International Frontier Science
Letters, ISSN: 2349-4484, Vol. 6, P. 6 – 15 (2015). http://dx.doi.org/10.18052/www.scipress.com/ifsl.6.6
Group
Function of Income Distribution in Society
Sergey G. Fedosin
Sviazeva Str. 22-79, Perm, 614088, Perm region, Russian Federation
email intelli@list.ru
Keywords: annual income; economic modeling; distribution of income.
Abstract. Based on the similarity of properties of photons and money, and on the formula for the density of distribution of photon gas by energies, the corresponding mathematical formula for distribution of annual income per capita is obtained. Application of this formula for the data analysis reveals several independent groups of population with different average levels of their income. In particular four main groups of population contribute to the distribution of income in the economy of the USA. JEL Codes: C51, E01, E66.
Introduction
Due to variety of
reasons, in every society there is inequality in income distribution, which
divides people into different groups. It sets a challenge to the economics,
which is trying to describe mathematically this
inequality, to understand its essence and to give such recommendations, which
could bring the society close to the optimal state. Among the achieved results
we shall note the Lorenz curve, based on which the dependence is built of the
total share of the society’s income 
in percentage (the vertical axis)
on the share of families 
in percentage (the horizontal
axis). If each share of families had the same income, we would have a linear dependence:
, (1)
where 
is the constant coefficient of
proportionality.
However, the income
of families in the different groups varies. Usually, the whole society is
divided into 10 shares equal by the number of people, and within each share the
family incomes do not differ much. These shares are called deciles. Let's plot
the Lorenz curve not for deciles but for quintiles, each of which contains 20 %
of the population. We shall start with the lowest income quintile and finish
with the highest income quintile. For the first quintile (and for all the rest)
the share of families is 
, and since the income share 
of the first quintile is small,
the slope 
of the segment from the origin to
the point A in Figure 1, the coordinates of which on the coordinate plane are 
, is small too. In order to obtain the second point of the Lorenz curve,
it is necessary to add the shares of families in the first two quintiles, which
gives 
. We also need to add the income shares of both quintiles and obtain the
number 
. After that the points 
and 
are connected on the graph with a
straight line. Obviously, due to the increased incomes in the second quintile
the slope of the second segment will increase: 
. Indeed 
, and 
.
Continuing this
procedure, we see that the Lorenz curve is bent upwards. If 
is the number of a segment of the
curve, then for each segment the slope will equal 
.
The greater is the
total deviation of the Lorenz curve from the straight line of the form (1),
which connects the origin and the end of the curve, the greater is the
difference in income for different groups of population.
For example, we
shall present the statistical data for Russia in 2006 – 2007 according to [1].
The incomes of 20 percent groups of population in 2007 were as follows: in the
first group – 5.1 % (in 2006 it was 5.2 %), in the second group – 9.8 % (9.9
%), in the third – 14.8 % (15 %), in the fourth – 22.5 % (22.6 %), in the fifth
group with the highest incomes, they were 47.8 % (47.3 %).
We can make the
following specification for the poorest and the richest deciles: in 2007 the 10
% share of the wealthiest population was equal to 31% of the total money income
(in 2006 – 30.6 %), and 10 % of the least wealthy population had only 1.9 %
(1.9 %). If we divide 31 % by 1.9 % we shall obtain the decile
coefficient equal to 16.3. This gap between the rich and the poor in Russia is
obscenely large because in all developed countries the decile
coefficient ranges from 6 to 9. For comparison in 1991 in the beginning of
“perestroika” (restructuring) in Russia, this coefficient was equal to 4.5 [2].
The Lorenz curve is
associated with another characteristic, which is called the Gini
coefficient (the index of income concentration). This coefficient is determined
as the ratio of the area of the figure lying between the segment 
and the curve 
to the area of the triangle 
in Figure 1. Obviously, in case
of uniform distribution of income the Gini
coefficient tends to zero, and in case of the extreme income inequality it
reaches unity.
The Gini coefficient in Russia in 2007 equaled 0.422 as against
0.416 in 2006. The Gini coefficient is the standard
tool for comparing countries with each other in the global economy. It turns
out that in terms of income inequality Russia is at the level of Latin American
countries [3]. If in Brazil the Gini coefficient is
0.57, in Venezuela – 0.486, then in Belarus it is 0.25, in the Czech Republic –
0.27, in Ukraine – 0.28, in Germany – 0.283, in France – 0.327, in Kazakhstan –
0.33, in the UK – 0.368, in India – about 0.37. Even lower values of the Gini coefficient are in the Scandinavian countries (Sweden
and Denmark – about 0.25).
Figure 2 shows
changing of the density of the distribution function of the per capita annual
income in the Soviet Union and the countries of the former Soviet Union from
1970 to 2000 according to [4].
The function is
shown with accuracy of the order of 1 %, as the variation range of the income
was divided into 100 intervals according to centiles
(each centile contains 1 % of the population, which
has only slight difference in the income), and the distribution function is
calibrated. This allows us by adding the number of people in 100 intervals to
find the total number of people that equals the number of population. In
addition, the sum of the products of the number of people and the average
income in the intervals gives the total income of the society. The vertical
line represents the income level of $ 1 a day, according to the prices in 1985,
as the level of absolute poverty according to the definition of the World Bank.
From Figure 2 we
see rather complicated structure of the distribution function. If we build this
function on the ordinary but not on the logarithmic scale, then we could see
the long falling tail in the region of high incomes. In 1897 the Italian
economist Vilfredo Pareto tried to present in terms
of quantity this decline with the help of the power function of the form [5]:
, (2)
where the index 
is of the order of
and less than 2,
is the income of the citizens or businesses.
Function (2) is
called the Pareto distribution for the number of people (companies) depending
on their income, and was intended to analyze the nature of the income
inequality in the society. Mathematicians and economists have also tried to
describe the tail of the income distribution function by exponential functions.
Thus according to the estimates in [6], in the UK and USA the income and the
property are distributed mainly exponentially, and only a small part of the
richest population satisfies the Pareto distribution.
However, the most
important is the analytical description of the entire distribution function and
not just a part of it. Some researchers modeled the general function of the
income distribution density by non-parametric methods. For example, in [7] the
method of nuclear estimates was used, in [8] and [9] – the diagram method, in
[10] – the method of Fourier series. In the conditions when the distribution function is theoretically unknown and therefore
can not be written by a simple mathematical formula
with a small number of parameters, non-parametric methods give a possibility to
structure large data arrays, to estimate the economic inequality and the levels
of poverty and wealth. In [11] we can find the nuclear estimates of
distribution densities of the logarithms of per capita incomes in nominal
calculation in Russia in 1994 – 1997. In the obtained dependences there are a
number of peaks, which are not seen in the estimates, based on the available
for the public average data of the State Committee on Statistics and the standard
log-normal distribution. The latter stands for such representation of the per
capita income distribution, which is given by:
, (3)
where
,
is the
number equal to the ratio of the perimeter of the circle to the diameter of the
circle,
is the
weight of the observed income 
,
is the
value of the income with respect to which the distribution is centered.
Formula (3) is
written by analogy with the Gaussian normal distribution of the random variable
for which the probability density is given by:
, (4)
where 
is the distribution center for
the random variable 
, which is also the point of maximum distribution and the center of
symmetry.
Expression (4) for
the function 
describes the bell-shaped curve.
The parameter 
is the distance from the vertical
line of symmetry, specified by the equation 
, to the inflection points that are on the right or left wings of the
curve.
In Soviet times,
the spread of the income of the population in Russia was small and was well
modeled by the normal distribution law. But in the market economy with a large
private sector the incomes of different groups differ significantly, depending
on the level of skills in the achieving the results, on various kinds of
talents, proficiency, the value of the employee for the employer, the
possibility of fair pay in the corresponding sector of the economy taking into
account various accepted norms of pay in different sectors, on the geographic
and demographic characteristics, health, or the presence of bringing profit
property, land, means of production, stocks and other tangible and intangible
rights. The income level is also influenced by discrimination in its various
forms, as well as by the objective phenomena (natural disasters, unemployment).
Studies show that the actual income distribution is far from the distribution of the form (4) with a single maximum. Using the sum in distribution (3) and the logarithms of the quantities instead of the quantities themselves improves the situation to some extent, allowing us to describe several peaks in the income distribution. However, we find it difficult to agree with the point of view that the incomes can be considered simple random quantities, in some way distributed around the average values. It is well known that the country’s economy is not just the sum of its sub-systems such as firms, organizations and households, but is something qualitatively different. The presence of logarithms in (3) also points to the special type of distribution, different in its essence from random distribution. Apparently, the quasi-normal logarithmic distribution is only one of the possible approximations to describe the situation, without revealing its internal features.
The general function of the income distribution density
Figure 3 shows the
normalized density function of the per capita annual income distribution in the
USA for the year 2000, taken from [4]. The range of changing of the income is
divided into 100 intervals, each interval corresponds
to one centile of population. How can we describe the
function of income distribution from Figure 3?
The presented
function has two clear maximums and a remarkable detail on the right wing in
the region of high incomes. Obviously, neither normal nor log-normal
distribution are suitable, as well as the Pareto distribution or simple
exponential distribution.
In search of a
suitable distribution function we shall refer to the Bose-Einstein function of
distribution of the particles by energies, which has
the following form:
, (5)
where 
is the number of particles in the
system, the energy of which is in the narrow interval from 
to 
,
is the number of corresponding
various quantum states of the particle,
is the
chemical potential per particle,
is the
Boltzmann constant,
is the
temperature of the system of particles.
Distribution (5) is
used for particles – bosons, which are, in particular, the neutral atoms and
photons. The initial idea in derivation of (5) in quantum mechanics is the
quantization of the possible energies of particles on the one hand, and the
exponential dependence of the probability of finding particles in the state
with the energy : 
.
As the consequence
of (5) and of zero chemical potential for the light, for the spectral
distribution of the intensity of the electromagnetic emission inside the black
box with temperature we obtain the Planck's law:
, (6)
where 
is the speed of light,
is the
Dirac constant (
is the Planck
constant),
is the
power of emission passing in one way through the surface area 
oriented perpendicular to the
direction of emission in the interval of the angular frequency of light from 
to 
.
At low energies of
photons 
as compared to the average energy
of atoms 
, we can expand the exponent in (6) into series and confine ourselves
with only the first expansion terms. This gives the following:
, 
. (7)
The last relation
in (7) is known as the Rayleigh–Jeans law and it approximates (6) in the low
frequency region. We shall suppose now that money and working people in a
certain sense have the properties of bosons and therefore can be described by
distributions found for bosons. In favor of this hypothesis is the fact that
money is the embodiment of the quantity, which is called in physics energy.
Electromagnetic quanta or photons transfer electromagnetic energy, and money is
the universal measure of the economic value. Changing the amount of energy in
physics is associated either with doing the work by the system or on the
system, or with accumulation or dissipation of energy. Similarly, changing of
the quantity of money in the subject of economic activity is associated with
performing paid work and services, and with the processes of accumulation and
distribution of money. Photons can be absorbed completely in case of
coincidence of their frequency with the resonant frequencies in the atom. The
emission from atoms is quantized, since the photons’ energies are equal to the
difference between the energies of the levels of atoms. In a solid body the
electromagnetic energy can be efficiently converted into other forms of energy.
Similarly, in the
society the incomes of employees are also discrete and correlated with the
known rates of wages and with the level of work performed. The surplus value of
goods and services produced by an employee is equal to the difference between
the price of their realization and the cost of paid labor, raw materials,
tools, third-party services needed on their production. At the moment of
realization of goods we can say that the energy of money (spent on the
production of goods and their realization) is converted into the internal
energy of goods in the form of its cost. The fact that the cost has not only
the objective but also the subjective property follows from the cost estimation
procedure by means of the market or expert evaluation. But the presence of
subjectivity in evaluating the cost of goods by the seller and the buyer does
not mean the absence of contained in the product corresponding energy of cost.
In respect of money, just as in many things related to the human society, there
is a significant subjective component in the evaluation and adoption of
equivalent relations between money, cost, price, etc.
The property of
atoms and photons as bosons is that a number of particles of the system can be
in the same state at a time. For example, in a small spatial volume a large
number of photons with the same energy can be concentrated (the example is
laser). In contrast to this, the fermions due to the Pauli principle should
have obligatory difference in the states of particles. It is obvious that tangible
and intangible values circulating in the society, expressed in the money
equivalent, are closer in the properties to bosons than to fermions.
Based on the stated
above, we shall assume money similarly to photons to be the carriers of
corresponding energy and referring by their properties to bosons. This property
will be true for incomes, as they can be expressed in money obtained for a
certain period. We shall now use the formula of the type (6) to describe the
density distribution function of the per capita annual income in the society.
In general terms, we can write it according to [12] as follows:
, (8)
where is the number of the population
referring to the range of income change 
, located in the interval from to 
,
is a
constant coefficient,
is the
quantity, similar in its meaning to the chemical potential in (5),
is the
quantity, specifying the characteristic "temperature" of the
considered sector of the economy.
With low incomes,
when tends to , 
will tend to zero in the case
when the exponent is 
. The constant in (8) is determined from the condition that:
, (9)
where is the total population of the
region.
With 
corresponding to the cubic degree
with in (6),
from (9) we find: 
. The sum of the products of the average income in the i-th income interval and the number of
people 
in this interval should be equal
to the total income of the population: 
. Moving to the integral, taking into account (8), we have:
. (10)
As it will be shown below, there are several groups in the society at
the same time, which differ significantly from each other in their income and
their work intensity. In this regard, we can apply the known in physics
superposition principle, assuming each group’s contribution into the total density of distribution 
of the annual
per capita income in the society relatively independent of each other. Then, instead of (8), we should use the sum of contributions of all groups:
, (11)
where the index 
denotes
the group number and it ranges from 1 to the number 
, equal to the number of population groups; the
coefficients 
are
proportional to the relative weight of the respective group; and the
coefficients 
by their
meaning specify the minimum income in each group. The coefficients 
correspond to the temperature in (5) and (6), and, according
to (7), the greater is in some
group, the larger are the cash flows in this group, all other conditions being
equal.
In case of several population groups, instead of (9) we can write:
, (12)
and if , we can estimate the number of people in each
individual group: 
. For the total income of the entire population,
instead of (10), we must have the following:
. (13)
Income
distribution in the USA in 2000
We shall now apply
the function (11) to the analysis of the density of the income distribution function in
Figure 3. As we can see, the curve in Figure 3 has at least two maximums. Therefore, we must
take the sum of several terms in (11), and find for each of them the quantities
. Doing this, we arrive at the following result for the income
distribution in the USA in 2000:
(14)
We obtained four terms in (14),
all of which have the same degree in the numerator . The total number of
unknown coefficients 
for the four terms equals 12, so
in order to find them it is enough to take 12 different points on the curve in
Figure (3). In this case, its own 
corresponds to each
, and we substituted these and in (11), and then solved a system
of 12 equations and determined the coefficients in (14).
The per capita
annual income in (14) is measured in thousands
of dollars, and – in the millions of people per
income interval, corresponding to one centile. In
Figure 4, we present four curves based on four functions in (14). The algebraic
addition of these curves gives exactly the same envelope as in Figure 3.
From (14) it
follows that the US population consists of four major groups, which mainly form the density of the total income distribution function.
The first group apparently consists of low-skilled people, or part-time
employees. The incomes of this group are small, on the average about $ 7,000
per year (we shall remind that we analyze available for us data for 2000). The
second group is more qualified employees who have obtained the minimum general
education and have a specialty. The income of this group at the peak of the
corresponding curve is about $ 22,000 a year. The third group mainly consists
of people with higher education. The income in this group varies around the
average of $ 43,000. Finally, the fourth group includes all highly skilled and
highly paid jobs, with an average income of $ 76,000 a year.
In all groups, to
this or that degree, there are also businessmen and owners of capitals, who get
additional income as interest on their business, dividends or rent. A more
precise analysis of these income groups is possible in those areas of income,
in which the curves in Figure 4 do not cross.
From (14) we can find
the coefficients of some distribution functions: 
, 
, 
, 
. These coefficients should have the dimension of a million people
divided by the fourth power of the income, according to the income scale,
divided into centiles. Taking the relations
, 
etc.,
we find that on the
average per one rich American there are 588 low-paid, 30 middle-paid and 6
well-paid Americans. However, these figures are not absolutely accurate, since
they would be valid only in case if the coefficients 
in all the groups would coincide.
If the coefficients are different, the more
informative are the ratios 
, 
, etc., where 
is from (12).
The constants 
, 
, 
, 
in (14) define the lower limit of
per capita annual income in thousands of dollars of the corresponding group.
The
fact that the found groups exist relatively independent on each other is
confirmed by different constants 
, 
, 
, 
in (14). These constants
are measured as a per capita annual income in thousands of dollars and similar in their
meaning to the temperature in (6).
Apparently, the job market offers such jobs and niches, which in different groups are fundamentally different in their ability to provide income to the employee.
Conclusion
Our goal was to
determine the mathematical formulas to describe the dependence of the
distribution of annual income per capita. This was achieved in equations (8) – (14).
These formulas are in good agreement with the statistical data for the
distribution of per capita annual income in the United States in 2000, and four
large economically independent groups were discovered in the data.
To some regret, the dependence of the density of the income distribution function presented in Figure 3 is not sufficient in general for the purposes of economic research. The fact
is that it was built based on the use of centiles.
Each centile contains 1 % of population, and the
average income in a given centile depends on the
position of the centile on the income axis.
Therefore, the width of the income interval 
corresponding to some centile is different in different parts of the income axis.
Now if we want to use the formulas (12) or (13), substituting in them from (14), the income intervals should be transformed
into differentials for further
integration. However, the information for such transformation is available only
in the statistical data, used by the authors of [4], so that the direct
integration by the formulas (12) or (13), without taking into account the
difference between and in Figure 3, is
inaccurate, giving too much deviation. In order to fully use the expansion of
the form of (14) into the groups for the density of income distribution
function, we must initially have other dependencies of the income distribution.
They should be built not on the basis of centiles,
but by determining the number of people with the same range of income change , while moving along the income axis . This will lead to some change in the parameters in (14),
without changing the quality of the general picture. The result is the ability
to use quickly and easily the distributions of the form of (14) in any economic
research concerning the obtaining, accumulating and spending money in the
society, as well as during the monitoring of the economy.
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Source: http://sergf.ru/fren.htm