## Introduction

Income distributive justice is a political subjective phrase related to an income distribution rather than to a scientific issue. Most people believe that income inequality should be as small as possible. Nevertheless, it is understood that a certain gap between the rich and the poor is necessary to stimulate competition between individuals. This competition is the invisible hand of any healthy economy. One may ask if there is an optimal inequality. This question is intriguing both from philosophical and practical points of view. Every society has a strong motivation to have a strong competitive economy on one hand and a social just on the other. These two factors are vital to the quality of life of the people. The governments regulate the net income distribution through taxation, and therefore it is of great importance to find if there is a theoretical criterion for an optimal wealth distribution. Moreover, history teaches us that a high income inequality may lead to political protests and even revolutions. In the words of philosopher Plutarch: “An imbalance between rich and poor is the oldest and most fatal ailment of all republics.”

The income inequality research which probably started with Pareto golden rule at the end of the 19^{th} century continues to these days (Ball, 2004). The contemporary physical approach to economy is based on statistical mechanics of ideal gas (Maxwell-Boltzmann), where as the distribution of incomeis compared to the distribution of energy-money among the particles-people (Dragulescu, & Petrova, 2000). However this approach that was applied by econophysicists (Ball, 2004) has not yield profound results. It was suggested previously (Kafri, 2014) that economy can be described more accurately as a network in which the money is a transient quantity exchanged between its nodes. In nature energy and transient energy, which is called heat, have different statistics. Energy has Maxwell Boltzmann statistics, and heat (i.e. photons, phonons and alike) obeys Planck statistics. In the network economy each node trades with the other nodes by transactions. Each transaction of money is represented by an integer number. The value of a number is the amount of money transferred and the sign of it is its direction. For example, if a transaction is +A it means that the node received A$, and if the transaction is –B its means that the node paid B$. In order to have an economy one need to add to this model a bank. The bank serves as the memory of the network in which all it transactions are registered. In addition of being a memory the bank is also aregulator. For example, the bank may decide that the balance of a given node, namely the sum of all itstransactions at any given time, cannot be negative. However, in order to have trade the bank should allow at least to some nodes to have a negative balance. In this case we say that the node receives a “credit” from the bank. When the bank issues the payment, it is registered as minus in the loaner-node’s account. But, since the loaner pays with the loan to other nodes, and they deposit this money back in the bank, the total balance of the bank remains zero. We see that the bank is not really affected by crediting the nodes. In fact, the bank generated money from nothing by crediting the nodes, and therefore we may conclude that money is not subject to a conservation law.

At a first glance it seems that in this toy model there is no room for recessions, crisis, economic booms and alike. However, the total amount of money, which reflects the sum of all the transactions between the nodes, is not conserved, and therefore it may be changed due to psychological reasons like fear, optimism or even long period of prosperity that is expected to end.When the total amount of transactions reduces, there is an economic recession, and when it increases there isan economicgrowth.

The network economy model enables us to calculate the distribution of money between people exactly as it was done with the distribution of links among nodes (Kafri, 2014) and the distribution of energy among photons. This distribution, which is called Planck Benford’s distribution (Kafri, 2016; Kafri, & Kafri, 2013), with accordance to the intuitive description of the network economy above, is also independent of the total amount of the money of the net or in the total amount of energy of the radiating object. That is to say; the ratio between the various income ranks is only a function of the number of the ranks. This is different from the normal distribution of energy between particles in ideal gas which varies with the total amount of energy of the gas.

The Planck-Benford distributionis basically a manipulation of Planck law (Planck, 1901) which describesthe equilibrium energy distribution in a finite number of radiation modes.The distribution of energy in the modes were calculated by maximizing the entropy (ME) of the radiating body (Kafri, 2016) namely,

ε(n)=ln〖(1+1/n)/ln(N+1) 〗 (1)

Where N is the number of the modes, which are interpreted here as the chosen number of income ranks (which might be deciles, percentiles, tenth percentiles or any other positive integer), n is a serial number called here the rank number of the nodes where n=1,2,…N. Therefore, the people are the nodes in rank n and ε(n) is their normalized wealth. If N=10 then ε(3) is the relative income of the third decile.

Eq. (1) was derived from Planck law (Planck, 1901) for photons, namely n=1/(exp(βε(n))-1);n is the occupation number which is the number of photons in a mode (mode is a radiation distinguishable state), β is a parameter related to temperature – which is determined by the total energy of the system, and ε is the energy-wealth of the photons. If we write Planck’s equation differently, namely,

ε(n)=β^(-1) ln〖(1+1/n)〗, and considering that the normalization factor, is the relative wealth distribution as expressed by Eq. (1). It is seen that smaller the rank number richer the people in it. Therefore when a number of people are divided randomly in N distinguishable groups, their wealth will decrease with the social rank n, according to Eq. (1).

## Gini Index

Gini Index is the standard measure of income inequality for countries. It is a single number (ranges from 0 to 1) that is obtained from the relative net income distribution function ε(n,N). If the income of x percent of the population is ε(x), then one defines the Lagrange function as L(x)=

Namely, L(x) is the total income of all the population up to the fraction x. If the income is distributed equally, then ε(x) is constant and L(x)=x.

Gini index is defined as . If ε(x) is constant then G is zero. Here we use a discrete version of the Gini index. We divide the population to 10 deciles according to the decreasing n, namely according to increasing income. We designate the fraction of the net income of the n decile by ε(n) and the discrete Gini index is defined as

(2)

L(11-n)is the discrete Lorentz curve, namely the fraction of the net income of all the deciles up to the 11-n decile, namely,

because ε is normalized L(1)=1 .

Now we calculate the Gini index for the Planck Benford’s distribution of wealth in 10 ranks. Each rank represents a decile of the population having similar income.

In Fig. 1 we see the result of the substitution of Eq. (1) in Eq. (2)

This calculation yields G=0.327. It is quiet surprising that the average Gini index of the 35 countries of the OECD in 2012 is almost identical to that obtained here theoretically for network economy in equilibrium, namely G=0.32. Moreover, it is counterintuitive to think that in the free world the highly regulated income inequality will be similar to that of energy inequality among photons. The reason for the surprise is the influence of the governments on Gini index by taxation in order to increase equality and decrease Gini index. Most countries in the world also compensate poor people by supplementary income in addition to taxation. Yet the Gini index is almost identical.

## The ratio between the incomes of the upper decile and the lowest decile

**Table 1***. The relative income of deciles of ME society where the average of a decile is 0.1*. *The numbers calculated from Eq.(1)**.* *The left column is ** and the right column is ** .*

0.289 | 1 |

0.169 | 2 |

0.120 | 2 |

0.093 | 4 |

0.076 | 5 |

0.064 | 6 |

0.056 | 7 |

0.049 | 8 |

0.044 | 9 |

0.040 | 10 |

## The Poverty

In Table 1 we see the equilibrium distribution of the wealth among the people according to their deciles. The median income which is given for a decile between the fifth and the sixth deciles is about 7 % of the total of the 10^{th} deciles. Half of this amount is 3.5%. Therefore, according to this definition, in country in equilibrium about 9% are poor. Indeed in the OECD countries the average percentage of poor is about this number. One should remember that the calculation of poverty as done by the countries’ institutions is not so simple as the calculation is done per capita while the income is calculated per family, therefore the number of children might change the numbers. Nevertheless, the equilibrium figures are with very good agreement with OECD economies (Murtin, & d’Ercole, 2015).

## The wealth of the rich as compared to the average

(3)

Which yields that; or

(4)

Using Eq. (4) we can calculate the ratio of the income of the richest and the average income.

(5)

It worth noting that is a function of . The higher is *N*, the higher the gap between the rich and the average. From Eqs. 5 and 4 we calculated table 2 which is the ratio between the upper fractions to the average. The left column is and the right is *R*.

**Table 2.** *The ratio between the upper fractions to the average. The left column is ** and the right is the wealth of richest fraction as compared to the average. As ** increases the ratio increases. *

2 | 10 |

7 | 100 |

22 | 1000 |

69 | 10000 |

219 | 100000 |

693 | 1000000 |

From Eq. (5) we can calculate the equilibrium net income of the richest. From Eq. (5) we can calculate the equilibrium net income of the richest. For example, if the average yearly income of a person is 30K$, we see from table 2 that for deciles in which *N=10*, the ratio between the upper decile and the average is 2, therefore the upper decile will make 60K$. Similarly, the upper percentile will make 210K$ and the upper tenth percentile annual income is 660 K$.

## CEO compensation

## Pareto Law

## Discussion

^{th}decile of the model appears in the OECD statistics but not here. On the other hand, if we divide the people’s wealth to percentiles instead of deciles, we count many poor people that are below the poverty that exists in the OECD. This limitation of “empty” percentiles of high rank number does not exist when calculating the CEO’s compensation of companies in which the rank’s number is low. The reason for it is that here we calculated the top salary in comparisonto the average salary which is substantially higher that the median salary. Generally, in the zone that the ratios of wealth will behave according to this model.

The same statistics was previously shown (Kafri, 2016) to be effective for voting. The distribution of the parliament seats among the 10 parties in Israel in the elections of 2015 is similar to that obtained by Eq.(1). In fact, the Gini index of inequality of seats among parties in the Israeli parliament, when is calculated according to Eq. (2) is 0.324. If so, one may ask whetherwe behave as a microcanonical ensemble after all. If we accept the assumption that the only physical law that causes irreversible changes in the universe is the second law, than the answer is that in equilibrium, maximum entropy distribution will be reached, and its probability should apply to economy which is a part of nature. As physicist Josiah Gibbs said (Kafri, & Kafri, 2013): “the whole is simpler than the sum of its parts”.

## Summary

## References

Ball, P. (2004). The physical modeling of human social systems – A Review, *ComPlexUs*, 1, 190-206. doi. 10.1159/000082449

Dragulescu, A. & Yakovenko, V.M. (2000). Statistical Mechanics of Money, *European Physical Journal*, 17, 723-729. doi. 10.1007/s100510070114

Gabarix, X. (1999). Zipf’s law for cities: An Explanation, *The Quarterly Journal of Economics*. 114(3), 739-767. doi. 10.1162/003355399556133

Kafri, O., & Kafri, H. (2013) *Entropy – God’s dice game*. CreateSpace, pp. 154-157.

Kafri, O. (2014). Money, information and heat in networks dynamics, *Mathematical Finance Letters* 4, 1-11.

Kafri, O. (2016). A novel approach to probability, *Advances in Pure Mathematics*, 6, 201-211. doi. 10.4236/apm.2016.64017

Ksenzhek, O., & Petrova S. (2008) Inequality and economic efficiency of society through the prism of thermodynamics, Hungarian Electronic Journal Economics, Manuscript No. ECO-080111-A.

Murtin, F., & d’Ercole, M.M. (2015). Household wealth inequality across OECD countries: New OECD evidence, *OECD Statistic Brief*, No.21.

Planck, M. (1901) On the law of distribution of energy in the normal spectrum, *Annalen der Physik*, 4, 553-562.

Zipf, G.K. (1949). *Human Behavior and the Principle of Least-Effort*, Addison-Wesley.