amount, barycentre and concentration (gini/lorenz who ... 26 may/paper...cannizzaro 278, 98122...

ABC - Amount, Barycentre and Concentration (Gini/Lorenz who?)

Gustavo De Santis

DESAGT (Dip. di Economia, Statistica e Analisi Geopolitica del Territorio), V. T. Cannizzaro 278, 98122 Messina, Italy E-mail: [email protected]

Summary

Concentration is frequently measured with Gini's index and represented graphically with Lorenz's concentration area. This paper introduces an alternative measure, ABC (Amount, Barycentre and Concentration), based on the measurement of the barycentre in an ordered, but not cumulated, frequency distribution. The ABC index leads to the same results as Gini's, but improves over it in a few noteworthy cases, because 1) it leads to consistent estimates for grouped and individual data; 2) is easier to calculate, especially with continuous distributions and infinite owners, 3) can be applied also when the character is not entirely transferable, and 4) in the presence of population subgroups, ABC makes it easier to decompose Gini's index into basic elements (between, within, and overlapping). Formal proofs and textbook examples are presented for each case. An application to actual data for contemporary Italy concludes the paper.

Key words

Concentration, Lorenz curve, decomposition of Gini's index.

Acknowledgements

My thanks to Carla Rampichini, for her encouragement and her useful critics on an earlier draft.

mailto:[email protected]

ABC - Amount, Barycentre and Concentration 2

Introduction

Every quantitative variable y may be distributed more or less equitably among potential "owners" x: for instance assets among a group of individuals, number of years lived by a cohort of newborn in a life table, proportion of ancestors of noble origin in a sample of respondents, etc. Given the total amount of the variable in that community, and therefore also the average amount, the most equitable distribution is undoubtedly the one that emerges when everybody possesses the same, average quantity. The opposite extreme, however, signalling the highest possible degree of concentration, is less obvious and may vary with the case at hand, as we will see shortly.

Concentration has been studied extensively, and Gini's index (G or, sometimes, R), is probably the most widely used indicator (Section 1). However, its applicability is somewhat limited, because Gini's index: 1. relies on the assumption that the variable y can be transferred indefinitely, i.e.

that the theoretical case of greatest possible concentration emerges when the richest individual (or class) owns everything, and all the others own nothing;

2. does not yield the same result with individual or grouped data (frequency distribution), even when two or more owners possess the same amount of y;

3. is cumbersome to apply with model distributions, continuous both in x and y; 4. cannot be decomposed easily, distinguishing in particular the part of G that

depends on the difference between and within subgroups. This papers sets out to show that the concentration problem can also be viewed

from a new perspective, leading to the set of measures described below, labelled ABC (Amount, Barycentre and Concentration), where the most important term is, obviously, C. The advantage of the proposed approach is twofold: it sheds new light on the concepts of inequality and concentration, and it permits to overcome the four shortcomings of Gini's index listed above.

1. Gini's concentration index: a reminder1

Let y be a quantitative, transferable variable and let x represent owners (N in total), in a broad sense, with the i-th owner (xi) possessing yi. In case of intensity distributions, let the jth class contain Xj=(∑i xji) owners, each possessing yj on average, and let Yj (=yjXj) represent the class total. Let us order owners xi in non-decreasing order with respect to the quantity of y that they own, with pi (=xi/N)

representing the simple, and Pi the cumulated, proportions of owners.

= ∑

=

i

jjp

0

1 The original contribution can be found in Gini (1939). Virtually all of the Italian

statistical textbooks with a part on descriptive statistics discuss Gini's index extensively: see, e.g., Leti (1983), Giusti (1983), and Piccolo (1998).


Further, let us consider the ratios hi=yi/Y and qj=Yj/Y (=yjXj/Y) where Y is the maximum quantity of y that an individual could theoretically possess. The standard practice with Gini is to assume that Y=∑yi, which means that the maximum possible concentration occurs when the Nth owner2 (the last and richest), owns everything and all the others own nothing. This is a rather extreme case that we will discuss shortly. In all cases, hi is the relative amount possessed by pi (relative to a maximum), qj the relative amount possessed cumulatively by

the members of the jth class3 and Qi the cumulated, relative amount

possessed by the first (i.e. poorest) Pi individuals. By definition, P0=Q0=0, and ∑pi=PI=∑qi=QI=1.

= ∑

=

i

jjq

0

∑=P

PQ

( ) ∑=+QQ'

For individual data, Gini's index can be calculated as

( )∑∑

∑ −−

=P

QPPP

G (1)

whereas, for grouped data, it is preferable to use the formula that gives the area of relative concentration (see figures 1 and 2). In this case, the index is frequently labelled R, and calculated as

( ) ∑∑ −−−= QPPQPPG '''1 (2)

where P' and Q' stand for Pi+1 and Qi+1, respectively. Both G and R range between 0 (perfectly equitable distribution) and 1 (the richest owns everything and the others own nothing), and therefore both assume that the variable is, at least in principle, totally transferable.

2 Or class of owners, with intensity distributions. 3 For individual data, obviously, qi=hj.


Fig. 1. Gini's index as the relative concentration area.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P(i)

Q(i)

Concentration area

R=.444

Source: Author's simulations (cf. tables 1 and 2, column 2).

Note, further, that equation (2) implicitly assumes that the number of individuals (or classes) is infinite: in this case, the maximum concentration area is 0.5, i.e. that of triangle OBC in figure 2. More generally, however, the area of maximum concentration is that of a slightly smaller triangle, OAC, so that

25.0max

IpR −= , where pI=(1-PI-1) is the share of the last and richest group, or

individual.


Fig. 2. Maximum concentration area for a finite and relatively small number of classes or individuals.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p(i)

q(i)

A B

C

O

Maximum concentration area, when the last class includes a fraction AB of

the total sample

Source: Author's simulations (cf. table 1, column 4).

2. A different graphical representation of the problem

Let us now consider a different graphical representation of the same problem, again with Pi (cumulated proportions of properly ordered owners) on the x axis, but this time with hi on the y axis, where hi = yi/Y is the simple proportion to the maximum possessed by the ith individual (or by each member of the ith class). The original distribution will now appear as an area located on the bottom, right-end side of the normalised square (side=1), as in figure 3, for instance.


Table 1. Income concentration (frequency distribution; hypothetical case)

Owners(x i ) Actual Max concentration Perfect equity(1) (2) (3) (4)2 0 0 42 2 0 42 4 0 42 6 0 42 8 20 4

Total 10 40 40 40Average 4 4 4

Owned variable y i

(e.g. income)

Fig. 3. Graphical representation of the ABC approach.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.3 0.5 0.7 0.9

Cumulative proportions of owners (Pi)

y(i)/

Y

A=.100B=.700C=.444


Three synthetic measures, among others, can be calculated here. The first is how large the darkened area is with respect to the total area of the square, equaling unity. This is the relative amount A of the variable


Y

yphpAA ii

iii∑∑∑ === (0<A<1) (3)

where Ai is the area of each rectangle, with base pi and height hi=yi/Y. Since the

average amount possessed is ∑∑∑ == ii

i

ii yppyp

y , A is also the ratio of the

average y to the maximum Y, Therefore, when Y=∑yi, A=1/N.

Yy

Yyp

AA iii === ∑∑ (3')

In the case of table 1, for instance, y =4, Y=∑yi=40, and A=10% (see the appendix for the details of the calculations of all ABC measures).

The second synthetic measure of interest to us is the abscissa of the barycentre of the distribution, B, indicated by the vertical arrow in all the figures of this paper. Its formula is

∑

∑=i

iiA

APB

* (4)

where 2

1* ++= ii

iPPP is the barycentre of each vertical bar, i.e. the abscissa of its

centre, and Ai its area. Since, by construction, potential owners appear in non-decreasing order of

possession, with the richest grouped on the right-end side of the distribution, B cannot be smaller than 0.5, a theoretical minimum signaling perfect equity (figure 4).


Fig. 4. Horizontal distribution (perfect equity).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.3 0.5 0.7 0.9


y(i)/

Y

A= .100B= .500C=.000


B has a maximum, too. In general, as figure 5 shows, this maximum is given by Bmax=(1-A/2), because the general barycentre of the darkened area cannot fall to the right of the barycentre of the rectangle representing the quantity owned by the richest group, or individual. However, if Y is very high with respect to all yi, which happens in particular when Y=∑y and there are several owners (N is high), the bias introduced by the assumption that A→0 and Bmax(= )→1 is small. *

1−IP


Fig. 5. Maximum concentration.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95


y(i)/

Y

A = .100B = .950C=1.000


The third and, for our purposes, most important measure is the concentration index C

( ) AB

AB

BBC

−−

=−−

−=

−−

=1

125.021

5.05.0

5.0

max (5)

which simplifies to C=2B-1 when Bmax→1 and A→0. How does the ABC approach compare with more traditional indexes, that is G

or R? The appendix shows that, in most cases, C=R, i.e. that the two measures of concentration coincide. In this sense, C is useful only in that it sheds new light on how to view the issue of concentration, and how to represent it graphically. The difference is that R and G focus on the relative extension of the concentration

area, using the non-negative elements i

iiP

QP −, while, in the ABC approach, the

problem is: "how far on the right does the barycentre fall, and how does this compare with its theoretical maximum?"

However, the ABC index also improves over Gini in some respects, considered in the next sessions.


3. Individual data vs. frequency distributions

When two or more owners possess the same amount of y, Gini does not yield the same result if the data are presented as a series or as a frequency distribution. The reason is that the elements of a given class i in eq. (1), i.e. (Pi-Qi) in the numerator and (Pi) in the denominator, appear only once with frequency distributions, but more than once with individual data, and since the ratio of the added elements

i

iiP

QP − does not necessarily coincide with G, including them more than once

affects the final result.

Table 2. Income concentration (individual data; hypothetical case)

Owners(x i ) Actual Max concentration Perfect equity(1) (2) (3) (4)1 0 0 41 0 0 41 2 0 41 2 0 41 4 0 41 4 0 41 6 0 41 6 0 41 8 0 41 8 40 4

Total 10 40 40 40Average 4 4 4

Owned variable y i

(e.g. income)

Compare tables 1 and 2 for instance: they describe exactly the same situation,

but the corresponding G's differ: for table 1, G(1)=0.5, while, for table 2, G(2)=0.44 (actually, the correct answer). Using eq. (2), instead of eq. (1), is not a solution: R(1)=0.6, while R(2)=0.55.

ABC measures, on the contrary, indicate in both cases that A=0.1 (average, relative to the maximum, i.e. 4/40 in this case), B=0.7 (barycentre), and C=0.44 (concentration). In short, contrary to Gini, ABC indexes are insensitive to how data are presented, either as a succession of individual data, or grouped in classes.


4. Proportions

The ABC approach works equally well with proportions. For instance, let us consider the concentration of "fuzzy" poverty, remembering that a fuzzy poverty index is normally the synthesis of several indicators of relative deprivation, concentrating on income, assets, amenities in the house, etc.4 All the elementary components are normalised, so as to range between 0 (no sign of poverty) and 1 (definitely poor). The synthetic result of such elaborations, a weighted average of elementary indexes, is itself and index ranging between 0 and 1 for each individual in the sample, and appears as in table 3 and in Figure 6 (hypothetical data), where 0 denotes individuals with absolutely no sign of poverty whatsoever, and 1 marks individuals who are totally poor, in all possible respects.

Table 3. Distribution of a fuzzy poverty index, and calculations of the ABC measures

Owners Average Total Base Height Barycentre Area(1) (2) (3) (4) (5) (6) (7) (8)

from to x i y i Y i =x i y i p i h i =y i /Y P i * A i =p i h i A i P i *0 0 500 0.00 0.0 0.5000 0.0000 0.2500 0.0000 0.00000 0.4 300 0.20 60.0 0.3000 0.2000 0.6500 0.0600 0.0390

0.4 0.6 100 0.50 50.0 0.1000 0.5000 0.8500 0.0500 0.04250.6 0.7 40 0.65 26.0 0.0400 0.6500 0.9200 0.0260 0.02390.7 0.8 30 0.75 22.5 0.0300 0.7500 0.9550 0.0225 0.02150.8 0.9 20 0.85 17.0 0.0200 0.8500 0.9800 0.0170 0.01670.9 1 10 0.95 9.5 0.0100 0.9500 0.9950 0.0095 0.0095

1000 185 0.185 0.153

RectanglesFuzzy povertyFuzzy measure

of poverty

Total Note. Y (highest possible fuzzy poverty measure) = 1. Source: Author's simulations.

Calculations are identical to those seen above (and in the appendix), except that the theoretical maximum for each owner is now Y=1 (or 100% poor), and not the sum of the observed values. Correspondingly, the height of each rectangle, normally the ratio yi/Y, simplifies in this case to yi, so that columns 2 and 5 coincide.

The diffusion of poverty in this hypothetical community is A=ΣAi=18.5%, because A is also the ratio Yy , or simply y in this case. The abscissa of the

barycentre is ∑

∑=i

iiA

APB

*= .153/.185 = .827. The minimum for this barycentre is

0.5 (as always), while its maximum is (1-A/2), or .908. This leads to

803.5.0908.5.0827.

=−−

=C , which (in this hypothetical case) signals a high degree of

4 Cerioli and Zani's (1990) seminal contribution on this issue has been subsequently refined

and applied empirically in a number of cases, by Lemmi and colleagues (e.g. Cheli and Lemmi, 1985).


concentration of poverty, or, in other words, a relatively sharp distinction between those who are better off and those who are worse off in economic terms.

Fig. 6. Concentration measures of "fuzzy" poverty

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cumulat ive proportions of individuals

Fuzz

y po

verty

inde

x

Source: Author's simulations (cf. table 3).

It is possible to view the same problem from the opposite perspective, i.e. to consider an index of non-poverty, and work on data like those of table 4, which mirror those of table 3. The results that one obtains in this case are perfectly consistent with those found before. The diffusion of non-poverty is A~ =.815 (and, therefore, A+ A~ =1); the barycentre is B~ =.468/.815=.574;5 and, finally, since the maximum for this barycentre is (1- A~ /2=) .592, the concentration index is C~ =.803 (or C= C~ ). And this holds more generally with the study of proportions:

5 This satisfies the condition that ( ) 5.0

~~1 =−+ ABBA , or, in other words, that the weighted sum of the barycentres of the two distributions gives the overall barycentre of the normalised square, 0.5. The reason why one needs to include (1- B~ ) instead of B~ in this formula is that, when both distributions are considered in the same figure (e.g. in figure 6), the two partial barycentres fall, by definition, one to the left and one to the right of 0.5.


ABC measures do not depend on whether one calculates them on those who have a certain characteristic, or those who do not have it.

Table 4. Fuzzy non-poverty index, and calculations of the ABC measures

Owners Average Total Base Height Barycentre Area(1) (2) (3) (4) (5) (6) (7) (8)

from to x i y i Y i =x i y i p i h i =y i /Y P i * A i =p i h i A i P i *0 0.1 10 0.05 0.5 0.0100 0.0500 0.0050 0.0005 0.0000

0.1 0.2 20 0.15 3.0 0.0200 0.1500 0.0200 0.0030 0.00010.2 0.3 30 0.25 7.5 0.0300 0.2500 0.0450 0.0075 0.00030.3 0.4 40 0.35 14.0 0.0400 0.3500 0.0800 0.0140 0.00110.4 0.6 100 0.50 50.0 0.1000 0.5000 0.1500 0.0500 0.00750.6 1 300 0.80 240.0 0.3000 0.8000 0.3500 0.2400 0.08401 1 500 1.00 500.0 0.5000 1.0000 0.7500 0.5000 0.3750

1000 815 0.815 0.468

RectanglesFuzzy measure

of poverty

Total

Fuzzy non-poverty

Note. Y (highest possible fuzzy non-poverty measure) = 1. Source: Author's simulations (cf. table 3).

4. Life tables

Intermediate cases exist between variables that can be considered entirely transferable (like income, for instance), and variables for which a given, well identified individual maximum exists (e.g. proportions, limited to unity). Life tables are an example: they furnish, among other things, the number of years lived by each individual in a cohort of newborn, under a certain set of assumptions that need not concern us here. The first and most interesting question to which a life table answers is: how long do these individuals live, on average? For instance, a hypothetical, typical Italian woman, subject all along her life to the mortality conditions prevailing in year 2000, would have lived about e0=82.5 years, where e0 stands for life expectancy at birth, or average duration of life (Istat, 2003).

What about the degree of concentration of this variable (number of years lived), and what about the hypothesized tendency toward "rectangularisaton" in the shape of the survival curve, i.e. the idea that, as e0 increases, concentration decreases and people tend to die more and more at the same (high) age? Gini's index does not seem (to me) immediately applicable here because "it is equal to zero if all individuals die at the same age, and equal to 1 if all people die at age 0 and one individual dies at an infinitely old age." (Shkolnikov, Andreev and Begun 2003, p. 311; emphasis added). For instance, the maximum degree of concentration in a life table like that of the Italian women in 2000 would emerge if 99 999 individuals died immediately after their birth, while one individual (Methuselah?) lived up to the age of more than 8.2 million years, thus preserving the total number of years lived by that hypothetical cohort of 100 000 individuals. This implies that Gini's index will always tend to 0 when applied to actual life tables, as shown, for instance, in Shkolnikov, Andreev and Begun (2003).


A possible alternative is to assume the existence of a maximum life span Y. Admittedly, this is arbitrary, but not more than the traditional practice of having "one individual dying at an infinitely old age". To this end, two approaches, among others, can be considered. One is to assume that Y is a constant, corresponding to the maximum ever reliably recorded for a human being, i.e. 122 years, the age at death of Jeanne Calment, a French woman who died in 1997.

The other is to assume that Y is the age by which the proportion still surviving is reasonably small, like 2.5% or 1%, for instance. The assumption here is that, given external conditions (medicine, economy, pollution, etc.), the human beings described in the life tables were not expected to exceed the age of Y, and it is only by chance that a small fraction of them actually survived, for a short time, past that threshold. In the latter approach, Y is not a constant, and varies with the case at hand, but readers will remark the same happens with Gini (cf. Table 5).

Both logics can be applied to an illustrative set of life tables and the results in terms of concentration are shown in table 5, together with the traditional version of Gini's index. The elaborations suggest that the C measure of concentration of the ABC approach does depend on the assumed upper age span, and is, in this sense, partly arbitrary. However, independently of the assumptions on the maximum possible age span, concentration consistently diminishes as e0 increases, implying less inequality in the distribution of the number of years lived by each individual. When the average length of life increases, people tend to die more and more at similar ages, and rectangularisation in the survival curve occurs.

There is widespread agreement about this conclusion, and only approaches to measurement differ: some researchers use to indicators like entropy (Keyfitz, 1977, pp. 64-66), or Anson's "MLS & U" (Anson, 1992), some other resort to Gini (Wilmoth and Horiuchi 1999). However, a closer look at Table 5 suggests that Gini, which is relatively close to 0 even in situations of high mortality, may bias the picture, not only at any given point in time, but also in dynamic terms. Consider, for instance the evolution from e0=35.1 (in 1880) to e0=81.8 (in 1999), with G decreasing from 37.6% to 7.9%. In absolute terms this is about 30 percentage points less, but in relative terms the decline in G approaches 80%. For the alternative indexes C' and C", absolute declines are higher (-45 and -37 points, respectively), but relative declines appear more moderate (between -64% and -44%), and more in line with the general impression conveyed by a look at survival curves (not reported here). In short, the C indexes discussed here (and especially C", in my opinion) may constitute a valid alternative to the synthetic measures of inequality in survival chances that are customarily used.


Table 5. Measures of concentration (G and C) of the number of years lived in a few model life tables (Italy, women, various historical periods)

e 0 35.1 44.9 56.7 62.6 75.1 81.8Years 1880-1884 1905-1909 1930-1934 1945-1949 1970-1974 1995-1999

G (Y=100.000 e 0 ) 37.6% 33.7% 27.8% 23.9% 11.6% 7.9%

C' (if Y =122) 70.8% 62.6% 52.0% 45.6% 29.3% 25.5%

C" (if Y =variable) 84.2% 81.2% 74.5% 69.0% 53.3% 47.2%variable Y 87.7 87.6 90.6 92.3 95.3 99.5

Life expectancy at birth (e0) and period

Note. Y (customarily indicated with ω, in demographic literature) is the highest possible age at death. With Gini, this is the sum of the years lived in the community, or N e0, where e0 is the average length of life, and N the number of individuals (here, 106). With C', Y is fixed at 122, the highest age at death ever reliably recorded. With C", Y is variable (reported in italics in the table), and calculated in such a way that only 2.5% of deaths occur after that age. Source: Human Mortality Database (http://www.mortality.org/, as of November 2004).

5. Continuous distributions

The ABC approach works equally well, and as simply, when the distribution of the quantity owned by each (properly ranked) owner is defined as a function y=f(x) [y'≥0], where x ranges between 0 and 1, and both y and x are continuous. In this case, the three basic measures of the ABC approach become

Y

dxyA ∫

=10 , with [ ]1;0∈x (6)

dxy

dxxyB

∫

∫= 1

0

10 again, with [ ]1;0∈x (7)

( ) AB

AB

BBC

−−

=−−

−=

−−

=1

125.021

5.05.0

5.0

max (8)

In this case, too, the last one simplifies to C=2B-1 when Y→∞, and A→0.

http://www.mortality.org/


Figure 7 presents a few simple functions, where all the synthetic measures refer to the case of proportions, with Y=1: for example, for y=x/2, A=1/4, B=2/3, and C=4/9 (panel a), while for y=√x, A=3/2; B=3/5, and C=3/5 (panel b)6.

It is worth noticing that the distribution y=x2, in panel c, produces the same

ABC measures as this

≥=<=

5.0,325.0,0.0

xifyxify

(panel d), that is A=1/3, B=3/4, and

C=3/4. In other words, distributions with the same ABC indexes (amount, barycentre and concentration) do not necessarily coincide, and may even differ sharply in some respects.

Fig. 7. Theoretical (continuous) functions representing distributions of y on x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.07

0.13 0.2

0.27

0.33 0.4

0.47

0.53 0.6

0.67

0.73 0.8

0.87

0.93 1

(c) y=x2

A=1/3, B=3/4, C=3/4.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.07

0.13 0.2

0.27

0.33 0.4

0.47

0.53 0.6

0.67

0.73 0.8

0.87

0.93 1

(d) y=0 se x<0.5; y=2/3 se x=0.5.A=1/3, B=3/4, C=3/4.

(a) y=xA=1/4, B=2/3, C=4/9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

(b) y=vxA=2/3, B=3/5, C=3/5.

6. Decomposition of ABC measures

Finally, the ABC approach makes it easier to decompose the concentration measure C into elementary components than traditional Gini (cf. e.g., Bhattacharya and Mahalanobis, 1967; Dagum, 1997; or Costa, 2004), because the whole process can be reduced to a succession of movements of the barycentre as conditions change, one by one.

6 The equality B=C occurs by definition when the distribution is of the type y=xz (see

appendix).


Figure 8 provides an example where a quantitative variable y (e.g. income) is distributed among possessors x, who can be classified in a certain number of subgroups (two, in this case)7. What we are confronted with is the general distribution of Figure 8 (panel 5), and we want to understand what caused the barycentre to move from its ideal starting point of 0.5 (thick arrow of panel 1) to its actual position (.694; thick arrow of panel 5). In order to do this, we first need to calculate income averages: one for the whole sample (5.2, in this example) and one for each subgroup (which turn out to be 4 and 6, respectively, for groups α and β, in our case). Then, the reasoning goes as follows:

Fig. 8. Graphical representation of the decomposition of ABC (movement of the barycentre)

0

2

4

6

8

10

12

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5

Group A Group B

0

2

4

6

8

10

12

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5

Group A Group B

0

2

4

6

8

10

12

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5

Group A Group B

0

2

4

6

8

10

12

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5

Group A Group B

0

2

4

6

8

10

12

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5

Group A Group B1. General average

2. Group average

3. Actual distribution for Group A; average for B 4. Actual

distribution within groups

5. Actual distribution

7 Tables and calculations are shown in the appendix.


1. in the ideal case of perfect equity, income distribution would be as shown in panel 1, with everybody earning the average income of 5.2. The barycentre would be at B0=0.5, by definition;

2. the second step is to imagine perfect equity within groups, ordered from the poorest to the richest (left to right). This would cause the barycentre to shift, from B0=0.5 to B1 (=.546, in this case - panel 2);

3. next we consider each group, in turn, and we let the distribution within that group become what it actually is, noticing that the result does not depend on the order with which we proceed. The barycentre moves from B1 to B2α (.573, in this case - panel 3), B2 β (.640 - panel 4), and so on, for as many groups as necessary;

4. finally, we consider the further rightward movement of B caused by the unconstrained ranking of individuals, which depends on the fact the subgroup distributions overlap, at least partially. We thus get to point B3 (.694, in this case - panel 5)

In summary, the general movement of B = B3-B0 (.194, in this case) can be

broken down into specific component: 1. between groups = B1-B0 (0.046, in this case, or 24% of the total) 2. within groups = B3-B1 (0.148, in this case, or 76% of the total), which can be

further detailed into: • within group α = B2α-B1 (0.027, in this case - 14% of the total) • within group β = B2β-B2α (0.067, in this case - 35% of the total) • overlapping of the series = B3-B2β (0.054, in this case - 28% of

the total) It is easy to prove that, for each component, the relative contribution to the

movement of B corresponds exactly to its relative contribution to the overall value of C, which can thus be decomposed into its elementary parts.

7. An application to Italy

7.1 Data

For an illustrative empirical application of the ABC measures, and in particular of the decomposition of section 6, let us consider Italy, using data the data of the SHIW, i.e. the Survey on Household Income and Wealth, carried out every other year by the Bank of Italy. The survey is well known, and I will therefore skip its description here: interested readers are referred to, for instance, Biancotti, D’Alessio, and Neri (2004).

I will use the (weighted) data of the two rounds of 2000 and 2002 and focus solely on income: personal income and household income, with two different


specification of the equivalence scale: simple per capita (each household member weighs 1), and OECD modified scale, where the first adult in the household is counted as 1, all subsequent adults as 0.7, and all children (up to 13 years) as 0.3.8 I will consider how income varies between 5-year age classes and genders, or, better, how concentrated it is, and how these two dimensions contribute to the total measure of concentration. In practice I work with data like those of Figure 9.

Fig. 9. Distribution of personal income by gender and 5-year age classes (Italy, 2002)

0

5

10

15

20

25

0 15 30 45 60 75 90 5 20 35 50 65 80 95Age classes, separately for women and men

Pers

onal

Y (E

uros

, 000

)

Women Men

Source: Own elaborations on SHIW data.

Note, however, that the representation of Figure 9 is incorrect because, by

giving each bar the same thickness, it conveys the wrong impression that there is the same number of persons in each age class. This is not true, obviously, but both Gini's index and the ABC measures discussed here duly keep frequencies into account, and are not biased in this respect. However, the organization of the data by cells (gender and 5-year age classes) introduces the implicit assumption that income is distributed equitably within cells, which is patently false. One could consider more detailed age classes, but this would magnify the randomness implicit in sample data, would make calculations more burdensome, and, most importantly, would not solve the problem, because income would still not be distributed equitably within cells. Readers must therefore be aware that the concentration measures of this analysis depend on the criteria used for forming groups.

8 In Italy, the semi-official equivalence scale is the one proposed by Carbonaro, increasing

with the household dimension with a constant elasticity of 0.7, i.e. [100, 167, 223, ...]. This lies somewhere in between the two scales considered in the text, and so do the results, not presented here.


7.3 Results

In order to simplify the exposition, it is probably better to follow the passages of Figure 8, except that we now have two classification criteria: gender and age. Let us start with net personal income, in 2002, which turns out to be slightly over 10 thousand Euros per capita, on average. With perfect equity in distribution, the barycentre would be B0=0.5, signalling absence of concentration (G=C=0; cf. equation 8). As a matter of fact, the barycentre falls, instead, at about B3=.7037, giving G=C=.407.

Table 6, below, decomposes this shift of the barycentre (from B0=0.5 to B3=.7037) into three parts. One, amounting to 42% of the total, is attributable to inequity between genders, because men earn more than women. Another 38% is attributable to inequity between age classes, because earnings are generally higher in the central stages of a man's life (although not so much for women; see again Figure 9), lower at old ages, and virtually zero before 20 years. Income inequality by age is much higher among men (27%) than among women (11%). Finally, there is a residual (19%), which depends on the overlapping of the two distributions.

Table 6. B and C measures for personal income, and their decomposition by gender and 5-year age classes (Italy, 2002)

B values Absolute % ExplanationB 0 0.5000B 1 0.5864 0.0864 42% Between gendersB 2a 0.6095 0.0231 11% Between age classes, women onlyB 2b 0.6642 0.0547 27% Between age classes, men onlyB 3 0.7037 0.0395 19% Overlapping

B 0-B 3 0.2037 100%

C 0.407

Delta B

Note: Calculations run on more decimal digits than presented here: rounding operates. Source: Own elaborations on SHIW data. The same calculations may be attempted on different types of income. Here, I selected household income, in two versions: simple per capita, i.e. dividing household income by the household size N, and adjusted per capita, i.e. dividing household income by the household equivalence scale ES (OECD edition). The latter should ideally take economies of scale into account, although the debate on the very existence and, in all cases, the correct values of an equivalence scale is still open (cf., e.g., De Santis, 2004). Finally, the same measures can be applied to different years: 2000, in our case - although the monetary values have been


inflated, so as to be expressed in their 2002 equivalent. The results of all this are in Table 7.

Table 7. B and C measures for personal and household income, and their decomposition by gender and 5-year age classes (Italy, 2000 and 2002). Monetary values in Euros 2002.

Sources of concentration Personal Hhld/N Hhld/ES Personal Hhld/N Hhld/ES

Between genders 44% 7% 18% 42% 4% 16%Between ages (women) 11% 25% 25% 11% 26% 25%Between ages (men) 27% 25% 23% 27% 24% 24%Overlapping 18% 44% 33% 19% 47% 35%

B3 0.707 0.547 0.529 0.704 0.547 0.529C (=Gini) 41.4% 9.4% 5.9% 40.7% 9.3% 5.7%

Av. income (Euros/year) 15 262 15 55010 155 10 380

year 2000 / type of income year 2002 / type of income

Note: Calculations run on more decimal digits than presented here: rounding operates. N=household dimension. ES=OECD modified equivalence scale. Source: Own elaborations on SHIW data.

In short, what emerges from this analysis is that: 1. virtually nothing changes from 2000 to 2002 with respect to the variables

considered here; 2. living in households is a very powerful vehicle of income redistribution

across ages and genders: Gini's index drops from about 41% to about 9%, even assuming absence of economies of scale, and it drops further to about 6% with the (strong) economies of scale implicit in the OECD equivalence scale;

3. living in households leads to a redistribution of income, but this happens more effectively across genders than across ages: inter-age variation contributes to about 50% of the total concentration measure with all types of income;

4. living in households with economies of scale lowers costs and corresponds to an increase in real income. This increase could be as high as 50% if OECD equivalence scale were correct.

Appendix

A.1. Details on the calculation of ABC

Table A.1 gives the details for calculating the ABC indexes on the data of table 1. Column (3) gives the total income within that group (not very useful, here, since we are considering individual data). Each rectangle in Figure 3 has the same base


(p=0.1), and a height hi=yi/Y, where yi is the average amount possessed by each individual in that class, and Y is the highest possible amount possessed by the richest individual (here Y=40, the total income in the sample). This gives the area Ai (pi hi)

of each rectangle (column 7), each with its own barycentre 2

1* ++= ii

iPPP

(column 6). The sum of all of the areas gives the total (relative) area A (=ΣAi), which is 0.100 in this example. The weighted average of the various barycentres

gives the general barycentre *iP

∑∑=

i

iiA

APB

*, equalling 0.7 in this example.

Finally, the value of B must be interpreted in relative terms, as the relative distance from the minimum (0.5) and the maximum (1-A/2). Since, in this case, the maximum is 0.95, C=.444.

Table A.1. Calculations of the ABC indexes on the example of table 1

Owners Average Total Base Height Barycentre Area(1) (2) (3) (4) (5) (6) (7) (8)x i y i Y i =x i y i p h i =y i /Y P i * A i = p i h i A i P i *1 0 0 0.100 0.000 0.050 0.000 0.00001 0 0 0.100 0.000 0.150 0.000 0.00001 2 2 0.100 0.050 0.250 0.005 0.00131 2 2 0.100 0.050 0.350 0.005 0.00181 4 4 0.100 0.100 0.450 0.010 0.00451 4 4 0.100 0.100 0.550 0.010 0.00551 6 6 0.100 0.150 0.650 0.015 0.00981 6 6 0.100 0.150 0.750 0.015 0.01131 8 8 0.100 0.200 0.850 0.0200 0.01701 8 8 0.100 0.200 0.950 0.0200 0.019010 4.0 40 0.100 0.070

RectanglesIncome

Note. Y (=∑Yi), is the highest possible amount possessed by the richest individual (here Y=40).

A.2 Proof that C coincides with Gini's index (R , or G)

We saw before that B, abscissa of the barycentre, is

∑

∑∑

∑ ==py

pyPA

APB

** (A2.1)

with Ai=area of each rectangle; pi=its base; hi=yi/Y=its height, or ratio of the average quantity owned to the maximum. Normalizing this abscissa with respect to its minimum and maximum gives the concentration index C


5.0

5.0

max −−

=B

BC (A2.2)

Now, assuming, for the sake of simplicity, Bmax=1, eq. (A2.2) transforms into

1212*

−=−=∑

∑py

pyPBC (A2.3)

Expressing all the variables in (A2.3) in terms of cumulative proportions P (for owners) and Q (for the owned variable), one gets

( )( )

−−

=

−=

+=

NPPYQQy

PPp

PPP

''

'2'*

(A2.4)

where P'=Pi+1; Q'=Qi+1, and N represents the total number of observations. Using (A2.4), and keeping in mind that Σ(Q'-Q)=1, equation (A2.3) may be re-

written as

( )( ) ( )( )

( ) ( )( )

( )( )

( )( ) 1''

1'

''1

'''

''''

21

2

−−+=

=−−

−+=−

−−

−

−−

−+=

∑

∑∑

∑

∑

QQPP

QQQQPP

NPPYQQPP

NPPYQQPPPP

C (A2.5)

Besides

( )( )

( ) ∑∑∑∑∑∑∑

−+=−+−=

=−+

QPPQQPPQPQQPQQPP

''1''''''

(A2.6)

because the only differences between the series (P'Q') and (PQ) are the terms (PN QN) in the first and (P0 Q0) in the second summation, and they are 1 and 0, respectively. In short, the concentration index C becomes

RGQPPQC ==−= ∑∑ '' (A2.7)

(Cf. equation 2 of the text)

A.3 Subpopulations, and the decomposition of ABC measures

Sometimes, the population can be partitioned into Z subgroups (1, 2, ..., z, ..., Z), and one may be interested in measuring the contribution of each subgroup to the overall value of B and C. In this case, the first thing to do is to identify the


individual theoretical maximum Y with respect to the entire distribution. Then calculations of the A and indexes B can proceed as usual, that is

Y

yphpAA z i

zizi

z izizi

z izi

∑∑∑∑∑∑ === (A3.1)

∑∑

∑∑=

z izi

z izizi

A

APB

*

(A3.2)

The only difference is that the procedure can now be applied repeatedly, changing the values of ygi at each round, preferably in the following order:

1. yyzi = , which leads to B=0.5, and serves as a starting point;

2. zzi yy = , and this, by difference with passage (1), measures the effect on B and C of an ideal situation in which there is no intra-group variability: all the individuals of group g possess the same (g-)average amount of y. Notice that, in doing this, we are forced to order groups in an increasing order of zy , with the poorest on the left;

3. ygi takes its actual value, but individuals must be ordered only within subgroups, while the succession of groups is the same as in passage (2), from the poorest to the richest9. This passage serves to measure the within-component of G, because it relaxes the assumption of perfect intra-group equity. It may be done for all subgroups at once, or one by one (as in section 6 of the text), so as to measure the within-component of each group. In all cases, results do not depend on the order with which one proceeds;

4. finally, we let each individual (or class) have its proper amount of y and we rank individuals (or classes) in the correct order, from the poorest to the richest. The difference with the final stage of passage (3) gives the effect on B and C of the overlapping of the various distributions.

This procedure may be made slightly speedier by simply calculating the value of B (instead of B and C) at each passage. Since

9 In some cases, this will lead to individuals of different groups not being properly ordered:

i.e. a relatively rich individual belonging to a relatively poor group will appear to the left of some relatively poor individuals who belong to a relatively rich group, as in panel 4 of Figure 8.


A

BC−

−=

112 (A3.3)

and A is a constant for each given distribution (by definition, since we use either actual or average values of y), C becomes a linear transformation of B, so that whatever causes B to vary by x% of its total variation will also cause C to vary by the same proportion.

A.4 Proof that B=C, in model distributions, when y=xz

Remembering that A

B−

−=

112C , the quantity B-C becomes

AABBCB

−−−

=−1

1 (A4.1)

which equals 0 for B+AB=1. Now, if y=xz, then 1

1+

== ∫ zyA and

21

++

==∫∫

zz

yxy

B , because we evaluate the integer between 0 and 1. Therefore, we

get

121

11

21

=++

++

++

=+zz

zzzABB Q.E.D. (A4.2)

Bibliography

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Costa M. (2004) "Notes on the Gini Index Decomposition", Atti della 42^ Riunione Scientifica della SIS, (Bari, 9-11 giugno), Padova, Cleup, pp. 187-190.

Cerioli A., Zani S. (1990) "A fuzzy approach to the measurement of poverty", in Dagum C., Zenga M. (eds) Income and wealth distribution, inequality and poverty. Springer-Verlag, Berlin-Heidelberg, pp 272-284.

Cheli, B., Lemmi, A. (1995), "A totally fuzzy and relative approach to the multidimensional analysis of poverty", Economic Notes, 24, pp 115-134.


Dagum C. (1997) "A new approach to the decomposition of the Gini income inequality ratio", Empirical Economics, 22, pp. 515-531.

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function: computation from discrete data, decomposition of differences and empirical examples", Demographic Research, 8, art. 11, pp. 305-357 (www.demographic-research.org).

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