Wealth distribution: how physics can help sociology

Wealth distribution in societies was interesting for economists and sociologists for centuries. Ideas from physics can help to study it...


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It is 1896, Italian economist Vilfredo Pareto publishes a work on the distribution of land among landowners, 80% of land belongs to 20% of landowners. The same rule seems to appear in all sorts of human activities from sports to computer software, from global economics to personal growth. Is it a coincidence? Can we get a reliable way to study the laws of society such as this one? What parts of physics might come to aid?

Even a single human being is extremally complicated, we cannot explain our own decisions and motivations well enough, let alone anything going on in the head of another person. It might seem that a product of millions of such complex agents interacting must be definitely intractable. However, it turns out that some laws of society are easier to figure out and more stable than the behavior of an individual.

Let’s think about the physical world. Each molecule has a complicated internal structure, atoms form various types of chemical connections, each atom consists of elemental particles held together by different physical forces, and even these particles are not as indivisible and staple as we often think. And are particles even real or are just the waves of probability that collapse as soon as we try to observe them? On the other hand, ice, water, and steam consist of precisely the same molecules yet look different; ice looks quite similar to glass although the particles in no way alike. It is an interaction type that matters, not the internal complexity of molecules.

1 liter of air contains 2.7*10^22 molecules, if we would like to store their positions and velocities we need about a zettabyte of storage, company like Facebook or Amazon can store about a one-hundredth of that amount. No hardware can solve the equations that describe this huge system, we are left with some integral characteristics. Temperature or pressure cannot be attributed to a specific molecule and energy conservation can be formulated for the entire system. Even greater power comes with the statistical idea of distributions, we cannot possibly measure the velocity of every molecule in even a mall container, but we can accurately tell how many particles have velocities within a given range.

Now we can try to apply these ideas to human society. First of all, we will ignore the internal complexity of a person and focus on interactions. The interactions will have the following form: at some point of time, two agents meet and participate in a transaction, during this transaction one agent loses some amount of wealth as it is transferred to another agent, the amount transferred depends on the initial amounts the agents possessed. This is the same dynamics as goes on in the gas, molecules collide and change velocity. And under this assumption physics can help us. 

While studying thermodynamics physicists have figured out differential equations that describe a lot of what is going on in gas, and with the help of advanced mathematical instruments, they found solutions to this equations. All of this work can be reused to study societies. The Pareto law is the solution to a Boltzmann equation as the time approaches infinity, the same Boltzmann equation describes the behavior of particles in a gas.

The analogy with a gas cannot fully describe society, humans do not obey the same laws as particles. Before the emergence of modern economics, it was believed that during trade wealth just changes hands, the only way to obtain wealth was to take it from someone else. On the other hand, nowadays the economic system relies on the idea that trade is beneficial for both sides as well as for the society. To take this into account we can assume that each transaction generates some amount of additional wealth that is distributed among participants, this means that the sum of wealth of agents grows after each transaction. In the gas analogy, wealth is equivalent to speed or energy of the particle and so total amount of wealth is the temperature. A society where transactions generate wealth is like the gas that defies the energy conservation and heats itself without any intervention from outside.

Differential equations are a powerful and fascinating instrument, but as the matter of study gets more and more complicated so do equations that describe it. Another mathematical tool might be easier to use in this situation. Let’s create a digital model of society, at each step we can pick a pair of agents and update their wealth according to some predefined rules, after some time as the system reaches an equilibrium we can measure different parameters of the system. Running such a model for different sets of parameters or initial conditions researchers can understand which factors are important.

With simulations, researchers can explore various ideas about our society. Chakraborti and Chakrabarti had used computer simulations in their paper on wealth distribution. They found studied the impact of three factors: the total amount of wealth in the system, the number of agents and saving factor. Saving factor is one more thing that distinguishes models of society from the models of gas, it describes that humans are not willing to risk all the wealth they have in a trade. The study has shown that the average amount of wealth per agent has no impact on the market dynamics, everyone gets richer, but the proportion of wealthy and poor stays exactly the same. On the other hand, changing the saving factor makes a difference, a propensity to save increases the amount of middle-wealth people increases, the fraction of extremally rich or poor people shrinks. 

Our everyday life looks nothing like that of people in the 19-century Italy, the world’s GDP has grown more than 50 times, but the 80/20 Pareto law stays in place. Is this a coincidence? The distribution of wealth we observe is likely the result of human habits, that endure centuries.


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