## Saturday, October 1, 2016

### Thinking about equilibrium and disequilibrium

On my flight back home last night, I started to try and understand David Glasner's post about sticky prices in terms of information equilibrium. Here's a long quote
All the theory of general equilibrium tells us is that if all trading takes place at the equilibrium set of prices, the economy will be in equilibrium as long as the underlying “fundamentals” of the economy do not change. But in a decentralized economy, no one knows what the equilibrium prices are, and the equilibrium price in each market depends in principle on what the equilibrium prices are in every other market. So unless the price in every market is an equilibrium price, none of the markets is necessarily in equilibrium.
Now it may well be that if all prices are close to equilibrium, the small changes will keep moving the economy closer and closer to equilibrium, so that the adjustment process will converge. But that is just conjecture, there is no proof showing the conditions under which a simple rule that says raise the price in any market with an excess demand and decrease the price in any market with an excess supply will in fact lead to the convergence of the whole system to equilibrium. Even in a Walrasian tatonnement system, in which no trading at disequilibrium prices is allowed, there is no proof that the adjustment process will eventually lead to the discovery of the equilibrium price vector. If trading at disequilibrium prices is allowed, tatonnement is hopeless.

So the real problem is not that prices are sticky but that trading takes place at disequilibrium prices and there is no mechanism by which to discover what the equilibrium prices are. Modern macroeconomics solves this problem, in its characteristic fashion, by assuming it away ...

Let me try and describe this in terms of information transfer (definitions for various terms at link). Let's start with a simplistic information equilibrium relationship between A and B that we will promote to an ensemble of markets with common factor of production B and different information transfer indices k.

The A growth states k are directly related to price (P) growth (change) states with k - 1. These distributions represent an ensemble of markets (see here for a summary of this idea). In general, no individual market (individual squares in a histogram) is required to be in information equilibrium (we can have non-ideal information transfer), so the price may fall anywhere below the ideal (information equilibrium) price. As I said, each box in the histogram is an individual market (as the number of markets approach infinity, the histogram approaches the distribution). Each market can be represented by a supply and demand diagram (partial equilibrium) with either an equilibrium price (at the intersection of the supply and demand curves) or a disequilibrium price (non-ideal information transfer) that falls below the curves.

Probably a better picture is this that shows the histogram better with hundreds of markets:

Let's simplify a bit and talk about only 4 markets, with 3 trading at disequilibrium prices. I drew the partial equilibrium view inside each box in the histogram (each box is a market, and the macroeconomy is a distribution of boxes):

We can see that partial equilibrium analysis can fail in the markets trading at disequilibrium prices:

Additionally, there may be correlations between markets and the entire distribution (histogram of k-1 states) represents a macroeconomic general equilibrium. You could think of the Walrasian auctioneer as establishing the distribution of these boxes. They could be correlated by e.g. common factors of production or coordination by government. Let's color-code the correlated markets:

This is the picture I was describing here with regard to national income accounting identities where I said that A = G + Y + P (output A is green markets plus yellow markets plus purple markets) can be thought of as "causal" if the components like G represent correlated markets (say, the financial sector).

Now the boxes will move around inside the (macro) distribution, so while a market might be in one k-1 state at one time, it can move to another k-1 state later. The average k-1 price growth state (that sets inflation) can be relatively constant over short periods of time (but seems to generally decline). However, as long as a market stays in one particular state, the ideal price will grow at the rate (k-1) β where β is the growth rate of the common factor of production B. The observed price will fall below this value (shown in the green market below) or be closer to ideal (purple market).

In terms of the information transfer model, we can see these boxes as containing ideal gases where the ideal (purple) box is in a maximum entropy state while the green box has agent correlations (say, a panic):

I've previous said that we should try to understand sticky prices not as individual prices being rigid, but rather as the resistance of the entire distribution to change.

Now that we have this picture, let's create a dictionary matching Glasner's quote with the language of information transfer ...
General equilibrium: Each market in information equilibrium and a stable distribution of markets
Tatonnement: The process of moving toward the entropy maximum. Importantly there are two of these -- the entropy maximum represented by information equilibrium in each market (think Gary Becker's irrational agents), but also the entropy maximum in the ensemble of markets (partition function).
Equilibrium price: Information equilibrium price p
Disequilibrium price: Non-ideal information transfer price p* < p
Equilibrium price vector: The set of all equilibrium prices for each market in each k-1 state
Excess demand/supply: the demand/supply (probability) distribution in an individual market doesn't match the (probability) distribution of supply/demand

Glasner's complaint is that we don't know if tatonnement will let us reach general equilibrium and we don't know that every market is even at its equilibrium price, much less that raising a price in a market with excess demand or lowering it in a market with excess supply leads to that general equilibrium.

In the information transfer picture, this is a valid complaint. We could easily have non-ideal information transfer in each market and therefore even if the macro distribution of markets was a (stable) maximum entropy distribution, there is no guarantee that would represent a general equilibrium. The case above with 4 boxes shows such a case -- only 1 of the markets is in information equilibrium:

However we have also connected the process of moving toward equilibrium in each individual market with the same process of moving toward the macro equilibrium: both are entropy maximizing processes. And entropy maximizing process in the individual markets will match up the supply and demand distributions (i.e. achieve information equilibrium). Additionally, the macro equilibrium (the distribution of price states) becomes a necessary but not sufficient condition for the entire system (and therefore a fortiori the individual markets) to be in information equilibrium -- a step towards Glasner's macrofoundations of micro.

In a sense, we have turned the resolution of Glasner's complaint into an assumption that agents do not stay in a stable correlated state in a large fraction of markets (either independently in each market, or even correlated across markets) for extended periods of time. One way to picture it is as Keynes' animal spirits:
Even apart from the instability due to speculation, there is the instability due to the characteristic of human nature that a large proportion of our positive activities depend on spontaneous optimism rather than mathematical expectations, whether moral or hedonistic or economic. Most, probably, of our decisions to do something positive, the full consequences of which will be drawn out over many days to come, can only be taken as the result of animal spirits—a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities.
That is to say animal spirits will drive humans from a stable correlation to explore the state space, becoming uncorrelated and maximizing entropy in the process. A box of crickets released in one corner of a room will lead to a state with a near-uniform distribution of crickets.

Basically, we assume information equilibrium is a good approximation most of the time. However, we then test that assumption using data. The assumption is justified because the theory that results from it is empirically accurate (to a given level of accuracy).

It is necessary to understand, however, this assumption breaks down in recessions where as a group our desires to move to less risky assets (financial crisis) or our outlook becomes uniformly pessimistic (via interest rate signals from the Fed or forecasts).

1. What if markets are related (I don't mean correlated) by multiple feedback loops?

"It is necessary to understand, however, this assumption breaks down in recessions...."

Doesn't this dose of reality pull the rug from underneath the information model?

1. If the markets X and Y are related they'll have to be correlated or anti-correlated ... or have a nonlinear relationship (and therefore e.g. X² and Y are correlated -- which is exactly the statement of information equilibrium where Xᵏ ~ Y).

Regarding your second question, not really. Most of the time an economy is not in recession (a few quarters out of 10s of quarters). But additionally it is the assumption of information *equilibrium* that breaks down, not information *transfer* (the general case).

Basically, the theory stops being easy ... but doesn't become invalid.

2. "Most of the time an economy is not in recession (a few quarters out of 10s of quarters)."

Isn't that when economic theory should count?

1. "It is necessary to understand, however, that you cannot pass the speed of light ...."

"Doesn't this dose of reality pull the rug from underneath physics?"

"Most of the time velocities are much less than light ..."

"But isn't physics about inventing warp drive?"

I think it might serve as a good corrective to to know that economics cannot understand recessions -- like telling engineers working on warp drive that it's impossible to exceed the speed of light. It would save people time.

Recessions might be sociological, and therefore looking for an economic answer could be a waste of time. Many macroeconomists start their "frameworks" with definitions of what a recession is -- much like you start your definition of what economics is by saying it must explain recessions. Maybe that is why it has been a dead-end so far? If you started your theory of physics by assuming you could exceed the speed of light, you'd never arrive at the right theory.

However, also let me quote the rest of my comment above:

But additionally it is the assumption of information *equilibrium* that breaks down, not information *transfer* (the general case). ... Basically, the theory stops being easy ... but doesn't become invalid.

So non-ideal information transfer (the general case of the theory) is valid during recessions. Information equilibrium (a specific limiting case of the theory) isn't necessarily valid during recessions.

3. I'm interested in your crickets in the box analogy.

For crickets to assume the stable end state of equidistant dispersion throughout the box takes time. During that time, crickets could lose limbs, feelers, eyes etc.. Maybe it's mating time and this might add another dynamic to the dispersion process. Why isn't getting to the final state relevant and not the final state? - because it may never be attained, because over time, the forces driving the equilibrium may not be stable.

It seems to me that economists (i.e. mainly neoclassical economists) are only interested in the stable end state otherwise known as "equilibrium" and ignore time and time to attain equilibrium.

Doesn't real world economics reside in the breach between the initial state and the final state?

And to borrow and bend a phrase from Keynes:

"In the long run we are all entropically dead."

1. Maybe it's mating time and this might add another dynamic to the dispersion process.

That would be a correlation -- in that case, you'd have a different effective degree of freedom (mating pairs), and the distribution of mating pairs would be uniform. A similar thing happens in superconductivity (electrons become Cooper pairs).

I generally acknowledge something like that could happen -- it is in no way ruled out by anything I said.

It is of course possible that the forces driving towards equilibrium might not be stable. They also might not be describable in terms of individual crickets, I mean, agents. There are all kinds of possibilities ... and they should be explored.

However, economics hasn't developed a good empirically useful definition of equilibrium yet, so first things first. You really can't have non-equilibrium when there's no definition of equilibrium. Even in physics there isn't a solid theory of non-equilibrium thermodynamics (in fact, I am actually applying information theoretic generalizations of the working bits). But at least there was a working definition of equilibrium first.

In fact, maybe I should quote the abstract of the paper this is all based on ...

Information theory provides shortcuts which allow one to deal with complex systems. The basic idea one uses for this purpose is the maximum entropy principle developed by Jaynes. However, an extension of this maximum entropy principle to systems far from thermodynamic equilibrium or even to non-physical systems is problematic because it requires an adequate choice of constraints. In this paper we discuss a general concept of natural information equilibrium which does not require any choice of adequate constraints. It is, therefore, directly applicable to systems far from thermodynamic equilibrium and to non-physical systems/processes ...

If you have some model that gets things empirically right and deals with non-equilibrium processes in a well-defined way, you should publish! But you should probably publish the equilibrium definition first (as Part I, maybe?) because that's the baby step that still hasn't been taken in economics -- and the non-equilibrium doesn't really make sense without a working definition of equilibrium.

4. "..economics hasn't developed a good empirically useful definition of equilibrium yet...."

Let's agree that there isn't one. How do you think this would look if there was one?

1. Not sure what the "this" refers to in the second sentence.

If you're saying hypothetically assume there isn't one that is fine. But we can't actually assume there isn't one because there is no oracle that tells us this is so.

5. "I think it might serve as a good corrective to to know that economics cannot understand recessions...."

I think this is a mighty stretch. Neoclassical economists just assume them away but other brands (Keynesians, Post-Keynesians, Minskyites etc) do have explanations.

1. Those are generally not explanations -- they tend to be assumptions about what recessions are. Minskyites define recessions as some kind of credit cycle. It leads to circularity -- what is a credit cycle? Building up credit and collapse in a recession. What is a recession? The end of a credit cycle. And 'round and 'round you go!

Post-Keynesians are a diverse lot, so I am not sure there is a single definition of what a recession is. It seems to me all economists do this. They define a recession and then work out a theory that follows that definition of a recession. The only ones that don't take them to be the result of exogenous shocks.

If you can point me to a model explanation of a recession that isn't assumed in the construction of the model, I'd be interested in seeing it!

6. The line of argument that you are taking is convenient for you. You argue, that because you assert that "normal" economics does not have an explanation for recessions, then it is OK for the IE model to not have to explain recessions. I don't think this follows.

To brush off non-neoclassical explanations for recessions the way you have is entirely facile and convenient. It is clear that there are multiple explanations for recessions in the various non-neoclassical schools. The neoclassical school conveniently assumes away recession and out of standard equilibrium behaviour, just as does the IE model.

As you have admitted above in the discussion about crickets, the path to equilibrium is important. I would argue that it is all that matters because it actually allows consideration of reality.

1. At the risk of repeating myself, my statement above was about information equilibrium and not the more general information transfer model (which can in fact address recessions, and which I do use as an explanation of recessions).

The key is that the IT framework doesn't define what recessions are (unlike most other economic theories) and in fact allows different possible explanations of recessions. The one linked in the previous paragraph makes the most sense in terms of the data.

The path to equilibrium is an important scientific question in economics, but much like whether string theory is the fundamental theory of matter, it may well not be important in terms of policy. And as we don't have a widely accepted operational definition of equilibrium, the path an economy takes from one undefined state to another undefined state is purely an abstract philosophical discussion.

7. "The path to equilibrium is an important scientific question in economics,........., it may well not be important in terms of policy."

I can't see how you can argue this. If an economy in reality spends most of if not all of its time in making its way to equilibrium ( and I would assert that it does) then it is important for policy to recognize this.

1. You basically have it! Just don't get locked into your assumptions ...

If an economy in reality spends most of if not all of its time in making its way to equilibrium ( and I would assert that it does) then it is important for policy to recognize this.

It follows that if an economy spends most if not all of its time in equilibrium, then it is not important for policy to recognize the path to equilibrium.

As you mention, you assert that economies spend most of their time making their way toward equilibrium.

This is just an assertion. Does it produce any useful models of empirical data? That's what makes an assertion reasonable. As far as I can tell, your assertion isn't based on anything.

Additionally, it depends on a particular definition of equilibrium -- one that apparently the economy does not actually attain very often (since you assert most of the time it is making its way towards equilibrium).

Is a definition of equilibrium where the economy is not in equilibrium most of the time a useful definition of equilibrium?

8. "Is a definition of equilibrium where the economy is not in equilibrium most of the time a useful definition of equilibrium?"

I'm not arguing this at all.

"It follows that if an economy spends most if not all of its time in equilibrium, then it is not important for policy to recognize the path to equilibrium."

I think this argument is perverse.

1. I'm not arguing this at all.

You literally said this. You said the economy spends most of its time on a path to equilibrium. Therefore it spends most of its time away from equilibrium.

How is your definition of equilibrium useful if the economy is not in it most of the time?

RE: Perversity ... It makes sense though, right? If I spend most of my time in the city, it really doesn't matter which road I took there ...

9. I am not offering a definition of equilibrium, I am asserting that the real world spends most of its time out of equilibrium, i.e on the road to the city, not in the city.

1. I am not offering a definition of equilibrium ...

Then how do you know you are out of equilibrium?

10. Yes, I figured you would go there and it is a fair point.

However, looking at a recession (and I guess you will want to raise the question of a definition again) when there are people who are unemployed who don't want to be unemployed, then market equilibrium is clearly not being maintained.

1. Sure, let's look at unemployment (U). It depends on what you define as the equilibrium. Now any given *level* (say U*) of unemployment is not going to be a useful definition of an unemployment equilibrium just by inspection of the data. U doesn't stay near to any potential U* for very long. There are some other options. One is to zoom out -- on a scale of 0% to 100%, unemployment is usually "near" U* = 6% (as opposed to 56%). This is one way of looking at the data (as fluctuations around a "natural rate" equilibrium).

I personally don't like that way of looking at the data -- it seems that if there was some natural rate U*, unemployment should stay closer to U* for longer, and the decline in U should either speed up or slow down as you get closer to/farther from U*.

However, the rate of decline is pretty constant after the initial increase across the entire postwar period, which made me think that the equilibrium was not a level of U, but a derivative dU/dt -- and found that the data is largely consistent with dU/dt ~ 0.04 percentage points per month.

If this is true, then the economy is in "disequilibrium" only when unemployment is rapidly increasing. Most of the time, unemployment is falling. This definition of recession (and equilibrium) matches up with the NBER recessions.

What you define as equilibrium is a case of unemployment level ("there are people who are unemployed who don't want to be unemployed"), but this is not a useful definition of equilibrium because the economy is always away from equilibrium.

My definition linked above is another possible definition in where the economy is usually in equilibrium. In more concrete language, the equilibrium is described by the net hiring rate (less separations) is positive. In your description above, you say the change from what you call disequilibrium (high unemployment) to equilibrium (low unemployment) -- the path -- is important. However the entire time you are on this path, you are in the rate of change equilibrium I describe. You are only on the path (from diseq to eq) for a few months during the heart of the recession.

This is why it is important to really understand what equilibrium is before saying you are in or out of equilibrium or that the path is important.

11. I can agree with a lot you say in your previous post, however, we are straying from my initial point which was that the information model is inadequate when it comes to explaining what happens in the real world, because as you seem to have admitted yourself in your previous post,"the economy is always away from equilibrium".

1. Nah, I said the economy is nearly always in (information) equilibrium.

There are definitions, like yours, where the economy is always away from equilibrium, but those aren't my definitions. I submit that those definitions are not useful.

12. Did you not say "because the economy is always away from equilibrium"?

1. That is part of a phrase, but misses the context:

What you define as equilibrium is a case of unemployment level ("there are people who are unemployed who don't want to be unemployed"), but this is not a useful definition of equilibrium because the economy is always away from equilibrium.

I said that using a particular definition, the economy is always away from equilibrium. But that particular definition is useless because it defines the economy as never in equilibrium. So we don't want to use it.