What exactly is inflation?

TL;DR: No one knows

Olivier Blanchard launched a bunch of macro threads with a thread of his own about inflation. It does the typical macro thing whereby you define what inflation is and then proceed to explain what inflation is using that definition:

Olivier Blanchard @ojblanchard1

1/8. A point which is often lost in discussions of inflation and central bank policy. Inflation is fundamentally the outcome of the distributional conflict, between firms, workers, and taxpayers. It stops only when the various players are forced to accept the outcome.

My summary of Olivier’s thread:

Inflation is the outcome of a competition between agents setting prices. Therefore firms and workers compete to set the prices of goods and wages which causes inflation. Therefore, to reduce inflation, you have to reduce the competition between agents. Which is to say to reduce inflation, you have to reduce inflation. Unfortunately, this does not happen.

Much like starting a discussion of recessions by defining what a recession is, much of macro seems to start discussions of inflation by defining what inflation is: wage-price spirals, too much medium of exchange, too many people with jobs (!), excess government spending. However, the truth is that we don’t exactly know and almost every macro model built to try and “explain” inflation defines an equation that is the mathematical equivalent of a just-so story. Inflation is a wage price spiral, so here are equations [pdf at dropbox] that generate a wage-price spiral.

New Keynesian DSGE models that Olivier recommends for exploring macro distortions link interest rates (which are defined to impact the aforementioned competition) to consumption and inflation via what is called an “Euler equation”, which might not be empirically accurate (and in fact might show the opposite effect on the aforementioned competition). This creates exactly the scenario where the central bank can affect inflation by raising and lowering interest rates. Make the expectations operators model-consistent and this becomes a tautology.

Do we even know if inflation is a thing? Sure, we can measure it and people see it at the individual level, but does that mean it contains useful information not measured by other quantities? In this old post, I was able to use a simple log-linear map log(y) = a log(x) + b to transform the unemployment rate, JOLTS hires, and wage growth onto each other — implying they were not independent, but rather different measures of the same thing. This is a formal mathematical statement in information theory — if y = f(x) then the process variables contain the same amount of information entropy. Simply putting an independent quantity called “inflation” in your model could be a modelling assumption.

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One of the more eccentric features of using information equilibrium at the macro scale is that to leading order, inflation is not a thing. Well, not an independent thing. Inflation is just another name for economic growth. In (information) equilibrium, if nominal economic growth is k γ then inflation is (k − 1) γ. This works remarkably well with k = 2 such that k − 1 = 1 and nominal growth is ~ 2x the inflation rate:

It’s really only a bad description for two periods in the 1970s (typically associated with energy crises), and just mediocre in the 60s as well as the late 90s (the latter is likely due to the dot-com bubble). Those periods would be considered candidates for non-equilibrium shocks while most of the post-war period would be close to equilibrium.

This description drops out of a simple dynamic information equilibrium version of the AD-AS model¹ with information transfer index k:

\(\displaystyle \frac{dAD}{dAS} = k \frac{AD}{AS}\)

If we identify aggregate demand AD as nominal output (NGDP) and say aggregate supply AS ~ exp(γ t) — background exponential growth — then

\(\begin{eqnarray} \frac{d}{dt} \log AD & \sim & k \gamma t\\ \frac{d}{dt} \log \frac{dAD}{dAS} & \sim & (k - 1) \gamma t \end{eqnarray}\)

Note that dAD/dAS is the exchange rate between an infinitesimal unit of aggregate demand and an infinitesimal unit of aggregate supply — the generic price of a widget (or consumption basket) a.k.a. the price level P in this simplified economy.

One simple trick can take aggregate supply and turn it into something more concrete. Ok, two tricks — multiplying by 1 and the chain rule². Let’s say AS is in information equilibrium with something measurable, like, say, the civilian labor force LF with information transfer index k′. Multiplying the AD-AS equation by the AS-LF equation then setting k* = k k′ we can write:

\(\begin{eqnarray} \frac{dAD}{dAS} \frac{dAS}{dLF} & = & k \; k' \frac{AD}{AS} \frac{AS}{LF}\\ \frac{dAD}{dLF} & = & k^{*} \frac{AD}{LF} \end{eqnarray}\)

This creates a kind of nominal Solow model with a single factor of production (labor) where³:

\(\begin{eqnarray} AD & \sim & LF^{k^{*}}\\ P^{*} \sim \frac{dAD}{dLF} & \sim & LF^{k^{*}-1} \end{eqnarray}\)

And if LF ~ exp(λ t) due to population growth, you get similar growth rate relationships as you do using AS except empirically k* ~ 4 (i.e. k′ ~ 2) such that γ ~ 2 λ and the nominal growth rate is ~ 4 λ. Here P* = P P′ ~ P². Like the AD-AS model, this relationship also works remarkably well for a leading order model (LF is a noisier series so doing semi-annual averaging):

It is possible to obtain a form of Okun’s law from this — real GDP growth is related to changes in employment (for where the actual Okun’s law appears to come from, see here). This is of course a simplified, leading order model but there are some things you can do to get models that are good enough to do some forecasting.

The main point here is that in this model inflation is not a fully independent quantity from economic growth and to a good approximation is not just another measure of it, but another measure of job growth⁴. That could mean the leading order explanations (that turn out to really be just definitions) in terms of some kind of specific price-setting mechanism⁵ are fundamentally misguided — like trying to explain why the sky is blue at leading order in terms of the air’s chemical composition instead of a basic consequence of Rayleigh scattering. To leading order, growth = inflation = jobs. Trying to reduce one means reducing the others. This seems to be why in the past the Fed has never reduced inflation without inducing a recession⁶ — and the idea of a “soft landing” is more about how big of a recession we get.

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Addendum:

I should add that the information equilibrium differential equations above were all written as “fundamental” relationships — a single state space on either side of the communication channel representing a single market for a single good. It would be somewhat analogous to microeconomic supply and demand mechanics. However, in my paper on the dynamic information equilibrium model, I show that you can obtain the exact same differential equation in terms of ensembles — summing up multiple markets results in a macro market that looks exactly like a single microeconomic market⁷:

\(\displaystyle \frac{d \langle AD \rangle}{dAS} = \langle k \rangle \frac{\langle AD \rangle}{AS}\)

The angle brackets mean an ensemble expectation value weighted by the partition function constructed from the information transfer indices k (“k-states” analogous to energy states in thermodynamics) of the individual markets:

\(\displaystyle \langle X \rangle \equiv \sum_{i} X_{i} \; e^{-\beta k_{i}}\)

The primary difference here is that while k is a constant, ⟨k⟩ can vary — in particular it will tend to fall slowly with the (log of the) size of the factors of production (or here, aggregate supply). If it changes slowly enough, then to leading order this ensemble “macro” differential equation has the same (approximate) solutions as the “micro” one:

\(\begin{eqnarray} \langle AD \rangle & \sim & AS^{\langle k \rangle}\\ P = \frac{d\langle AD\rangle}{dAS} & \sim & AS^{\langle k \rangle - 1} \end{eqnarray}\)

So if after reading the section above about the simple model and are thinking “sure, but that’s too simple” the whole argument can be made to capture much more complexity by instead referring to ensembles of microeconomic markets⁸.

1. If you hold either AD or AS constant (AD varies slower than AS or vice versa) this equation generates (short run) aggregate supply (SRAS) and aggregate demand curves respectively as well the mechanics of the diagram (see my paper Sec. 2.1 and Sec 3.1 here). Plus, letting them both vary generates the LRAS vertical line. Essentially a full AD-AS model.

2. You can use this same trick to add in money, but that means “money” would be kind of a made-up abstraction like “aggregate supply”.

3. You can add multiple factors of production if you take the derivatives as partial derivatives, which generates a Cobb-Douglas form (see my paper Sec. 3.5 here).

4. There is also the possibility that the deviations of inflation from this simple model might be related to a second order effect I have called “gravity waves” associated with periods of rapid labor force growth such as women entering the workforce or post-war or post-pandemic returns to work.

5. It’s not so much of a “mechanism”, but in the information equilibrium picture as economic growth expands the state space while individual goods and service realizations of that state space wander into the comparably expanded price state space. In the AD-AS model, you can think of GDP as a 2D disk of states and the price level as the 1D bounding circle with its own states. Growing the disk exponentially with a growth rate of 2 γ means growing the boundary at a rate of γ. While the model is often constructed in terms of random agents, these agents are actually “so complex they appear random” and so move into newly available states. Prices and wages go up because they can whenever the economy grows. And in a market economy, they kind of have to in order to maintain information equilibrium.

6. In the information transfer picture (transfer because we’re talking more generally than equilibrium now), the Fed’s megaphone and interest rate signals can cause what are otherwise complex agents following their own plans to coordinate their actions (e.g. sell stock at the same time), temporarily becoming less complex (or, equivalently, less random) per footnote 5. Using a thermodynamic analogy, this results in a spontaneous drop in entropy and smaller fraction of the state space being explored. Since the size of an economy is directly related to the size of that explored state space, we get a recession. (Side note: that capacity for a spontaneous drop in entropy due to intelligent, complex agents is what separates economics from thermodynamics in the information transfer picture.)

7. It’s an interesting self-similarity between micro and macro. It didn’t have to work out this way as the derivation is non-trivial. Not hard, but non-trivial. See the paper.

8. The ensemble average also resolves the whole “what are the units of capital” aspect of the “Cambridge Capital Controversy” by simply making the capital operator into a linear combination of individual capital operators. Apples + Oranges doesn’t make sense, but Apples ⨯ Oranges does. (It also resolves Samuelson’s re-switching capitulation saying well, actually, Samuelson and Sraffa were wrong in general if you have a market with more than a couple firms — a well-behaved production function is emergent.)