Blog de Paul Jorion

Billet invité. La version française se trouve ici. Un grand merci à Serge Boucher pour la traduction.

On the subject of wealth concentration and rising inequalities, Thomas Piketty tells us that there is indeed a growing rift, and that the fifties were only an exception. One can always pretend that the Gilded Age gave us several decades of only occasionally rusty capitalism, hence reviving those levels of inequality is nothing to scoff at, especially if millions of people concurrently rise out of poverty.

Entertaining that view requires that we ignore many aspects of the Great Depression, which is highly difficult to understand, having taken place between two world wars, and in a period mixing technical progress, colonisation and then decolonisation. In any case, one can conceivably believe that history as a whole is so chaotic that what we’ll get at the end of the current rise in inequality need not be exceptionally bad.

Many reasons, which one might wish to file under “our planet’s survival”, suggest that now is a horrible time for deadlocking a system already made rusty by wealth concentration and the mass poverty that it implies: even though a sizeable and growing middle-class manages to get by, a world where even in rich countries 30% of inhabitants are poor, with poor countries doing much worse, can’t be expected to make the right choices regarding our planet’s resources.

I’ve come up with a model, which suggests that even a rather slight Piketty-esque reduction of inequality is enough to bring about enormous gains in stability. The fundamental reason is that, in this model – which is intentionally simplistic and intended only as inspiration – there’s “coupling” between the fluctuations among the fortune of the hyper-rich and the irreversible downfall of the more modest into survival-level misery. Moderating hyper-wealth is thus a need, not just a societal choice among others.

In this model, you’ll only find very simple things: the “wealth” of individuals, who will be lucky or unlucky in their daily exchanges, and end up after a while with more or less “fortune”.

There’s also a poverty level, because one can only risk what lies above that threshold: the poor won’t risk any more than a few euros, the rich only a few millions.

Global wealth evolves with those exchanges, but I’ve chosen to put that aside, to “normalise” it as the mathematically-inclined say. This circumvents much more complex debates on demography, technology, sociology, “value creation”, etc. This only affirms that each society defines financial well-being locally, that one is only rich or poor in comparison with his neighbours.

Every day, one risks a part of his surplus (in every form, including time and effort) with varying success. This will be described as an independent random process. In principle, he exchanged with other people, and some form of compensation should exist, but that can only be true on average – and sometimes not even so, as an especially bad bet involving a huge sum of money can make everyone poorer. In contrast, one who invents something especially useful but only extracts a fraction of the societal gain (this being springtime, I’m hoping for a silent lawnmower) can make everybody substantially richer.

I’ll now give you the maths, but if this disturbs you feel free to jump ahead:

Let K be the wealth of an individual and Kp the poverty level. Wealth will change from day to day according to a bounded random value:

K’=K+alpha*(K-Kp)*RAND

Where K_extra=alpha*(K-Kp) is that day’s “bet”, and RAND is a random number between -1 and 1. (Computationally, it’s -1+2*rand, where ‘rand’ is the usual random function in [0,1]).

With a fixed population of N actors (numbered i from 1 to N), we do this independently for each i:

K(i,D+1)=K(i,D)+alpha*(K(i,D)-Kp)*RAND_i     (N bets are made every day).

This gives us N temporal series or “random walks”, which will all eventually slow down as they approach Kp, which is a lower bound. (We start with K=1000 “euros” for all K_i on the first day, and Kp=400 “euros”).

We see that in this model total wealth varies, and that there is no higher bound for K. This lack of a higher bound is what I’ll call the “FT effect”, as financial journalists have shown very little concern over the rising gap between the wealthy and the ones they employ, and it is this single fact that leads to paralysis, as we’ll see, despite the independence between random trajectories.

I’ve chosen alpha=0.06, for a simple reason: nobody really risks 6% of his surplus every single day, but maybe 50% over three months, which is about the same.

As for global wealth, in the beginning (where there is no “super-rich” making huge bets), evolution is slow. With many actors (N>>1), the winners nearly compensate for the losers. We’ve chosen to normalise this anyway, for reasons explained above. After each day, each actor’s wealth is slightly adjusted so that the mean remains 1000, using an elementary and proportional operation (K’_norm = K’*mean/1000 for each actor)

This ends most of the maths.

In this first post, I’ll simply show that a few “rather rich” appear far quicker than one would intuitively assume, and mostly familiarise the reader with the color histograms I use to present the results. I’ll also talk about the “Gini coefficient”, often used as a “synthetic” measure of wealth inequality. An Appendix at the end of the second post explains the movie in detail using selected frames, 6 of them, from chosen movies. It tells you “what it means and what you should and can see”, hoping to train you to feel what the model may tell us on inequality, and what weapons it could help us design. These 6 commented frames will be labelled: ((1)), ((2)),…((6)). For those unwilling to browse between two html windows, a pdf version of the Appendix is here.

In the second section, I’ll explain the “Piketty vs FT” remedy, showing how paralysis occurs, and how a slight bias can counterbalance the “rich-gets-richer” syndrome and avoid deadlock. This is a mathematical version of an ill condition that the Greeks already wanted to cure, “pleonexia”, or immoderate appetite for wealth. Non-western anthropology is full of examples where, every time wealth is accrued by a member of society, it is then “socialised” to avoid the lucky rich ending up gravely hurt. (See « Conversation sur la naissance des inégalités », by Christophe Darmangeat.)

Here’s the first movie:

I use 4 panels, all showing time as the « x » coordinate, which I’ll do in each subsequent movie. This simulation only lasts 2000 days: we’re showing the “micro” picture, which is much easier to grasp at the beginning of the simulation (a few days or weeks). Also, these movies show currency as ‘$$’ instead of ‘euros’, which doesn’t seem all that important to me.

Let’s review the panels [See ((1)) in the appendix for more details]:

Firstly, remember that time is measured in days. For convenience, I’ve set t0 as the first day of 2015, and I show each semester as Q1 Q2 Q3 Q4, so that smaller timescales remain visible. Thus, after two years we find ourselves in Y2016Q4.

Top left, each individual’s wealth is shown by a color averaging light blue. (The color bar on the right of this panel shows the colours between 0 and 3000.) The graphs show daily results, (which aren’t that important), and we can see twelve random walks taking place. The normalisation process has very little influence over the short timescale of 2000 days.

Bottom left, we show the Gini index computed over our population of 12, starting at 0 for perfect equality, and rising slowly until day 100, where it reaches 0.2 and only fluctuates around that value from then on.

Top right, we show wealth trajectories graphically. It’s actually a wealth histogram going from $$400 to $$2000: we display a yellow dot on cells that represent 10% of the population, bluer if less, redder if more. Since N is small in this run, there are only one or two individuals in each cell, 8 or 16%, which isn’t very readable. This graph will become very helpful with bigger populations, but you can safely ignore it for now. Cell distribution is logarithmic: each cell represents an interval between A and 1.005*A (or a similar factor) so that 300 cells cover our range of interest. (1.005^300 is about 4.5).

Bottom right is applied Pikettism: we display a cumulative histogram, showing “percentiles” that help us better understand wealth distribution: at each moment, 100% of people have more than the minimum (which tends towards Kp for the least fortunate), 50% have more than about 800-900, and 10% have more than 2000. As for the 1% or 0.01% where we’ll later find our “paralysis agents”, we don’t see any of them now, as with N=12 each individual represents 8.3% by himself.

We can see an evolving distribution that starts as “nearly equalitarian” in the first days, where the median (the 50%) is very close to the mean (that we keep at 1000). Then a few divergences appear, but nothing dramatic for the moment. Actually, I’ve already had to slightly constrain wealth accumulation for this to look as benign as it does. I’ll explain this along with “Pikettisation” in the next post.

To close for today, let’s run the simulation further: what happens if we wait for 35 000 days, slightly less than a century, with the same number of actors. Here’s the movie:

Top right looks similar, except that the map now shows reversals of fortune in great numbers. I’ve really let randomness have its way, and daily normalisation starts playing a role, as the richest alone often gets a huge section of the 12 000 cake. (Thus 12000+today’s fluctuation becomes substantially different from 12 000).

Bottom left, the Gini curve grows chaotic, showing some “jungle justice”, which doesn’t seem nice to me. Top right and bottom right, we get a first glimpse of the “damage” inflicted by the richest. As soon as “the” rich among our 12 actors reaches a fortune of 4000 or 5000, all others become poor in comparison, and frustrated. One might object that letting 8.3% of the population capture 50% of total wealth isn’t realistic. To the contrary, this is pretty much the current situation in the United Kingdom. A stronger objection is that this population seems too small to really understand what’s going on. True indeed.

In the next part, we’ll go to N=3600, giving the 0.1% their five minutes of fame.