incorrect expectations lead to a negative impact regardless if they are positive or negative jibes with Milton Friedman's "plucking model" where there is a natural path of e.g. NGDP and deviations are all negative deviations.

This idea that there are only negative deviations traces back to the idea in the information transfer model that the information in the demand (D) represents a bound on the amount of information transferred so that the information received by the supply (S) is always less, i.e. I(S) ≤ I(D). We call the case of equality "ideal information transfer" and inequality "non-ideal". Solving some differential equations and using Gronwall's inequality leads us to the following supply and demand diagram:

The solid supply and demand curves represent ideal information transfer intersecting at the ideal price P*, and the real world price P would appear somewhere in the lower triangle. If the difference between the ideal and non-ideal cases was stable, the price would follow the dotted supply and demand curves. The key takeaway is that the price P always falls below the theoretical model for the ideal case.

In the earlier post on expectations, I showed a plot of the information loss due to assuming an incorrect probability distribution. I want to connect that graph to the "plucking model" of the short term interest rate. Here is the model calculation (as an upper bound, blue) and the empirical 3-month interest rate (green):

Zooming in on the gray box, and reproducing the graph from the earlier post on expectations, I indicate how the information loss connects to the deviation of the empirical data from the theoretical model assuming ideal information transfer:

The typical case is shown in gray. Sometimes expectations are more accurate and the empirical price matches more closely to the theoretical price (blue). Sometimes expectations are less accurate and the empirical price falls well below the theoretical price (red).

...

In thinking about this while writing this post, maybe the theoretical price should be fit to the empirical data (as is done here) instead of being fit to an upper bound of the empirical data as is done above. This solution would represent the gray "least informative prior" solution running through the data and the blue more accurate expectations would rise above the theoretical curve and the inaccurate expectations would still fall below the theoretical curve. I will update with this version in a follow-up post.

A neo-Fisherite rebellion? Yes, please.

I miss all the good stuff when I'm on vacation. I missed most of the Steve Williamson debate, and now I missed the neo-Fisherite rebellion. The rebellion is linked to secular stagnation and deflationary monetary expansion, to both of which I think the information transfer model lends support. But let me tackle Noah Smith's characterization rather directly. First, here's the characterization:

... the Neo-Fisherite idea is NOT that a fall in interest rates causes the price level to jump up and then drift back down to its original level. If the Neo-Fisherites are right, holding interest rates at low levels for a long time will cause a long-term deflationary trend that will eventually push the price level lower than if interest rates had been kept at the old, higher level

And here's the tackling (in graph form**):

The red lines represent constant interest rates, the black line represents the "information trap"/"liquidity trap" criterion ∂P/∂M0 = 0 (it is the top of a ridge in the price level function P = P(M0, NGDP) that is graphed in 3D here, the gray lines in the graph above are the level curves of that surface). On the left side of the ∂P/∂M0 = 0 line in the graph, the price level has a positive relationship to changes in the monetary base (printing currency causes inflation) but on the right side printing currency causes deflation.

You can see that lower interest rates (red lines in the graph) run into the ∂P/∂M0 = 0 line at a lower monetary base relative to higher interest rates. This is exactly the neo-Fisherite claim! Holding interest rates low runs runs you into low inflation faster than holding rates higher.

Noah Smith also asked for some microfoundations for this effect -- I have them here. It is the competition between the unit of account and medium of exchange functions of money that leads to a diminishing marginal utility of monetary expansion. Eventually there is enough currency to base transactions on and more currency just means each unit is smaller relative to NGDP and doesn't carry as much information.

** This graph is just the 2D projection of the 3D graph that is at the upper right of every blog page. The red lines are the level curves of the red "interest rate surface" r = r(M0, NGDP) and the gray lines are the level curves of the white "price level surface" P = P(M0, NGDP).

Expectations destroy information

I have a beef with the use of expectations in economics. They seem to be so powerful as to make them useless as an explanation; they also seem to lack empirical impact where they originally were designed to have an impact.

Here, I am going to describe the role expectations play in the economy using information theory -- it's not diving too deep, mostly just an argument using Shannon information and measuring information loss with the KL divergence.

The KL divergence measures the extra message length for a given amount of data that must be sent if you have a code optimal for the wrong distribution relative to the true distribution. In general, it represents the information loss (measured in nats or bits depending on which logarithm you use) by being wrong about a distribution relative to being right.

Relating this to information transfer economics, the KL divergence represents the extra information that must be sent in the market (lower information efficiency) given the wrong expected economic state distribution relative to the true future economic state distribution. I previously hypothesized that lower information efficiency was related to recessions.

In this post, I want to show that economic "expectations" generally destroy information unless they are right. I put together some simulations illustrating this point using a relatively simple model. Imagine there are 10 states the economy can be in (think a hidden semi-Markov model). There is a probability distribution that gives the chance it is in one of the state in the next time period, for example, if all states are equally likely, this distribution would be

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

i.e. a 10% chance it is in any of the ten states.

If the economy was going to be in state #2 with 100% probability, then it looks like this:

0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

We will compare two of these distributions, call them the actual (A) and the expected (E) distributions with the KL divergence. The KL divergence of the two distributions above is

0.0 log(0.0/0.1) + 1.0 log(1.0/0.1) + 0.0 log(0.0/0.1) + ... + 0.0 log(0.0/0.1)

= 0 + 2.303 + 0 + 0 + ... + 0

= 2.303

That is, there is about 2.3 nats (3.3 bits) more information in the second sequence relative to the first. (Which makes sense: specifying a number from 1 to 10 -- in our case state #2 --  requires 3.3 bits.)

In the following, I actually show the negative of the KL divergence (partially because I accidentally did it that way and am too lazy to change it, and partially because it emphasizes that we are looking at information loss).

In the simulations below, I randomly generated 10,000 probability distributions on the 10 states and looked at the KL divergence. I did three cases: one, the expected distribution was a small perturbation from the actual distribution (E = A + δA), two, the expected distribution was the first distribution given above .. all states equally likely aka the least informative prior (E = 0.1), and three, the expected distribution was another randomly generated distribution (E and A are uncorrelated). Here are the results:

We can see that if there aren't any big changes in the distribution (the small perturbation, blue), the information loss is minimal. If there are big changes (the new distribution is uncorrelated, red) then information loss is not only large, but has a long tail. Interestingly, the least informative prior (gray) is not only in the middle of these, but doesn't have much of a tail (there is a sharp cut-off). You'd imagine the blue histogram showing the typical case where past performance gives an indication of future performance, with only small deviations. The red histogram shows what it's like if you're wrong about the future.

What does this mean? Well, if expectations are accurate (people can accurately predict the future), then there is a limited amount of information loss. It may be reductio ad absurdum, but I'd say this pretty much proves that there is information loss in the market mechanism. Who can accurately predict the future? I'll take it a step further and say that expectations are the cause of that loss of information, and information loss appears to be a primary driver of recessions.

When people are especially bad at predicting the future, you not only get massive information loss on average, but there is a long tail of even greater loss (red histogram).

The average case for information loss, falling between the two extremes, is pretty well described by the least informative prior. This is potentially a reason why the information transfer model, which assumes this as a starting point, can do a good job with trends. This least informative prior is also effectively the efficient markets hypothesis in the real-world sense. There are trends and momentum, but price movements are unpredictable. They are not completely unpredictable; the information loss (red histogram) would be large. This least informative prior is also the assumption that expectations do not affect the long run trends.

Ah, but, you say, if you kind of know what you are talking about (you're able to predict the future), then you can do better than the least informative prior. That's why I have this next graph: it shows that the ability to do better than the least informative prior is very sensitive to how wrong you are -- you only need to be a little bit wrong for the tail risk to be a serious negative impact on your average performance. I plot the histograms for a 10% perturbation (used in the graph above) up to a 40% perturbation (in the last one, I show the 100% perturbation from the graph above in red):

The takeaway is that you basically always have to be right otherwise the potential losses from being wrong will add up and cause you to lose more information than the least informative prior over time.

Ah, but, you say, wouldn't an expectation based on the information transfer model give you at least some capability to predict the future -- contradicting your assertion that people are bad at predicting the future? Nope. The information transfer model is built on the least informative prior -- you are assuming maximum ignorance about the future, that is to say, you are assuming as much ignorance about the future as I am claiming people are already in possession of! That is to say, maximum ignorance.