More on Japanese inflation

I recently said that Noah Smith (and Scott Sumner in a comment on his blog here) had been hasty in their declarations of victory over deflation in Japan. Noah credits both monetary and fiscal policy, while Sumner (naturally) credits just monetary policy.

There was a large jump in the price level that started in April that I was quite shocked by. However it appears as though it is due to the increase in the VAT from 5% to 8% in April. In fact, Sumner predicted a jump in the price level because of the VAT increase almost a year ago (he warned: "Don't be fooled").

Where does this put us with regards to "Abenomics"? I'll still say things are inconclusive. And I'll say it with a graph:

The data points are the blue dots. I show the "starting point" of Abenomics along with the deflationary trend (gray dashed) and 0%, 1% and 2% inflation paths (light blue, with 2% being the "Abenomics" target). The information transfer (IT) model was fit to pre-2013 data (shown in orange) and used subsequent evolution of NGDP and the currency component of the monetary base to show where the model predicts the price level to be (red). I also show a counterfactual path without the stimulus spending during 2013 (the Keynesian component of Abenomics, assuming a multiplier of 1, i.e. ΔNGDP = ΔG) as dotted red line.

It appears as though the data is consistent with 1% inflation, the IT model is consistent with 0% inflation and the IT model without stimulus is consistent with the pre-2013 deflationary path. The post-2013 data are within the model error, though (gray band, 1-σ errors). We'll just have to see more data. However I think the rest of 2014 can potentially be decisive.

PS I made an attempt to subtract out the VAT increase (red points) by linear extapolation from the last two pre-VAT points to the first post-VAT point and subtracting the difference from the last two points. I can't say the result changes much -- the data is still consistent with 1% inflation.

A challenge to macroeconomists

In light of my comments on this post by Scott Sumner who declares victory without having specified anything quantitative at all, and pursuant to a discussion with Tom Brown on this post where I brought up something I have noticed in my efforts to build up a macroeconomic model, I'd like to make a point. It appears no economist has ever actually compared a theoretical model of inflation to empirical data. I've tried to find other cases with which to compare my efforts (my efforts are here, here and here, for example), and the best I've come up with is this analysis from 1983:

It's from Principles of Economics, Volume 1 by Mankiw. Even this is taking "theoretical model" fairly loosely. First off, these graphs don't even appear in the original paper by Sargent. Second, the graphs only loosely imply that P = k M, however economists don't actually believe the k in that equation is constant. Third, this only covers a four years of the post WWI hyperinflation.

This puts not just DSGE fans in the realm of being like string theorists [1] in physics (theoretical models with no contact to data), but in fact all macroeconomists since, well, Walras.

My challenge/request/bleg is simple:

Produce a graph with an empirical measure of the price level (or inflation rate*) and a theoretical curve.

It can be any measure of the price level (here is an example from FRED). Just give me some data points with a line near it (it doesn't even have to go through the data points).

Maybe there are a bunch of such graphs in gated economics journals. I highly doubt it. I don't believe they exist. Why? Because when I do a Google image search on "price level model" (without the quotes, even) I get back my own #$%@ graphs!!!

You might want to send me this graph from Nick Rowe:

I see your ad hoc monetary-policy-can-hit-any-inflation-target-it-likes model (that isn't even accurate) and raise you the interest rate (link):

I'd really prefer something that isn't a straight line. Actually, I'd really like to see results for Japan. Here are mine (oh, with an interest rate model of course):

Did I mention that it's the same model as the one used for the Canadian graphs above? [2]

[1] Don't get me wrong. I love string theory.

[2] Hopefully this post is taken as good-natured ribbing, or at least good-natured snark.

*Update 7/3/2014. Nick Edmonds makes a good point (H/T Tom Brown) that economists tend to look at the inflation rate rather than the price level. To that end, I'll expand the challenge to price level or inflation rate. Here is one for the US from me:

Here are results for Japan. Here are some "out of sample" results for the US (predicting today's low inflation from data in the 1970s). Here are some more general considerations, extracting the general trend towards lower inflation.

Macroeconomists, you're "On Notice" :)

Update 7/3/2014 This one (on inflation) is pretty good from the NY Fed:

Update 7/3/2014 Not for nothing, the NY Fed model has 42 (!) parameters. The information transfer model (ITM) has 3. Additionally, the NY Fed results were "smoothed", whereas the ITM results were not.

Output and price level behavior across several economies

I thought I'd aggregate the price level vs monetary base (which I've shown before at this link) and nominal output vs monetary base data into graphs and compare them with their expected behavior in the information transfer model (using 100 aggregated economies each built from 100 random markets based on the results at this link).

The result is pretty remarkable:

Note that this not only reproduces the relative curvatures of the theoretical graphs, but their variance (the nominal output vs monetary base being a tighter curve than the price level vs monetary base). Another interesting aspect is that the overall behavior only becomes apparent when looking at the time series for many countries.

This model unifies "the Great Stagnation"/"secular stagnation" and "neo-Fisherite" models in a larger synthesis. The slowing of economic growth and prolonged low interest rates leading to lower inflation are two facets of the same general behavior of economies. The concept of convergence (smaller economies experiencing "catch-up" growth) also falls under this general behavior.

Hard core information transfer economics

Noah Smith had a link that reminded me of something from the philosophy of science in his post from yesterday. It inspired me to lay out the "hard core" of the information transfer economics research program since it is fairly simple:

1. Demand is a source of information that is transferred to the supply, and a price is a detector of information transfer.

2. The dynamics of supply, demand and the price are governed by the differential equation (and definition):

$$ p \equiv \frac{dD}{dS} = \frac{1}{\kappa} \; \frac{D}{S} $$

There are two approaches to macroeconomics I've been taking. One is that macro is just like micro: you can write down aggregate demand and aggregate supply functions just like you'd do for a single good market. The second is that macro is the sum of micro: i.e. macroeconomic observables are expectation values of microeconomic observables in an ensemble of micro markets. The former may be a good approximation to the latter within some realm of validity. The first approach is an "auxiliary hypothesis" in the Lakatosian sense. The second adds some assumptions around the partition function as auxiliary hypotheses.

The idea of a changing $\kappa$ in the price level is another "auxiliary hypothesis"  (this has some support from the sum of micro approach). The idea that the "price" of a treasury bond with interest rate $r$ is $r^{c}$ is another auxiliary hypothesis.

Is the supply curve flat?

Answering the question in the title, no, but during the course of writing the past few posts, I'd looked at the wikipedia article on general equilibrium. I saw this random bit about Sraffa:

Anglo-American economists became more interested in general equilibrium in the late 1920s and 1930s after Piero Sraffa's demonstration that Marshallian economists cannot account for the forces thought to account for the upward-slope of the supply curve for a consumer good.

What follows is a just-so story mechanism about how changes in output should affect the price. It appears as though Sraffa's argument entirely ignores the premise of Marshallian supply and demand diagrams (there is a single good in a single market) by asserting that there are other goods and factors of production. The use of "first order" in the wikipedia explanation of the mechanism is also pretty laughable since Sraffa didn't include a single equation much less any scales which we could use to say that anything is "first order". I tried to read Sraffa but was instantly filled with a grim sense of philosophers talking about what change is. I then Googled around and found this, which helped a bit. I concluded that information theory is the proper way to deal with the entire situation.

What are we talking about when we draw supply and demand curves anyway? Let's go back to the beginnings of this blog. Supply (S) and demand (D) are related by the equation:

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

where P is the price. The general solution (general equilibrium) to this is:

$$\text{(2) } \frac{D}{D_{ref}} = \left( \frac{S}{S_{ref}} \right)^{1/\kappa} $$

A demand curve is what you get when you look at the partial equilibrium by holding demand constant $D = D_{0}$ (an "exogenous" constant information source) and relating it back to the price to get a pair of equations:

$$ P = \frac{1}{\kappa} \; \frac{D_{0}}{\langle S \rangle}$$

$$ \Delta D \equiv D - D_{ref} = \frac{D_{0}}{\kappa}  \log \frac{\langle S \rangle}{S_{ref}} $$

Symmetrically, a supply curve is what you get when you look at the partial equilibrium by holding supply constant (an "exogenous" constant information destination)

$$ P = \frac{1}{\kappa} \; \frac{\langle D \rangle}{S_{0}}$$

$$ \Delta S \equiv S - S_{ref} = \kappa S_{0}  \log \frac{\langle D \rangle}{D_{ref}} $$

What are the angle brackets for? They're there to remind us that the variable is "endogenous" (the expected value inside the model) while the other variable is exogenous. These angle bracket variables are important to the discussion above because they parameterize our position along a supply or demand curve. Here are the supply (red) and demand (blue) curves along with the "fully endogenous" general equilibrium solution (gray dotted):

The demand curve seems to make intuitive sense -- at least at first. If the price goes up, the quantity of a good demanded goes down. But you get into trouble if you try this reasoning the other way: if the price goes down, the quantity demanded goes up? Maybe. Maybe not. If you weren't getting enough at the current price, it might. That depends on your utility, though. Sometimes this is referred to as the diminishing marginal utility of a good: the price you are willing to pay goes down the more widely available a good is. But there we've gone and switched up the independent variable again. The first half (effect of a price change) looks at price as independent, while the second half (diminishing marginal utility) looks at $\langle S \rangle$ as the independent variable [1]. 

Both of these get the partial equilibrium analysis in the wrong order mathematically. What we have is an exogenous (independent) change in demand. If demand increases and is satisfied (which we assume by taking $\langle S \rangle$ as endogenous/dependent), the price goes down.

This seems like a totally non-intuitive way to think about it. What is really going on here?

What we have is a system in contact with a "demand bath" [2] (or better yet, an "aggregate demand bath"). You could also call it an "information bath". If that bath didn't exist, adding (satisfied) demand would make the price go up, per equation (2) above (and it would move along the gray dotted curve in the figure above). What the bath is doing is sucking up any extra demand (source information) that we are creating by moving along the demand curve, so that "demand" is the same before and after the shift. Since there is "no change" in demand (any change is mitigated by the bath), the next variable in the chain, $\langle S \rangle$ effectively becomes the changing independent variable. This means the explanation is that decreasing/increasing the supply increases/decreases the price at constant demand (in the presence of a demand bath).

So why does the supply curve slope upwards? This time we're in contact with a "supply bath", so "the supply" is basically the same after we move along the curve. Moving along the supply curve is a change in $\langle D \rangle$. This means the explanation is that decreasing/increasing the demand decreases/increases the price at constant supply (in the presence of a supply bath).

Therefore there is no actual change in the supply along a supply curve so there is no bidding up factors of production or lack thereof per Sraffa. What we're doing is increasing the demand, so the price goes up.

That is to say supply and demand curves are kind of misnomers. A supply curve is the behavior of the price at constant supply, but is parameterized by increasing or decreasing demand. A demand curve is the behavior of the price at constant demand, but is parameterized by increasing or decreasing supply. [3]

[1] Yes, economists take price to be the independent variable, but in the formulation above it is more natural that either supply or demand (or both) are the independent variable(s). The price is the derivative of the demand with respect to the supply (the marginal change in demand for a marginal change in supply).

[2] This whole description is based on an isothermal expansion/compression of an ideal gas in contact with a thermal bath.

[3] Shifts "of" the supply and demand curves (as opposed to shift "along") are effectively changes in the information bath

Towards Arrow-Debreu-McKenzie equilibrium, part N of N

After the results of this post on the macroeconomic partition function, I'm abandoning Arrow-Debreu-McKenzie equilibrium. Not because it is hard, but because it is likely meaningless for macroeconomics.

Let's look at what the ADM equilibrium says with regards to a partition function in thermodynamics. It effectively says there exists some set of occupation numbers so that the energy of the system is the total energy, or more generally, there exists a microstate consistent with an observed macrostate. The SMD theorem then tells us that there are only limited properties of that microstate that survive to the macrostate. In some sense, the SMD theorem should be intuitive: if you have a system with N degrees of freedom, but is described by n << N degrees of freedom at the macro scale, then the subset of properties of N degrees of freedom that follow as properties of the n degrees of freedom, is likely to be smaller. 

The other consequence of the SMD theorem should also be intuitive. If your macro system appears to be described by n << N degrees of freedom, then it seems highly likely that among the total number of microstates, large subsets of the microstates are going to be described by a given macro state -- i.e. the equilibrium (the microstate satisfying macro constraints) is not going to be unique. For example, in an ideal gas, you can reverse the direction of the particle velocities and obtain another equilibrium (actually, all spatial, rotational and time-reversal symmetries lead you to other equilibria).

The reason economists think ADM is useful is probably due to their obsession with initial endowments. The ADM theorem goes part way to answering the question: Which set of prices let households and firms reach their final desired endowments given their initial endowments? The theorem says that there exists a set of prices that do that, and that is good to know! But these prices clear the markets in period two and they've finished their job [1]. This is a bit like worrying about how the energy gets redistributed to each atom of in gas when two gasses are mixed.

In an economy, the equilibria are more restricted than energies among the atoms in a gas and it's not trivial to show that they exist (or that they are Pareto efficient). I'm not knocking ADM. However, the existence seems meaningless for a real economy. As soon as a new product is invented, you're heading to another equilibrium. As soon as someone gets paid, you're heading for another equilibrium (if that someone would like to have more goods and services instead of holding cash). In reality, there may be a detailed balance that keeps the equilibria in an equivalence class described by e.g. a given NGDP growth rate. But that's the rub! Macroeconomics is the study of the behavior of those equivalence classes, not the instances of them!

That is to say macroeconomics is the study of the properties of ensemble averages (equivalence classes of microstates). Or another way, what we're interested in is:

$$
\langle P \rangle = \langle a m^{a-1}\rangle = \frac{\sum_{i} a_{i} m^{a_{i}-1} e^{-a_{i} \log m}}{\sum_{i} e^{-a_{i} \log m}}
$$

$$
= \frac{\sum_{i} a_{i} m^{-1}}{\sum_{i} e^{-a_{i} \log m}} = \frac{1}{m}  \frac{\sum_{i} a_{i}}{Z(\log m)}
$$

not the particular configuration of the $i^{th}$ market.

This is not to say the individual configurations are meaningless in general. You might have very small number of markets. You might have a strongly interacting system. You might care about the effect of some policy or other in a particular market. But inasmuch as you are studying macroeconomics, the existence of an ADM equilibrium does not help you reach understanding.



[1] Footnote added 10/3/2014: David Glasner quotes Franklin Fisher: "To only look at situations where the Invisible Hand has finished its work cannot lead to a real understanding of how that work is accomplished.", which is similar to my sentiment.

The macroeconomic partition function and the information transfer index

I've made some remarks in the past about how analogies between physics and economics only have merit inasmuch as they are useful. You might think: here he goes down the rabbit hole with partition functions. And that may be so. However, this is going to be a very useful analogy. Continuing from this post where I realized that the sum (defining nominal output or NGDP)

$$
\text{(1) } N(m) =\sum_{i} n_{i} =  \sum_{i} m^{a_{i}} = \sum_{i} e^{a_{i} \log m}
$$

had the form of a partition function

$$
Z(\beta) =  \sum_{i} e^{-\beta E_{i}}
$$

If you check out the link to the previous post, you'll realize there is a change to the first equation (I traded $-\log 1/m$ for $\log m$). The reason for this was because $N(m)$ should be the expectation value of an operator, not the partition function itself. I'll define the macroeconomic partition function to be:

$$
\text{(2) } Z(m) \equiv \sum_{i} \frac{1}{n_{i}} =  \sum_{i} m^{-a_{i}} = \sum_{i} e^{-a_{i} \log m}
$$

Where the $n_{i}$ are the demands in the individual markets and $m$ is the money supply (it doesn't matter which aggregate at this point). The individual markets are the solutions to the equations:

$$
\text{(3) }\frac{d n_{i}}{d m} = a_{i} \frac{n_{i}}{m}
$$

as was shown in this post. One interesting thing is that the defining quality of these micro markets -- equation (3) leads to supply and demand diagrams -- is homogeneity of degree zero in the supply and demand functions, which is one of the few properties that survive aggregation in the Sonnenschein-Mantel-Debreu theorem (see my notes here).

The expectation value of the exponent $a_{i}$ is:

$$
\langle a \rangle = -\frac{\partial \log Z(m)}{\partial \log m} = \frac{\sum_{i} a_{i}e^{-a_{i} \log m}}{\sum_{i} e^{-a_{i} \log m}}
$$

which is related to the information transfer index $\kappa = 1/\langle a \rangle$. Additionally, the nominal economy will be the number of markets $N_{0}$ [1] times the expectation value of an individual market $m^{a_{i}}$, i.e.

$$
\langle N(m) \rangle = N_{0} \frac{\sum_{i} m^{a_{i}} e^{-a_{i} \log m}}{\sum_{i} e^{-a_{i} \log m}}
$$

$$
= N_{0} \frac{\sum_{i} 1}{\sum_{i} e^{-a_{i} \log m}} = \frac{N_{0}^{2}}{Z(m)}
$$

First there is an interesting new analogy with thermodynamics: $\log m$ is playing the role of $\beta = 1/kT$. As $m$ gets larger the states with higher $a_{i}$ (high growth markets) become less probable, meaning that a large economy (with a large money supply) is more like a cold thermodynamic system. As an economy grows, it cools, which leads to slower growth ("the great stagnation" or "secular stagnation") and as we shall see a bending of the price level vs money curve (low inflation in economies with large money supplies).

Let's take $N_{0} =$ 10 (left) and 100 (right) random markets with uniformly distributed $a_{i} \in [0,2]$ (this basically allows any $\kappa \geq 1/2$, see for instance here) and plot the information transfer index $1/\langle a \rangle$, the price level $\langle a m^{a -1}\rangle$ and the nominal output $\langle N(m) \rangle$:

In the first pair of graphs we can see the economies start out well described by the quantity theory ($\kappa = 1/2$) and move towards higher $\kappa$ as the money supply increases. However, the first part of that observation is because the lowest value of $\kappa$ allowed was 1/2 (i.e. $a_{max} = 2$); the move toward higher $\kappa$ is a more robust observation. We can see the bending of the price level vs money supply in the second pair of graphs. In the third pair of graphs, we can see the trend towards lower growth relative to the growth in the money supply.

The question I had was: how well does this oversimplified picture work with real data? We have $N_{0}$ markets of equal size at $m = 1$ (arbitrary units), how well could it possibly work? Pretty well, actually. In fact, after normalizing the price level and scaling the money supply, the function $P = \langle a m^{a-1} \rangle$ almost exactly matches "the" information transfer model I've been using that incorporates an information transfer index that was only "motivated" by information theory, not actually derived from it. In the current post, $\kappa$ arises out the individual microeconomic markets.

Here is the price level with the basic information transfer model and the partition function version:

There is only a small deviation in the 1960s. Finally, here is the information transfer index:

The green line represents "data", i.e. if we assume the functional form of $P(NGDP,M0)$, what value of $\kappa$ gives us the exact CPI.

With the partition function approach, we can see that reduced inflation with a large money supply (a thermodynamically colder system) as well as reduced growth are emergent properties. They do not exist for the individual markets. An economy with a larger money supply is more likely to be realized as a large number of lower growth states (higher entropy) than a smaller number of high growth states. This approach also resolves some of the issues at the end of this post (the relationship between $\kappa$ and the exponents $a_{i}$).

[1] In the future, I might want to try to treat and economy where the number of markets is changing -- much like in the case of particle number in a quantum field.

Random markets and partition functions

Continuing from this post, I realized that the sum (defining nominal output or NGDP)

$$
N(m) =\sum_{i} n_{i} =  \sum_{i} m^{a_{i}} = \sum_{i} e^{- a_{i} \log 1/m}
$$

has the form of a partition function

$$
Z(\beta) = \sum_{i} e^{- \beta E_{i}}
$$

Where $\beta = 1/k T$ corresponds to $\log 1/m$ and the energy of the states $E_{i}$ corresponds to the $a_{i}$, which corresponds to the maximum entropy probability distribution. If we take that analogy at face value, the expected value of the random $a_{i}$ with maximum entropy would be

$$
\langle a \rangle = - \frac{\partial \log Z(\beta)}{\partial \beta} = \frac{\sum_{i} a_{i} m^{a_{i}}}{\sum_{i} m^{a_{i}}}
$$

Since $N \sim m^{\langle a \rangle}$, we now have an exponent that varies with $m$ -- exactly we have observed with the exponent $\kappa(N, M)$! (see here or here). The resulting function $N \sim m^{\langle a \rangle}$ now overestimates the result of adding the markets together. Here are the results for uniformly distributed $a_{i}$ where $a_{i} \in [0,1]$, $[0,2]$ and $[0,4]$ (the plot of $\langle a \rangle$ appears alongside the corresponding graph of $N \sim m^{\langle a \rangle}$):

One thing is that in statistical mechanics, higher temperature means that more of the higher energy states are occupied, it appears as though the observation in economics is that higher $m$ (money supply) means that more of the lower $a_{i}$ states are occupied in order to produce this figure. I'll have to look into this a little more to fully understand how this works. However this may be a pretty important result, at least for information transfer economics. Stay tuned!

A thousand more random markets

A quick post extending the previous post: if the exponents are log-normally distributed instead of uniformly distributed, you get the same deviation at high and lower values of m for the same reasons. However, the deviation is much smaller:

A thousand random markets

In this post [1], I set up a framework with a large number of markets $p_{i}: n_{i} \rightarrow s_{i}$ mediated by money so that we obtain (for the individual markets) the differential equations

$$
\frac{dn_{i}}{dm} = a_{i} \frac{n_{i}}{m}
$$

The solution to these differential equations are

$$
\frac{n_{i}}{n_{i}^{ref}} = \left( \frac{m}{m^{ref}} \right)^{a_{i}} + c_{i}
$$

In  [1], I made the approximation that the $a_{i}$ could be replaced by their average $\bar{a}$ and therefore the sum of the markets obeyed the approximate differential equation

$$
\frac{dN}{dm} \simeq \bar{a} \frac{N}{m}
$$

where

$$
N = \sum_{i} n_{i}
$$

Now I ask the question: how well does this work? First, here is the sum of 10 random markets (with 10,000 random evaluations, blue points) where we take $n_{i} \sim m^{a_{i}}$ with a uniformly distributed $a_{i} \in [0,1]$. The approximate aggregate differential equation has solution $N \sim m^{\bar{a}}$ (shown in red):

A region that represents 10% variation is shown in gray. We can see that this solution works pretty well, but I found that if we summed up 1000 random markets a systematic deviation for higher values $a_{i}$ and higher/lower values of $m$ begin to show up. Here are the results for $a_{i} \in [0, 0.5]$, $a_{i} \in [0,1]$, $a_{i} \in [0,2]$, and $a_{i} \in [0,4]$ which have $\bar{a} = $ 0.25, 0.5, 1.0 and 2.0, respectively.

A systematic deviation appears for small/large values of $m$ that is more apparent for larger values of $a_{i}$.  The source of this is not mysterious: for larger values of $m$, $m^{\text{max } a_{i}}$ tends to dominate while for smaller values of $m$, $m^{\text{min } a_{i}}$ tends to dominate. Still, for 10% shifts from the reference point $(m^{ref}, N^{ref})$, it remains a remarkably good approximation.

The markets with high values of $a_{i}$ would, in the long run, come to dominate the economy (e.g. if the market for apples went as $n_{apples} \sim m^{4}$, the entire economy would quickly become just apples). This doesn't appear to happen in diversified economies [2] (it might be true of e.g. oil-based economies), which implies there is a constraint on the values of the $a_{i}$. The interesting thing is that this constraint appears to make the macro formulation (the aggregate market) more accurate than the sum of individual markets -- i.e. there is an enforcement mechanism that makes the individual markets behave more like the average in diversified economies.

Is there an effect due to

$$
a_{i} = \frac{\log \nu_{i}}{\log M}
$$

so that we should use

$$
\bar{a} = \frac{\log \bar{\nu}}{\log M}
$$

instead? Does that then have a relationship with $\kappa (M, N)$? A uniform logarithmic distribution is related to e.g. Benford's law. I will look into all of this in a future post.

[2] This is almost a circular definition: diversified economies are economies that haven't had one commodity or product take over their economy. However, it seems that diversified economies stay diversified -- that is the sense of the statement.