Guest post. The French version can be found here. Special thanks to Serge Boucher for the translation

We’ve shown in the first part of this post that a population of actors effecting “random walks” by betting their wealth daily, or more precisely a modest part of their fortune that lies above the poverty level, will eventually see a small group of “very rich” emerge, making everybody else comparatively poorer.

We now increase the number of actors to N=3600, all other hypotheses remaining the same, to get a more clear distinction between a continuum of the society and a few dominant elements. We’ll show how a small dose of “Pikettisation” avoids paralysis.

First, let’s study the situation as is, to identify the problem. Here’s a typical simulation with N=3600 actors during 35000 days. [See ((2)) in the appendix for a more detailed explanation, … even if you have already read ((1))!]

From the start, we see that the top right histogram is now “dense”, as many actors populate all the cells surrounding the median. Percentiles are displayed bottom right. They start diverging symmetrically from the $$1000 starting point, then the poor “land” right above $$400, while the middle-class (the 10% rose-purple line in the yellow region) and even the 1% (green-blue) briefly go through a maximum before everybody from the 1% on down fall to the poverty level, slowly at first, then at crisis speed around the days 6000 to 12 000 and later at 17 000.

However, as the scale on the right side now goes from $$100 to $$1 000 000, we see that immense fortunes are indeed possible for a happy few. And as they go through $$10 000, then $$100 000, we see the median closing on Kp, without any remission [see ((3))]. In this “paralysis” state, most of the population form a huge miser class that can never rise up, because they never bet big enough to escape poverty. It’s obviously the big “risk taking” of the rich that condemns every one else to poverty, through the coupling induced by forcing down the mean (which is analogous to GDP/capita) to a constant $$1000.

There are occasional “happy days”: the rich get a chance to “jettison” their fortune, as in the collective bump occurring around 12 500-14 000 days and later at 17 500 19 000 days [see ((4))]. They too get bad days that benefit everybody else, putting another metaphorical coin in the jukebox so that everybody can start dancing again. An analogous process regularly happens in the real word, and needs involve no cruelty: without any one stealing anything from the rich, there just happens to be circumstances where their investments prove extremely profitable, and they don’t capture all the benefit. In that case, everybody gets wealthier, and the wealthy only miss an opportunity to become hyper-rich.

The panel top right exacerbates these different phases: the histogram’s color is computed from the densest population in its wealth quantile (the “bins” or “cells” of the histogram), that is to say it depends on the ratio “quantile’s population / largest quantile’s population”. When the largest quantile’s population grows suddenly, all color references adjust: the higher cells (that are bluer) don’t become that empty, but they represent smaller fractions of the largest cell, that of the “proletariat”, the poorest of the population. The cumulative histogram (bottom right) shows that percentile variations are not humungous, only very large.

However, this model suggests that the creation of a huge class of people living in poverty is a typical effect of the huge fluctuation among the rich’s fortunes: as they violently shake the system, many “near-poor” end up exactly “poor”, so close to the subsistence level that their hope of escaping it becomes vanishingly small. This is the moment where their employers typically make amends by coming up with a “social plan”.

Thus paralysis occurs, brought about by the unbridled action of immense fortunes, authorised to risk a lot and fluctuate a lot.

As a reader of Piketty and Jorion, I now propose a factor that mitigates hyper-wealth. To that end, I introduce for each actor a measure of “status”, “S”, in the Aristotelian fashion that Paul Jorion follows in his theory of pricing, quite different from the law of supply and demand. [See “Le Prix”, in French, or Aristotle’s theory of price revisited in English.]

This status is defined as S_i = K_i/(K_i+Kavg) where Kavg, the mean K, is essentially forced to 1000 as is remembered. Status thus varies between a value below 1/2 for the poorest (K~Kp < Kavg) to a value above 1/2, getting close to S=1 for the richest (K_i >> Kavg). The lower bound in our case is Sp=0.286=Kp/(Kp+Kavg)=4/14 as Kp=400 and Kavg=1000.

This status S is shown as a continuous histogram in the small gold insert in a corner of the bottom right panel, shown in every movie.

Using this S, we can bias the rich’s “fortune”, their probability of gain, to suit the taste of either Piketty, or the FT.

We need a little more math here:

RAND, which used to be uniformly distributed between -1 and 1, is now replaced by RANDS:

RANDS = RAND – (1+RAND)*cPik*S

Don’t be scared. (1+RAND) is a number between 0 and 2, which is weighted by (cPik*S <1) and added to RAND: thus it only affects the higher bound. The minus sign makes cPik especially evocative of a parameter that “Pikettises” society and limits the rich’s impact: with cPik>0, as S nears 1, huge gains become increasingly unlikely.

More explicitly, with a wager defined as (alpha*(K_i-Kp)), the daily “gains” coefficient is uniformly distributed in [-1, 1-2cPik*S]. For the richest, this tends towards [-1, 1-2*cPik], which has a mean expectancy of -cPik. For the poorest (Smin<1/2, down to Sp=0.286 for K=Kp), the interval is [-1, 1-2*cPik*Smin], which is on average more favourable. The gain from the wager no longer falls in a uniform [-1,1] distribution, but the only consequence is that each day sees on average a little aggregate loss. As we “normalise” results back to a mean of $$1000 at the end of each day, the effect is only modest wealth redistribution in favour of those closest to the poverty level, which is just a very timid form of social policy.

Finally, to allow for cPik<0 (which gives us the “reinforced FT” effect) without the model breaking down, we introduce a “safety net” for the poor. As is, the general trend would be to make the poor increasingly miserly. An actor can easily cross the subsistence level and reach K < Kp=400 through normalisation: if one ends the day at $$401 but normalisation takes 0.5% from everyone, which can happen when a super-rich wins big, the poor fellow will be left with $$399. To avoid a downward spiral, we forbid amounts below Kp before normalisation, by defining K(D) = MAX[ K(D-1)+K_ex*RANDS , Kp]. This way, one can end up just below Kp after normalisation, but at the first positive normalization, she will again jump above Kp and be allowed back into the game. One can verify that this construction neutralises edge effects that would otherwise occur around K~Kp.

To summarise, these new rules with cPik>0 restrain the unlimited growth of the largest fortunes, and the negative shocks they cause. We could also take the opposite action, and amplify the rich’s potential gain by choosing a negative cPik. The latter would model the well-known aphorism: “the rich get richer”.

With these tools in hand, we’ll run three different simulations to demonstrate our “medical breakthrough”, which essentially says that a little trimming is all that’s needed to prevent catastrophic paralysis (nicknamed “FT” fits). Which will hopefully give a few people pause.

I first need to explain, as promised, that the last post’s example with N=12 during 2000 days had already been “Pikettised”, although I only barely mentioned it at the time. Why? Because I wanted to show you things “as you’d naturally expect them”, and ease yourself, dear reader, in a “gentle” version of inequality, the one that makes you think: “of course, rich people exist, and they’re quite rich indeed, but there’s nothing obscene about it and no reason to worry”. That psychological trick required that I either strongly restrain the secular ambition of my 12 actors, or keep re-running my simulation hoping for an extremely (un)lucky random run that ends up not generating hyper-wealth — which did indeed start to show up near the end of the 2000 days run.

Let’s now study this restraining mechanism, on a large N=3600 population, which is much more illustrative.

We’ll start with cPik=0.01, which is to say the hyper-rich (status S~=1) are restrained by a probability of 0.98, and give back on average 1% of their daily bet. This may seem like a lot, but it comes to only about 10% every three months. (Probabilities don’t sum linearly, but through their “variance” which is proportional to their values squared; as I explained in the first post about the 6% daily bet, computing the resulting value over a quarter or a year confirms that these estimates are indeed realistic.) This is in line with how big companies are supposedly taxed in many developed countries, where huge gains incur about a 40% marginal tax rate. (In practice, many multinational companies can significantly reduce this burden through “fiscal optimisation”.)

As you can see below, this 1% “Pikettisation” causes significant smoothing: the society experiences occasional small turbulences (short dips associated to the richest’s fluctuations) but never “crashes”, at least not in a reasonable time frame. Inequality is significantly restrained, and “big bets” have a more limited impact. [see ((5)) for more details.]

The 1% still get about six times more than the average ($$6000), and 15 times more than the poor ($$400), but higher up, even the truly rich, Bill Gates types rarely get over $$200 000.

If we want to go hyper-Scandinavian, and crack down on everyone who looks like they might engage in rent-seeking, we can use cPik=0.075, which is 7.5 stronger than above. This limits the super-wealthy’s average gain to 1-2*cPik=0.85 instead of 1.

This is what we get [see ((6)) for more details]:

The Gini coefficient gets pushed down to G=0.19, and the 1% earn two-and-a-half as much as the 50%. This is significantly more Scandinavian than the actual Nordic countries, where the Gini after taxes and transfers is currently about 0.25.

For a final flourish, let’s unleash our inner FT contributor and liberate those oppressed job creators by helping the rich get richer. That this is what currently happens has been observed many times: Piketty gives as example the 800 or so American universities: their accounting is public record and most of them manage some kind of wealth fund. The unsurprising and inescapable result shows that simply having more capital gives one access to significantly higher return rates. Notably, through access to better, and more expensive, wealth management: Harvard spends about 100 million on the management of its 30 billion (!) endowment, and gets returns above 9%. The next university, significantly “poorer”, has to make do with “only” 7-8%.

“Pikettisation” can be equally applied in both directions. Here’s the result when we use it to help the hyper-wealthy: a true explosion of civilisation through unbridled capitalism. Who wouldn’t want that?

The picture is clear: a high and strong Gini coefficient, only rarely interrupted by ill-advised attempts by the poor to escape subsistence-level income, always crushed within one or two years. But the rich are indeed magnificent! Behold the heroic risk-taker brave enough to embrace the ecstasy of creative destruction! Atlas shrugs indeed.