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Вестник РУДН. Серия: Экономика
RUDN Journal of Economics
ISSN 2313-2329 (Print); ISSN 2408-8986 (Online) 2021 Vol. 29 No. 3 595-605
http://journals.rudn.ru/economics
DOI 10.22363/2313-2329-2021-29-3-595-605 UDC 330.564.2+316.34
Research article / Научная статья
Group averaging and the Gini deviation
Oleg I. Pavlov1 13, Olga Yu. Pavlova2
1 Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
2All-Russian Correspondence Multidisciplinary School, B-234 Vorob'evy Gory, Moscow, 119234, Russian Federation
Abstract. It is known that partitioning a society into groups with subsequent averaging in each group decreases the Gini coefficient. The resulting Lorenz function is piecewise linear. This study deals with a natural question: by how much the Gini coefficient could decrease when passing to a piecewise linear Lorenz function? Obtained results are quite illustrative (since they are expressed in terms of the geometric parameters of the polygon Lorenz curve, such as the lengths of its segments and the angles between successive segments) upper bound estimates for the maximum possible change in the Gini coefficient with a restriction on the group shares, or on the difference between the averaged values of the attribute for consecutive groups. It is shown that there exist Lorenz curves with the Gini coefficient arbitrarily close to one, and at the same time with the Gini coefficient of the averaged society arbitrarily close to zero.
Keywords: Gini coefficient, Lorenz curve, Gini deviation
Article history: received 21 April 2021; revised 1 June 2021; accepted 15 June 2021.
For citation: Pavlov, O.I., & Pavlova, O.Yu. (2021). Group averaging and the Gini deviation. RUDN Journal of Economics, 29(3), 595-605. http://dx.doi.org/10.22363/2313-2329-2021-29-3-595-605
Усреднение по группам и изменение коэффициента Джини
Аннотация. Известно, что разбиение общества на группы с последующим усреднением в каждой группе приводит к уменьшению коэффициента Джини. Изучается вопрос, насколько может уменьшиться коэффициент Джини при переходе к данной кусочно-
© Pavlov O.I., Pavlova O.Yu., 2021
[¿S 0 I This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.Org/licenses/by/4.0/
И pavlov_oi@rudn.ru
О.И. Павлов1 н, О.Ю. Павлова2
1 Российский университет дружбы народов, Российская Федерация, 117198, Москва, ул. Миклухо-Маклая, д. 6
2Всероссийская заочная многопредметная школа, Российская Федерация, 119234, Москва, Воробьевы горы, В-234 И pavlov_oi@rudn.ru
2-
линейной функции Лоренца. Получены весьма наглядные (так как они выражены через геометрические параметры графика кусочно-линейной функции Лоренца, такие как длины ее звеньев и углы между последовательными звеньями) оценки сверху на максимально возможное изменение коэффициента Джини при ограничении на величину доли групп либо на величину разности между усредненными значениями признака по данной группе и по предшествующей группе. Показано, что существуют кривые Лоренца с коэффициентом Джини, сколь угодно близким к единице, и при этом с коэффициентом Джини усредненного общества, сколь угодно близким к нулю.
Ключевые слова: коэффициент Джини, кривая Лоренца, отклонение коэффициента Джини, кусочно-линейная функция
История статьи: поступила в редакцию 21 апреля 2021 г.; проверена 1 июня 2021 г.; принята к публикации 15 июня 2021 г.
Для цитирования: Pavlov O.I., Pavlova O.Yu. Group averaging and the Gini deviation // Вестник Российского университета дружбы народов. Серия: Экономика. 2021. Т. 29. № 3. С. 595-605. http://dx.doi.org/10.22363/2313-2329-2021-29-3-595-605
Introduction:
the Gini coefficient of a piecewise linear Lorenz function
The Gini coefficient is one of the most important (income) inequality measures. Originally due to Gini (1912), also see (Ceriani, Verme, 2012), it is defined as twice the area of the region between the Lorenz curve and the diagonal OD (Figure 1). The Lorenz curve is the graph of the Lorenz function, that is, the function L(x) defined on [0; 1] such that L(x) is equal to the fraction of the total income or wealth, owned by the poorest fraction x of the population.
The Lorenz function is necessarily non-decreasing and concave up, i.e.
X—Xq
is a non-decreasing function of x for every x0 E [0; 1].
Figure 1. The Lorenz curve
Throughout the paper, the Gini coefficient corresponding to the Lorenz function L(x) is denoted by J(L) or simply by J. The Gini coefficient is widely used in economics and other fields of knowledge (see more details in: Arnold, 2007; Farris, 2010; Astashenko, Malykhin, 2012; Pavlov, Pavlova, 2016, 2018), but most often for measuring income inequality. For example, the Gini coefficient
for the total volume of monetary income of population in Russia in 2019 was 0,411.1 In the European Union in 2019 the Gini coefficient for equivalised disposable income ranged from 0,228 for Slovakia to 0,408 for Bulgaria according to Eurostat.2
If the Lorenz curve is a polygonal chain (which corresponds to the division of society into groups with uniform distribution in each group) then the Gini coefficient of the society is (Kampke, Radermacher, 2015):
n
/ = 1-^0 i-Xi_1)(yi+yi_1), (1)
¿=1
where (x0;y0) = (0; 0), (x1;y1), (x2;y2),..., (xn;yn) = (1; 1) are the vertices of this chain.
Consider a typical oval-shaped Lorenz curve L (with no segment of population having a uniform income distribution). Let x0 = 0 x^ ^ ^ Xji—\ ^ Xyi — 1 be a finite increasing sequence on the x-axis. If we average out the income in each group [x0;x1), [x1;x2),..., [xn_1;xn), then the resulting Lorenz curve L' will be piecewise linear and inscribed into original Lorenz curve L (Figure 2). Since L is concave up and does not contain a non-trivial segment, the only common ponts of L and L' are (x0;L(x0)), (x1;L(x1)),_, (xn;L(xn)). Hence U is above L everywhere except these points, so the area of the region between L' and the diagonal of the square is strictly less than between L and this diagonal. Therefore J(L') < /(L).
The procedure of partitioning the population into 5 or 10 equal groups (i.e. into quintiles or deciles) is a standard one and is used "for the convenience of computations and to increase the analyticity of the data."3 Such data modification can lead to computational errors. How much does the Gini coefficient decrease when such a procedure is used?
1 Federal State Statistics Service. (2020). Russian Statistical Yearbook 2020 (p. 161). Moscow.
2 Eurostat. (2021). Gini coefficient of equivalised disposable income - EU-SILC survey. Retrieved July 15, 2021, from http://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=ilc_di12
3 Minashkin, V.G. (2016). Gini index. The Great Russian Encyclopedia. Electronic Version. (In Russ.) Retrieved July 15, 2021, from https://bigenc.ru/economics/text/1952567
In the above mentioned total monetary income in Russia in 2019 the combined income of the five 20 percent groups is, respectively, 5,3; 10,1; 15,1; 22,6, and 46,9 percent of the total monetary income. It follows from (1) that this partition (with subsequent averaging in each group) corresponds to the Gini coefficient of 0,319, which is considerably less than 0,411!
Literature review
Estimation of the Gini coefficient from incomplete/grouped data has been studied extensively. Since grouped data leads to Lorenz curve which is a polygonal chain, and the definite integral over the latter is computed with the trapezoidal rule exactly, any estimation formula for this rule (e.g., see: Zorich, 2019) produces an estimate for the Gini coefficient J(L). However, there are better alternatives. For example, Gastwirth (1972) presented several methods for calculation of the lower and upper bounds using a probability density function. Golden (2008) proposed an easy to compute estimate for data presented in cumulative income quintiles using a single point of the Lorenz curve. Farris (2012) gave interval estimates using aggregate parameters (such as the number of individuals/families in a percentile and their overall wealth). Fellman (2012) surveyed further publications and concluded that "no method is uniformly optimal, but the trapezium rule is almost always inferior, and Simpson's rule is superior. Golden's method is usually of medium quality."
In this note we obtain upper bound estimates for the maximum possible change in the Gini coefficient with a restriction on the group shares, or on the difference between the averaged values of the attribute for consecutive groups. These estimates are as illustrative as in (Golden, 2008) because they are expressed in terms of the geometric parameters of the polygonal Lorenz curve (such as the lengths of its segments and the angles between successive segments), but some of them are more accurate. Let L be an arbitrary Lorenz function/curve and P - some finite increasing sequence on the x-axis x0 = 0<x-l <••• <xn_1 <xn = 1. Then LP or simply V will denote the Lorenz function/curve which is obtained from L by linearization on segments whose endpoints are consecutive elements of P. We will call V a piecewise inearization of L corresponding to P and call the difference J(L) —J(L') the deviation of the Gini coefficient or simply the Gini deviation. We show that the deviation of the Gini coefficient can be arbitrary close to 1 (i.e., similtaniously J(L) can be arbitrary close to 1, and J(L') to 0).
A Lorenz curve with a large Gini deviation
Example 1. Consider a Lorenz curve L which is a polygonal chain OABCD, where O = (0; 0), D = (1; 1), A = (s; s2), C = {1-s2;1-s) and s is small (so that A is much closer to the x-axis than to the y-axis and C is much closer to the vertical line x = 1 than to the horizontal line y = 1, see Figure 3).
1 E —1
The lines OA and CD have equations y = sx and y = - x + — respectively; we denote their intersection by B. By solving the system of these equations, we get B = As B is close to the vertex (1; 0), the area of the region
between L and the diagonal OD is close to the area of the triangle with the vertices
О, D and (1; 0), and the latter area equals 1/2. This means that ]{L) is close to 1. ]{L) could be calculated exactly using (1), but it is easy to do so by doubling the difference between 1/2 and the area of the region under L. This region is a union of a rectangle (actually, a square) and two triangles. Its area is equal to
11 £ 1 ( 1 \f S \ / 1 \ S £+£
+ 2 l1-—К1-—И1-—)r
л
2 1 + S 1 + S 2 \ 1 + SJ\ 1 + S V 1 + SJ1 + S 1 + S
(1 s + s2\ 2S + 2S2
№ = 21- -——) = 1-
K2 1 + S2) 1 + S
Thus, ]{L) is indeed close to 1 for small s
8
О £ 1-S2
Figure 3. Lorenz curve L
2
8
Let P = {0;s;1 — s2;1}, then LP is a polygonal chain OACD. Since the points A and C are close to the diagonal, J(LP) is close to 0. We calculate it using (1). The differences xt — xi_1 equal £ — 0, 1—e2 — s, 1 — s2 for i=1, 2, 3 respectively; the sums y{ +yi-± equal s2 + 0, 1 — s+s2, 2 — s respectively.
J(LP) = 1-[S^£2 + (1-Е2 -s)(1-s+s2) + s2(2 — s)] = 2S- 3s2 + .
Therefore, both /(L) and 1—J(LP) are of order 0(s) for the considered Lorenz curve L. This means that]{L) is close to 1, and]{LP) is close to 0 for a small s.
Note the following features of the considered L: the share of one of the groups (the second one) is close to 1 and the angle between some segments of L (namely, between AB and BC and also between AB and CD) is close to 90°.
These features are not accidental. We show below that both the smallness of the share of each group or the smallness of the angle between every two consecutive segments of L (or of every two segments standing the next but one) implies the smallness of the Gini deviation.
The Gini deviation and the shares of groups
Theorem 1. Let d be the largest share of groups in a society (i.e. the maximum of values xt — xi_1 for 1<i < n). ThenJ{L) —J(L") < d.
Proof. Let U be a piecewise linearization of the Lorenz curve L corresponding to a sequence P = x0 = 0<x± <^<xn_± <xn = 1. For every i< n, y{ will denote the common value of L(x{) and L'{x{).
If n= 1, then the society is "divided" into a single group whose share is equal to 1. The theorem is valid in this case as the Gini deviation never exceeds 1. So we can assume that n> 1.
Consider two consecutive vertices At_± = (xi_1;yi_1) and At = (x^yi) of L'. Since L is a graph of a non-decreasing function, its entire piece lying between At_± and At is contained in the right triangle Ai_1AiOi, which is bounded by the hypotenuse Ai_1Ai from above and by a horizontal leg Ai_1Oi and a vertical leg Afti from below (Figure 4).
1
Figure 4. The Gini deviation and the shares of groups
Consequently, the part of the region between L' and L which lies between At_± and At is also included in this triangle. Therefore, the whole region between L' and L is included in the union of such right triangles. Hence it is enough to show that the combined area of these triangles does not exceed d.
In the triangle Ai_1AiOi, the leg Ai_1Oi is equal to xt —xi_1 and the leg
AiOi is equal to yt — yi-i, so its area is equal to - (xt —xi_1)(yi —yi-i). The deviation J(L) —J(L') is equal to twice the area of the region between L' and L, which does not exceed i(.Xi—xi_1)(yi—yi_1) according to the previous calculations. But Z]i=i(xi -xi_1)(yi -y^) < if=i d(yt -yi-1) <d^1 = d. Theorem 1 is proved.
If a society is divided into n equal groups, then share each of them is equal
i
to - Then, according to Theorem 1, the Gini deviation does not exceed 1/n.
n
Therefore, the maximum possible Gini deviation in the above mentioned total monetary income distribution in Russia in 2019 is 1/5 = 0,2. The real deviation for this distribution is equal to 0,411 — 0,319 = 0,092, which is a significant fraction of 0,2. We obtained J(L') = 0,319 using (1) and the data from Federal State Statistics Service.4
4 Federal State Statistics Service. (2020). Russian Statistical Yearbook 2020 (p. 161). Moscow. 600 ECONOMIC AND SOCIAL TRENDS
The Gini deviation estimate given by Theorem 1 is sharp for every d. For example, in case of perfect inequality in a society (all the income/wealth belongs to a single person), / = 1. The corresponding Lorenz curve L is a two-segment polygonal chain ABD, where B almost coincides with (0;1). Divide such a society into several groups so that the share of the richest of them is equal to an arbitrary d. Then V will consist of a horizontal segment OB', where
B' = (this segment may correspond to several groups of the society)
and an inclined line segment B'D. The doubled area between L' and the diagonal OD is equal to 1-d which is /(¿'). Thus, /(L) -J(L') = 1-(1-d) = d as required.
The Gini deviation and the angles between segments
Every two consecutive segments of a polygonal Lorenz curve are at the angle from 0 to 90° to each other. The more segments, the smaller this angle can be uniformly given a fixed Gini coefficient. It turns out that a constraint on the magnitude of such an angle also limits the Gini deviation. It is also possible to consider a constraint on segments standing the next but one. Since the slope of the segment5 with the vertices (Xj;yj) u (xi+1;yi+1) is the average income in the group [XjjXj+1], the proximity of the slopes of two consecutive groups is equivalent to the closeness of the average incomes in these groups. We denote the points (—1;0) and (1;2) by A_1 and An+1, respectively, and will consider A_1A0 and DAn+1 as new segments of L'.
Theorem 2. Let a be the largest of the angles between the segments of V standing the next but one. Then
a) if l1, l2,..., ln denote the lengths of the segments A0A1, A1A2,..., An_1An,
then/(L)-/(L') <i2f=iCi£)2tg(f);
b) if p1, p2,. ., pn denote the lengths of the projections of the segments A0A±, A1A2,...,An_1An on the diagonal OD, then/(L) -/(Z/) < |2f=i(Pi)2 tg(f);
c) if - is the largest fraction of the projections of the segments of L' on the diagonal OD, then J(L) -J(L') < ^tg(f);
d)/(L) -/(L') <tg(f).
f
In each item a-d, - can be replaced by the largest of the angles between any
two consecutive segments of L'. Proof:
a) we use the following well-known fact from geometry (e.g. see: Boltyan-skij et. al., 1974): if the length of a side S of a triangle is equal to I, and the other two sides make angles with S that in total do not exceed f, then the area of such
a triangle does not exceed ^2tg(f). Consider an arbitrary segment Ai_1Ai of L',
as well as the previous and subsequent segments (Figure 5).
5 More precisely, of a straight line containing this segment. ЭКОНОМИЧЕСКИЕ И СОЦИАЛЬНЫЕ ТРЕНДЫ
D
° A-
Figure 5. The Gini deviation and the angles between segments
Due to concavity of L, every point lying between the arc Ai_1Ai of L (the arc is not shown in Figure 5) and the segment Ai_1Ai, must be not lower than both lines Ai_2Ai_1 and AtAi+1, i.e. it belongs to triangle Ai_1AiP, where P is the intersection of the lines Ai_2^i-1 and AiAi+1. By condition, the angle between the lines Ai_2Ai_1 and AtAi+1 is at most k. Then, by the mentioned fact, the area of the triangle Ai_1AiP does not exceed
If the lines Ai_2Ai_1 and AtAi+1 coincide, then triangle Ai_1AiP degenerates into segment Ai_1Ai having zero area. Therefore J(L) —J(L') < tg (j).
Item a is proved;
b) let p(Ai_1) and p{Aj) denote, respectively, the projections of points Ai_1 and At on the diagonal OD. Then the length of the segment Ai_1Ai does not exceed the length of segment p(Ai_1)p(Ai) multiplied by —2. Thus <Pi—2 and further
n n
rn -rn < ^(.P,-^2 tg (f) = j^p,)2 tg (f).
¿=i ¿=i
Item b is proved;
c) it follows directly from the condition of item c that Pi<- —2 whence
1 l / 2
l V"1 N2
d) follows from c as tg ^ since n>l.
If P is the largest of the angles between any two consecutive segments of L', then the largest of the angles between the segments of L' standing the next but
OC
one does not exceed 2fi, so - can be replaced by P in each item a—d. Theorem 2 is proved.
A comparison with Golden's approach
Golden (2008) proposed a simple way to compute the upper bound Gini coefficient estimate for data presented in cumulative income quintiles. We compare the Gini deviation corresponding to this estimate, obtained for a particular Lorenz function L(x) = x2, with the estimates from our Theorem 2 and find that the estimates given in items a, b, c of Theorem 2 are more accurate.
Golden called his method for approximating the upper bound of the Lorenz curve's L Gini coefficient "the Z-gradient rule". The estimate given by this rule is the Gini coefficient of a three-segment polygonal chain OPQD (which we further denote by LG), where 0 = (0; 0), D = (1; 1), and points P and Q lie on the x-axis and y-axis, respectively, so that the line PQ is tangent to L at a point (z0,L(z0)) of L farthest from the diagonal OD. It is known that L'(z0) = 1, see Hoover (1936) and Kakwani (1980). Since L(x) =x2, L'(x) = 2x, so that z0 = 0,5 andL(z0) = 0,225. The tangent line to the Lorenz curve at the point (0,5; 0,25) has an equation
y - 0,25 = 1 • (x — 0,5),
or y = x- 0,25, thus P = (0,25; 0) and Q = (1; 0,75). Therefore, J(LG) = ^
by (1). Let x0 = 0, x1 = 0,2, x2 = 0,4,..., xs = 1 be the first coordinates of the vertices of L' (which is the linearization of L(x) =x2 corresponding to the division of the segment [0; 1] into five equal subsegments). Then J(L') = 0,32 by (1), therefore
J(LG)-J(L')= 16- 0,32 = 0,1175,
which is the error of Golden's approximation.
We organize the calculations of the estimates from Theorem 2 in the Table.
Here kt is the slope of the i-th segment of the linearized Lorenz curve L' and is its angle with the positive direction of the x-axis (measured in radians). The largest half-angle a/2 between the segments of L' standing the next but one is 0,294 rad (the angle between the first and the third segments), its tangent is equal to 0,303. The sum of the squares of the segments of L' is equal to 0,464, the sum of the squares of the projections of these segments on the diagonal OD is 0,432.
Calculations of the estimates from Theorem 2
Xi Vi &xt bVi kt = Ayi/Axi & Ч pI
0 0
0,2 0,04 0,2 0,04 0,2 0,197 0,042 0,029
0,4 0,16 0,2 0,12 0,6 0,54 0,343 0,054 0,051
0,6 0,36 0,2 0,2 1 0,785 0,294 0,245 0,08 0,08
0,8 0,64 0,2 0,28 1,4 0,951 0,205 0,165 0,118 0,115
1 1 0,2 0,36 1,8 1,064 0,139 0,113 0,17 0,157
£ 0,464 0,432
max 0,294 0,343
Consequently, the estimates obtained in items a, b, c of Theorem 2 are
5
¿=i 5
2YJ(-pi)2tg(2)=o'o6S'
¿=i
1 K ^(l) = 0,061
respectively (rounded to the nearest thousandth), which are considerably more accurate, than 0,1175 obtained by applying the Golden's Z-gradient rule. The largest angle p between any two consecutive segments of V is 0,343 rad (the angle between the first and the second segments). Its tangent is equal to 0,357, which is 18% greater, than tga = 0,303. Hence replacing k/2 with p in the latter three displayed formulas increases the estimates by 18% which still leaves them less than 0,1175.
Conclusion
We obtained upper bound estimates for the maximum possible change in the Gini coefficient with a restriction on the group shares, or on the difference between the averaged values of the attribute for consecutive groups (geometrically -in terms of the geometric parameters of the polygon Lorenz curve, such as the lengths of its segments and the angles between successive segments). For a considered particular Lorenz curve these estimates are much more accurate than derived by Golden (2008). Further research may include deriving still better (ultimately, sharp) estimates, e.g. such that for a given s>0 the Gini deviation could not be multiplied by 1 — s. The case of equal groups is of particular interest.
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Сведения об авторах I Bio notes
Павлов Олег Иванович, кандидат физико-математических наук, доцент кафедры экономико-математического моделирования, экономический факультет, Российский университет дружбы народов. E-mail: pavlov-oi@rudn.ru
Павлова Ольга Юрьевна, кандидат физико-математических наук, доцент кафедры высшей математики, Всероссийская заочная многопредметная школа. E-mail: lolgau@ yandex.ru
Oleg I. Pavlov, PhD, Associate Professor of Economic and Mathematic Modelling Department, Economic Faculty, Peoples' Friendship University of Russia (RUDN University). E-mail: pavlov-oi@rudn.ru
Olga Yu. Pavlova, PhD, Associate Professor at the Department of Higher Mathematics, All-Russian Correspondence Multidisciplina-ry School. E-mail: lolgau@yandex.ru
