chaos and economy and weather

Economies are multi-variate chaotic systems.

What’s a chaotic system?

A chaotic system is a function. It’s arithmetic. But it’s a function in which its variables future state depends on its current state. For example:

f(z) = z² + C

C is a constant. Every iteration you take the old value of z and square it, then add C and that’s your new value of z. Doesn’t look dangerous, right? Well if z is a complex number (so it’s really two parameters, not one — the real part and the imaginary part) and you map the number of iterations before it explodes to some huge number or to zero, you get this;

File:Mandel zoom 00 mandelbrot set.jpg
Created by Wolfgang Beyer with the program Ultra Fractal 3. – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=321973

And that’s just two variables. Well, one complex one. Which is like two. Even that lovely image doesn’t do justice to the complexity of this result. If you zoom in on the regions that border on the black area of certainty, drilling down into variations in starting conditions at the fifth, tenth, sixteen thousandth decimal place, you will see an explosion of new complexity. Not random, but chaotic. With tiny islands of stability, regions of periodicity, and a whole lot of places that cannot be determined without another decimal place.

Just two variables.

Let’s look at just one. Let’s use f(x) =  + 0.21. Just one variable, no complex math. And let’s use a spreadsheet to see what happens between, say 0.01 and 1.3 or so. I won’t paste in my whole sheet but you can do it for yourself. Here are some features:

Screenshot 2019-09-26 17.02.54
It’s tiny so you can see the convergence patterns. The #NUM! errors is because Excel doesn’t like to work with numbers that have more than around 300 digits. So we’re calling that infinity.

If we start with x less than 0.3, the function trends up towards 0.3.

Between 0.3 and almost 0.7 (in fact infinitely close to 0.7), the function trends down towards 0.3.

At 0.7 the function just always returns 0.7

After 0.7 the function explodes faster and faster towards infinity.

0.3 is clearly some kind of attractor, an orbit in this simple (so very simple) system. And something is magic about 0.7 — it’s perfectly, utterly stable over time and yet it is so very precise. A millionth of a millionth more or less than 0.7 and it either hugs 0.3 eventually or it spins very rapidly indeed off to infinity.

In any chaotic system there may be regions of stability, called “attractors” or “orbits” where state fluctuates around some point on the map, never quite leaving the region (for the period we simulate). You see some perfect attractors in the super simple system above — 0.3 is obviously an attractor. 0.7 isn’t but it’s a stable point. A tiny tiny one. In more complex systems these orbits might not be so reliably convergent — they might diverge suddenly as they approach. Maybe at the fiftieth iteration. Maybe later.

The important thing here is…well there’s two. They are:

Unpredictability. You can’t guess from the current state what the next state will be. You have to crunch the numbers. In simpler systems you can make a guess of course, and even likely be right, but it’s hard (or in more complex systems impossible) to prove your guess analytically.

Sensitivity. You can’t simulate the future state because tiny variations in a variable’s value can have dramatic impact on future state: your simulation has to be perfect to be useful. 0.6999999999999999 converges on 0.3. 0.7000000000000001 goes to infinity. A rounding error can kill you.

Our economy has thousands if not millions of variables. Just three would need a three dimensional image to display, and one we can see inside of. Four variables would need a space we are not equipped to visualize. The economy is mind bogglingly complex. It’s at least as bad as the weather. And while we have all kinds of tricks for predicting the weather they all pretty much boil down to this: tomorrow will be like today only slightly different, with a sprinkling of last time it was like this the next day was like that.

A free market economy (a perfect one — a spherical cow in a vacuum) is a chaotic system in which participants have faith that this algorithm will result in the best possible world for the most people. This algorithm was not, however, designed in any fashion. It was just set loose. No one ever has understood all the variables and it’s intrinsically impossible to predict its behaviour. It has no knowledge nor interest in us or itself. It’s just a huge chunk of arithmetic. This is a weird place to put your faith.

Variations on the free market are an attempt to change key variables to get local effects that are desirable. Experimentally (and even analytically) we can find ways to reduce interdependency, to force certain variable states, and even to make certain variables irrelevant to the calculations. It’s fundamentally an attempt to simplify the chaotic system, to make it less chaotic. Or to push people into stable regions with positive effects.

You are already aware of some of the more stable regions. Being incredibly wealthy is a fairly stable region. Few variables impact you meaningfully and you have to make extreme moves to put you in a position where you won’t just be nudged back into this comfortable orbit no matter what you do.

Being incredibly poor is also a very stable region. There is very little that will shift this orbit short of the random injection of a ton of money. It’s why lotteries are so popular even though the odds are bad. It’s worth being worse off even for the hope of being catapulted out of this region of space, whether or not it comes true. Because it’s pretty much the only game in town once you orbit this dark star.

But without deliberately manipulating the system, without trying to control variables based on past experience, you are at the mercy of the winds. There are no guarantees. No large body of math cares about you. Don’t put your faith in it. Put your faith in people.