Hi Steemians, I was thinking to write another math story but it seems that today's topic is more suited for an article-type blog post. Don't be worried about complicated equations there are none. It should be an easy ready for everyone! So there is no excuse not to read it. In addition, there are pretty gifs!

Today I want to use math to present a phenomenon which almost everybody has come across:

Wealth inequality

Wealth inequality is in itself not very surprising since people are not equal. So you expect that there are people who are better at making money than others which results in wealth inequality. However, wealth inequality goes hand in hand with the following

Inequality observation: a small percentage of the richest population controls an extremely large percentage of the total wealth.

I think most of us have experienced this at a local level. This observation is the reason why there are only few kids with rich parents in your school/university. It is also why in an air-plane there are so many people in second class compared to first class.

The inequality observation is true in many hierarchical systems all over the world. Consequently, it is a popular universal topic in political debates. Unfortunately, when I hear politicians discuss wealth inequality they fail at pointing out the root of the problem. The left-wing groups say that the evil big banks and large corporations rob the poor and the right-wing groups say that the poor are poor because they are unmotivated and uncompetitive. Both of these explanations pertain to human psychology. More specifically, left-wing groups blame human greed and right-wing groups blame human laziness. Who is right? Of course both could be valid causes and I suspect that both of them affect wealth inequality. But are these causes the root of the problem?

Let us suppose that we could fix the human psyche by removing greed and laziness does this imply that wealth inequality becomes a problem of the past? Let's construct a mathematical model for this case!

A simple wealth model

If you are not keen on modelling feel free to skip this section and continue with the simulation section.

In the real word there are many variables to take into account. However, our aim will not be to model the real world but to capture the crucial aspects which drive the extreme inequality.

So for our model let us take a group of 1000 people. I want to start from a state of pure equality. So I am going to give everyone the same starting wealth: 100 dollars.

How are we going to model the evolution of their wealth? A lot of events in life are random so I want to take a random approach to modelling the wealth evolution of each individual. I also want to take into account the effect that rich people are differently affected by the randomness of life than the poor. In an informal sense, the more money you have the more money you can loose. To clarify this let's treat an example: suppose the house market crashes then your wealth (in an absolute sense) will take a much greater hit if you would live in a mansion than if you would live in a tiny shed. This is of course a great simplification of the real world but we are using a broad brush to create our mathematical model. So I will model the wealth evolution using the following rule:

Wealth evolution rule. Let the percentage change of wealth per week of a single individual be given by a random number between 50 and 150.

So wealth can both decrease and increase. This wealth evolution rule completes our model. Hence, we are ready to run the simulation.

The wealth inequality simulation

Before you check out the gif below you first need to think about what the simulation will display. On the x-axis I will put our test group of 1000 people. Every week I will order these from richest to poorest (number 1 being the richest). You might expect that I will put the wealth of person x on the y axis but it turns out that this gives bad pictures since wealth can change greatly over time. So I will put relative wealth to the richest person on the y-axis which is the wealth of person x divided by the richest person. So person 1 will have relative wealth 1 because person 1 is the richest person. Now you can watch the gif:

The simulations seem to suggest that after many weeks a great inequality in wealth arises. To get a clearer understanding of who controls the total wealth it is helpful to compute how many of the richest people control most of the total wealth. Of course we need to specify what most means. I will let most mean 80% of the total wealth. I made another program just for this:



So the cyan domain indicates the domain of the people that control 80% of the total wealth.
Now we clearly see the earlier inequality observation. Only a few people control almost all the wealth.

Fighting wealth inequality

What can we do against this extreme inequality? Let's introduce tax. Taxing everybody an equal amount of course will not help. We need to tax the rich more to create more equality. I am going to implement the following crude tax law: the 10 richest individuals give half their wealth to the 10 poorest individuals where the poorest receives half of the wealth of the richest and the second poorest receives half the wealth of the second richest etc. So this is really more like a redistribution of wealth law than an actual tax. Here is the simulation:

As you can see the total wealth is much better distributed. We can also apply this tax law to the richest 100:

This gives us an even better distribution of wealth.

What can we conclude from this experiment?

In the case that there is no redistribution of wealth this simple model confirms the inequality observation. Since the model does not consider human psychology it gives evidence that wealth inequality naturally arises in society independent of the presence of greed or laziness (assuming that the wealth evolution rule is true). It is gives evidence that wealth inequality is a fact of life. The model does not say if this is a good or bad thing. More generally, the model does not say anything about what the best economical/political system is. So an important problem that remains is how should we deal with extreme inequality. We saw that redistribution of wealth creates more equality but is more equality a good thing or a bad thing?

With a great concentration of wealth an individual can move society forward with his/her vision. However, a great concentration of wealth can also corrupt an individual which leads to practices which can destroy or enslave society. These problems mainly relate to the human psyche/brain. And we humans are not even close in having a good understanding of the human psyche/brain. So lots more of interesting research awaits!

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How I got this idea and additional sources

I never learned about the mathematics driving this phenomenon in university. A long time ago I learned about this phenomenon from this non-mathematical video by youtube god zefrank:

So the underlying maths of this post is the Pareto distribution or Power-Law (here is the wiki page if you want to learn more). I very recently came across the Pareto distribution in a video by Prof. Peterson:

(I really like Prof. Peterson but the coin flip example he gives in the video has no relation to the Pareto distribution.) Anyway, I thought it would be cool to write a post about the math governing extreme inequality since it is still a little known fact.

The model and program I created in this post are my own creation. (If you want the code let me know) From a mathematical viewpoint it is a discrete random dynamical system. If you are interested you can check the wiki page (WARNING: only click if you like advanced math).

Top picture by Geralt - Pixabay - CC0 Creative Commons

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Owl tax

Photo by Kridsadar Sanyear - Pixabay - CC0 Creative Commons

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