## Thursday, May 25, 2017

### Scale invariance and wealth distributions

In a conversation with Steve Roth, I recalled a paper I'd read a long time ago about wealth distribution:
We introduce a simple model of economy, where the time evolution is described by an equation capturing both exchange between individuals and random speculative trading, in such a way that the fundamental symmetry of the economy under an arbitrary change of monetary units is insured.
That's how econophysicists Bouchaud and Mezard open their abstract. Their approach is a good example of an effective field theory approach (write down the simplest equation that obeys the symmetries of the system). But interestingly, the symmetry they chose is exactly the same scale invariance that leads to the information equilibrium condition (see here or here). I hadn't payed much attention to this line before, but now it has more significance for me. The scale invariance is also related to money: money is anything that helps the scale invariance hold.

The equation Bouchaud and Mezard write down simply couples (creates a nexus between) wealth of each agent and some field that exhibits Brownian motion with drift (i.e. a stock market). It also couples the wealth of agents to each other (i.e. exchange):

As you can see, taking W → α W leaves this equation unchanged.

This scale invariance is probably what allows their model to generate wealth distributions with Pareto (power law) tails.

1. The problem that I see is that the scale actually does change. How do I justify that claim?

Just this morning I commented on another blog on this subject. I liked my comment so much that I turned it into a short post.

The title is "S = I + (G - T) and National Gift Certificates".

If the national money supply changes, then the scale we are using actually changes. In my comment, I argue that new government debt, wherein government borrows from itself, makes that scale change (without actually discussing "scale", just money).

1. If I was to move the decimal place over on all quantities measured in terms of currency, are you saying economic forces would be different?

In terms of the example in your blog post, if the merchant issued a gift certificate that was 1 Triganic Pu then the situation would be different if the merchant instead issued 8 Ningis?

Now I can imagine a *behavioral* violation of scale invariance (18,000 Lira sounds like a lot of money), but you seem to be making an accounting argument.

All accounting is scale invariant.

Even if S = I + (G - T), then

α S = α I + (α G - α T) ... divide through by α
S = I + (G - T)

→ scale invariant.

2. Isn't the scale varying over time? The first day we measure value and wealth not using the merchant gift certificate. The second day, a new gift certificate has been issued so we have more in existence.

An economist counts two different amounts of gift certificates on two different days. Both counts are definite numbers that do not change once recorded. I agree that both numbers are plots on a scale of money.

The balance for savings is investment. Investment is also a scale. Investment (on the average) is agonizingly stable, creating a scale that is much more stable than money when money is measured by money supply.

Of course I admit that we measure the value of investment using a scale of money. We do that for convenience, not for the stability of economic theory.

I can see why you might think that a scale is invariant. The numbers are the numbers. It is the investment scale that changes far slower. Using a physics metaphor, it is as if the speed of light changed year to year, depending upon how many FEET are in a meter per second velocity measurement.

That is why I claim that the money scale is variant, even allowing that the accounting IS invariant.

And that is the trouble with prices as an information exchange. From a manufactures perspective, today's price of sale must purchase all of the components of next year's products. He is forced to use last years scale on next year's investment. If the two scales have relative movement, someone has a problem.

3. Then there must be some other equation you are not telling me about that lacks scale invariance that you are using.

What is that equation?

2. Good question and a perceptive question. It makes me think this issue through in more detail.

I mentioned how investment is "agonizingly stable". A commodity scale would be "agonizingly stable". An intelligent commodity scale would reference one commodity against all other commodities. Then we could write Scale Commodity = f(housing, machinery, farm land, equities, etc). I would not place either labor or money in that equation because both have very different characteristics.

OK, assume we have that commodity scale. Where do we fit a "money scale" with a commodity scale? I think that discussion would take us to (perhaps) a gold standard type of discussion. Instead, let's ask ourselves if money is a commodity or is it something else.

Perhaps money is "information". If we think of "mercantile money" as a reference, a gift certificate is nothing more than information that the holder has been promised goods in a store, with the actual value to be converted to whatever exchange-unit prices are in use. A merchant routinely converts the gift certificate into the monetary scale in use at the moment of presentation.

So can money become a "commodity". Apparently "yes". Gift certificates and money are physical in form (as well as electronic). The use of a gift certificate in trade could be considered as one commodity being traded for another. On the other hand, the merchant would not consider it a commodity. He would consider the return of a gift certificate to be the return of information that he had a debt to cover.

What about labor? I think of labor as a renewable resource owned only by the laborer himself. I really don't know how to value my own future labor. I certainly would not use it as a scale, but I probably would scale against future labor to make a purchase decisions.

What about scaling money against money? Of course, that is done all the time. And it is a constantly moving relative comparison.

All that said, we should still be able to scale money against money, to find it invariant against itself.

1. As scale invariance is a mathematical property, the question should have a mathematical answer. If I said a system has a gauge invariance or a conformal invariance, you can't really prove or disprove that with words.

For example, the equation of exchange is scale invariant:

PY = MV
α (PY) = (αM)V
PY = MV

Writing a function f = f(x) tells us nothing about its properties under the transformation x → α x. In fact, there are various different possibilities including what's called homogeneity of degree n where

f(αx) = α^n f(x)

I'll ask again: what is the equation you are using to claim there is a lack of scale invariance?

The answer should be in the form of an equation. This is economics, not philosophy.

3. If I said

PY = MV

is scale dependent, I think you would agree. We would point to two currencies and agree that we would get different results by using two different scales to measure one economy.

I think this is what you are saying with

PY = MV
α(PY) = (αM)V

Two equations for the same economy, measured with two scales.

Is that what you mean by "scale invariance"?

If that is what you mean, the fact that each currency changes in relative value between measurements would force term "a" to vary from period to period. Despite the changing "a" term, both measurements would be of the single economy.

I am afraid my lack of math skill is frustrating to you. I don't mean to frustrate you.

1. "If I said PY = MV is scale dependent, I think you would agree."

Nope.

It is scale invariant as the equation stays the same under a redefinition M and PY (measured in nominal dollars). If you "scale" the nominal dollar by a factor α, you obtain the same equation (or an equation that is only trivially different) -- that is what it means to be scale invariant.

I'm never frustrated by a lack of math skills. I find it a little bit funny in an "ain't life strange" kind of way when people make try to make mathematical claims about scale invariance without understanding the math.

If someone asked me about a chemical reaction, I would probably throw my hands up and say "I don't know". I worked in a chemistry lab in high school and spent some time trying to grow diamond films on a silicon substrate. But except for a few basic facts, my chemistry knowledge is pretty limited. As such, I know that I probably wouldn't know what I'm talking about when asked a chemistry question and so would not make any claims.

Some people do try to make claims about subjects they don't understand ... in psychology it's known as the Dunning-Kruger effect. I probably suffer from it a bit myself in economics (this blog may be Exhibit A). But I don't suffer from DK in chemistry because I know the limits of my knowledge and understanding; I wouldn't make claims contradicting someone with a Phd in chemistry.

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5. I understand "scale invariance" now (I think). You are considering the underlying mathematical formulation and find it correct no matter what scale we use. The fact that two scales measure at a different level of fineness results in a trivial difference in the scheme of things.

I certainly suffer from the Dunning-Kruger effect or perhaps it's corollary.

My claim is that the economy and macro-economics is mechanical in character. If so, then all the parts must be connected and must mesh coherently. It follows that everything must fit logically.

Knowing little about past economic theory, I must take what I observe and fit the pieces into a coherent framework. I know of no existing framework that is mechanically correct so I find myself inventing terms and struggling for analogies. The term "mercantile money" is an example of a recent grasp for framework elements.

Usually your work is such that I cannot fit it into anything. "Scale invariant" (as a term) seemed like a path towards my learning your system and testing my framework.

I have learned a lot from this exchange. Thanks.