Sunday, April 25, 2021

Implicit assumptions in Econ 101 made explicit

One of the benefits of the information equilibrium approach to economics is that it makes several of the implicit assumptions explicit. Over the past couple days, I was part of an exchange with Theodore on twitter that started here where I learned something new about how people who have studied economics think about it — and those implicit assumptions. Per his blog, Theodore says he works in economic consulting so I imagine he has some advanced training in the field.

The good old supply and demand diagram used in Econ 101 has a lot of implicit assumptions going into it. I'd like to make a list of some of the bigger implicit assumptions in Econ 101 and how the information transfer framework makes them explicit.

I. Macrofoundations of micro

Theodore doesn't think the supply and demand curves in the information transfer framework [1] are the same thing as supply and demand curves in Econ 101. Part of this is probably a physicist's tendency to see any isomorphic system in terms of effect as the same thing. Harmonic oscillators are basically the same thing even if the underlying models — from a pendulum, to a spring, to a quantum field [pdf] — result from different degrees of freedom.

One particular difference Theodore sees is that in the derivation from the information equilibrium condition $I(D) = I(S)$, the supply curve has parameters that derive from the demand side. He asks:

For any given price you can draw a traditional S curve, independent of [the] D curve. Is it possible to draw I(S) curve independent of I(D)?

Now Theodore is in good company. A University of London 'Econ 101' tutorial that he linked me to also says that they are independent:

It is important to bear in mind that the supply curve and the demand curve are both independent of each other. The shape and position of the demand curve is not affected by the shape and position of the supply curve, and vice versa.

I was unable to find a similar statement in any other Econ 101 source, but I don't think the tutorial statement is terribly controversial. But what does 'independent' mean here?

In the strictest sense, the supply curve in the information transfer framework is independent of demand independent variables because you effectively integrate out demand degrees of freedom to produce it, leaving only supply and price. Assuming constant $S \simeq S_{0}$ when integrating the information equilibrium condition:

$$\begin{eqnarray}\int_{D_{ref}}^{\langle D \rangle} \frac{dD'}{D'} & = & k \int_{S_{ref}}^{\langle S \rangle} \frac{dS'}{S'}\\ & = & \frac{k}{S_{0}} \int_{S_{ref}}^{\langle S \rangle} dS'\\ & = & \frac{k}{S_{0}} \left( \langle S \rangle - S_{ref}\right)\\ \log \left( \frac{\langle D \rangle}{D_{ref}}\right) & = & \frac{k}{S_{0}} \Delta S \end{eqnarray}$$

If we use the information equilibrium condition $P = k \langle D \rangle / S_{0}$, then we have an equation free of any demand independent variables [2]:

$$\text{(1)}\qquad \Delta S = \frac{S_{0}}{k} \log \left(\frac{P S_{0}}{k D_{ref}}\right)$$

There's still that 'reference value' of demand $D_{ref}$, though. That's what I believe Theodore is objecting to. What's that about?

It's one of those implicit assumptions in Econ 101 made explicit. It represents the background market required for the idea of a price to make sense. In fact, we show this more explicitly by recognizing the the argument of the log in Eq. (1) is dimensionless. We can define a quantity with units of price (per the information equilibrium condition) $P_{ref} = k D_{ref} / S_{0}$ such that:

$$\text{(2)}\qquad \Delta S = \frac{S_{0}}{k} \log \left(\frac{P}{P_{ref}}\right)$$

This constant sets the scale of the price. What units are prices measured in? Is it 50 € or 50 ¥? In this construction, the price is set around a market equilibrium price in that reference background. The supply curve is the behavior of the system for small perturbations around that market equilibrium when demand reacts faster than supply such that the information content of the supply distribution stays approximately constant at each value of price (just increasing the quantity supplied) where the scale of prices doesn't change (for example, due to inflation).

This is why I tried to ask about what the price $P$ meant in Theodore's explanations. How can a price of a good in the supply curve mean anything independently of demand? You can see the implicit assumptions of a medium of exchange, a labor market, production capital, and raw materials in his attempt to show that the supply curve is independent of demand:

The firm chooses to produce [quantity] Q to maximize profits = P⋅Q − C(Q) where C(Q) is the cost of producing Q. [T]he supply curve is each Q that maximizes profits for each P. The equilibrium [market] price that firms will actually end up taking is where the [supply] and [demand] curves intersect.

There's a whole economy implicit in the definition profits $= P Q - C(Q)$. What are the units of $P$? What sets its scale? [4] Additionally, the profit maximization implicitly depends on the demand for your good.

I will say that Theodore's (and the rest of Econ 101's) explanation of a supply curve is much more concrete in the sense that it's easy for any person who has put together a lemonade stand to understand. You have costs (lemons, sugar) and so you'll want to sell the lemonade for more than the cost of the lemons based on how many glasses you think you might sell. But one thing it's not is independent of a market with demand and a medium of exchange.

Some of the assumptions going into the Theodore's supply curve aren't even necessary. The information transfer framework has a useful antecedent in Gary Becker's paper Irrational Behavior in Economic Theory [Journal of Political Economy 70 (1962): 1--13] that uses effectively random agents (i.e. maximum entropy) to reproduce supply and demand. I usually just stick with the explanation of the demand curve because it's far more intuitive, but there's also the supply side. That was concisely summarized by Cosma Shalizi:

... the insight is that a wider range of productive techniques, and of scales of production, become profitable at higher prices. This matters, says Becker, because producers cannot keep running losses forever. If they're not running at a loss, though, they can stay in business. So, again without any story about preferences or maximization, as prices rise more firms could produce for the market and stay in it, and as prices fall more firms will be driven out, reducing supply. Again, nothing about individual preferences enters into the argument. Production processes which are physically perfectly feasible but un-profitable get suppressed, because capitalism has institutions to make them go away.

Effectively, as we move from a close-in production possibilities frontier (lower prices) to a far-out one (higher prices), the state space is simply larger [5]. This increasing size of the state space with price is what is captured in Eqs. (1) and (2), but it critically depends on setting a scale of the production possibilities frontier via the background macroeconomic equilibrium — we are considering perturbations around it.

David Glasner [6] has written about these 'macrofoundations' of microeconomics, e.g. here in relation to Econ 101. A lot of microeconomics makes assumptions that are likely only valid near a macroeconomic equilibrium. This is something that I hope the information transfer framework makes more explicit.

II. The rates of change of supply and demand

There is an assumption about the rates of change of the supply and demand distributions made leading to Eq. (1) above. That assumption about whether supply or demand is adjusting faster [2] when you are looking at supply and demand curves is another place where the information transfer framework makes an implicit Econ 101 assumption explicit — and does so in a way that I think would be incredibly beneficial to the discourse. In particular, beneficial to the discussion of labor markets. As I talk about at the link in more detail, the idea that you could have e.g. a surge of immigration and somehow classify it entirely as a supply shock to labor, reducing wages, is nonsensical in the information transfer framework. Workers are working precisely so they can pay for things they need, which means we cannot assume either supply or demand is changing faster; both are changing together. Immediately we are thrown out of the supply and demand diagram logic and instead are talking about general equilibrium.

III. Large numbers of inscrutable agents

Of course there is the even more fundamental assumption that an economy is made up of a huge number of agents and transactions. This explicitly enters into the information transfer framework twice: once to say distributions of supply and demand are close to the distributions of events drawn from those distributions (Borel law of large numbers), and once to go from discrete events to the continuous differential equation.

This means supply and demand cannot be used to understand markets in unique objects (e.g. art), or where there are few participants (e.g. labor market for CEOs of major companies). But it also means you cannot apply facts you discern in the aggregate to individual agents — for example see here. An individual did not necessarily consume fewer blueberries because of a blueberry tax, but instead had their own reasons (e.g. they had medical bills to pay, so could afford fewer blueberries) that only when aggregated across millions of people produced the ensemble average effect. This is a subtle point, but comes into play more when behavioral effects are considered. Just because a behavioral explanation aggregates to a successful description of a macro system, it does not mean the individual psychological explanation going into that behavioral effect is accurate.

Again, this is made explicit in the information transfer framework. Agents are assumed to be inscrutable — making decisions for reasons we cannot possibly know. The assumption is only that agents fully explore the state space, or at least that the subset of the state space that is fully explored is relatively stable with only sparse shocks (see the next item). This is the maximum entropy / ergodic assumption.

IV. Equilibrium

Another place where implicit assumptions are made explicit is equilibrium. The assumption of being at or near equilibrium such that $I(D) \simeq I(S)$ is even in the name: information equilibrium. The more general approach is the information transfer framework where $I(D) \geq I(S)$ and e.g prices fall below ideal (information equilibrium) prices. I've even distinguished these in notation, writing $D \rightleftarrows S$ for an information equilibrium relationship and $D \rightarrow S$ for an information transfer one.

Much like the concept of macrofoundations above, the idea behind supply and demand diagrams is that they are for understanding how the system responds near equilibrium. If you're away from information equilibrium, then you can't really interpret market moves as the interplay of supply and demand (e.g. for prediction markets). Here's David Glasner from his macrofoundations and Econ 101 post:

If the analysis did not start from equilibrium, then the effect of the parameter change on the variable could not be isolated, because the variable would be changing for reasons having nothing to do with the parameter change, making it impossible to isolate the pure effect of the parameter change on the variable of interest. ... Not only must the exercise start from an equilibrium state, the equilibrium must be at least locally stable, so that the posited small parameter change doesn’t cause the system to gravitate towards another equilibrium — the usual assumption of a unique equilibrium being an assumption to ensure tractability rather than a deduction from any plausible assumptions – or simply veer off on some explosive or indeterminate path.

In the dynamic information equilibrium model (DIEM), there is an explicit assumption that equilibrium is only disrupted by sparse shocks. If shocks aren't sparse, there's no real way to determine the dynamic equilibrium rate $\alpha$. This assumption of sparse shocks is similar to the assumptions that go into understanding the intertemporal budget constraint (which also needs to have an explicit assumption that consumption isn't sparse).

Summary

Econ 101 assumes a lot of things — from the existence of a market and a medium of exchange, to being in an approximately stable macroeconomy that's near equilibrium, to the rates of change of supply and demand in response to each other, to simply the existence of a large number of agents.

This is usually fine — introductory physics classes often assume you're in a gravitational field, near thermodynamic equilibrium, or even a small cosmological constant such that condensed states of matter exist. Econ 101 is trying to teach students about the world in which they live, not an abstract one where an economy might not exist.

The problem comes when you forget these assumptions or try to pretend they don't exist. A lot of 'Economism' (per James Kwak's book) or '101ism' (see Noah Smith) comes from not recognizing the conclusions people drawn from Econ 101 are dependent on many background assumptions that may or may not be valid in any particular case.

Additionally, when you forget the assumptions you lose understanding of model scope (see here, here, or here). You start applying a model where it doesn't apply. You start thinking that people who don't think it applies are dumb. You start thinking Econ 101 is the only possible description of supply and demand. It's basic Econ 101! Demand curves slope down [7]! That's not a supply curve!

...

Footnotes:

[1] The derivation of the supply and demand diagram from information equilibrium is actually older than this blog — I had written it up as a draft paper after working on the idea for about two years after learning about the information transfer framework of Fielitz and Borchardt. I posted the derivation on the blog the first day eight years ago.

[2] In fact, a demand curve doesn't even exist in this formulation because we assumed the time scale $T_{D}$ of changes in demand is much shorter than the time scale $T_{S}$ of changes in supply (i.e. supply is constant, and demand reacts faster) — $T_{S} \gg T_{D}$. In order to get a demand curve, you have to assume the exact opposite relationship $T_{S} \ll T_{D}$. The two conditions cannot be simultaneously true [3]. The supply and demand diagram is a useful tool for understanding the logic of particular changes in the system inputs, but the lines don't really exist — they represent counterfactual universes outside of the equilibrium.

[3] This does not mean there's no equilibrium intersection point — it just means the equilibrium intersection point is the solution of the more general equation valid for $T_{S} \sim T_{D}$. And what's great about the information equilibrium framework is that the solution, in terms of a supply and demand diagram, is in fact a point because $P = f(S, D)$ — one price for one value of the supply distribution and one value of the demand distribution.

[4] This is another area where economists treat economics like mathematics instead of as a science. There are no scales, and if you forget them sometimes you'll take nonsense limits that are fine for a real analysis class but useless in the real world where infinity does not exist.

[5] For some fun discussion of another reason economists give for the supply curve sloping up — a 'bowed-out' production possibilities frontier — see my post here. Note that I effectively reproduce that using Gary Becker's 'irrational' model by looking at the size of the state space as you move further out. Most of the volume of a high dimensional space is located near its (hyper)surface. This means that selecting a random path through it, assuming you can explore most of the state space, will land near that hypersurface.

[6] David Glasner is also the economist who realized the connections between information equilibrium and Gary Becker's paper.

[7] Personally like Noah Smith's rejoinder about this aspect of 101ism — econ 101 does say they slope down, but not necessarily with a slope $| \epsilon | \sim 1$. They could be almost completely flat. There's nothing in econ 101 to say otherwise. PS — had a conversation about demand curves with our friend Theodore as well earlier this year.

Saturday, April 24, 2021

Eight years of blogging

I spent the past week on Twitter putting up a list of my favorite posts on the blog as a kind of Irish wake for the form for the blog's 8th anniversary. As is typical on these anniversaries, a bit of statistical analysis or visualization — this time, the years of the selected favorites:

Looks like 2016-2017 was my peak by the number in my own opinion. It's not just bias by the quantity of posts, either. The most prolific year was 2015 with an average rate of a post per day. Favorites per total number has been increasing steadily — at least in terms of my own opinion, quality has been rising:

Here's the list (not in order) for posterity:

"Keen"

Sunday, February 28, 2021

Ongoing evolution of time series after the COVID-19 shock

I haven't been blogging much the past two years — due to a) work taking nearly all of my energy, and b) the near daily update of data around COVID-19 being more conducive to Twitter than blogging. However, I thought it was time for a longer-term assessment of the economic time series in the context of Dynamic Information Equilibrium Models (DIEMs).

First, let's look at the consumer spending data from tracktherecovery.org (Raj Chetty and John Friedman's project using with proprietary credit card data and only bulk credit to the "OI team" of undergrad RAs) — beginning with a little history (and links to Twitter). I originally put together several models back in early June of 2020 to describe the shock to consumer spending data. About two months later (end of July 2020) I added in long run growth because it would start to become an important factor as the recovery dragged on. Towards the end of October, I decided to rank the performance of the four different models. Basically, all of them performed about equally — except for the "entropic shock" with a complete return to equilibrium. This means that there was some persistent gap in spending that wasn't made up until after the most recent round of stimulus checks in January of 2021.

Here are the four models (click to enlarge).

 Positive + negative shock

 Step response (see here)

I illustrate both the "entropic" shock models in this post on evaporating information. The basic idea is that there's a shock to the time series and it either evaporates entirely or leaves some residual:

While the full return to equilibrium model, if looked at in a vacuum, does look like it describes the entire series more accurately; however, the errors are correlated and it did poorly at forecasting up until a specific event (the January stimulus payments). We can chalk that up to luck.

The entropic shock with return to a different equilibrium is the most parsimonious model (fewer independent parameters, lower RMS error, so lower AIC), but the two shock model is probably the best explanation — if we attribute the January 2021 deviation to stimulus payments we should include some identifiable effect from the April 2020 stimulus.

Speaking of that January 2021 stimulus, I posited a couple weeks ago that it looked like an evaporating shock itself. That seems less likely once we include more data:

In fact, a classic DIEM shock seems more appropriate:

Time will tell.

In the aggregate PCE data, it's hard to tell if we can see the January 2021 shock at all:

It's only updated monthly, so doesn't have the temporal currency or resolution of the Opportunity Insights data. In this time series we can also see the effect of the TCJA implemented at the end of 2017. The other impact of the TCJA seems to have been a negative shock to median sales prices of houses (last update here). There has been a recent bump up that may reflect lack of inventory since the COVID shock hit, but again that's another place where time will tell.

I never got around to turning the reasoning for the TCJA (and its changes to the mortgage interest deduction) to be the source of that shock into a blog post after posting it on Twitter at the end of January of 2020 — just before the COVID shock. The shock begins about a year before the Fed rate increase of December 2018, the other (at least theoretically) plausible culprit. I'm sure there are just-so models where people expected the future rate increase because of the higher deficit spending brought on by the TCJA, but I personally prefer causality in my models.

On to another four letter acronym: ICSA (Initial Claims Seasonally Adjusted). I pretty much accurately described this "entropic shock" path back in June of 2020 (see previous post here), but now we have some additional evidence for some slight deviations in the model. First the big picture:

And now zooming in a bit, we can see two correlated deviations in the summer of 2020 and fall/winter of 2020 into 2021 pretty much match up with the surges in COVID-19 cases in the US (and the accompanying layoffs):

Although I talked about it in the previous post, I don't think I've shown my latest view of the unemployment rate based on Jed Kolko's "core" unemployment rate (U5 minus temporary layoffs) that I refined in a Twitter thread (U3 minus temporary layoffs, in order to compare to "headline" unemployment). New data is coming out next week, so now would be a good time to document my forecast.

Here's U3 and the U3 "core" rate with temporary layoffs removed:

As you can see, the relationship between "core" U3 and U3 has been relatively stable until the COVID-19 shock with its mass "temporary" (?) layoffs. The core rate continues to be well-described by the DIEM with its standard logistic (approximately Gaussian derivative) shocks, but the temporary layoffs required — like many of the time series here — an "entropic" shock in order to be able to describe headline (U3) inflation. I believe this forecast will be pretty accurate barring any additional shocks:

That's it for the updates. One of the major lessons for modeling economic time series in the COVID era has been accounting for these rapidly evaporating "entropic" shocks — everything from the S&P 500 to the unemployment rate. Recession and demographic shocks are extremely slow moving by comparison.