=Paper= {{Paper |id=Vol-2177/paper-03-c004 |storemode=property |title= Some Characteristic Properties of Resource Distribution Nonuniformity in Economic Systems |pdfUrl=https://ceur-ws.org/Vol-2177/paper-03-c004.pdf |volume=Vol-2177 |authors=Leonid A. Sevastianov,Alexander V. Kryanev,Daria E. Sliva,Valentin V. Matokhin }} == Some Characteristic Properties of Resource Distribution Nonuniformity in Economic Systems == https://ceur-ws.org/Vol-2177/paper-03-c004.pdf

UDC 519.248.6
Some Characteristic Properties of Resource Distribution
Nonuniformity in Economic Systems
Leonid A. Sevastianov* , Alexander V. Kryanev† ,
Daria E. Sliva† , Valentin V. Matokhin‡
*
Department of Applied Probability and Informatics,
Peoples’ Friendship University of Russia (RUDN University),
6 Miklukho-Maklaya str., Moscow, 117198, Russia
†
National Research Nuclear University “MEPhI”
31 Kashirskoe shosse, Moscow, 115409, Russian Federation
‡
ZAO “TEKORA”
65 Profsoyuznaya st., Moscow, 117997, Russian Federation
Email: sevastianov_la@rudn.university, avkryanev@mephi.ru, desliva@mephi.ru,vmatokhin@tekora.ru

In this article the research on two-parameter Lorenz curve approximants is continued.
Several characteristic properties of one such family of two-parameter functions used to approx-
imate the Lorenz curve is studied. Several interpretations of the functional parameters are
introduced. Distribution density functions that correspond to various values of parameters
of the class of functions used to approximate the Lorenz curve are analysed. The implica-
tions and the value brought to the study of nonuniformity of resource distributions by this
introduction is discussed. The ties between the values of the parameters and statistic estima-
tors are tested. It is concluded that the values of the approximant parameters are connected
to the coefficient of kurtosis and the value of the interquartile range. An interpretation of
the approximant parameters with regards to economic inequality is suggested. The special
case corresponding to uniform distribution of the shares of the resource allocated to separate
agents is distinguished and discussed. It is concluded that it is of special value with regards
to solving the problem of finding the optimal distribution of a resource within an economic
system.

Key words and phrases: resource distribution, Lorenz curve, economic inequality.

Copyright © 2018 for the individual papers by the papers’ authors. Copying permitted for private and
academic purposes. This volume is published and copyrighted by its editors.
In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VIII Conference
“Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Moscow, Russia, 20-Apr-2018, published at http://ceur-ws.org
20 ITTMM—2018

1. Introduction
One possible representation of a empiric distribution of resources within an economic
system is a Lorenz curve. Various single parameter and multiple parameter approxi-
mants of the Lorenz curve, the corresponding probability distribution functions and the
probability density functions have been studied extensively [1–8]. As the Lorenz curve
is currently the predominant apparatus used to characterise and simulate inequality in
economic and social systems, both its analysis and applications are of great interest.
The present paper constitutes an inquiry into one of its many proposed approximants
and its properties. This is a follow-up research from previous elaboration on the class of
two-parameter functions [3, 4].

2. Main section
Let 𝐺 be a resource distributed among 𝑁 economic agents. Let 𝑖 = 1, . . . , 𝑛, . . . , 𝑁
be the indices of 𝐺𝑖 — the amount of resource 𝐺 allocated to the,
𝑁
∑︁
𝐺𝑖 = 𝐺.
𝑖=1
A resource distribution 𝐺𝑖 , 𝑖 = 1, . . . , 𝑁 may be represented by Lorenz curve [7, 8].
For that purpose the original series is ranged in ascending order: 𝐺1 6 𝐺2 6 . . . 6 𝐺𝑁 .
Next the accumulated sums
𝑛
∑︁
𝑆𝑛 = 𝐺𝑖
𝑖=1
are calculated and plotted on a plane with axes 𝑥 = 𝑛/𝑁 and 𝑦 = 𝑆𝑛 /𝑆𝑁 .
Previously [3,4] it has been demonstrated that the Lorenz curves may be approximated
my the two-parameter curve belonging to the class of functions

𝑦(𝑥, 𝛼, 𝛽) = 1 − (1 − 𝑥𝛼 )1/𝛽 ,
where 1 6 𝛼 < ∞, 1 6 𝛽 < ∞.
It has been previously demonstrated [5, 6] that parameters 𝛼 and 𝛽 may serve as
indicators of resource distribution non-uniformity. At 𝛼 = 𝛽 = 1 the accumulated
sums 𝑦(𝑥, 𝛼, 𝛽) = 𝑥 and the corresponding resource distribution is absolutely uniform,
which means that each economic agent acquires possesses exactly 𝐺/𝑁 of the total
accumulated wealth. In the opposite scenario, while 𝛼 → ∞, 𝛽 → ∞ the resource
distribution approaches an extremely non-uniform distribution, where the agent of the
highest rank has exclusive control over the entire stock of wealth and the agents of lower
rank have nothing.
By manipulating the values of the parameters 𝛼 and 𝛽 various distribution density
functions may be attained. It has previously been shown [7, 8] that even the single
parameter class of curves given by 𝑦(𝑥, 𝛼, 𝛽) = 1 − (1 − 𝑥𝛼 )1/𝛼 , demonstrate a good fit
for empiric wealth distributions. However, the implementation of the two-parameter
class of functions 2 enables to improve the quality of the fit by allowing for asymmetric
resource distributions, while the class of functions ?? can only simulate a symmetric
(with regards to the main diagonal of the Lorenz square) distribution [9, 10]. Besides,
the use of the two-parameter approximant allows to more closely fit the intervals that
correspond to groups of the highest and the lowest rank, while sustaining the ease of
interpretation and calculation that were part of the incentive for the use of the single
parameter approximant [11–14].
Fig. 1–2 illustrate the correspondence between selected values of the parameters 𝛼
and 𝛽 and the corresponding frequency histograms. In this paper the discrete values
case is considered, for the continuous case and the analytical results see [3, 4]. Here we
Sevastianov Leonid A. et al. 21

scrutinise the distribution of the total resource 𝐺 = 1000 among 𝑁 = 1000 economic
agents with regards to variation of parameters 𝛼 and 𝛽. In fig. 1-2 the relative shares of
the resource 𝐺 are plotted along the horizontal axis and the relative number of agents
that possess that portion of the shared resource are plotted along the vertical axis.
The histograms are standardised so that the 1000 values of 𝐺𝑖 , 𝑖 = 1, . . . , 𝑁 are always
grouped into 20 equal-sized intervals.

Figure 1. The resource distribution histograms for various values of 𝛼 (𝛽 = 1)

Figure 2. The resource distribution histograms for various values of 𝛽 (𝛼 = 2)
22 ITTMM—2018

It is evident that for 𝛼 = 1, 𝛽 = 1 all agents acquire an equal share of the resource
𝐺 [15]. While 𝛼 increases while the value of 𝛽 is fixed, the uniformity of the distribution
is compromised as the portion of the total resource allocated to single agents are varied.
Provided that every percentile of the resource distribution in the context of societal
wealth represents a class, at 𝛼 → 2, 𝛽 = 1, probability that any given agent 𝐺𝑖 is
positioned within any given class is uniform. This does not mean that the shares 𝐺𝑖 are
equal. In fact, when 𝛼 = 2, 𝛽 = 1, 𝐺𝑖 varies from 0 to 2 · 𝐺.
It may be noted that the distribution that corresponds to the Lorenz curve

𝑦(𝑥, 𝛼, 𝛽) = 1 − (1 − 𝑥𝛼 ), 𝛼=2
is of special interest for two reasons:
– the maximal share 𝐺𝑖 that may be allocated to an agent or a group of agents is
limited by 𝐺𝑖 = 2 · 𝐺;
– the probability that an agent will be positioned within any percentile, decile,
quartile is equal.
The density function of the considered distribution with respect to 𝐺 is as follows
⎧ 1
⎨ , 𝑥 ∈ [0, 2 · 𝐺],
𝑓 (𝐺𝑖 ) = 2 · 𝐺
𝑥∈/ [0, 2 · 𝐺].
⎩
0:
The Gini coefficient 𝐾𝐺 — an extensively implemented measure of distribution
2
non-uniformity [16–21] — for the curve given by 2 is equal to 1 − 𝛼+1 , or, for the case
in consideration 𝐾𝐺 = 13 . It must be noted that high (𝐾𝐺 > 0.5) values of 𝐾𝐺 are
rarely observed with regard to societal economic inequality and at present time only
occur in undeveloped and some developing countries.
It is further proposed that the values of parameters 𝛼 and 𝛽 of the class of functions
2 specify distinct statistical properties of empiric distributions of total accumulated
wealth within a society among its members.
The parameter 𝛼 is a characteristic of peakedness of the probability density function.
The resource density function will become less flat and more pointed as the value of
parameter 𝛼 increases. For lower values of 𝛼 the values of 𝐺𝑖 are concentrated closer to
the mean value
𝑁
∑︁
𝐺= 𝐺𝑖 = 𝐺/𝑁,
𝑖=1
while for higher values of 𝛼 the values of 𝐺𝑖 are scattered further from 𝐺. This means
that as the value of 𝛼 increases, less and less agents are allocated the ’fair’ portion of the
resource 𝐺. As to the statistical properties, the leptokurtic probability density functions
are classified by a higher 𝛼 coefficient of kurtosis, calculated as
𝜇4
𝛾2 = − 3,
𝜎4
where 𝜇4 = E[(𝐺 − E𝐺)4 ] is the kurtosis of the stochastic value 𝐺, and 𝜎 is its standard
deviation.
On the other hand, the parameter 𝛽 is characteristic of the disparity between the
proportion of the resource belonging to the agents of the lowest and the highest rank.
From an economic point of view the increase in the value of 𝛽 is accompanied by societal
polarity which is characterised by the formation of two segregated groups of ultra-rich
and ultra-poor. The value of the interquartile range (IQR) is determined primarily by
the value of 𝛽. 𝐼𝑄𝑅 is the difference between the median value of the resource allocated
to the 50% of the highest ranking agents and the median value of the resource allocated
to 50% lowest ranking agents. With regards to measuring economic inequality the
interquartile range is a measure of the gap between the median wealth of the 50% of the
Sevastianov Leonid A. et al. 23

richest and the median wealth of the 50% of the poorest. The value of the interquartile
range is a robust measure of variation in comparison with integral range, a characteristic
of spread of a stochastic value.

0.15
800

0.1 600
IQR

2
400
0.05
200

5 5
4 5 4 5
3 4 3 4
3 3
2 2 2 2
1 1 1 1

Figure 3. Plots of kurtosis and IQR for various values of 𝛽 and 𝛼 = 2

Fig. 3 illustrates the correspondence between the values of the interquartile range
and the coefficient of kurtosis with regards to various values of parameters 𝛼 and 𝛽. The
values of the parameters 𝛼 and 𝛽 are plotted along the axes of abscissa and ordinates,
while the values of the measures of distribution in question are plotted along the applicate
axis.
It is clear that the value of the coefficient of kurtosis of the considered class of
distributions is chiefly determined by the value of the parameter 𝛼. It can be concluded
that the parameter 𝛽, in comparison to the weight of the parameter 𝛼, scarcely affects the
value of the coefficient of kurtosis of the distribution density and is largely insignificant
with regards to measuring the flatness of the distribution. It is also shown that the
maximal value of the interquartile range is attained at large values of the parameter
𝛽 while 𝛼 = 1. Therefore we propose that the parameter 𝛼 may be interpreted as a
characteristic of the gap between the wealth of those of the highest rank and those of
the lowest rank, while the parameter 𝛽 is a measure of variation.

3. Conclusions
The attained results are useful for solving various problems associated with finding
optimal resource allocation among several objects or agents within an economic system.
The proposed interpretation of the parameters 𝛼 and 𝛽 of the two-parameter Lorenz
curve approximant makes it possible to subsequently define classes and clusters of
empiric economic resource distributions and the corresponding economic and and societal
distributions. The question whether there exists an empiric relationship between the
resource distribution that corresponds to the special case of the studied Lorenz curve
approximant and the performance of real economic systems is a subject of further
investigation.
24 ITTMM—2018

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