Thermodynamics of economic inequality

Vladimir Pokrovskii111Corresponding author: Vladimir Pokrovskii<EMAIL_ADDRESS>
Moscow State University of Economics, Statistics and Informatics

Abstract

An heuristic model of the society, as an assembly of weakly interacting
individuals, is discussed. The model allows to connect macroscopic phenomena
with features of relations between individuals. Addressing to the problem of
inequality, a non-equilibrium situation is considered. The calculated income
distribution function, which coincides with the Pareto distribution at large
incomes, can be interpreted as a strongly deformed Gauss distribution
function. The external, in relation to the system of interacting individuals,
force is necessary to maintain the strong non-equilibrium in a stationary
state. The model representation of the society allows us to explain the
mechanism of the emergence and maintenance of economic inequality, the
universal cause of which is asymmetry of elementary exchanges.

Key words: inequality, income distribution, Pareto distribution, social
thermodynamics, society model, sociophysics.

##  1 Introduction

The modern human economic activity is based on the flows of solar energy,
which passes through plants, animal bodies, the work of machines and, in the
end, is used by man to maintain his life. The production system was developed
by the human populkation on the Earth. The current pattern of economy have
been leading to significant inequality in the distribution of income and
wealth [1]. The typical pattern was uncovered by the sociologist and economist
Wilfredo Pareto, who found ([2], pages 299-345) that the distribution density
function of the individuals $p(x)$ over income $x$ in the region of the large
values is described by a power function

$p(x)=Ax^{-(1+\alpha)}.$ (1)

The power law has been verified over and over again since then for different
countries and at different times; the reliable estimates of the empirical
values of index $\alpha$ in the power distribution (1), according to ([3]
p.84), range from 1.2 to 3.2.

Since Pareto’s time, many researchers have sought to understand whether the
power distribution is a law of nature, or whether it can be reduced to more
fundamental principles. Numerous investigations, devoted to description and
explanation the mechanism of emergence and existence of the inequality,
recently are reviewed by a Brazilian researcher [3]. However, the problem of
finding the shape of the distribution function for those who do not belong to
the rich class (the vast majority of people in modern societies), according to
([3], p. 64), is remained basically open. This statement is confirmed in the
paper [4]: ’Despite decades of intense research and data analysis, the exact
physical mechanisms of wealth distributions are not yet fully understood’.

To reconsider the problem and to try to understand the original of inequality
we use a new approach, defining the society to be an open thermodynamic
system. The addressing to the a simplest heuristic model of a market economy,
as an assembly of interacting economic agents, allows us, referring to the
empirical Pareto index $\alpha$, to restore the entire distribution density
function. The model allows to explain the mechanism of emergence and existence
the wealth inequality.

##  2 Heuristic model of the society

The productive activities apparently underlies the entire human life, so that,
under the desire to construct an heuristic model of the society, we are
encounting, first of all, the problems of description of economic activity,
which involves the coordinated activity of a huge number of people, who unite
in large groups to manufacture needed things. Investigating the production-
consumption phenomena, the total population should be taken into account, but
theory, nevertheless, identifies a group of people who are directly involved
into production — economically active individuals; we consider an assembly of
$N$ such active individuals, who jointly create the entire social product.

The assembly of interacting individuals, together with their artificial
environment, ought to be considered as an open thermodynamic system; the
existence of the system is supported by streams of matter and energy coming
into the system through the bodies of individuals and various appliances.
There are two fluxes of external energy, which, during the process of
production, feed efforts of individuals, the total of which is denoted as $L$,
and substitutive work of production equipment $P$. These two quantities
determine production of value in money units [5, 6]

$Y=Y_{0}\,\frac{L}{L_{0}}\left(\frac{L_{0}}{L}\frac{P}{P_{0}}\right)^{\alpha},\quad
0<\alpha<1,$ (2)

where the zeroes indicate the values of output, labor and substitutive work in
the base year. The time-dependent quantity $\alpha$ — the technological index
— is an internal characteristic of the production system, and is expressed
through the technological characteristics of the production equipment222Note
that the relation (2) has the form of the famous Cobb-Douglas production
function, but in which, after reasoning about the mechanism of substitution of
labor by capital [5], a variable $P$ is used instead of a variable capital $K$
that is the estimate of value of all production equipment. This led to a
accurate determination of the production of value through production factors
without any corrections and arbitrary parameters [6]. Capital (production
equipment) $K$ is the means, by which the active factors of production, $L$
and $P$ can unrestrictedly substitute each other under modification of
production processes. The equipment itself is dead until it is not animated by
streams of human efforts and external energy. .

The incoming energy has been dissipated in the production processes, and, it
is assumed for simplicity, the created product is completely consumed by the
individuals, so that the stream of value $Y$ is coming through the system. As
in every open thermodynamic system [7], the conditions for creating of
internal structure of the system appear, and, indeed, in following sections,
we will follow a mechanism of emergence of non-equilibrium of the system.

The two production factors: the activities of people $L$ and substitutive work
of production equipment $P$ appears to be essential in production of value
that is an estimate of all produced goods and services. People’s work is used
directly, but in large-scale machine production, people’s work is alienated:
this means that a person must perform prescribed actions without realizing the
true goals of the work, for which he receives a reward, the correspondence of
which to the efforts expended is not obvious. Substitutive work is based on
the preliminary activity of people for centuries in the study of the laws of
nature, the design of technological schemes, the manufacture of production
equipment and everything else. The results of this activity are a common
heritage that we have received from previous generations.

The individual contributions into the production of value are various; in the
simplest case, one can assume that the input of each agent is equal and
corresponds to the mean estimate of the individual efforts $Y/N$. In a more
complicated case, the individual contributions could be described by the Gauss
distribution. Each participant receives a share of the created product — the
income of an individual $x$, which should be considered as a random variable,
and, therefore, it is possible to introduce a distribution function of agents
over income $p(x)$, such that the value of $p(x)\Delta x$ represents the
number of people with income is situated between $x$ and $x+\Delta x$.
Obviously, the following relations have to be fulfilled

$\int_{0}^{\infty}p(x)dx=N,\quad\int_{0}^{\infty}xp(x)dx=Y$ (3)

The market mechanism presupposes free exchange between various economic
agents: the efforts of the workers for money, money for products, products for
other products, and so on. We assume that any interactions could be reduced to
exchange of value and, consequently, each individual is prescribed with the
only status variable — the income of an individual $x$, which changes its
value in a random way. Considering the chaotic interactions, one can introduce
the probability of the change of agent’s status from $x$ to $x-a$, whereas the
alteration $a$ is supposed to be distributed according to the standard
Gaussian function

$w\left(a,\langle a\rangle,\langle
a^{2}\rangle\right)=\left(\frac{1}{2\pi\langle
a^{2}\rangle}\right)^{\frac{1}{2}}\exp{\left[-\frac{(a-{\langle
a\rangle})^{2}}{\langle a^{2}\rangle}\right]}$ (4)

It is assumed that the transition probability function could depend on the
variable $x$ through the mean value ${\langle a\rangle}$ and standard
deviation ${\langle a^{2}\rangle}$.

This simple heuristic model takes into account the multitude of exchanges
between economic agents according to market rules in the processes of
production and distribution of value and allows us to find the distribution
function of economic agents over the income received, which allows to clear up
the mechanism of occurrence and maintenance of economic inequality.

##  3 Derivation of the distribution function

###  3.1 Kinetic equation

We assume that society is represented by the system of weakly interacting
economic agents, the macroscopic state of which is determined by the
distribution density function of the individuals over income $p(t,x)$, which
depends also on time $t$. To find this function by the implementation of the
described model of the assembly of individuals, changing their states due to
the random elementary transitions from one state to another, we use a known
method that allows us to write a kinetic equation for the distribution
function (see, for example, ([8], chapter 2). In this case, the change in the
distribution function can be written as

$\frac{\partial p(t,x)}{\partial
t}=\int_{-\infty}^{+\infty}\left[w(a,x+a)p(t,x+a)-w(a,x)p(t,x)\right]da$ (5)

We assume that the change in the income $a$ is much smaller than its current
value, $a\ll x$; based on this, the first term in the integrand (5) can be
replaced by the first terms of expansion

$w(a,x+a)p(t,x+a)\approx w(a,x)p(t,x)+a\left.\frac{\partial(wp)}{\partial
x}\right|_{a=0}+\,a^{2}\left.\frac{\partial^{2}(wp)}{\partial^{2}x}\right|_{a=0}.$

The variables $a$ and $x$ are considered independent, so that the kinetic
equation for the distribution function (5) can be written as

$\frac{\partial p(t,x)}{\partial t}=\frac{\partial}{\partial
x}\left[\left(\langle a\rangle+\frac{\partial\langle a^{2}\rangle}{\partial
x}\right)p+\langle a^{2}\rangle\frac{\partial p}{\partial x}\right].$ (6)

Under these assumptions, the equation is valid in the region of positive
values of $x$. If the function $w(x,a)$ is symmetric with respect to the
variable $a$, $\langle a\rangle=0$, and the probability variance does not
depend on $x$, the first term on the right side of equation (6) disappears,
and the kinetic equation reduces to a one-dimensional diffusion equation. To
describe more complex situations, it should be assumed that the transition
function could be asymmetric with respect to $a$, and the average value and
variance may depend on the variable $x$.

The square brackets on the right side of the equation (6) contain two terms:
the diffusion leveling is compensated by the flow. The equation describes a
balanced internal motion and defines a non-equilibrium distribution function,
what is supported by external streams of matter and energy.

###  3.2 Steady-state distribution

In the steady-state case, the terms of expression (6) enclosed in square
brackets does not depend on $x$, and therefore we can introduce a constant $C$
and write the equation

$\frac{dp}{dx}=\frac{1}{\langle a^{2}\rangle}\left[C-\left(\langle
a\rangle+\frac{d\langle a^{2}\rangle}{dx}\right)p\right].$ (7)

We are looking for a solution of this equation, which, for large values of
$x$, should correspond to the Pareto distribution (1). The limit value of the
function $p(\infty)$ is not known, but an expression for the derivative can be
set as

$\left(\frac{dp}{dx}\right)_{\infty}=-\hat{p}(1+\alpha)x^{-(2+\alpha)},\quad
p=p(\infty)+\hat{p}x^{-(1+\alpha)}.$ (8)

The mean value ${\langle a\rangle}$ and the variance of ${\langle
a^{2}\rangle}$ in equation (7), depend on income in the region of positive
values of $x$. It can be expected that these quantities are increasing
functions of the values of $x$ (otherwise, exchanges of agents with high
income may be inefficient), which are conveniently represented with power
function. To satisfy the asymptotes (8), we choose the dependencies

$\langle a\rangle=\langle a\rangle_{0}+rx^{1+\alpha},\quad\langle
a^{2}\rangle=\langle a^{2}\rangle_{0}+kx^{2+\alpha}.$ (9)

With these values of the indices, equation (7) has the desirable asymptotic
(8), and value of the constant in equation (7) could be found

$C=\hat{p}\,[1+r+(1+k)(1+\alpha)],\quad p(\infty)=0.$ (10)

Figure 1 Distribution density function

The empty circles show empirical values of the density of distribution of the
Russian population by income in 2018, calculated according to available data
[9]. The slight line represents the Pareto distribution (1) at $\alpha=1.65$.
The solid line shows the solution of equation (7) at the values of the
parameters (11) and (12). The distribution density function is normalized by
100 units. Income is shown in thousands of rubles.

Equation (7) with relations (9) - (10) appears to be an ordinary differential
equation of the first order with the initial condition $p(0)=0$, which is
impose additionally.

Without looking for an analytical expression of the solution, we turn to
numerical methods that we use for the specific case of the distribution of
individuals over income for Russia in 2018, taken as an example. In Figure 1,
the empty circles represent the values of the probability density of the
distribution estimated according to Rosstat data [9]. One can notice that the
dependence represents typical income distributions, as could be seen from the
graphics on pages 93 and 98 of the review [6]. In Figure 1, a slight line
shows the Pareto power law (1) with index $\alpha=1.65$, which reproduces
empirical results above the income $x>50$.

Equation (7) is considered as an ordinary differential equation, which we
solve numerically by a standard method, taking into account relations (9). The
simple consideration allows us to choose the parameters: for zero values of
income, the mean values are close to zero, whereas the variance is not zero.

$\langle a\rangle_{0}=0;\quad\langle a^{2}\rangle_{0}=0.02.$ (11)

The quantities increases with increasing income according to the law (9) with
the parameters, which must obey to some conditions. Both quantities did not
exceed the total amount of income, which determines the small values of the
parameters

$r=0.0000014;\quad\quad k=0.0000005.$ (12)

The numerical values in the ratios (11) and (12) are selected in such a way
that the maxima of the theoretical and empirical dependencies coincide. The
value of constant $C$ is determined by the relation (10), while the parameter
$\hat{p}$ can take an arbitrary sufficiently large value; calculations are
carried out at the value of $C=36.6$.

We are looking for a solution of equations (7) - (10) with the values of the
parameters (11) and (12) that are specific values for every sample. The result
of calculating the distribution function is shown in Figure 1 in comparison
with the empirical values and the Pareto distribution. This result convinces
us that the entire distribution function could be restored according to the
Pareto index and empirical values of the maximum of income and its position in
relation to the mean income.

##  4 The anatomy of inequality

Returning to the consideration of an assembly of individuals, each of whom,
according to our simplest assumption, produces the $Y/N$ units of value (see
Section 2), so that distribution of individuals over participation in
production could be represented by delta-function. More exatly, the
distribution could be represented by the standard Gaussian distribution about
the mean value of income $Y/N$.

The distribution of individuals over income could be different depending on
the circumstances of the allocation. In a simple case, when there are only
random interactions of individuals and other chaotic uncontrolled influences,
the probability function of the distribution of individuals over income takes
the form of the standard Gaussian distribution about the mean value of income
$Y/N$

$p\left(x\right)=\left(\frac{1}{2\pi\langle(\Delta
x)^{2}\rangle}\right)^{\frac{1}{2}}\exp{\left[-\frac{(\Delta
x)^{2}}{\langle(\Delta x)^{2}\rangle}\right]}.$ (13)

Figure 2 Income distribution functions

The slight curve represents the natural distribution of individuals over
income, calculated for the average value of contribution by an individual
$Y/N=38.1$ and variance $\langle(x-Y/N)^{2}\rangle=90$. The solid curve
represents the real distribution of the Russian population by income in 2018,
reproduced from Figure 1. The distribution functions are normalized by 100
units. Income is presented in thousands of rubles. Here it is denoted $\Delta
x=x-Y/N$. The graph of the distribution (13) is shown in Figure 2 by the
slight line.

In reality, the situation appears to be more complicated. The share of the
created product, which receives by an individual, does not correspond,
generally speaking, to the contribution of the individual to production of
value. The graph of the real distribution function of individuals by income,
reproduced in Figure 2 with the solid line, shows that the real distribution
function of individuals over income differs significantly from the Gaussian
distribution and has the form (1) in the region of big incomes. The real
distribution looks like a strongly deformed Gaussian distribution. Both one
and the other distributions shown in Figure 2 are stationary non-equilibrium
distributions.

Turning to the situation in more detail, we note that, according to the
assumption, each individual, regardless of his real income, produces
$Y/N=38.1$ units of value. It can be easily estimated that about 70% of
individuals have a real income less than the average value $Y/N=38.1$; in
total they receive about 35% of the total product. The remaining part of
individuals, respectively, receives an income exceeding the mean value, which
means that there is a mechanism that transfer some amount of the value from
individuals with lower income to individuals with higher income. Such a flow
of value cannot arise naturally, without some influences; there is an embodied
mechanism, which supports the trends towards the non-equivalence of elementary
exchanges: in each act, the agent who already receives the higher income has
an advantage. Formally, the non-equivalence of the elementary exchanges is
related to the asymmetry of the transition function (4) between states with a
non-zero mean transition value.

The description of inequality in income distribution were discussed by many
autors [6]. Yakovenko and his collaborates, for example, noticed that the
empirical distribution functions can be represented as a composition of two
functions [10]: for small values of the variable, the distribution is
described by the Boltzmann-Gibbs exponential function, while for the large
values of the variable, the distribution corresponds to the Pareto power law.
This gives them a reason to present the totality of individuals as quasi-
independent communities of the poor, the number of which, according to Ludwig
and Yakovenko [10], surpasses 90% in the USA, and the rich (about 4% in the
USA). Not to mention the fact that the two-component representation of
distribution causes an feeling of professional dissatisfaction, the quasi-
equilibrium two-component scheme of Ludwig and Yakovenko omits the essence of
the phenomenon: the mechanism of interchange between the poor and the rich, in
the process of which the wealth flows from the poor to the rich, and, one can
think that the representation of distribution by a single function gives a
more adequate picture of the phenomenon; in any case, it is useful to look at
the entire function.

##  5 Conclusion

The representation of the society as an assembly of weakly interacting
individuals appears to be a conventional beginning of all models, aspiring to
connect macroscopic properties of the society with features of relations
between individuals [3, 4]. A peculiarity of the presented approagh is
considering the assembly of individuals to be an open thermodynamic system
that cannot exist without external streams of energy through bodies of the
individuals and production equipment. The law of production (2), determining
relationship between the production of value and production factors is
objective law, this relation cannot be changed according to the desires of
people, and there is nothing in tht scheme of production that hints on the
inequality. The inequality is connected with peculiarities of behavior of
economic agents; the distribution of income can be explained by asymmetry of
elementary exchanges that is external in relation to production processes. In
contrast to the laws of production, the distribution of products is determined
by rules that are established by people based on ideas about the production of
value, taking into account production factors. The formulated rules are then
reformulated into laws, approved by Parliament, written into the constitution
and implemented: the rules for the distribution of social product are
legitimized, and the life of society (social relations) is organized in such a
way as to guarantee the implementation of the written rules. The difference in
social systems is significantly related to the rules of distribution of the
public product, which need to be revised from time to time.

In the simple case, when streams of matter and energy into and out the society
are ignored, the model system is considered to be in the equilibrium state,
which allows to define equilibrium distribution functions. However, one ought
to be conscious: there is the exchange of matter and energy with the
environment, and the society ought to be considered as an open thermodynamic
system that cannot exist without external streams. This paper demonstrate how
the usage of a thermodynamic model helps to uncover mechanism of the emergence
and maintenance of economic inequality. The knowledge of the mechanism allows
us to determine recipes for mitigating the inequality, but this is a different
problem that could be considered on the basis of the results of this
investigation.

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