# A. Kononovicius: Socialism and capitalism in the kinetic exchange models

Worldwide and in Lithuania, due to historical context, there is one very popular topic for the economic-based discussion. It is based on the standard political conflict between the "right" and "left" - capitalism vs socialism. This ideological debate might be rather interesting, but it is not quite clear who is right and who is wrong. It would rather interesting to see if and how these general ideas work. In my opinion these generalized ideas might be easily introduced into some simple agent-based models. Previously considered kinetic exchange models appear to be one of the best candidates for the job. Thus in this text I will discuss the implementation of the simplistic sketches of these economic ideologies into the kinetic exchange models.

## Brief introduction into the kinetic exchange models

Let us recall that in kinetic models only the colliding particles are interacting. During a single interaction two randomly selected particles exchange energy. In the economic scenario one can see this interaction as a simple buy-sell transaction, while the energy in such case is analogous to the money or wealth. It is assumed that energy and money are conserved. Thus one particle will increase its energy by \( \Delta w_{ij} \), while the other will lose \( \Delta w_{ij} \):

\begin{equation} w_i(t+1) = w_i(t) - \Delta w_{ij} , \quad w_i(t+1) =w_i(t) + \Delta w_{ij} , \end{equation}

here time is measured in discrete time steps (\( t=0,1,2,\ldots \)). One of the simplest and most well known kinetic exchange models is the reshuffling model. In this model the total energy of both particles after the interaction is randomly redistributed between these particles:

\begin{equation} \Delta w_{ij} = (1-\varepsilon) w_i(t) - \varepsilon w_j(t) , \end{equation}

here \( \varepsilon \) is random variable uniformly distributed in \( [0,1] \). In this fair, physical laws as we all know have absolutely no emotion, model the stationary distribution is the Boltzmann-Gibbs. Will it change under the influence of "ideologies"?

## Extreme simplification of economic ideologies

Internet folklore, Fig. 1, reveals how people tend to see these two different ideologies. The question is how formally correct and also as simply as possible translate these ideas into the kinetic exchange models? Without taking into account for some small details we aim to capture the essence of these ideologies and attempt to appropriately modify the expression for \( \Delta w_{ij} \).

## Socialism

As we can see socialists are seen as people who take away money from people who have money. In the kinetic exchange models we could introduce this as a fixed tax to particles, which interact:

\begin{equation} \Delta w_{ij} = (1-\varepsilon) [w_i(t)-w_0] -\varepsilon [w_j(t)-w_0] . \end{equation}

In the above \( w_0 \) is a fixed "solidarity" tax. In order for a model to work correctly we must require that particles should have energies larger than this tax in order to interact. If particles have smaller energies, then their energy does not change after the interaction (and they are not taxed). If the tax was collected, then it is evenly redistributed among all particles:

\begin{equation} w_n(t+1) = w_n(t) + \frac{2 w_0}{N} , \quad \forall n . \end{equation}

Below you can find HTML5 applet, which illustrates that socialism, as defined, does not have any noticeable impact on the stationary distribution. The Boltzmann-Gibbs distribution is still obtained. Though it is worthwhile to note that ideally "lazy", avoiding interaction, particles would increase their energy indefinitely, despite the fact they do nothing to earn it. But in reality this ideal "laziness" is impossible, because certain level of economic activity is mandatory to survival.

## Capitalism

Capitalism on the other hand might be related to investments and borrowing. Previously we have already allowed particles to "borrow" energy. The problem is that one has to repay loans. Thus in case of capitalism, we will allow particles to borrow certain amount of energy, \( w_{max} \):

\begin{equation} \Delta w_{ij} = (1-\varepsilon) [w_i(t)+w_{max}] -\varepsilon [w_j(t)+w_{max}] . \end{equation}

Note that now after interaction it is possible to reach negative energy state, which we would like to avoid. Thus if energy of any interacting particle goes below zero (\( w_i(t)<0 \) or \( w_j(t)<0 \)), then all particles will have to repay its debt collectively (lets say it equals \( w_{neg} \)):

\begin{equation} w_m(t+1) = 0 , \end{equation}

\begin{equation} w_n(t+1) = w_n(t) - \frac{w_{neg}}{N-N_{bankr}} , \quad\forall n , \label{capitalismkitos} \end{equation}

here \( N_{bankr} \) is a number of bankrupt particles (the energy smaller than \( -w_{max} \)). Why is it possible to go bankrupt in this model? Note that losses are compensated only if the particles were interacting - if they were not interacting, then it is their own problem. Observe, in the HTML5 applet below, that the number of bankrupt particles (in the distribution the remain fixed at \( -w_{max} \)) steadily increases and the distribution becomes flatter and flatter.

We refer to this model as "simple" capitalism model, because it is possible to obtain better taxation rules. If we would like to completely avoid bankruptcy we could require that the debt would be repaid by particles with positive energy. In such case we can rewrite \eqref{capitalismkitos} as:

\begin{equation} w_n(t+1) = w_n(t) - \frac{w_{neg}}{N_{w>0}} , \quad\forall n: w_n(t)>0 , \end{equation}

here \( N_{w>0} \) is a number of particles with positive energy. It appears that the problem of bankrupt particles is solved, because debt is not repaid by those who are already in debt. The problem remains that the tax can still put the particles in debt (those for whom \( w_n(t) < \frac{w_{neg}}{N_{w>0}} \) is true).

Note that the number of particles with negative energies (we refer to them as "poor" particles) fluctuates around certain fixed mean, which depends on the maximum allowed size of loan. While the Boltzmann-Gibbs distribution is stable and holds for the particles with positive energies. It is also worthwhile to note that in this case the "lazy" particle is once again treated unfairly - but now its energy would decrease despite the fact it does nothing wrong.

We could invent a better taxation mechanism by solving the following condition:

\begin{equation} w_{min} \geq \frac{w_{neg}}{N_{w>w_{min}}} , \end{equation}

but the direct numerical solution of this problem is somewhat complicated and thus hardly we could see this mechanism working in reality. Furthermore even if we have solved this precisely, the would have to deal with unfair inequality of particles with energies around \( w_{min} \). Ones with \( w_n(t) \lesssim w_{min} \) would not pay any tax, while those with \( w_n(t) \gtrsim w_{min} \) would pay to full amount. It is highly doubtful that the radical adherents of capitalism would like such outcome.

## Instead of conclusions

In this text we have review two possible modifications of elementary kinetic exchange models. We have seen how the simplistic sketches of the main economic ideologies affect the energy distribution in these models. It appears that the model for "socialism" works well, though it somewhat encourages laziness. The model for "capitalism" on the other hand severely punishes for laziness and encourages mindless risk taking. Improvements were made for the "capitalism" model and some of the most severe problems were solved.

To conclude we could say that neither of the extreme cases of socialism or capitalism works as it should. One should look for a certain combinations of the features offered by these ideologies.