Not all inequality measures were created equal : the measurement of wealth inequality, its decompositions, and an application to European household wealth.
Date: 2019-09-01
Institutions: European Central Bank
Subjects: economic indicator, household income, social inequality, wealth

Rita Neves Costa, Sébastien Pérez-Duarte 

Statistics Paper Series 

Not all inequality measures were 
created equal 

The measurement of wealth inequality, its 
decompositions, and an application to European 
household wealth 

No 31 / December 2019 

Disclaimer: This paper should not be reported as representing the views of the European Central Bank (ECB). 
The views expressed are those of the authors and do not necessarily reflect those of the ECB. 

 
Household Finance and Consumption Network (HFCN) 

This paper contains research conducted within the Household Finance and Consumption Network (HFCN). The HFCN consists of survey 
specialists, statisticians and economists from the ECB, the national central banks of the Eurosystem and a number of national statistical 
institutes. 

The HFCN is chaired by Ioannis Ganoulis (ECB) and Oreste Tristani (ECB). Michael Haliassos (Goethe University Frankfurt), Tullio 
Jappelli (University of Naples Federico II) and Arthur Kennickell act as external consultants, and Juha Honkkila (ECB) and Jiri Slacalek 
(ECB) as Secretaries. 

The HFCN collects household-level data on households’ finances and consumption in the euro area through a harmonised survey. The 
HFCN aims at studying in depth the micro-level structural information on euro area households’ assets and liabilities. The objectives of the 
network are: 

1) understanding economic behaviour of individual households, developments in aggregate variables and the interactions between 
the two; 
2) evaluating the impact of shocks, policies and institutional changes on household portfolios and other variables; 
3) understanding the implications of heterogeneity for aggregate variables; 
4) estimating choices of different households and their reaction to economic shocks; 
5) building and calibrating realistic economic models incorporating heterogeneous agents; 
6) gaining insights into issues such as monetary policy transmission and financial stability. 

The refereeing process of this paper has been co-ordinated by a team composed of Pirmin Fessler (Oesterreichische Nationalbank), 
Michael Haliassos (Goethe University Frankfurt), Tullio Jappelli (University of Naples Federico II), Juha Honkkila (ECB), Jiri Slacalek 
(ECB), Federica Teppa (De Nederlandsche Bank) and Philip Vermeulen (ECB). 

The paper is released in order to make the results of HFCN research generally available, in preliminary form, to encourage comments and 
suggestions prior to final publication. The views expressed in the paper are the author’s own and do not necessarily reflect those of the 
ESCB. 

ECB Statistics Paper Series No 31 / December 2019 1 
 
 
 
 
 
Contents 

Abstract 

Non-technical summary 

1 

2 

3 

4 

Introduction 

The measurement of inequality 

2.1  Main properties of inequality measures 

2.2 

Indicators selected to measure inequality 

2.3  Other measures 

2.4  Relationships between the measures 

2.5  Computations in the case of negative values 

The measurement of changes in inequality 

3.1  Atkinson’s comparison criterion 

3.2  Decomposing changes into Pigou-Dalton transfers 

Decompositions of inequality measures 

4.1  Notation 

4.2  Decomposition of Generalised Entropy indicators by population 

sub-groups 

4.3  Decomposition of the Gini coefficient by population sub-groups 

4.4  Non-overlapping population sub-groups given by the intersection 

of Lorenz curves and their link to the Gini index decomposition 

4.5  Contributions of wealth components: the case of the Gini 

coefficient 

4.6  Shorrocks’ systematic approach to decompositions by variable 

components 

5 

Applications to European household wealth 

5.1 

The Household Finance and Consumption Survey 

5.2  Household wealth in the euro area: general facts 

5.3  Evolution of wealth inequality in the euro area between 2010 and 

2014: investigation into the reported modest increase 

4 

5 

7 

10 

10 

11 

15 

16 

17 

20 

20 

21 

25 

25 

26 

27 

28 

29 

31 

33 

33 

34 

35 

ECB Statistics Paper Series No 31 / December 2019 25.4  Differences in inequality trends by indicator: contributions to total 

change from Pigou-Dalton transfers 

5.5  Sub-group dynamics: analysis using the difference between 

Lorenz curves 

5.6  Contributions from wealth components 

6 

Conclusion 

References 

Appendix 

Expressions for inequality indicators 

Decomposition of inequality changes by Pigou-Dalton transfers 

Expressions for decompositions of inequality indicators 

Complementing charts 

38 

40 

43 

47 

49 

52 

52 

52 

54 

56 

ECB Statistics Paper Series No 31 / December 2019 3Abstract 

Much of the literature on inequality, both that on the theoretical features of inequality 
measurement and that on the discussion of the results of empirical analysis, has 
preferred to focus on income inequality. This paper looks into the analysis of wealth 
inequality, which can be performed by carefully adapting the techniques used in the 
case of income distributions. The paper focuses on the measurement of inequality 
itself and includes an application to European data on wealth. We summarise the main 
inequality measures used in the economic literature, expanding the focus to lesser 
known but relevant ones, grounding their use in socio-economic theory and 
highlighting the connections between them. In particular, we investigate how each 
measure captures the same movement in the wealth distribution and why different 
measures can lead to differences in the observed change in inequality over time or 
across countries. In the main theoretical contribution of the paper we obtain a novel 
decomposition of changes in inequality measures as a set of equalising and 
disequalising factors, which sheds some light on the different results across indicators. 
We complement the analysis by focusing on the decomposition of wealth inequality 
measures, gaining an understanding of the contributions of inequality by wealth 
component and socio-demographic characteristics. The distribution of wealth of 
European households obtained by the Household Finance and Consumption Survey 
(HFCS) in 2010 and 2014 is used for empirical analysis and application of our 
methods. 

JEL codes: D31, D63 

Keywords: wealth, inequality, Atkinson, Lorenz curve, Gini coefficient, Pietra index, 
decomposition 

ECB Statistics Paper Series No 31 / December 2019 4Non-technical summary 

The growing interest in inequality, along with the widespread perception that inequality 
has been increasing in developed countries, merits a detailed discussion of how it is 
measured. In this vein, there are a number of considerations to take into account, 
namely i) the different kinds of inequality – wealth, income, well-being, 
multidimensional (a combination of these factors) or another concept; ii) whether an 
inequality analysis should be performed on a set of indicators or by means of a 
comparison of the distributions themselves; and iii) the specific indicators that could 
be used. 

Much of the literature on inequality, both that on the theoretical features of inequality 
measurement and that on the discussion of the results of empirical analysis, has 
preferred to focus on income inequality. However, in this paper we are interested in 
the analysis of wealth inequality, which can be performed by carefully adapting the 
techniques used in the measurement of income inequality. As a practical application of 
our methods, we use the results of the second wave of the Household Finance and 
Consumption Survey (HFCS), published in December 2016, to understand the 
changes in the distribution of wealth in the euro area and to address the apparent 
issue that arises as a result: that inequality increases for most measures but 
decreases for some. 

Our first building block focuses on the definition of inequality measures, which starts 
with the reasonable properties that such measures should have. We then present 
indicators from different branches of the literature: Atkinson’s social welfare-based 
approach, which led to the Atkinson index; the Generalised Entropy indices, 
including Theil’s measure; the Lorenz curve; the well-known Gini coefficient and 
the less well-known Pietra index. These different measures behave differently when 
there are negative values of wealth – which is not a concern in the analysis of income 
inequality – so we study which measures can be used for wealth distributions and 
under what conditions. We also show how the different families of measures are 
related, also touching briefly on the popular “top percent shares” and quantile ratios 
measures used in other strands of inequality research. 

Our second building block concerns the measurement of changes in inequality and, 
specifically, Atkinson’s absolute comparison criterion, which allows two distributions to 
be compared without reference to a specific inequality measure. We apply Atkinson’s 
idea of an absolute criterion to the analysis of elementary transfers that always 
decrease inequality. This results in a novel decomposition of the change in inequality 
that takes a series of redistributive (“equalising”) or anti-redistributive (“disequalising”) 
moves. These two types of move lead to a unique decomposition of the changes in the 
Gini coefficient, which also carries over to the other inequality measures. This allows 
us to view a change between two distributions as the combination of transfers – some 
increasing the equality and others increasing the inequality. We tie this decomposition 
to the intersection of the Lorenz curves and show how the two are linked, with each 

ECB Statistics Paper Series No 31 / December 2019 5segment between two intersection points corresponding to a strictly equalising or 
disequalising change. 

Our third and final building block is to bring together the other decompositions of 
inequality measures proposed in the literature and highlight how they can be used 
based on the inequality measure and the type of decomposition (for example, by 
wealth component or by population sub-group). The Generalised Entropy indicators 
can be uniquely decomposed by population sub-groups into between and within 
components. The Gini coefficient can only be decomposed into these between and 
within components when the population sub-groups are ranked by wealth level and 
are non-overlapping on this variable; otherwise there will be a residual term. As 
regards wealth components, general results make it possible to decompose the 
inequality measures through marginal effects. From this we can see how the change 
in the value of a wealth component affects inequality. 

These different building blocks are then applied to the results of the Household 
Finance and Consumption Survey, which collected granular information on the wealth 
of households in most euro area countries in 2010 and 2014. In the intervening period 
wealth inequality was found to have increased by a positive but modest amount for 
most inequality measures at the euro area level. There was, however, variation across 
countries and the discrepancies between measures were larger at the country level 
than at the euro area level. In these countries the presence of redistributive and 
anti-redistributive factors becomes apparent in our decomposition and we link the 
differences in the amount of change to the inequality measures’ varying levels of 
sensitivity to the different parts of the distribution and to the presence of different 
sub-groups in the changes in the distribution of wealth. To investigate further, we 
estimate the sensitivity of the different measures to various wealth components and 
find that an increase in the value of the household’s main residence or safe financial 
assets tends to reduce inequality, while an increase in the value of liabilities or 
self-employment businesses increases inequality. 

ECB Statistics Paper Series No 31 / December 2019 61 

Introduction 

The purpose of inequality analysis is to compare the distribution of an underlying 
resource – in its changes over time, across countries or groups, or both – or to provide 
a measure of inequality of a distribution, with possibly a specific but not always explicit 
concept of “equality”. The pursuit of both goals, following rigorous logical reasoning 
that starts from accepted requirements with socio-economic justification, has been 
described in a large literature; we refer to the seminal contributions of Dalton (1920) 
and Atkinson (1970) among many others, and point to the recently updated volume by 
Cowell (2011). It has nevertheless left many practitioners confused about the proper 
indicators to use, their properties and their interpretation, as evidenced by several 
papers that attempt to explain these elements (Jurkatis and Strehl, 2013; Kimhi, 
2011). 

The default choice in many papers on inequality has been the Gini coefficient. 
Although the Gini coefficient has many good properties, it does not lend itself well to 
proper decompositions over time or across groups and it may not be as sensitive to 
changes in the distribution as one could wish. A lot of work has also been carried out 
recently on the “top x% share” of income or wealth but the intrinsic properties of this 
indicator are rarely discussed. An extreme approach – and one we find unfortunate –is 
to reject an inequality measure either on (misplaced) theoretical grounds or because 
of practical difficulties in obtaining an accurate measurement. 

We propose returning to the definition of the concept of inequality proposed by 
Atkinson (1970) and transposing it to a mathematical indicator or ranking criterion that 
will measure the level of inequality of a distribution or allow us to compare two 
distributions along the dimension of inequality. The level of inequality can be 
understood, on the one hand, as the distance to equality: in the case of a wealth 
distribution, measuring inequality is choosing a way of measuring the distance from 
the observed distribution to the distribution in the case of complete equality, i.e. the 
distribution when all households have the same wealth. On the other hand, comparing 
the inequality of two distributions may not always produce a clear answer, so we will 
determine the conditions under which it is possible to provide an answer. This is 
particularly the case when evaluating the changes in inequality over time, where 
different conclusions can be reached depending on the measure used; when this 
occurs it is important to determine why. 

First it is necessary to review the main techniques for inequality analysis that have 
been proposed in the literature and study the similarities and differences between 
them. This will allow a better understanding of the varied results that may be obtained 
when measuring inequality using the different methods. 

This investigation on the measurement of inequality requires the selected indicators, 
including Gini, Pietra, Atkinson, Theil and the class of Generalised Entropy indices, to 
be defined. These indicators can be linked to the Lorenz curve, which we will use to 
understand and compare distributions. 

ECB Statistics Paper Series No 31 / December 2019 7Of particular interest is the analysis of changes in inequality over time, which answers 
questions such as “why did inequality decrease?” or “who benefited the most from the 
change in the distribution of wealth?” One major difficulty is that the different inequality 
indicators will not show changes of the same sign or magnitude, and the answers to 
the questions will depend upon the specific indicator used. By returning to Atkinson’s 
(1970) absolute principle, it is possible to devise a novel technique for decomposing 
the changes over time into equalising and disequalising components, i.e. those that 
contribute to decreasing inequality and those that contribute to increasing it. Again, the 
components will be linked to the Lorenz curves of the two distributions by way of their 
intersection points. This approach clarifies the differing results that arise between 
indicators owing to their particular sensitivities to transfers along the wealth 
distribution. Using the inequality indicators it is also possible to apply decompositions 
and related techniques to assess the contributions to total inequality of population 
sub-groups or components that make up the analysed variable. 

Our main testing ground for these measures will be the Household Finance and 
Consumption Survey (HFCS). The HFCS – an initiative of the ECB and the 
Eurosystem national central banks – provides data on the distribution of wealth of euro 
area households in two different periods, the first around 2010 and the second around 
2014, and is therefore an excellent source for analysing recent trends in wealth 
inequality in a large set of European countries. 

The results of the second wave of the survey, which were published in December 
2016, reported a modest increase in wealth inequality in the euro area between 2010 
and 2014. Several recent contributions (Lenza and Slacalek, 2018; Ampudia et al., 
2018) either investigate the impact of monetary policy in general, and quantitative 
easing, in particular, on household inequality in the euro area, or investigate the 
drivers of wealth inequality in the short and in the long term (e.g. Lieberknecht and 
Vermeulen, 2018). Our approach here is different, and clearly in the camp of 
measurement theory, as we focus on the measurement of inequality in itself, and not 
on the possible driving factors behind it. A careful application of several techniques 
introduced in the first chapters of this paper will be used to investigate the changes in 
equality between 2010 and 2014 and highlight the different situations across 
countries. 

The paper is organised as follows: Chapter 2 is devoted to the problem of measuring 
inequality. It starts by listing the basic requirements that an inequality measure should 
satisfy then explains the processes for obtaining different indicators, giving definitions 
and providing indications for interpretation. Other measurement attempts, 
relationships between the measures, and procedures for dealing with negative values 
of the variable are also discussed. Chapter 3 presents different techniques for 
measuring changes in inequality over time and introduces the main theoretical 
development of the paper: a method to decompose the differences between two 
distributions into equalising and disequalising components. Chapter 4 presents and 
discusses different decompositions of the inequality indicators and related techniques 
for assessing the contributions to inequality of population sub-groups or wealth 
components. These include the property of additive decomposability, the 
computations of marginal effects for some indicators, and the decomposition of the 

ECB Statistics Paper Series No 31 / December 2019 8change in time of the Gini coefficient. The relation between intersections of Lorenz 
curves and decompositions by population sub-groups is also investigated. Finally, in 
Chapter 5 we illustrate some of the techniques of inequality analysis by using HFCS 
data on the wealth distribution of euro area households. In particular, we investigate 
the reported modest increase in wealth inequality in the euro area between 2010 and 
2014 by inspecting the results in each country and using different inequality analysis 
techniques. 

ECB Statistics Paper Series No 31 / December 2019 92 

The measurement of inequality 

Several measures of inequality have been introduced in the literature, mostly in the 
context of income distributions. This section presents several indicators and describes 
their use as tools for measuring the inequality of wealth distributions. 

In this section, we first list the four properties that an inequality indicator should have. 
Then, we describe three different approaches for obtaining inequality measures: 
i) using social welfare notions; ii) looking for a convenient mathematical structure; and
iii) building on the Lorenz curve. These lead to the definitions of Atkinson (1970),
Generalised Entropy (including the Theil index, 1967), Gini (1914) and Pietra (1915)
indicators. In the third part we introduce other widely used measures in the analysis of
inequality, including the “top shares”, or shares of wealth owned by specific groups at
the top of the distribution, and the ratios of percentiles. In the fourth part we present
the relationships between all the inequality measures, before detailing how to deal
with negative and zero values of the distribution for each of the inequality measures.

2.1 

Main properties of inequality measures 

The main properties that appear in the literature as basic requirements for an 
inequality indicator are as follows.1 

Anonymity: the indicator remains unchanged if the individuals of the 
distribution are permuted. In other words, all individuals are interchangeable and all 
that matters is their wealth, not their name. 

Population principle (Dalton, 1920): the indicator remains unchanged if the 
distribution is replicated a finite number of times. 

Principle of transfers (Pigou, 1912; Dalton, 1920): if a transfer of value
made from an individual with wealth 

  to another with wealth 

is 

  such that 
 𝑡𝑡 > 0 

, the indicator decreases. In other words, a transfer that takes from the 

𝑦𝑦𝑖𝑖
rich and gives to the poor, but the rich remain richer than the poor, can only cause 
𝑦𝑦𝑖𝑖 − 𝑡𝑡 > 𝑦𝑦𝑗𝑗 + 𝑡𝑡
inequality to decrease. This transfer is known as a Pigou-Dalton transfer. This 
principle could be termed “strict”, while the “relaxed” version only requires that the 
indicator not increase in the presence of a positive Pigou-Dalton transfer. 

𝑦𝑦𝑗𝑗

Scale invariance: the indicator remains unchanged if the distribution is scaled 
by a constant factor. In other words, there is no monetary illusion in the 
measurement of inequality.2 

These four properties ensure that the indicators behave in a reasonable manner. In 
particular, scale invariance means that the units used to measure wealth can be 

1    Definitions adapted from Cowell (1998) and Shorrocks (1980). 
2    This can be contrasted with the principle of additive invariance, which is used in some of the literature. 

ECB Statistics Paper Series No 31 / December 2019 10disregarded. However, these properties are not sufficient to single out a measure or a 
family of measures, and additional properties may be assumed in order to fix the 
uniqueness of the measure (see Section 2.2.2). Finally, indicators that do not 
necessarily respect these principles, or that respect other principles, have been 
proposed. We will briefly touch upon these in the sections below. 

2.2 

Indicators selected to measure inequality 

Different approaches can be followed when selecting specific indices from among 
those displaying the properties introduced above. We detail three approaches in the 
following subsections. First, we present the welfare approach described by 
Atkinson (1970), which is based on the use of a social welfare function and leads to 
the definition of the family of eponymous inequality measures. Second, we consider 
the requirement that the indicator possess the property of decomposability. This 
approach results in the definition of the Generalised Entropy family of indices, which 
includes Theil’s entropy measure. Finally, we investigate the Lorenz curve of the 
distribution, which is a natural way of obtaining the Gini and Pietra indicators. 

2.2.1 

Atkinson’s measures and the social welfare approach 

Let 

  be a distribution of wealth. The equally distributed equivalent level of wealth, 
  is the level of wealth per household which, if equally distributed, would give the 
𝐹𝐹
same level of social welfare as the distribution 
𝜉𝜉(𝐹𝐹)
where 
function. Let 

  is a social welfare function given by 

  denote the mean of distribution 

, where 
𝑊𝑊�𝜉𝜉(𝐹𝐹)� = 𝑊𝑊(𝐹𝐹)

. It is such that 

  is a utility 

𝐹𝐹

. 

𝑊𝑊 = ∫ 𝑢𝑢(𝑦𝑦)𝑑𝑑𝐹𝐹(𝑦𝑦)

𝑢𝑢

The Atkinson index (Atkinson, 1970) is, in the most general sense, defined as the 
normalised ratio of the equally distributed equivalent level of wealth to the mean of the 
actual wealth distribution. 

𝐹𝐹

𝑊𝑊

𝜇𝜇(𝐹𝐹)

Atk

= 1 −

𝜉𝜉(𝐹𝐹)
𝜇𝜇(𝐹𝐹)

It depends on the choice of the social welfare function. To ensure that the properties 
introduced in Section 2.1 are displayed, the choice of the form of the utility function is 
limited. The family of Atkinson indices referred to in the literature is obtained by taking 

the utility function 

1−𝜀𝜀

𝑦𝑦

−1

  when 

  and 

  when 

. The

parameter 

  is interpreted as accounting for the level of aversion to inequality. 

𝜀𝜀 ≠ 1

𝑢𝑢(𝑦𝑦) = ln 𝑦𝑦

𝑢𝑢(𝑦𝑦) =

1−𝜀𝜀

𝜀𝜀 = 1

𝜀𝜀 > 0

In the case of an empirical distribution with 
of household 

  denotes the wealth 
  elements, where 
  the sample average, the Atkinson indices can be expressed as: 

  and 

𝑛𝑛

𝑦𝑦𝑖𝑖

Atk

𝑖𝑖

𝑦𝑦�

𝑛𝑛

1
1−𝜀𝜀 if

(𝜀𝜀) = 1 −

1
𝑦𝑦�

�

1
𝑛𝑛

1−𝜀𝜀

� 𝑦𝑦𝑖𝑖
𝑖𝑖=1

�

 𝜀𝜀 ≠ 1

ECB Statistics Paper Series No 31 / December 2019 11 
 
 
Atk

𝑛𝑛

(1) = 1 −

1
𝑦𝑦�

�� 𝑦𝑦𝑖𝑖
𝑖𝑖=1

1
𝑛𝑛 

�

The Atkinson index lies between zero and one and increases with inequality. It is equal 
to zero in the case of complete equality, i.e. when all individuals have the same 
(positive) wealth, and tends to one in the case of complete inequality, i.e. when one 
individual has all the wealth and all others have nothing at all. 

The parameter 
welfare function. A low 
more weight is attached to transfers at the lower end of the distribution and less weight 
𝜀𝜀
to transfers at the top and thus the level of inequality-aversion increases. 

  accounts for the level of aversion to inequality in the social 
  rises, 

  implies a low degree of inequality-aversion. As 

𝜀𝜀 > 0

𝜀𝜀

In general, Atkinson indices can only be computed for positive values of the variable. 
, they can also be computed for zero and negative values of 
For some values of 

the variable. 

𝜀𝜀 > 0

The value of the Atkinson index can be interpreted in terms of the equally distributed 
equivalent described above. For example, an Atkinson index equal to 0.7 means that, 
if wealth was equally distributed, the same level of social welfare could be achieved 
with only 30% of the actual total wealth. 

The indicators that make up the family of Atkinson indices respect the four properties 
presented in Section 2.1: anonymity, the population principle, the principle of 
transfers, and scale invariance. 

2.2.2 

Theil’s measure and the Generalised Entropy indices 

An inequality indicator is said to display the property of additive decomposability, 
defined by Shorrocks (1984), if it can be decomposed by population sub-groups and 
expressed as a weighted sum of a within-group and a between-group component. 

The Generalised Entropy (GE) measures constitute the only family of indicators (up 
to a transformation) that display additive decomposability as well as anonymity, the 
population principle, the principle of transfers, and scale invariance 3. The GE 
measures depend on a parameter 
different parts of the distribution. The special cases of 
𝛼𝛼
the Theil index and the mean log deviation, respectively. The case of 
𝛼𝛼 = 0
equal to half the squared coefficient of variation. 

  that expresses the sensitivity of the indicator to 
  are known as 

𝛼𝛼 = 1

  and 

  is 

𝛼𝛼 = 2

In the case of an empirical distribution with 
of household 

  elements where 
  the sample average, the Generalised Entropy indices can be 

  denotes the wealth 

  and 

𝑛𝑛

𝑦𝑦𝑖𝑖

expressed as: 
𝑖𝑖

𝑦𝑦�

3    More precisely, a continuous inequality measure respecting anonymity and the population principle 
satisfies the principle of transfers, scale invariance and decomposability if and only if it is ordinally 
equivalent to a Generalised Entropy indicator for some parameter alpha. See Cowell (2000) for details. 

ECB Statistics Paper Series No 31 / December 2019 12Indicator 

Formula 

Generalised Entropy (

) 

α ∉ {0,1}

Theil (GE(

)) 

α = 1

Mean log deviation (GE(

)) 

α = 0

GE

(𝛼𝛼) =

GE

𝑛𝑛

1
𝛼𝛼(𝛼𝛼 − 1)

1
𝑛𝑛

� ��
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

𝛼𝛼

�

− 1�

𝑛𝑛

1
𝑛𝑛

�
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

𝑛𝑛

ln �

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

(1) =

GE

(0) = −

1
𝑛𝑛

� ln �
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

The Generalised Entropy indicators are equal to zero in the case of complete equality, 
i.e. when all individuals have the same wealth. A larger value of the index indicates 
larger inequality in the distribution. The GE is unbounded when considered for a 
theoretical distribution but is bounded in the case of a finite independent and 
identically distributed sample (for example, in the case of the Theil index, the bound is 

, where 

  is the sample size, and it corresponds to the case of complete 

inequality). 
ln 𝑛𝑛

𝑛𝑛

The Generalised Entropy indices can be computed for all values of the variable when 
  is a positive integer. They can only be computed for non-negative values of the 

variable in certain cases, depending on the parameter 
𝛼𝛼 ≠ 1

  chosen. 

The special case of Theil’s measure (GE
consequence of Shannon’s information theory. Theil’s measure is given by the 
difference between the maximum possible entropy and the observed entropy of the 
wealth distribution. Indeed, the entropy of the distribution of net wealth shares, where 
the vector 
  represents the wealth share of all observed households 

) was introduced by Theil (1967) as a 

(𝛼𝛼 = 1)

𝛼𝛼

  and where each household has the same weight, is given by 

𝑧𝑧 = (𝑧𝑧1, 𝑧𝑧2, … , 𝑧𝑧𝑛𝑛)
𝑛𝑛
𝑖𝑖=1

𝑧𝑧𝑖𝑖 ≔ 𝑦𝑦𝑖𝑖/ ∑ 𝑦𝑦𝑖𝑖

𝑛𝑛

1
𝑧𝑧𝑖𝑖� 

𝑖𝑖=1

ln �

𝐻𝐻(𝑧𝑧) = � 𝑧𝑧𝑖𝑖
The maximum possible entropy corresponds to complete equality in the distribution, 
i.e. where all households have the same (positive) wealth. In this case 
  and 

thus 
has all the wealth and all others have nothing at all, we have 

. In the opposite case, complete inequality, i.e. when one household 

𝑧𝑧𝑖𝑖 =

1
𝑛𝑛 ∀𝑖𝑖
  and 

𝐻𝐻(𝑧𝑧) = ln 𝑛𝑛

  and thus 

. 

∃𝑖𝑖 𝑠𝑠. 𝑡𝑡. 𝑧𝑧𝑖𝑖 = 1

𝑧𝑧𝑗𝑗 = 0 ∀𝑗𝑗 ≠ 𝑖𝑖
Theil’s measure is, then, the difference between the maximum possible entropy and 
the observed entropy of the system: 

𝐻𝐻(𝑧𝑧) 𝑧𝑧→0�⎯� 0

Theil

GE

𝑛𝑛

=

(𝛼𝛼 = 1) = � 𝑧𝑧𝑖𝑖

𝑖𝑖=1

ln(𝑛𝑛𝑧𝑧𝑖𝑖) =

𝑛𝑛

1
𝑛𝑛

�
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

ln �

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

2.2.3 

Gini’s and Pietra’s measures and Lorenz curves 

The Lorenz curve of a distribution of wealth plots the proportion of total wealth 
belonging to the bottom 

  of the population. The Lorenz curve is increasing, convex 

𝑥𝑥%

ECB Statistics Paper Series No 31 / December 2019 13 
 
 
 
 
 
and lies in the first quadrant of the Cartesian plane in the case of positive values of the 
variable. It is equal to the 45-degree line in the case of complete equality, i.e. it 
represents the distribution that would arise if all the households had the same wealth. 
In all the other cases it lies below the 45-degree line. Intuitively, the closer the Lorenz 
curve is to the 45-degree line, the more equal the distribution is. 

More formally, in the case of a distribution of wealth 
functional is defined as 

, for 

  the quantile 
  and the cumulative wealth 

functional is defined as 

is the normalised cumulative wealth functional and plots the quantiles 

  against 

  where 

  and 

  is the mean of the distribution 

𝐹𝐹
𝑄𝑄(𝐹𝐹, 𝑞𝑞) ≔ inf{𝑦𝑦|𝐹𝐹(𝑦𝑦) ≥ 𝑞𝑞} ≔ 𝑦𝑦𝑞𝑞
𝐶𝐶(𝐹𝐹, 𝑞𝑞) ≔ ∫

𝑄𝑄(𝐹𝐹,𝑞𝑞)
−∞

𝑦𝑦 𝑑𝑑𝐹𝐹(𝑦𝑦)

0 ≤ 𝑞𝑞 ≤ 1

. The Lorenz curve of the distribution 

𝑞𝑞

. 

𝐹𝐹

𝐿𝐿(𝐹𝐹, 𝑞𝑞)
Figure 1 
Lorenz curve of a distribution 

𝐿𝐿(𝐹𝐹, 𝑞𝑞) = 𝐶𝐶(𝐹𝐹, 𝑞𝑞)/𝜇𝜇(𝐹𝐹)

𝜇𝜇(𝐹𝐹)

h
t
l
a
e
w

l

a
t
o
t

f
o
e
r
a
h
s

Lorenz curve 

share of population 

The Lorenz curve gives an intuition for the definition of both the Gini indicator and the 
Pietra indicator. The Gini coefficient (Gini, 1912) corresponds to the normalised area 
between the Lorenz curve of the distribution and the 45-degree line: 

1

𝐺𝐺 = 1 − 2 � 𝐿𝐿(𝐹𝐹, 𝑞𝑞)𝑑𝑑𝑞𝑞
0
The Gini coefficient can be expressed in several equivalent forms, such as the 
normalised average absolute difference between all pairs of wealth in the population 
or the covariance formula: 

. 

2
𝑦𝑦� 𝑐𝑐𝑐𝑐𝑐𝑐�𝑦𝑦, 𝐹𝐹(𝑦𝑦)�

𝐺𝐺 =

In the case of an empirical distribution with 
household 
observed households and 

, the vector

𝑖𝑖 ∈ {1, … , 𝑛𝑛}

, we define 
𝑦𝑦�

𝑖𝑖

𝑛𝑛

𝑋𝑋𝑖𝑖 =

𝑦𝑦1 ≤ 𝑦𝑦2 ≤ ⋯ ≤ 𝑦𝑦𝑛𝑛

  elements, 

  denotes the wealth of 

  represents the wealth of all 

  is the sample average. In the case where 

𝑛𝑛
 𝑦𝑦 = (𝑦𝑦1, 𝑦𝑦2, … , 𝑦𝑦𝑛𝑛)
  as the cumulative share of population and 

  is such that 

𝑦𝑦𝑖𝑖

𝑦𝑦

ECB Statistics Paper Series No 31 / December 2019 14 
 
 
 
 
 
 
 
  as the cumulative share of wealth. The Gini coefficient can then be 

𝑖𝑖
∑
𝑗𝑗=1
𝑛𝑛
𝑌𝑌𝑖𝑖 =
expressed as: 
𝑗𝑗=1

𝑦𝑦𝑗𝑗

𝑦𝑦𝑗𝑗

∑

𝑛𝑛−1

𝑖𝑖=0

𝑛𝑛
2 ∑ 𝑖𝑖 𝑦𝑦𝑖𝑖
𝑖𝑖=1
𝐺𝐺 = 1 − �(𝑋𝑋𝑖𝑖+1 − 𝑋𝑋𝑖𝑖)(𝑌𝑌𝑖𝑖+1 + 𝑌𝑌𝑖𝑖)
𝑛𝑛
𝑖𝑖=1 −
𝑛𝑛 ∑ 𝑦𝑦𝑖𝑖
The Pietra index (Pietra, 1915), also known as the Ricci index (Ricci, 1916), the 
Schutz index or the Hoover index, is calculated as the mean relative absolute 
deviation to the mean, or as the maximum vertical distance between the Lorenz curve 
of the distribution and the 45-degree line. 

𝑛𝑛 + 1
𝑛𝑛

=

The Pietra index has a simple interpretation; it is the proportion of total wealth that 
would have to be redistributed in order to achieve complete equality of the wealth 
distribution, i.e. when all individuals have the same wealth. 

In the case of an empirical distribution with 
of household 

  and 

  the sample average, the Pietra index can be expressed as: 

𝑛𝑛

𝑦𝑦𝑖𝑖

  elements where 

  denotes the wealth 

𝑖𝑖

𝑦𝑦�

𝑛𝑛

1
2𝑛𝑛

|𝑦𝑦𝑖𝑖 − 𝑦𝑦�|
𝑦𝑦�

�
𝑖𝑖=1

𝑃𝑃 =
Both the Gini and the Pietra indices are equal to zero in the case of complete equality, 
i.e. when all individuals have the same (positive) wealth. When calculated for 
non-negative values of the variable, both indicators tend to one in the case of 
complete inequality, i.e. when one individual has all the wealth and all others have 
nothing at all. The Gini and Pietra measures can be calculated for all values of the 
variable but in that case are not bounded above. A larger value of the measures 
indicates larger inequality in the distribution. 

The Gini index respects anonymity, the population principle, the principle of transfers 
and scale invariance. The Pietra index respects anonymity, the population principle 
and scale invariance but it does not respect the (strict) Pigou-Dalton principle of 
transfers. This can easily be seen by taking the definition of the indicator as the 
maximum vertical distance between the Lorenz curve and the 45-degree line. A 
Pigou-Dalton transfer can be made by modifying only one part of the Lorenz curve, 
which does not affect the point of the maximum vertical distance. This reduces the 
area between the Lorenz curve and the 45-degree line and, consequently, the value of 
the Gini coefficient diminishes without the Pietra index being affected. The Pietra 
index would, however, respect a weaker version of the Pigou-Dalton principle of 
transfers, that is, a non-strict decrease. 

2.3 

Other measures 

The indicators presented in the previous sections, namely the Atkinson and 
Generalised Entropy families and the Gini coefficient, have the main properties that, 
according to the literature, a useful inequality measure should possess (the Pietra 
index does not respect the Pigou-Dalton principle of transfers in the strict sense). 

ECB Statistics Paper Series No 31 / December 2019 15 
 
There are, however, other indicators that are widely used in the analysis of wealth 
inequality; the most popular include the shares of wealth owned by specific groups of 
households, such as the wealthiest 5%, and the ratios of quantiles of the distribution. 

The top wealth share indicators (popularised by Piketty and co-authors mainly for 
income distributions) are defined as the share of total wealth owned by the households 
that are above the corresponding percentile of the wealth distribution. For example, 
the top 10% share gives the proportion of total wealth owned by the households that 
lie to the right of the 90th percentile. Top shares are bounded below by the share itself 
and would be bounded above by 100% in the case of non-negative values of net 
wealth. A higher value of the top share indicator indicates higher concentration of 
wealth. 

The ratios of quantiles are defined as the ratio of the corresponding percentiles of 
the distribution. A P90/P50 value of four (4) means that the household at the 90th 
percentile of the distribution is four times wealthier than the household at median 
wealth. In the case of a distribution of wealth, the lower percentiles can be negative or 
very close to zero. This directly affects the denominator of the percentile ratios and 
undermines the interpretation of the value of the indicator. The ratios of quantiles are 
bounded below by 1 (in the case of complete equality, i.e. when all the households 
have the same wealth, all quantile ratios are equal to 1). Higher values of the quantile 
ratios indicate a wider gap in net wealth quantiles. 

The top wealth shares and quantile ratio indicators respect anonymity, the population 
principle and scale invariance but they do not respect the (strict) Pigou-Dalton 
principle of transfers. For this reason, they are seldom considered as inequality 
measures in the literature. For example, the top 10% indicator equal to 50% means 
that half of total wealth is owned by the 90% poorest households and the other half by 
the 10% wealthiest but no information is given on the distribution of wealth within each 
of the two groups. If half of the wealth was equally distributed among the 90% poorest 
or if the household at the 90th percentile alone owned half of total wealth, the value of 
the top 10% indicator would remain the same. 

2.4 

Relationships between the measures 

The Generalised Entropy and Atkinson indicators were presented independently in the 
previous sections but they are in fact formally related. The Atkinson indicators can be 
transformed into GE indicators via a strictly increasing transformation, given by the 
following expression: 

GE

�1 −

(1 − 𝜀𝜀) =

Atk

1−𝜀𝜀

(𝜀𝜀)�
𝜀𝜀(𝜀𝜀 − 1)

− 1

This relationship implies that the Atkinson index with parameter 
corresponding GE index with parameter 
distributions.4  However, this transformation does not preserve the additive 

ε > 0

  induce the same ranking of 

  and the 

𝛼𝛼 = 1 − ε < 1

4   

In particular, 

. 

GE(0) = − ln �1-Atk(1)�

ECB Statistics Paper Series No 31 / December 2019 16 
decomposability property of the GE indices. The Atkinson indices do, however, 
respect a weaker property of decomposability (Shorrocks, 1984). 

Figure 2 
Relationships between indicators and Lorenz curve 

The other measures presented in this paper (Gini, Pietra, top shares, ratios of 
percentiles) can all be linked to the Lorenz curve of the distribution. 

The Gini corresponds to the normalised area between the Lorenz curve of the 
distribution and the 45-degree line,  Gini

. 

The Pietra index gives the maximum vertical distance between the Lorenz curve and 
the 45-degree line,  Pietra
, with the distance obtained at the point 

1
0
= 1 − 2 ∫ 𝐿𝐿(𝐹𝐹, 𝑞𝑞)𝑑𝑑𝑞𝑞

in the Lorenz curve that corresponds to the household with mean wealth. 
= max𝑞𝑞{𝑞𝑞 − 𝐿𝐿(𝐹𝐹, 𝑞𝑞)}

The share of wealth owned by the top x% is given by the distance from the Lorenz 
curve at point 
. 
𝑞𝑞 = 1 − 𝑥𝑥

  to the horizontal line at ordinate = 1,  top 

𝑥𝑥% = 1 −

𝐿𝐿�𝐹𝐹, 𝑞𝑞(1 − 𝑥𝑥)�
The ratio of percentiles, 
points 

, 

, 

𝑝𝑝1
𝑝𝑝2 =

d𝐿𝐿(𝑝𝑝1)

d𝑞𝑞 /

𝑝𝑝1/𝑝𝑝2
d𝐿𝐿(𝑝𝑝2)
d𝑞𝑞  

𝑝𝑝1

𝑝𝑝2

, is the ratio of slopes of tangents to the Lorenz curve at 

when the derivatives exist. 

Besides the relationship with the Lorenz curve, the Gini coefficient can be shown to 
relate to the average of top shares, 
Gini converges to 

; as the size of the population goes to infinity, the 

. 

𝑡𝑡̅

2𝑡𝑡̅ − 1

2.5 

Computations in the case of negative values 

Some of the Generalised Entropy and Atkinson indices are not properly defined for 
negative and zero values of the variable. The Lorenz curve of the distribution is also 
affected by the presence of negative and zero values of the variable. Indeed, in this 
case, as the lower percentiles of the distribution are negative or zero, the Lorenz curve 

ECB Statistics Paper Series No 31 / December 2019 17 
 
will lie below zero and will be strictly decreasing for the first percentiles of the wealth 
distribution. 

Nevertheless, the Gini coefficient and the other measures related to the Lorenz curve 
can be computed taking into account all values of the variable5. They will be affected in 
some way, for example in the boundaries of the possible values assumed by the 
indicators. For ratios of percentiles with a denominator that is negative or very close to 
zero, the value of the ratio is severely affected. The shares of total wealth owned by 
the bottom x% of the distribution will also be negative or zero in certain cases. 

Different ad hoc procedures can be applied when relevant for analysis, such as 
deleting observations with non-positive values, replacing non-positive values with €1 
or replacing negative values with €0. These are, however, outside the scope of this 
paper. The detailed restrictions on the computations along with the possible 
procedures applied, depending on the choice of parameter, are provided in the tables 
below for the Generalised Entropy and the Atkinson indicators. 

Table 1 
Generalised Entropy – variable restrictions 

Parameter 

Restriction on the variable 

α ∈ ℕ, α ≠ 1, α ≠ 0
  or 

α = 0

α = 1

−α ∈ ℕ

α ∈ ℚ\ℤ, α > 0

In other cases, 

α ∈ ℚ\ℤ, α < 0

α ∈ ℝ\ℚ 

Table 2 
Atkinson – variable restrictions 

Parameter 

ε = 1

ε ∈ ℕ, ε ≠ 1

ε ∈ ℚ\ℕ, 0 < ε < 1

ε ∈ ℚ\ℕ, ε > 1

No restriction, 

x ∈ ℝ

x > 0

x ≠ 0

x ≥ 0

x > 0

x > 0

Restriction on the variable 

x > 0

x ≠ 0

x ≥ 0

x > 0

These procedures for dealing with non-positive values of net wealth have different 
effects on the overall distribution. Taking the example of the second wave of the 
HFCS, we can see that net wealth is negative for around 5% of euro area households, 
with important differences across countries (the percentages range from 1% in Malta 
to 14% in the Netherlands in 2014)6. The value of the indicator as a measure of the 
inequality of the entire distribution is thus affected differently in each country. 
However, no important differences arise in the measurement of the change in time of 

5    The only exceptions are the ratios of percentiles that cannot be computed when the percentile in the 

denominator of the ratio is equal to zero. 

6    According to the second wave of the Household Finance and Consumption Survey. 

ECB Statistics Paper Series No 31 / December 2019 18 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
the indicator provided that the share of households with negative or zero wealth is 
similar in both cases. 

ECB Statistics Paper Series No 31 / December 2019 193 

The measurement of changes in 
inequality 

The previous section discussed different measures of the inequality of a distribution 
and, more specifically, a wealth distribution. However, the analysis of wealth inequality 
does not only cover the levels of inequality of a distribution, as given by the value of 
one of these indicators, it also involves a comparison of several wealth distributions. 
When looking at comparisons of distributions referring to the same population at 
different moments in time, an analysis of the changes can provide interesting results. 

In the first part of this section we introduce the criterion established by Atkinson to 
compare two distributions according to inequality using the Lorenz curves of the 
distributions. In the second part we propose a method for analysing the changes in the 
distribution by only analysing Pigou-Dalton transfers between convenient 
transformations of the analysed distributions. The different impact each transfer has 
on each specific inequality indicator is then assessed. 

3.1 

Atkinson’s comparison criterion 

Instead of using a measure of inequality like the indicators presented above, it would 
be useful to have a way of comparing two given distributions without having to assign 
a numeric value to each considered distribution. The comparison criterion proposed 
by Atkinson (1970) follows a reasoning similar to the deduction of the eponymous 
indices presented in Section 2.2.1. The indicators depend on a social welfare function 
relying on a specific form of a utility function. In the case where a wider class of utility 
functions is considered, Atkinson shows that two distributions can be compared by 
considering their Lorenz curves, provided that these curves do not intersect. 

This comparison criterion is absolute in the sense that it does not depend on the 
social welfare utility function (provided that it is increasing and concave). 

The ranking of distributions according to the Atkinson criterion is always possible if 
the Lorenz curves of the distributions do not intersect. Three situations can occur: 
i) the Lorenz curve of distribution A always lies above the Lorenz curve of distribution 
B, meaning distribution A is more equal than distribution B; ii) the Lorenz curve of 
distribution A always lies below the Lorenz curve of distribution B, meaning distribution 
A is more unequal than distribution B; or iii) the Lorenz curves of distributions A and B 
intersect, meaning it is not possible to rank them according to this absolute notion of 
inequality. 

Atkinson further interprets the result, stating that two distributions can be compared 
independently of the utility function (other than being increasing and concave) if and 
only if one distribution can be obtained from the other by redistributing wealth from the 
richer to the poor, that is, exclusively by means of Pigou-Dalton transfers. This also 
implies that two distributions can be compared by Atkinson’s Lorenz curve criterion if 

ECB Statistics Paper Series No 31 / December 2019 20and only if all the inequality indicators adhering to the principle of transfers rank 
them in the same way. These results will be further developed in the next sections. 

3.2 

Decomposing changes into Pigou-Dalton transfers 

The inequality measures differ in their sensitivity to the shape of the wealth distribution 
and these differences may lead to opposite conclusions being drawn in practice on the 
change in wealth inequality over time. Indeed, the latest results of the Household 
Finance and Consumption Survey show that, in some countries, one indicator points 
towards an increase in wealth inequality while another suggests a decrease. 

To understand the origins of these differences, we express the total change between 
any two wealth distributions as a finite sequence of transfers between pairs of 
elements of the distribution, such that each transfer is either a Pigou-Dalton transfer or 
a “reverse”, or “anti-” Pigou-Dalton transfer, that is, either a redistributive (or 
equalising) or an anti-redistributive (or disequalising) factor. These transfers always 
lead to an unequivocal change in inequality in all inequality measures that conform to 
the principle of transfers described above. We will then see how each indicator reacts 
differently to the same transfer, depending on its amount, the ranks of the two 
elements, and their wealth. 

3.2.1 

Measuring the contribution from a Pigou-Dalton transfer 

A Pigou-Dalton transfer is defined as the positive wealth transfer from the richer to the 
poorer of two individuals such that the richer individual is still richer than the poorer 
individual after the transfer has occurred. If distribution 
through a Pigou-Dalton transfer, we say that 
“reverse” or anti-Pigou-Dalton transfer. A Pigou-Dalton transfer always decreases 
inequality according to the (strict) Pigou-Dalton principle of transfers (see Section 2.1) 
and will thus cause the inequality indicator to decrease (or remain unchanged, when 
the non-strict Pigou-Dalton principle of transfers is respected). Similarly, an 
anti-Pigou-Dalton transfer will cause the indicator to increase or remain unchanged. 

  is obtained from 
𝑦𝑦

  is obtained from 

𝑖𝑖
  through a 

𝑖𝑖+1

𝑖𝑖+1

𝑦𝑦

𝑦𝑦

𝑦𝑦

𝑖𝑖

  denote the transfer of amount 
. 
  to individual 

performed on distribution7 
  denotes a Pigou-Dalton transfer when 

  from 

𝑡𝑡 
  and an anti-Pigou-Dalton transfer when 

𝑦𝑦
𝑐𝑐𝑡𝑡

𝑦𝑦𝑗𝑗

�𝑦𝑦𝑘𝑘, 𝑦𝑦𝑗𝑗�

𝑦𝑦

Let 
𝑦𝑦
individual 
𝑐𝑐𝑡𝑡

(𝑦𝑦𝑘𝑘, 𝑦𝑦𝑗𝑗)
𝑦𝑦𝑘𝑘
𝑦𝑦𝑗𝑗 < 𝑦𝑦𝑗𝑗 + 𝑡𝑡 < 𝑦𝑦𝑘𝑘 − 𝑡𝑡 < 𝑦𝑦𝑘𝑘
𝑦𝑦𝑘𝑘 − 𝑡𝑡
Let 

  be obtained from 

. Total wealth is unchanged. 

  by a Pigou-Dalton transfer 

, 

. The effects 

𝑖𝑖+1

𝑖𝑖

of this transfer on the different inequality indicators introduced in the previous sections 
are as follows8: 

�𝑦𝑦𝑘𝑘, 𝑦𝑦𝑗𝑗�

𝑗𝑗 < 𝑘𝑘

𝑦𝑦

𝑦𝑦

𝑖𝑖
𝑐𝑐𝑡𝑡

Gini

Gini

Gini

𝑖𝑖
�𝑐𝑐𝑡𝑡

� ≔

Δ

𝑖𝑖+1

(𝑦𝑦

) −

𝑖𝑖

(𝑦𝑦

) =

2𝑡𝑡(𝑗𝑗 − 𝑘𝑘)

𝑛𝑛∑𝑦𝑦𝑖𝑖 =

are sorted ascendingly, i.e., 

𝑗𝑗 − 𝑘𝑘
𝑛𝑛

2
𝑛𝑛

𝑡𝑡
𝑦𝑦�
. 

7    The individuals 
8    We note the transfer 

𝑦𝑦𝑘𝑘

  for a lighter reading. 

𝑦𝑦𝑘𝑘 ≤ 𝑦𝑦𝑘𝑘+1 ∀𝑘𝑘

(𝑦𝑦𝑘𝑘, 𝑦𝑦𝑗𝑗)

𝑖𝑖
𝑐𝑐𝑡𝑡

𝑖𝑖
= 𝑐𝑐𝑡𝑡

𝑦𝑦𝑗𝑗 + 𝑡𝑡 > 𝑦𝑦𝑗𝑗 > 𝑦𝑦𝑘𝑘 >

ECB Statistics Paper Series No 31 / December 2019 21 
 
GE

Δ

GE

𝑖𝑖
(0)�𝑐𝑐𝑡𝑡

� ≔

GE

GE

GE

𝑖𝑖+1

𝑖𝑖

(0)(𝑦𝑦

) −

(0)(𝑦𝑦

) = −

GE

𝑖𝑖+1

𝑖𝑖

1
𝑛𝑛

�ln �

𝑦𝑦𝑗𝑗 + 𝑡𝑡
𝑦𝑦𝑗𝑗 � + ln �

𝑦𝑦𝑘𝑘 − 𝑡𝑡
𝑦𝑦𝑘𝑘 ��

𝑖𝑖
(1)�𝑐𝑐𝑡𝑡

� ≔

Δ

GE

𝑖𝑖
(𝛼𝛼)�𝑐𝑐𝑡𝑡

� ≔

Δ

ΔPietra

𝑖𝑖
�𝑐𝑐𝑡𝑡

� ≔

(1)(𝑦𝑦
𝑦𝑦𝑗𝑗
1
�
𝑦𝑦�
𝑛𝑛

=
GE

) −

(1)(𝑦𝑦
)
𝑦𝑦𝑘𝑘
𝑦𝑦�

𝑦𝑦𝑗𝑗 + 𝑡𝑡
𝑦𝑦𝑗𝑗 � +
GE

ln �

𝑖𝑖+1

𝑖𝑖

ln �

𝑦𝑦𝑘𝑘 − 𝑡𝑡
𝑦𝑦𝑘𝑘 � +

𝑡𝑡
𝑦𝑦�

ln �

𝑦𝑦𝑗𝑗 + 𝑡𝑡
𝑦𝑦𝑘𝑘 − 𝑡𝑡

��

(𝛼𝛼)(𝑦𝑦
𝛼𝛼

)

(𝛼𝛼)(𝑦𝑦
1
𝛼𝛼(𝛼𝛼 − 1)

) −
𝑦𝑦𝑗𝑗 + 𝑡𝑡
1
��
𝑦𝑦�
𝑛𝑛
Pietra

=
Pietra

�

− �

𝛼𝛼

�

+ �

𝑦𝑦𝑗𝑗
𝑦𝑦�

𝑦𝑦𝑘𝑘 − 𝑡𝑡
𝑦𝑦�

𝛼𝛼

�

− �

𝛼𝛼

�

�

𝑦𝑦𝑘𝑘
𝑦𝑦�

𝑖𝑖+1

) −

(𝑦𝑦
1
2𝑛𝑛

�𝑦𝑦𝑗𝑗 + 𝑡𝑡 − 𝑦𝑦��
�
𝑦𝑦�

𝑖𝑖

(𝑦𝑦

)
�𝑦𝑦𝑗𝑗 − 𝑦𝑦��
𝑦𝑦�

|𝑦𝑦𝑘𝑘 − 𝑡𝑡 − 𝑦𝑦�|
𝑦𝑦�

|𝑦𝑦𝑘𝑘 − 𝑦𝑦�|
𝑦𝑦�

=

−

+

−
For transfers of a fixed amount 
, the impact on the Gini coefficient only depends on 
the distance between the relative positions of the individuals in the distribution, not on 
their absolute wealth. The Gini attaches more weight to transfers between elements of 
the distribution whose rankings are more distant, irrespective of their wealth or their 
absolute rank. 

𝑡𝑡

�

, the Generalised Entropy with parameter 

Again for transfers of a fixed amount 
attaches more weight to transfers at the bottom of the distribution, that is, when the 
𝛼𝛼 = 0
  is large when compared to the wealth of the individuals. The 
amount of the transfer 
effects on  GE
  depend on a combination of factors. The ratio of the wealth of each 
individual concerned to the mean plays a role and, the larger the ratio, the larger the 
impact on the  GE  if 

  is large. 

(1)

𝑡𝑡

𝑡𝑡

𝛼𝛼

The impact on the Pietra index depends on the wealth of the individuals, before and 
after the transfer, in relation to the mean. If the wealth of all individuals concerned is 
smaller or larger than the mean, the Pietra index remains unchanged. In the other 
cases, three situations can occur for a Pigou-Dalton transfer, producing an impact on 
the Pietra index of: 

if

Pietra

Δ

𝑖𝑖
�𝑐𝑐𝑡𝑡

� =

𝑦𝑦𝑗𝑗 − 𝑦𝑦�
𝑛𝑛𝑦𝑦�
𝑡𝑡
−
𝑛𝑛𝑦𝑦�
𝑦𝑦� − 𝑦𝑦𝑘𝑘
𝑛𝑛𝑦𝑦�

⎧
⎪
⎪

⎨
⎪
⎪
⎩

 𝑦𝑦𝑗𝑗 < 𝑦𝑦� < 𝑦𝑦𝑗𝑗 + 𝑡𝑡 < 𝑦𝑦𝑘𝑘 − 𝑡𝑡 < 𝑦𝑦𝑘𝑘,

if 

𝑦𝑦𝑗𝑗 < 𝑦𝑦𝑗𝑗 + 𝑡𝑡 < 𝑦𝑦� < 𝑦𝑦𝑘𝑘 − 𝑡𝑡 < 𝑦𝑦𝑘𝑘,

if

 𝑦𝑦𝑗𝑗 < 𝑦𝑦𝑗𝑗 + 𝑡𝑡 < 𝑦𝑦𝑘𝑘 − 𝑡𝑡 < 𝑦𝑦� < 𝑦𝑦𝑘𝑘,

in all other cases.

0,  

3.2.2 

Obtaining a sequence of Pigou-Dalton transfers 

We now present an algorithm with 
starting from the initial distribution 
distribution 

𝑀𝑀
1
𝑦𝑦

  can be obtained from distribution 

  in the previous step by a 

𝑀𝑀

𝑦𝑦

Pigou-Dalton or an anti-Pigou-Dalton transfer. 

𝑖𝑖+1

𝑦𝑦

𝑖𝑖

𝑦𝑦

  steps for obtaining a sequence of distributions 
  such that 
  and ending in the final distribution 

ECB Statistics Paper Series No 31 / December 2019 22 
 
 
 
 
 
 
First, the two distributions being compared are transformed so that they both have the 
same mean and number of elements. These transformations do not affect the 
inequality of the distributions because our definition of inequality respects scale 
invariance and the population principle. Second, the distributions are sorted in 
ascending order by their wealth, in line with the property of anonymity. 

Each step of the algorithm9  consists of the following actions: 

• 

• 

• 

the difference between the current distribution and the final distributions is 
computed, giving the sorted list of individuals who need to increase their wealth 
and those that need to decrease it; 

the first individual who needs to decrease their wealth gives the most they can 
give (either all they can give or what the other needs to receive) to the first 
individual who needs to increase their wealth; 

the elements of the distribution are then sorted. 

Applying the algorithm starting from distribution 
will result in a sequence of distributions 

. 

  until distribution 

  is reached 

1

𝑦𝑦

𝑀𝑀

𝑦𝑦

𝑆𝑆
The total change in the inequality indicator can then be expressed as a sum of the 
contributions to the change from each step. The set of all the Pigou-Dalton transfers in 

  is denoted 

  and the set of all the anti-Pigou-Dalton transfers is denoted 

. In the case of the Gini coefficient the expressions are as follows: 

𝑃𝑃𝑃𝑃(𝑆𝑆)

Gini

𝑀𝑀−1

Gini

Gini

Gini

𝑀𝑀

(𝑦𝑦

) −

1

(𝑦𝑦

) = � Δ
Gini
𝑖𝑖=1

Gini 

𝑖𝑖
�𝑐𝑐𝑡𝑡

�

= � Δ
∈𝑃𝑃𝑃𝑃(𝑆𝑆)

𝑖𝑖
𝑐𝑐𝑡𝑡

𝑖𝑖
�𝑐𝑐𝑡𝑡

�

+ � Δ
∈𝑎𝑎𝑃𝑃𝑃𝑃(𝑆𝑆)

𝑖𝑖
𝑐𝑐𝑡𝑡

𝑖𝑖
�𝑐𝑐𝑡𝑡

�

= 𝐶𝐶𝑃𝑃𝑃𝑃

+ 𝐶𝐶𝑎𝑎𝑃𝑃𝑃𝑃

The total contributions to the change in the Gini from Pigou-Dalton and 
Gini
anti-Pigou-Dalton transfers are 
  and 
contributions can be equal to zero when there are either no Pigou-Dalton transfers or 
no anti-Pigou-Dalton transfers in the sequence. In this case, the two distributions can 
be compared according to Atkinson’s comparison criterion introduced in Section 3.1 
and one of the distributions will be more equal than the other. 

, respectively. Each of the 

𝐶𝐶𝑎𝑎𝑃𝑃𝑃𝑃

𝐶𝐶𝑃𝑃𝑃𝑃

≤ 0

≥ 0

Gini

More generally, for any inequality indicator 
distributions transforming 

, there exists a sequence 
  by Pigou-Dalton and anti-Pigou-Dalton 

  into 

  of 

transfers such that the change in the indicator can be decomposed as follows: 

𝑆𝑆

1

𝑦𝑦

𝑀𝑀

𝑦𝑦

𝐼𝐼

𝑀𝑀

1

𝐼𝐼
) = 𝐶𝐶𝑃𝑃𝑃𝑃

𝐼𝐼
+ 𝐶𝐶𝑎𝑎𝑃𝑃𝑃𝑃

) − 𝐼𝐼(𝑦𝑦

𝐼𝐼(𝑦𝑦
As seen in Section 3.2.1, the same Pigou-Dalton (similarly, anti-Pigou-Dalton) transfer 
will be measured differently by each inequality indicator, resulting in different values 
for the total contribution from Pigou-Dalton transfers, 
change in inequality will be measured differently by each of the indicators, reflecting 

. Consequently, the total 

𝐼𝐼
𝐶𝐶𝑃𝑃𝑃𝑃

9    The algorithm is detailed further in Appendix section entitled “Decomposition of inequality changes by 

Pigou-Dalton transfers”. 

𝑆𝑆
𝑎𝑎𝑃𝑃𝑃𝑃(𝑆𝑆)
Gini

ECB Statistics Paper Series No 31 / December 2019 23 
the indicators’ sensitivity to the distribution and resulting in changes in opposite 
directions in certain cases. 

Theorem (minimum Pigou-Dalton sequence): the sequence of Pigou-Dalton and 
anti-Pigou-Dalton transfers following from the algorithm minimises the absolute value 
of the total contribution to the Gini coefficient from Pigou-Dalton transfers (

Gini). 

Proof: see the section entitled “Decomposition of inequality changes by Pigou-Dalton 
transfers” in the Appendix. 

𝐶𝐶𝑃𝑃𝑃𝑃

The theorem ensures that the Pigou-Dalton sequence, although not necessarily 
unique, provides a unique decomposition of the total change in the Gini in two terms. 

∎

ECB Statistics Paper Series No 31 / December 2019 24 
4 

Decompositions of inequality measures 

Having presented different inequality measures (in Chapter 2) and shown how the 
changes between distributions can be decomposed (in Chapter 3), in this chapter we 
look at how decomposition techniques can be useful for investigating the contributions 
to total inequality of different sub-groups of the population or of components that add 
up to the analysed variable. The population can be partitioned into relevant 
sub-groups according to, for example, age groups, net wealth quintiles, number of 
household members or countries. In the analysis of net wealth, the disaggregation of 
the variable into different types of real and financial assets can be performed. 

Not all the inequality indicators make it possible to obtain decompositions with 
satisfactory socio-economic interpretations as it is often unclear how the 
decomposition should account for different changes in the population sub-groups or 
variable components. Another main issue is to deal with terms that are not 
unambiguously assigned to a specific sub-group or component and are then qualified 
as residuals or interaction terms. Moreover, in some cases different decompositions 
may be deduced for the same inequality indicator, without a strong justification for 
using one instead of another. Nevertheless, one can find decompositions and related 
techniques that are useful in practice. 

This section is organised as follows: after a brief first section on notation, in 
Section 4.2 we provide a decomposition of the Generalised Entropy indicators by 
population sub-groups, followed in Section 4.3 by different decompositions of the Gini 
coefficient by population sub-groups. In Section 4.4 we propose a useful and novel 
way of partitioning the total population into sub-groups for analysing the changes in 
inequality, motivated by the relationship between the Gini decomposition by 
non-overlapping population sub-groups and the intersections of Lorenz curves. 
Section 4.5 deals with the assessment of the contributions of the components adding 
up to total net wealth. In the case of the Gini coefficient, three different techniques are 
introduced: decomposition by variable components, the elasticity of the wealth 
components in terms of marginal effects, and a decomposition of the changes of the 
indicator over time. We finish in Section 4.6 with a discussion on a systematic 
approach, initiated by Shorrocks, for obtaining decompositions by variable 
components for a class of inequality indicators, complemented with the elasticities of 
other inequality indicators. 

4.1 

Notation 

Let 

  represent the net wealth of household 

. 

  is the sample average. 

𝑦𝑦𝑖𝑖

Population sub-groups: let 
households that can be partitioned into 
that 
  and 

𝑋𝑋

. 

𝐾𝐾
𝑋𝑋 = ⋃ 𝑋𝑋𝑘𝑘
𝑘𝑘=1

𝑛𝑛 = ∑

𝐾𝐾
𝑘𝑘=1

𝑛𝑛𝑘𝑘

𝑦𝑦𝑘𝑘���

  represent the complete population formed by 

𝑖𝑖 ∈ {1, … , 𝑛𝑛}

𝑦𝑦�

  sub-groups of 

  households each, such 

  is the sample average of sub-group 

𝐾𝐾

𝑛𝑛𝑘𝑘

𝑛𝑛
. 

𝑋𝑋𝑘𝑘

ECB Statistics Paper Series No 31 / December 2019 25 
Variable components: the variable net wealth can be expressed as a sum of 
components such that 
  where 
household 
  is the sample average of component 
𝑦𝑦𝑖𝑖𝑖𝑖
distribution of the wealth component 

  is the value of component 

𝑦𝑦𝑖𝑖 = ∑

𝑅𝑅
𝑖𝑖=1

𝑦𝑦𝑖𝑖𝑖𝑖

. 

. 

. 

𝑟𝑟

  for 
𝑅𝑅

  denotes the theoretical 

𝑟𝑟

𝐹𝐹𝑖𝑖

𝑖𝑖

𝑦𝑦𝑖𝑖�

In the theoretical sections that follow, we take the case in which all households have 
the same weight 

. Small adjustments in the computations are needed to take into 

𝑟𝑟

1

account different household weights; these are presented in Appendix section entitled 
“Expressions for decompositions of inequality indicators”. 

𝑛𝑛

4.2 

Decomposition of Generalised Entropy indicators by 
population sub-groups 

The Generalised Entropy indicators have the property of additive decomposability as 
introduced in Section 2.2.2. Therefore they can be decomposed by population 
sub-groups and expressed as a weighted sum of a within-group and a between-group 
component. The within component accounts for the inequality that exists inside each 
group whereas the between component accounts for the inequality across the groups. 
The Generalised Entropy computed over the entire population,  GE
, can then be 

expressed as: 

GE

𝐾𝐾

𝛼𝛼 GE

𝑛𝑛𝑘𝑘
𝛼𝛼(𝑋𝑋) = �
𝛼𝛼(𝑋𝑋𝑘𝑘)  
�
within component
𝑛𝑛
𝑘𝑘=1
���������������

𝑦𝑦�𝑘𝑘
𝑦𝑦�

�

𝛼𝛼(𝑋𝑋)

GE
𝐾𝐾
between component
𝛼𝛼(∪𝑘𝑘=1
𝑋𝑋𝑘𝑘)
���������

+

  is the value of the GE indicator computed for the households 

where  GE
belonging to sub-group 
𝛼𝛼(𝑋𝑋𝑘𝑘)

: 

𝑋𝑋𝑘𝑘
𝑛𝑛𝑘𝑘

GE

𝛼𝛼(𝑋𝑋𝑘𝑘) =

and  GE

1
𝛼𝛼(𝛼𝛼 − 1)

1
𝑛𝑛𝑘𝑘 � ��

𝑖𝑖=1

𝛼𝛼

𝑦𝑦𝑖𝑖
𝑦𝑦�𝑘𝑘�

− 1�

  is the between-group component, given by: 

𝐾𝐾
𝛼𝛼(⋃ 𝑋𝑋𝑘𝑘
𝑘𝑘=1

)

GE

𝐾𝐾

𝛼𝛼

𝑛𝑛𝑘𝑘
𝑛𝑛

𝑦𝑦�𝑘𝑘
𝑦𝑦�

𝐾𝐾
𝛼𝛼(∪𝑘𝑘=1

𝑋𝑋𝑘𝑘) =

1
𝛼𝛼(𝛼𝛼 − 1)

�

��

�
𝑘𝑘=1

− 1�
The within component is calculated as the weighted sum of the value of the indicator in 
  sub-groups. The between-group component is calculated as the value 
each of the 
of the indicator of a distribution with 
  elements, each having as net wealth the mean 
of net wealth in the corresponding group and as weight the population share of the 
corresponding group. 

𝐾𝐾

𝐾𝐾

ECB Statistics Paper Series No 31 / December 2019 26 
 
 
 
4.3 

Decomposition of the Gini coefficient by population 
sub-groups 

Unlike the Generalised Entropy measures, the Gini coefficient cannot, in general, be 
decomposed by population sub-groups as a sum of a within and a between 
component. However, a decomposition into two components, within and between, 
does arise when the population is partitioned into “non-overlapping” groups in terms of 
wealth, i.e. when 
. The expression (adapted from Dagum 

(1997) and Alvaredo (2011)) is as follows: 

min𝑦𝑦𝑖𝑖(𝑋𝑋𝑘𝑘) > max𝑦𝑦𝑖𝑖(𝑋𝑋𝑘𝑘−1) ∀𝑘𝑘

Gini

within

between

𝐾𝐾

𝐾𝐾

𝑘𝑘−1

=

+

= � 𝐺𝐺𝑘𝑘 𝑛𝑛𝑘𝑘

𝑛𝑛𝑘𝑘𝑦𝑦𝑘𝑘���
within component + � � 𝐺𝐺𝑘𝑘ℎ �
𝑛𝑛𝑦𝑦�
𝑛𝑛
𝑘𝑘=1
���������

�
+
between component
𝑘𝑘=1
���������������������

𝑛𝑛ℎ𝑦𝑦ℎ���
𝑛𝑛𝑦𝑦�

𝑛𝑛𝑘𝑘𝑦𝑦𝑘𝑘���
𝑛𝑛𝑦𝑦�

𝑛𝑛ℎ
𝑛𝑛

𝑛𝑛𝑘𝑘
𝑛𝑛

ℎ=1

The within component is the weighted sum of the Gini coefficients of the sub-groups, 
, and the between component is the weighted sum of the factors accounting for the 

relationships between each pair of sub-groups, 
𝐺𝐺𝑘𝑘
sub-groups. 

, taking 

  and sorted 

𝐺𝐺𝑘𝑘ℎ

𝑘𝑘 > ℎ

𝐺𝐺𝑘𝑘 =

𝑛𝑛𝑘𝑘
𝑖𝑖=1

𝑛𝑛𝑘𝑘
𝑗𝑗=1

𝑦𝑦𝑘𝑘���

∑ ∑ �𝑦𝑦𝑖𝑖𝑘𝑘 − 𝑦𝑦𝑗𝑗𝑘𝑘�
2
2𝑛𝑛𝑘𝑘
𝑛𝑛ℎ
𝑗𝑗=1

∑ ∑ �𝑦𝑦𝑖𝑖𝑘𝑘 − 𝑦𝑦𝑗𝑗ℎ�
𝑛𝑛𝑘𝑘𝑛𝑛ℎ(𝑦𝑦𝑘𝑘��� + 𝑦𝑦ℎ���)

𝑛𝑛𝑘𝑘
𝑖𝑖=1

𝑦𝑦𝑘𝑘��� − 𝑦𝑦ℎ���
𝑦𝑦𝑘𝑘��� + 𝑦𝑦ℎ���

=

𝐺𝐺𝑘𝑘ℎ =
In general, in any population sub-group decomposition, a term that can be qualified as 
a residual will arise. It is argued to account mostly for interactions between the 
sub-groups but its allocation is ambiguous. Different decompositions of the Gini 
indicator by population sub-groups have been proposed and discussed in the 
literature, including decompositions by Bhattacharya and Mahalanobis (1967) or 
Dagum (1997). The decomposition proposed by Dagum writes the Gini as a sum of 
three factors, as follows: 

+ 𝐺𝐺𝑛𝑛𝑛𝑛 + 𝐺𝐺𝑡𝑡
�����
𝑛𝑛𝑏𝑏𝑡𝑡𝑤𝑤𝑏𝑏𝑏𝑏𝑛𝑛

𝐺𝐺 = 𝐺𝐺𝑤𝑤
�𝑤𝑤𝑖𝑖𝑡𝑡ℎ𝑖𝑖𝑛𝑛
Dagum (1997) gives a socio-economic interpretation to each of the three factors in 
terms of their contributions to total inequality.10  The term 
  is equal to zero in the 
case of non-overlapping population sub-groups and, in that case, this decomposition 
is equivalent to the decomposition into two terms presented at the beginning of this 
subsection. 

𝐺𝐺𝑡𝑡

Another decomposition of the Gini coefficient by population sub-groups has been 
proposed by Podder (1993). It is similar to the decomposition by variable components 
of Lerman and Yitzhaki (1985) and expresses the Gini as a sum of 
corresponding to the 

  sub-groups that add up to the complete population: 

  components 

𝐾𝐾

𝐾𝐾

10    The details are given in Dagum (1997). The three factors can be named as follows: 

component; 
contribution of the income intensity of transvariation between sub-populations. 

, net contribution of the extended Gini inequality between sub-populations; 

𝐺𝐺𝑤𝑤

, within 
, 

𝐺𝐺𝑡𝑡

𝐺𝐺𝑛𝑛𝑛𝑛

ECB Statistics Paper Series No 31 / December 2019 27 
 
 
 
𝐾𝐾

𝐺𝐺 = �
𝑘𝑘=1
where 

𝑛𝑛𝑘𝑘𝑦𝑦𝑘𝑘���
𝑛𝑛𝑦𝑦�
  is the concentration coefficient of the 

𝐶𝐶𝑘𝑘

-th population sub-group vector 

. 

𝐶𝐶𝑘𝑘

The elements of the sub-group are 

  and the concentration coefficient is 
𝑥𝑥𝑘𝑘
𝑘𝑘
computed as the Gini coefficient of the curve that plots the cumulative proportions of 
𝑥𝑥𝑖𝑖𝑘𝑘 = 𝑦𝑦𝑖𝑖𝟙𝟙{𝑖𝑖∈𝑋𝑋𝑘𝑘}
vector 

  against the cumulative proportions of the total population ordered in 

𝑥𝑥𝑘𝑘

ascending order according to their wealth. The concentration coefficient provides an 
indication of the relative position of the wealth distribution of the specific sub-group 
when compared to the wealth distribution of the complete population. 

4.4 

Non-overlapping population sub-groups given by the 
intersection of Lorenz curves and their link to the Gini 
index decomposition 

According to Atkinson’s comparison criterion introduced in Section 3.1, distribution 
can be considered more unequal (respectively, equal) than distribution 
if the Lorenz curve of 
distribution 

1
  if and only 
y
  always lies below (respectively, above) the Lorenz curve of 

  (excluding of course the points (0,0) and (1,1)). 

𝑦𝑦

2

1

2

𝑦𝑦

𝑦𝑦

In all other cases, the Lorenz curves of the distribution intersect at least once. These 
intersections define non-overlapping population sub-groups and the intersection 
points correspond to the individuals for which the cumulative difference between the 
distributions is zero. We name these Lorenz curve groups (LCG). 

Figure 3 
Lorenz curve groups 

l

a

t

o

t

f

o

e
r
a
h
S

h

t
l

a
e
w

Lorenz Curve 
Group 2 

Lorenz Curve 
Group 1 

Share of 
population 

ECB Statistics Paper Series No 31 / December 2019 28 
 
 
 
 
 
 
 
 
Indeed, the Lorenz curve of the distribution 

𝑖𝑖
∑
𝑗𝑗=1
𝑛𝑛
  to be sorted, to have 
Without loss of generality, we consider distributions 
𝑗𝑗=1
the same means and the same number of elements. We write the difference between 
two distributions 
cumulative difference as 

  and define the 

  where 

1
𝐿𝐿𝐶𝐶𝑖𝑖

  and 

1
𝑦𝑦𝑗𝑗
1
𝑦𝑦𝑗𝑗

=

𝑦𝑦

𝑦𝑦

𝑦𝑦

. 

∑

1

1

2

  can be written as 

. 

1
𝑑𝑑𝑖𝑖 = 𝑥𝑥𝑖𝑖

2
− 𝑥𝑥𝑖𝑖

 ∀𝑖𝑖 ∈ {1, … , 𝑛𝑛}

𝑑𝑑 = (𝑑𝑑1, … , 𝑑𝑑𝑛𝑛)

𝑖𝑖
𝑃𝑃𝑖𝑖 = ∑ 𝑑𝑑𝑗𝑗
𝑗𝑗=1

It follows that the two Lorenz curves intersect when the cumulative difference 
equal to zero: 

  is 

𝑃𝑃𝑖𝑖

𝑖𝑖

𝑖𝑖

𝑖𝑖
1
𝑗𝑗=1
∑ 𝑥𝑥𝑗𝑗
𝑛𝑛
1
𝑗𝑗=1 =
∑ 𝑥𝑥𝑗𝑗

𝑖𝑖
2
𝑗𝑗=1
∑ 𝑥𝑥𝑗𝑗
𝑛𝑛
2
𝑗𝑗=1
∑ 𝑥𝑥𝑗𝑗

⟺  

� = 0

2
− 𝑥𝑥𝑗𝑗

2
= 𝐿𝐿𝐶𝐶𝑖𝑖

⟺ � 𝑑𝑑𝑗𝑗
𝑗𝑗=1

= 0  ⟺   𝑃𝑃𝑖𝑖 = 0

1
  ⟺   ��𝑥𝑥𝑗𝑗
𝑗𝑗=1

1
𝐿𝐿𝐶𝐶𝑖𝑖
The LCG sub-groups – defined by taking all the individuals in between those for which 
the cumulative difference is equal to zero – have the same total wealth and the same 
number of elements and thus the same means. As seen in the expressions introduced 
in the previous sections, the between components of the decompositions by 
population sub-groups of both the Gini and the Generalised Entropy indicators only 
depend on the means and number of elements of the sub-groups; therefore, they will 
remain unchanged when transforming 
that distribution 
1
of the LCG sub-groups. Moreover, each sub-group can be taken separately, 
producing a situation in which Atkinson’s comparison criterion allows for a comparison 
of the distributions restricted to the sub-group. We conclude that, when transforming 
, wealth transfers only occur inside each of the LCG sub-groups and that, in 

  via transfers occurring exclusively inside each 
𝑦𝑦

  for LCG sub-groups. This implies 

  is obtained from 

  into 

  into 

𝑦𝑦

𝑦𝑦

𝑦𝑦

1

2

2

1

2

each of these sub-groups, either exclusively Pigou-Dalton or anti-Pigou-Dalton 
𝑦𝑦
transfers take place11. 

𝑦𝑦

4.5 

Contributions of wealth components: the case of the Gini 
coefficient 

4.5.1 

Decomposition by wealth components 

Taking the covariance formula for the Gini coefficient, 

cov

2

, the 

decomposition by variable components (Lerman and Yitzhaki, 1985) is expressed as: 

𝑦𝑦�

𝐺𝐺 =

�𝑦𝑦, 𝐹𝐹(𝑦𝑦)�

𝑅𝑅

𝑅𝑅

𝑅𝑅

𝐺𝐺 = � 𝐶𝐶𝑖𝑖𝑆𝑆𝑖𝑖 =
𝑖𝑖=1

� 𝑃𝑃𝑖𝑖𝐺𝐺𝑖𝑖𝑆𝑆𝑖𝑖
𝑖𝑖=1

= �
𝑖𝑖=1

cov
cov

(𝑦𝑦𝑖𝑖, 𝐹𝐹)
(𝑦𝑦𝑖𝑖, 𝐹𝐹𝑖𝑖)
�������𝑃𝑃𝑟𝑟

cov

(𝑦𝑦𝑖𝑖, 𝐹𝐹𝑖𝑖)
2
𝑦𝑦𝑖𝑖����������𝐺𝐺𝑟𝑟

𝑦𝑦𝑖𝑖�
𝑦𝑦�⏟𝑆𝑆𝑟𝑟

  and 

  correspond to the concentration coefficient (or 

The two terms 
“pseudo-Gini”) and the share of total wealth of component 
, respectively. The 
concentration coefficient is the product of the “Gini correlation” between wealth 
component 

, and the real Gini of component 

  and total wealth, 

𝐶𝐶𝑖𝑖

𝑆𝑆𝑖𝑖

, 

. 

𝑟𝑟

11    We note nonetheless that, in practice, the intersection points of Lorenz curves may not correspond 

𝑟𝑟

𝑃𝑃𝑖𝑖

𝑟𝑟

𝐺𝐺𝑖𝑖

precisely to an individual of the distribution. 

ECB Statistics Paper Series No 31 / December 2019 29 
 
𝑟𝑟

𝑟𝑟

, the concentration coefficient of a wealth component 

  alone when compared to the distribution of total wealth. It is defined as 

, also known as the 
“pseudo-Gini”, gives an indication of the relation of the distribution of wealth 
𝐶𝐶𝑖𝑖
component 
one minus twice the area under the concentration curve, i.e. it is calculated in the 
same way as the Gini but with reference to the concentration curve instead of the 
Lorenz curve. The concentration curve plots the cumulative proportions of the wealth 
component 
ascending order according to their total net wealth. It can thus lie above the 45-degree 
line, for example when a wealth component is owned mostly by the households at the 
bottom of the distribution. The concentration coefficient is then bounded between -1 
and 1, in the case of non-negative net wealth. 

  against the cumulative proportions of the population organised in 

𝑟𝑟

This decomposition is a valid mathematical identity but its interpretability can be 
questioned, as will be discussed in Section 4.6. Nevertheless, the expression of this 
decomposition will be shown to be the most useful for obtaining results in the following 
section. 

4.5.2 

Marginal effects of wealth components 

The decomposition of the Gini by wealth components presented in the previous 
section can be useful in obtaining the elasticity of the indicator, which leads to the 
analysis of marginal effects, i.e. investigating how small changes in specific net wealth 
components will affect total net wealth inequality. The Gini elasticity with respect to the 
mean of component 
inequality as measured by the Gini coefficient if component 
  was uniformly 
increased by 1%. It was derived by Lerman and Yitzhaki (1985) following the 
decomposition by variable components presented in Section 4.5.1: 

  measures the percentage change that would arise in total 

𝑟𝑟

𝑟𝑟

𝑆𝑆𝑖𝑖(𝐶𝐶𝑖𝑖 − 𝐺𝐺)
𝐺𝐺

𝜂𝜂𝑖𝑖(𝐺𝐺) =
The sign of the elasticity of a component indicates whether it has an equalising or 
disequalising effect. Indeed, if 

, component 

  has a 

disequalising effect in the distribution. 

η𝑖𝑖(G) > 0 ⟺ C𝑖𝑖 − G > 0

𝑟𝑟

4.5.3 

Contributions of wealth components to the changes in time 

Following from the expressions of the Gini decomposition by variable components and 
the marginal effects presented in the precedent sections, a procedure for analysing 
the changes in time in the Gini coefficient can be obtained. The changes in time are 
then decomposed into two factors: i) the change in inequality within each component 
that makes up net wealth (“change in inequality”); and ii) the change in inequality that 
comes from a change in the allocation of the different components, i.e. the change in 
the share of each component in total net wealth (“change in share”). 

Redistribution of wealth within one component (e.g. financial assets) will affect the 
change in inequality factor but keep the relative shares unchanged, while a 

ECB Statistics Paper Series No 31 / December 2019 30 
proportional increase of the value of a component will keep all within-component 
inequalities constant but will affect the shares. A disproportionate increase of business 
assets for the wealthy households, for example, will affect both factors and thus 
combine to increase inequality as a whole. 

The expression has been proposed by Jurkatis and Strehl (2013) 

𝐺𝐺̇ = �(𝐶𝐶𝑖𝑖 − 𝐺𝐺)

change in share + � 𝑆𝑆𝑖𝑖𝐶𝐶𝑖𝑖̇
𝑆𝑆𝑖𝑖̇
change in inequality
𝑖𝑖
𝑖𝑖
�����
���������

The expression can be adapted for computing the changes between the time periods 

  and 

  as follows: 

𝑡𝑡 − 1

𝑡𝑡

Δ𝐺𝐺𝑡𝑡 = � �

� 𝛥𝛥𝑆𝑆𝑖𝑖,𝑡𝑡
change in share
𝑖𝑖
���������������������������

�𝐺𝐺𝑖𝑖,𝑡𝑡 + 𝐺𝐺𝑖𝑖,𝑡𝑡−1�
2

𝐶𝐶𝑖𝑖,𝑡𝑡 + 𝐶𝐶𝑖𝑖,𝑡𝑡−1
2

−

𝑆𝑆𝑖𝑖,𝑡𝑡 + 𝑆𝑆𝑖𝑖,𝑡𝑡−1
𝛥𝛥𝐶𝐶𝑖𝑖,𝑡𝑡
+ �
change in inequality
2
𝑖𝑖
�������������

4.6 

Shorrocks’ systematic approach to decompositions by 
variable components 

Shorrocks (1982) applied a systematic approach to the problem of obtaining 
decompositions by variable components for most inequality measures 12. The analysis 
includes all the indicators that can be expressed as 
  is the coefficient 
corresponding to a specific inequality measure. Gini, Pietra, Generalised Entropy and 
Atkinson are therefore included. 

, where 

𝑖𝑖
∑ 𝑎𝑎𝑖𝑖𝑦𝑦𝑖𝑖

𝑎𝑎𝑖𝑖

𝑟𝑟

𝐹𝐹𝑖𝑖

∑ 𝐹𝐹𝑖𝑖𝑖𝑖

  where 

. The main technical issue is to decide on the component to 

The goal is to find a decomposition that satisfies the condition “one source, one term”, 
i.e. to find a decomposition that can be written as 
  is the contribution of 
the wealth component 
which terms arising owing to interactions between several components should be 
allocated. There are many ways to perform this allocation, giving rise to distinct 
decomposition expressions. To narrow down the possible decomposition expressions 
following a logical set of socio-economic assumptions, Shorrocks proposes a list of 
requirements that the decomposition should meet. These would give a theoretical 
justification for the choice of factors by allocating the interaction term. One of these 
requirements is, however, controversial, as it states that the contribution of a 
component to total wealth inequality is zero if this component is equally distributed 
among all households. This contradicts the fact that the addition of the same positive 
quantity to all the elements of the distribution decreases inequality, which is a property 
displayed by all the inequality indicators that obey to the principle of transfers and 
scale invariance. This critique has lead Paul (2004) to propose an assumption of 
negativity, which states that the contribution of a component to total wealth inequality 
is negative if this component is equally distributed among all households. 

12    The original reasoning by Shorrocks refers to decompositions by income components. The same applies 

to decompositions by wealth components. 

ECB Statistics Paper Series No 31 / December 2019 31 
 
From a combination of requirements set by both Shorrocks and Paul, decomposition 
rules can be obtained for some indicators but there is no consensus in the literature on 
the proper use of these decompositions. Still, the expressions of the decompositions 
are valid mathematical identities that can be used to derive the elasticities of the 
indicators, which have clear interpretations and are useful in practice. Other than the 
case of the Gini coefficient, already presented in Section 4.5.2, the elasticities for the 
Pietra and Generalised Entropy indices are derived as follows, adapting the reasoning 
of Paul: 

𝜂𝜂𝑖𝑖(𝐺𝐺𝐺𝐺(0)) =

(𝑆𝑆𝑖𝑖 − 𝑆𝑆𝑖𝑖� )
𝐺𝐺𝐺𝐺(0)

𝜂𝜂𝑖𝑖�𝐺𝐺𝐺𝐺(𝛼𝛼)� =

′
𝛼𝛼𝑆𝑆𝑖𝑖�𝐺𝐺𝐺𝐺(𝛼𝛼)𝑖𝑖

− 𝐺𝐺𝐺𝐺(𝛼𝛼)�

𝐺𝐺𝐺𝐺(𝛼𝛼)

, 0 < 𝛼𝛼

′
𝑆𝑆𝑖𝑖(𝑃𝑃𝑖𝑖
𝑃𝑃

− 𝑃𝑃)

𝜂𝜂𝑖𝑖(𝑃𝑃) =
With the “pseudo-indicators” defined as: 

′
𝐺𝐺𝐺𝐺(1)𝑖𝑖

  =

𝑛𝑛

1
𝑛𝑛

�
𝑖𝑖=1

𝑦𝑦𝑖𝑖𝑖𝑖
𝑦𝑦𝑖𝑖�

ln �

�

𝑦𝑦𝑖𝑖
𝑦𝑦�

𝑛𝑛

′
𝐺𝐺𝐺𝐺(𝛼𝛼)𝑖𝑖

  =

1
𝛼𝛼(𝛼𝛼 − 1)

1
𝑛𝑛

� �
𝑖𝑖=1

𝑦𝑦𝑖𝑖𝑖𝑖
𝑦𝑦𝑖𝑖�

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

′
𝑃𝑃𝑖𝑖

=

1
𝑛𝑛

�
𝑖𝑖,𝑦𝑦𝑖𝑖>𝑦𝑦�

𝑦𝑦𝑖𝑖𝑖𝑖 − 𝑦𝑦𝑖𝑖�
𝑦𝑦𝑖𝑖�

𝛼𝛼−1

�

− 1�

, 0 < 𝛼𝛼 < 1

ECB Statistics Paper Series No 31 / December 2019 32 
 
 
 
 
 
5 

Applications to European household 
wealth 

We now apply some of the techniques of wealth inequality analysis introduced above 
using the distribution of the net wealth of European households as measured by the 
Household Finance and Consumption Survey. 

The first section of this chapter describes the main features of the HFCS, a 
harmonised cross-country survey providing household-level microdata that are 
comparable across countries and thus able to be used for the analysis of the wealth 
distribution in the euro area in the first half of the 2010s. In the second section we 
present some stylised facts about wealth in the euro area. Third, using the HFCS we 
investigate the modest increase in wealth inequality in the euro area between 2010 
and 2014 shown in the results of the second wave of the survey, which were published 
in December 2016. Several inequality measures are used in the analysis and country 
differences are also assessed. In the fourth section we use the decomposition of the 
total change in the distribution into Pigou-Dalton and anti-Pigou-Dalton transfers, 
presented in Section 3.2, to further investigate the different country results by 
indicator. In the fifth section, we use intersections of Lorenz curves to perform 
sub-group analysis and understand the different inequality trends across countries. In 
the sixth and final section, we analyse the contributions of the components making up 
net wealth to overall inequality and its change in time. 

5.1 

The Household Finance and Consumption Survey 

The HFCS is a harmonised cross-country survey that collects household-level data on 
households’ wealth, income and consumption along with related economic and 
demographic variables. The micro-dataset provides insights into the economic 
behaviour and financial situation of households. 

The HFCS is conducted by the Household Finance and Consumption Network 
(HFCN), which is composed of researchers, statisticians and survey specialists from 
the ECB, European national central banks, some national statistical institutes and a 
number of experts in the field of household finances. 

The survey is conducted every three years and the results for the first two waves were 
published in 2013 and 2016, with fieldwork mainly carried out in 2010/11 and 2013/14, 
respectively13. In the first wave, more than 62,000 households from 15 euro area 
countries were surveyed. All the 18 euro area countries (at the time) plus Hungary and 
Poland participated in the second wave, with over 84,000 households being 
interviewed. 

13    See the Methodological report for the second wave of the HFCS (2016) for country details. 

ECB Statistics Paper Series No 31 / December 2019 33The HFCS is a good source of data for estimating the wealth distribution of euro area 
households. Since wealth distribution is highly uneven, with certain financial 
instruments being held almost exclusively (and in large quantities) by the wealthiest 
households, the sampling method has been designed accordingly. Indeed, most of the 
countries participating in the HFCS oversample the wealthy via different methods 14. 
This partially addresses the fact that wealthier households tend to be more difficult to 
contact and less likely to respond. In addition, the most important variables (including 
net wealth) have been subject to multiple imputation to correct item non-response, 
i.e. plausible values based on the information collected from other households were 
assigned to a variable when it was not collected or not collected correctly. 

The resulting micro-dataset used for the computations in this paper is weighted at the 
household level and multiply imputed. The weights take into account the unit’s 
probability of selection, coverage issues, unit non-response and an adjustment of 
weights to external data. The sum of the weights is equal to the total number of 
households in the population. The HFCN also provides replicate weights to be used in 
the estimation of the variance following the Rao-Wu rescaled bootstrap method. 

5.2 

Household wealth in the euro area: general facts 

This section briefly describes some of the main features of the distribution of assets, 
liabilities and net wealth in the euro area, as available in the HFCS. Interested readers 
are invited to refer to the HFCS reports on the results from the first and second wave 
(HFCN 2013 and 2016b). 

Chart 1 
Average portfolio by net wealth quintile, euro area 2014 

(EUR thousands) 

Source: HFCS, reproduced from HFCN (2016b). 
Note: Hungary and Poland are not included. 

14    Idem. 

ECB Statistics Paper Series No 31 / December 2019 34 
Regarding concentration of assets, 48% of all assets are owned by the wealthiest 10% 
of the population. Moreover, the composition of wealth strongly varies by wealth level, 
with the poorest households having their wealth mostly in vehicles and valuables, the 
median households mostly in their main residence, and only the wealthiest 
households having a sizeable share of their wealth in their own businesses. 

Average wealth levels vary by country (see in particular Box 4.1 of HFCN, 2016b), but 
the heterogeneity of households’ wealth within a country dwarfs the heterogeneity 
between countries. 

In terms of inequality, Table 3 shows the different measures described in Section 2. 

Table 3 
Wealth inequality indicators 

2010 wave 

2014 wave 

Measure 

Standard error 

Measure 

Standard error 

Inequality measures 

Atkinson index (

) 

Atkinson index (

𝜺𝜺 = 𝟏𝟏

) 

0.708 

0.395 

Mean log deviation – Generalised Entropy (

𝜺𝜺 = 𝟎𝟎. 𝟓𝟓

)  1.232 

Theil index – Generalised Entropy (

𝜶𝜶 = 𝟎𝟎

) 

0.931 

Generalised Entropy (

) 

𝜶𝜶 = 𝟏𝟏

Gini coefficient 

𝜶𝜶 = 𝟐𝟐

Pietra index 

Share indicators 

Top 5% share 

Top 10% share 

Quantile ratio indicators 

P90/P50 ratio 

P80/P20 ratio 

5.186 

0.680 

0.495 

37.2 

50.5 

4.7 

40.1 

0.006 

0.007 

0.020 

0.037 

1.276 

0.006 

0.004 

1.0 

0.8 

0.1 

2.0 

0.713 

0.399 

1.250 

0.942 

4.411 

0.685 

0.501 

37.8 

51.2 

4.8 

41.0 

0.005 

0.006 

0.018 

0.032 

0.655 

0.005 

0.004 

0.8 

0.7 

0.1 

1.8 

Sources: HFCS and authors’ calculations. 
Notes: Atkinson(0.5) computed for non-negative values of wealth only. Atkinson(1), GE(0) and Theil index computed only on positive 
values of wealth, following the conditions described in Table 1 and Table 2. Standard errors computed taking into account the multiple 
imputation and bootstrap weights. 

5.3 

Evolution of wealth inequality in the euro area between 
2010 and 2014: investigation into the reported modest 
increase 

In the results of the second wave of the HFCS in 2016 the Gini coefficient and other 
indicators pointed to a modest increase in wealth inequality in the euro area in the 
period 2010-2014. 15  These results should be complemented with the changes taking 
place in each of the euro area countries and using different measures of inequality. 
The changes in the Gini, Pietra, Theil and Generalised Entropy (with parameter 
) indicators in selected countries are given in Chart 2 (with the confidence 

intervals shown in Chart 3 and in Appendix section entitled “Complementing charts”). 
𝛼𝛼 = 0

15    As the only exception, the Generalised Entropy (

) showed a decrease in inequality (see Table 3).   

𝛼𝛼 = 2

ECB Statistics Paper Series No 31 / December 2019 35 
 
 
 
 
 
 
 
 
 
 
 
 
Chart 2 
Percentage change in wealth inequality in euro area countries between 2010 and 
2014 

Gini

Pietra

Theil (GE(α=1))

GE(α=0)

25%

15%

5%

-5%

-15%

r
o
t
a
c
d
n

i

i

e
h
t

n

i

e
g
n
a
h
c
e
g
a
t
n
e
c
r
e
p

euro
area

BE

DE

GR

ES

FR*

IT

CY

LU

MT

NL

AT

PT

SK

FI

Sources: HFCS and authors’ calculations. 
Notes: FR*: results for France corrected for the lack of information on the value of vehicles in the first wave of the survey. Observations 
with non-positive values have been dropped from the calculations of the Theil and GE(0) indicators. Changes between 2010 and 2014 for 
most countries (see exceptions in HFCS (2016)). 

Results show that the changes in the indicators in the different countries are, in 
general, more pronounced than the aggregated euro area changes. More 
interestingly, the changes occur in different directions: the selected indicators point 
towards a decrease in wealth inequality in Belgium and Luxembourg and towards an 
increase in Greece, Spain, France, Malta, Slovakia and Finland. As for the other 
countries, the results are mixed, reflecting important differences in the nature of the 
change in the distribution. The Generalised Entropy index with parameter 
  is 
more sensitive to transfers at the bottom of the distribution (when compared to the 
same indicator with a larger parameter 
inequality in Austria and an increase in Portugal, whereas the Theil index (relatively 
more sensitive to transfers at the top of the distribution) gives the opposite results: an 
increase in inequality in Austria and decrease in inequality in Portugal. 

) and, as an example, points to a decrease in 

𝛼𝛼 = 0

𝛼𝛼

These results should, however, be interpreted with care as the estimation of standard 
errors of the measures implies that the changes are only statistically significant at the 
5% level in a few cases. 

Chart 3 shows that the changes in the Gini coefficient are only statistically significant 
in Greece and Slovakia. The same holds for Atkinson’s indicator with parameter 
(shown in Chart A.1 in the Appendix), where the increase in Portugal is also 
statistically significant. None of the changes as measured by Theil’s index are 
statistically significant (shown in Chart A.2 in the Appendix). 

𝜀𝜀 = 1

ECB Statistics Paper Series No 31 / December 2019 36 
 
 
 
 
 
Chart 3 
Absolute change in Gini with 95% confidence interval 

i

i

n
G
n

i

e
g
n
a
h
c
e
t
u
o
s
b
a

l

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

euro
area

BE

DE

GR

ES

FR*

IT

CY

LU

MT

NL

AT

PT

SK

FI

Sources: HFCS and authors’ calculations. 
Notes: FR*: results for France corrected for the lack of information on the value of vehicles in the first wave of the survey. Changes 
between 2010 and 2014 for most countries (see exceptions in HFCS (2016)). 

Another aspect to consider is the composition of the euro area used for computing the 
indicators. Indeed, the euro area results of the HFCS are in general computed 
considering the euro area with a changing composition, i.e. it is defined as the 
15 participating countries in the first wave of the survey and the 18 participating 
countries in the second wave. The use of a fixed composition of the 15 countries 
participating in both waves of the survey gives slightly different results, with even 
smaller changes in the indicators, when compared with the euro area with a changing 
composition (see Chart 4). 

Chart 4 
Percentage change in wealth inequality: different euro area compositions 

Gini

Pietra

Theil (GE(α=1))

GE(α=0)

r
o

t
a
c
d
n

i

i

e
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1.5%

1.0%

0.5%

0.0%

euro
area

euro
area 15

Sources: HFCS and authors’ calculations. 
Note: Changes between 2010 and 2014 for most countries (see exceptions in HFCS (2016)). See also footnote to Chart 2. 

ECB Statistics Paper Series No 31 / December 2019 37 
 
 
 
 
 
 
 
 
5.4 

Differences in inequality trends by indicator: contributions 
to total change from Pigou-Dalton transfers 

The results in the previous subsection show that, in some countries, the inequality 
indicators point in opposite directions as regards the changes in the inequality of the 
wealth distribution between the first two waves of the survey. 

To see how the choice of specific indicator influences the result, we express, for each 
country, the difference between the distributions of the first and second waves of the 
HFCS as a sequence of Pigou-Dalton and anti-Pigou-Dalton transfers, applying the 
method introduced in Section 3.2. The absolute change in the inequality indicators is 
expressed as a sum of the contributions of Pigou-Dalton and anti-Pigou-Dalton 
transfers to the overall indicator. 

Chart 5 illustrates how the same Pigou-Dalton and anti-Pigou-Dalton transfers are 
accounted for differently by the inequality indicators.16  In Greece there are only 
anti-Pigou-Dalton transfers, that is, there was an increase in inequality according to all 
the indicators. In all the other countries, both Pigou-Dalton and anti-Pigou-Dalton 
transfers occur, even if in most cases the contribution to the change in the Gini 
indicator is barely measurable from one of the two transfer types. The exceptions are 
the Netherlands, Germany, Finland, and Portugal. The chart also shows how the 
Pietra index remains unchanged when some of the Pigou-Dalton or anti-Pigou-Dalton 
transfers are performed. More interestingly, for most countries, the total contribution to 
the Pietra from one of the two transfer types is zero, meaning that these changes 
occur on the same side of the distribution relative to the mean. Using the Generalised 
Entropy indicators to account for the same transfers gives yet another picture, shown 
in Chart 6 below. 

Chart 5 
Contributions of Pigou-Dalton transfers to total change in inequality – Gini and Pietra 
indicators 

Pigou-Dalton transfers

anti-Pigou-Dalton transfers

total change

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Sources: HFCS and authors’ calculations. 
Notes: FR*: results for France corrected for the lack of information on the value of vehicles in the first wave of the survey. Changes 
between 2010 and 2014 for most countries (see exceptions in HFCS (2016)). 

16    Owing to the complexity of the calculation of the components, the confidence intervals for the 

decomposition into Pigou-Dalton and anti-Pigou-Dalton components have not been shown in these 
graphs. 

ECB Statistics Paper Series No 31 / December 2019 38 
 
 
 
Chart 6 
Contributions of Pigou-Dalton transfers to total change in inequality – Generalised 
Entropy indicators 

Pigou-Dalton transfers

anti-Pigou-Dalton transfers

total change

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AT

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FR*

MT

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SK

Sources: HFCS and authors’ calculations. 
Notes: FR*: results for France corrected for the lack of information on the value of vehicles in the first wave of the survey. Observations 
with non-positive values have been dropped from the calculations of the Generalised Entropy indicators. Changes between 2010 and 
2014 for most countries (see exceptions in HFCS (2016)). 

Unlike the contributions to the Gini coefficient, the contributions of Pigou-Dalton and 
anti-Pigou-Dalton transfers to the total change in the Generalised Entropy indicators 
are, in most cases, not negligible for both types of transfers. The type of transfer giving 
the biggest contribution can change depending on the parameter 
  chosen, resulting 
in a total change in the Generalised Entropy indicator, with it pointing in opposite 
directions in some cases, like in Austria, Italy, Germany or Portugal. 

𝛼𝛼

  when using the Generalised 

Chart 7 also shows how the choice of the parameter 
Entropy index affects the inequality trend for a selected number of countries. A higher 
value of 
  corresponds to a stronger importance of transfers at the top of the 
distribution, as discussed in Sections 2.2.2 and 3.2. In Germany and Italy, an 
increasing parameter 
for Greece the change is strongly positive in all cases. In Belgium, it is likely that a 
slightly negative value of the parameter would correspond to a positive change in the 
indicator. 

  leads to a decrease in the change in the GE indicator, while 

𝛼𝛼

𝛼𝛼

𝛼𝛼

ECB Statistics Paper Series No 31 / December 2019 39 
 
 
 
Chart 7 
Change in the Generalised Entropy indicator depending on the parameter 

EA

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IT

𝛼𝛼

15%

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10%

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-10%

0

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1

parameter (alpha)

Sources: HFCS and authors’ calculations. 
Notes: Observations with non-positive values have been dropped from the calculations. Changes between 2010 and 2014 for most 
countries (see exceptions in HFCS (2016)). 

5.5 

Sub-group dynamics: analysis using the difference 
between Lorenz curves 

The change in wealth inequality can also be explained by the groups of households 
among which wealth transfers would need to be made to obtain the most recent 
distribution from the initial distribution. This analysis considers each household’s 
wealth share. As described in Section 4.4, the intersections of the Lorenz curves of the 
two distributions being compared divide the population into sub-groups such that only 
one kind of wealth transfer (either Pigou-Dalton or anti-Pigou-Dalton) occurs within 
each of the sub-groups. 

At the euro area level, the Lorenz curves of the two waves of the survey are relatively 
close to each other, intersecting close to the percentiles P30 and P55 and dividing the 
population into three sub-groups (Chart 8a). This means that it is possible to move 
from one distribution to another by redistributing wealth within these sub-groups, 
namely among the 30% poorest households, then among those between the 30% and 
55% poorest, and finally among the 45% richest households. In each of these 
sub-groups, the share of total wealth has stayed the same across waves, and only the 
distribution within the sub-group may have changed. The difference between the 
Lorenz curves is positive for the “middle” sub-group, meaning that the wealth 
distribution within this sub-group has become more equal between the two waves of 
the survey. Within the two remaining sub-groups, that is, those comprising the poorest 
30% and the richest 45%, inequality has increased. As the difference in the area 
between Lorenz curves is negative and directly related to the changes in the Gini, the 
charts allow us to conclude that most of the changes in the Gini between 2010 and 
2014 were driven by changes in the top 45% of the distribution. 

ECB Statistics Paper Series No 31 / December 2019 40 
 
 
 
 
 
Chart 8b shows the relative wealth increase or decrease between the two waves of 
the survey for each percentile group. 17  In particular, we can see that the transfers 
occurring within the sub-group containing the poorest 30% to 55% of households are 
Pigou-Dalton transfers taking place from households in the percentiles P40 to P55 to 
those in the percentiles P30 to P40, therefore decreasing inequality in this population 
sub-group. Chart 8b further highlights the importance of the top of the distribution in 
the analysis of wealth inequality, as the changes at the top have a strong impact on the 
overall analysis. 

Chart 8 
Difference between Lorenz curves and normalised percentile difference: euro area 

a) Difference between Lorenz curves

b) Normalised percentile difference

0.001

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Sources: HFCS and authors’ calculations. 
Notes: Observations with non-positive values have been dropped from the calculations. Changes between 2010 and 2014 for most countries (see exceptions in HFCS (2016)). 95% 
confidence intervals are shown in light grey (Charts 8a and 8b) and truncated at [-0.20, 0.20] (Chart 8b). 

The same analysis made at the country level again shows the differences across 
countries that are hidden in the aggregated euro area results. The differences 
between Lorenz curves are plotted in Chart 9 for a selection of countries. 

17    This quantile difference chart, inspired by the work of Kennickell (2009, 2011), normalises the wealth in 
the two periods to focus only on redistribution effects; in some sense the quantile difference chart is the 
first derivative of the difference between the Lorenz curves. 

ECB Statistics Paper Series No 31 / December 2019 41 
 
 
 
 
 
Chart 9 
Difference between Lorenz curves of the 2010 and the 2014 wealth distributions – selected countries 

EA

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Sources: HFCS and authors’ calculations. 
Notes: Observations with non-positive values have been dropped from the calculations. Changes between 2010 and 2014 for most countries (see exceptions in HFCS (2016)). 95% 
confidence intervals are shown in grey. 

Different patterns arise: in Germany the population is divided into four sub-groups, 
with inequality increasing in the group made up of the poorest half of the population 
and in the group comprising households between percentiles P82 and P98. In the 
other two sub-groups, the first made up of households between P48 and P82 and the 
second of the 2% richest households, inequality has decreased. 

The situation is different in Italy, with only two sub-groups arising: the poorest 51% 
households and the richest 49%. Between 2010 and 2014, wealth inequality increased 
among the households comprising the former sub-group and decreased among those 
in the second sub-group. 

In Belgium the wealth distributions of the two waves of the survey are almost 
overlapping until the household at the 14th percentile. Inequality then decreased in the 
sub-group comprising the households between the 14th and 99th percentiles. 

In Greece the difference between Lorenz curves is always negative as the curves do 
not intersect; the curve corresponding to the second wave of the survey always lies 
below that of the first wave. Therefore, according to Atkinson’s comparison criterion 
and to any definition of wealth inequality consistent with the principle of transfers, 
wealth inequality has increased in Greece over the time period considered. 

  has tended to increase. For Generalised Entropy 

These results can be further compared to those of Section 5.4. Charts 6 and 7 show 
how, in Germany and Italy, inequality measured with Generalised Entropy indicators 
with the lower parameter 
indicators, the lower the parameter, the greater the weight attached to transfers at the 
bottom of the distribution. Indeed, Chart 9 shows that, in Germany and Italy, inequality 
increased at the bottom of the distribution. Yet, when using values of the parameter 
equal to or greater than 0.5, these GE indicators show that, overall, inequality has 
decreased for these countries. On the contrary, with low values of the 

  parameter, 

𝛼𝛼

GE indicators report an increase. 

𝛼𝛼

ECB Statistics Paper Series No 31 / December 2019 42 
 
 
 
 
5.6 

Contributions from wealth components 

5.6.1 

Marginal effects 

Net wealth is measured in the HFCS as the difference between the household’s total 
assets and liabilities. Assets can then be split into real assets – the value of the 
household main residence, other real estate, vehicles and valuables and 
self-employment businesses – and financial assets. Financial assets can be further 
decomposed into safe, risky and other financial assets18. Lindner (2015) presented a 
decomposition of wealth along these components. We conduct a similar analysis for 
the Gini and for the other main inequality indicators, starting with the second wave of 
the HFCS before considering the changes between the two waves in the following 
section. 

The elasticities of the Gini coefficient with respect to the mean for each component of 
total wealth were introduced in Section 4.5.2 and are presented in Chart 10. By 
construction, the sum of the elasticities is null: an increase of all components of wealth 
by 1% leaves inequality unchanged overall. Since the total effect is equal to the sum of 
the effects of the individual components, for the total to be zero, some effects have to 
have opposite signs, meaning there will always be equalising and disequalising 
components under this decomposition of marginal effects. 

Chart 10 
Elasticity of the Gini coefficient, second wave – wealth components 

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FI

HMR

other real estate

self-employment business

vehicles and valuables

financial assets

liabilities

Sources: HFCS and authors’ calculations. 

Across all countries, the household main residence and vehicles and valuables have 
an equalising effect (i.e. a proportional increase in the value of HMR for all households 
decreases the Gini coefficient, other things being equal) whereas liabilities and 
self-employment businesses have a disequalising effect in all countries. It might 
appear surprising that an increase in the value of an asset that is only held by a 
fraction of the population decreases inequality. However, this is explained by two 

18    HFCS variables, with reference to the derived variables in HFCS (2016): Safe financial assets = deposits 
(DA2101) + voluntary pension/whole life insurance (DA2109); Risky financial assets = mutual funds 
(DA2102) + bonds (DA2103) + shares, publicly traded (DA2105); Other financial assets = value of non 
self-employment private business (DA204) + managed accounts (DA2106) + money owed to households 
(DA2107) + other assets (DA2108). 

ECB Statistics Paper Series No 31 / December 2019 43 
facts, namely that i) the ownership of both the HMR and vehicles and valuables is 
more widespread than for most other components; and ii) there is less inequality in the 
distribution of these two types of assets than, for example, self-employment 
businesses, of which the ownership is concentrated and the range of values of 
businesses is much wider, especially at the top of the wealth distribution (see Chart 1). 
For most less-wealthy households, the HMR is the main or even the only asset while 
for wealthier households the HMR makes up a smaller fraction of assets. In countries 
where ownership of the HMR is less prevalent (Austria and Germany) the marginal 
effect of the HMR is both smaller and of comparable size to that of financial assets, 
while in other countries the marginal effect of financial assets is smaller. However, we 
should point out that this analysis on wealth inequality does not take into account the 
fact that housing is both an asset and a consumption item and that the increase in the 
value of HMR also increases the imputed consumption of housing services of 
households renting their main residence; nevertheless these considerations are not 
pursued here further. 

Financial assets have an equalising effect in most countries but not in Belgium and 
Spain, for example, although the effect is moderate in these two countries. Other real 
estate has a disequalising effect in all countries except Cyprus, the Netherlands and 
Latvia. Chart 11, which presents the Gini elasticities of financial assets, shows that 
safe financial assets have equalising effects in all countries. Risky and other financial 
assets mostly have small, disequalising effects in most countries. 

Chart 11 
Elasticity of the Gini coefficient, second wave – financial assets 

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other

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all financial assets

Sources: HFCS and authors’ calculations. 

The elasticities computed for other indicators (Charts 13 and 14) described in 
Sections 4.5.2 and 4.6 show the contributions of each of the components. 
Interestingly, all elasticities are of the same sign as the Gini elasticities, although the 
magnitude varies. The Pietra elasticities are very close to those of the Gini, while the 
Theil index GE(1) and the GE(0) index show more variation. The equalising role of 
financial assets and, in particular, safe financial assets is much stronger for the GE(0) 
measure. 

ECB Statistics Paper Series No 31 / December 2019 44 
 
Chart 12 
Elasticity of other inequality indicators – total wealth 

a) Total wealth 

b) Financial assets 

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Pietra

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HMR
self-emp. business
financial assets

other real estate
vehicles and valuables
liabilities

other

risky

safe

all financial assets

Sources: Second wave of the HFCS, euro area, and authors’ calculations. 
Note: Observations with non-positive values have been dropped from the calculations of the Theil and GE(0) indicators. 

5.6.2 

Decomposition of change in Gini 

The change in the Gini coefficient between the two waves can be decomposed into 
two factors (as described in Section 4.5.3), taking as components the constituents of 
total wealth described in the previous section, namely household main residence, 
other real estate, vehicles and valuables, self-employment businesses and financial 
assets. One factor accounts for the change in inequality within each component 
(“change in intrinsic inequality”) and the other accounts for a change in the allocation 
of the different components (“change in share”). Chart 13 shows how these two 
factors add up to the total change in inequality, as measured by the Gini coefficient for 
the euro area and for each country. 

ECB Statistics Paper Series No 31 / December 2019 45 
Chart 13 
Absolute change in the Gini coefficient between waves and its decomposition 

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change in
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change in
total Gini

AT BE LU NL IT FR DE EA FI MT PT ES CY GR SK

Sources: HFCS and authors’ calculations. 
Notes: Changes between 2010 and 2014 for most countries (see exceptions in HFCS (2016)). Countries are sorted by increasing change 
in total Gini. The components adding up to net wealth are household main residence, other real estate, vehicles and valuables, 
self-employment businesses and financial assets. 

The “change in inequality” factor only contributed positively to the change in total Gini 
in Germany, Greece, Malta and Slovakia. In all the other countries where the Gini 
coefficient increased, the increase can only be explained by a change in the allocation 
of the wealth components. Furthermore, the change in the allocation of the wealth 
components mainly contributed to an increase in total inequality, with the only 
exceptions being Austria, Italy and Germany. 

In the euro area, the “change in inequality” factor also contributed to the increase in 
inequality, albeit with a very small contribution when compared with the “change in 
allocation” factor. Moreover, this factor is even smaller when considering the euro area 
with unchanged decomposition between waves (Chart 14). 

Chart 14 
Change in Gini coefficient between waves and its decomposition for the euro area 

0.006

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0.000

change in
inequality
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change in
share
factor

change in
total Gini

EA

EA15

Sources: HFCS and authors’ calculations. 
Notes: Changes between 2010 and 2014 for most countries (see exceptions in HFCS (2016)). The components adding up to net wealth 
are household main residence, other real estate, vehicles and valuables, self-employment businesses and financial assets. 

ECB Statistics Paper Series No 31 / December 2019 46 
 
 
6 

Conclusion 

A change in an abstract indicator usually hides the significant complexity of its 
interpretation and explanation. The modest increase in wealth inequality over the first 
two waves of the HFCS is a case in point for this observation. 

This paper started as an attempt to shed some light in the correct measurement of 
wealth inequality and, more importantly, understand its determinants through 
appropriate sub-indicators. It took a very brief path through the rich literature on the 
axiomatic specification of inequality indicators, pointing out the relationship between 
the Gini coefficient, the Pietra index, top shares, and quantile ratios, with the last two 
indicators lacking some important characteristics desired in inequality measures. It 
also introduced Atkinson’s measure, Theil’s measure, and their family of Generalised 
Entropy indices. All these different measures are, however, sensitive to different parts 
of the distribution and can lead to different measures of changes – over time or over 
space. 

Atkinson’s comparison criterion is an important stepping stone to address this problem 
and makes it possible to state “absolute” comparisons over a large family of inequality 
measures verifying the principle of transfers (stated loosely, “transferring wealth down 
the distribution always decreases inequality”). We then built a novel decomposition of 
changes between two distributions with a succession of Pigou-Dalton transfers that 
were either all equalising or all disequalising within contiguous sub-groups of the 
population, ensuring uniqueness of the decomposition in two non-negative factors. 
Atkinson’s comparison criterion translates in to one of these two factors being exactly 
zero; any deviation from this implies that there is an inequality measure that would 
show a change in the opposite direction. 

The literature has also proposed a large number of decompositions of inequality 
measures or of their changes, some of which can be applied to several of the different 
measures studied in the paper, which we briefly introduced. 

Using the results of the first two waves of the HFCS allowed us to achieve two 
objectives: to illustrate how the various indicators and their decompositions can be 
applied to real wealth and to return to the matter that motivated the paper, namely 
understanding the nature of the changes in wealth between 2010 and 2014. The small 
increase in wealth inequality at the euro area level, and the relative agreement 
between the different inequality measures, is not replicated in several of the countries. 
Our decomposition into Pigou-Dalton transfers shows that in most countries the 
change in the Gini coefficient can only be understood as a set of transfers of the same 
kind, either equalising or disequalising. In the few countries where this is not the case 
(Netherlands, Germany, Finland, Portugal, and to a lesser degree Italy and France), 
the different inequality measures do lead to changes of opposite sign. As seen with 
Atkinson’s comparison criterion and the decomposition into Lorenz sub-groups, the 
increase in inequality in the euro area is almost entirely due to the increase in wealth 
inequality of the top 50% of the population, although results vary strongly by country. 

ECB Statistics Paper Series No 31 / December 2019 47The different decompositions show the elasticity of the different components of wealth, 
with the household’s main residence and safe financial assets being an equalising 
component (i.e. a proportional increase in the value of the main residence or of the 
safe financial assets decreases inequality, all other things being equal). Other real 
estate, self-employment businesses, and liabilities are disequalising, as they are held 
only by a smaller fraction of the population, and correlated with total wealth. Looking at 
the changes between the two waves of the survey, the change in the allocation of the 
components of wealth has tended to increase inequality, while the change in the 
distribution of each component of wealth drives the overall change in inequality: 
countries with a decrease in within-wealth component inequality are also the countries 
with the strongest decrease in wealth inequality. 

Summing up, the changes in wealth between 2010 and 2014 in the euro area can be 
described as a modest increase, with a sign that is robust to most inequality 
measures. At the country level, measurement precision is lower and points to 
ambiguous changes in several countries (Germany, France, Italy, Netherlands, 
Portugal, Finland) while in others the results are more clear cut. 

ECB Statistics Paper Series No 31 / December 2019 48References 

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ECB Statistics Paper Series No 31 / December 2019 51Appendix 

Expressions for inequality indicators 

  represent the wealth of household 

Let 
represents the wealth of all observed households. Let 
𝑦𝑦𝑖𝑖
𝑖𝑖 ∈ {1, … , 𝑛𝑛}
household 

. 

. The vector 

  represent the weight of 

𝑦𝑦 = (𝑦𝑦1, 𝑦𝑦2, … , 𝑦𝑦𝑛𝑛)

𝑤𝑤𝑖𝑖

In the case where 

𝑖𝑖

  is such that 

, we define: 

𝑦𝑦

𝑦𝑦1 ≤ 𝑦𝑦2 ≤ ⋯ ≤ 𝑦𝑦𝑛𝑛

  as the cumulative weight and 

𝑤𝑤𝑗𝑗

𝑖𝑖
∑
𝑗𝑗=1
𝑛𝑛
𝑗𝑗=1

∑

𝑤𝑤𝑗𝑗

𝑋𝑋𝑖𝑖 ≔
The table below gives one possible expression for each indicator in two cases: 1) 
generic weight and 2) in the case where all the individuals have the same weight. 

𝑌𝑌𝑖𝑖 ≔

𝑦𝑦𝑗𝑗𝑤𝑤𝑗𝑗

∑

  as the cumulative wealth. 

𝑦𝑦𝑗𝑗𝑤𝑤𝑗𝑗

𝑖𝑖
∑
𝑗𝑗=1
𝑛𝑛
𝑗𝑗=1

Table A.1 
Weighted and unweighted formulas for the different inequality indicators 

Indicator 

Generic weight 

Gini 

Generalised Entropy (

) 

𝛂𝛂

Theil (GE(

)) 

𝛂𝛂 = 𝟏𝟏

Mean log deviation (GE(

)) 

𝛂𝛂 = 𝟎𝟎

Atkinson (

) 

𝛆𝛆 ≠ 𝟏𝟏

Atkinson (

) 

𝛆𝛆 = 𝟏𝟏

Pietra 

𝒘𝒘𝒊𝒊

𝑛𝑛−1

𝐺𝐺 = 1 − �(𝑋𝑋𝑖𝑖+1 − 𝑋𝑋𝑖𝑖)(𝑌𝑌𝑖𝑖+1 + 𝑌𝑌𝑖𝑖)

𝑖𝑖=0

GE

𝑛𝑛

(𝛼𝛼) =

1
𝛼𝛼(𝛼𝛼 − 1)

�
𝑖𝑖=1

𝑤𝑤𝑖𝑖
𝑛𝑛
∑ 𝑤𝑤𝑖𝑖
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

��

𝛼𝛼

�

− 1�

GE

𝑛𝑛

(1) = �
𝑖𝑖=1

GE

𝑦𝑦𝑖𝑖
𝑦𝑦�

𝑤𝑤𝑖𝑖
𝑛𝑛
∑ 𝑤𝑤𝑖𝑖
𝑖𝑖=1
𝑛𝑛

ln �

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

(0) = − �
𝑖𝑖=1

𝑤𝑤𝑖𝑖
𝑛𝑛
∑ 𝑤𝑤𝑖𝑖
𝑖𝑖=1

ln �

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

(𝜀𝜀) = 1 −

1
𝑦𝑦�

��
𝑖𝑖=1

𝑤𝑤𝑖𝑖
𝑛𝑛
∑ 𝑤𝑤𝑖𝑖
𝑖𝑖=1

1−𝜀𝜀
𝑦𝑦𝑖𝑖

�

Atk

exp

𝑛𝑛

(1) = 1 −

1
𝑦𝑦�

𝑤𝑤𝑖𝑖
𝑛𝑛
∑ 𝑤𝑤𝑖𝑖
𝑖𝑖=1

��
𝑖𝑖=1

ln 𝑦𝑦𝑖𝑖

�

𝑛𝑛

𝑃𝑃 =

1
2

� �
𝑖𝑖=1

𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1 −
∑ 𝑤𝑤𝑖𝑖

𝑦𝑦𝑖𝑖𝑤𝑤𝑖𝑖
𝑛𝑛
∑ 𝑦𝑦𝑖𝑖𝑤𝑤𝑖𝑖
𝑖𝑖=1

�

Constant weight 

𝒘𝒘𝒊𝒊 =

𝟏𝟏
𝒏𝒏 ∀𝒊𝒊

𝑛𝑛
2 ∑ 𝑖𝑖 𝑦𝑦𝑖𝑖
𝑖𝑖=1
𝑛𝑛
𝑖𝑖=1 −
𝑛𝑛 ∑ 𝑦𝑦𝑖𝑖

𝑛𝑛 + 1
𝑛𝑛

𝐺𝐺 =

GE

(𝛼𝛼) =

GE

𝑛𝑛

1
𝛼𝛼(𝛼𝛼 − 1)

1
𝑛𝑛

� ��
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

𝛼𝛼

�

− 1�

𝑛𝑛

1
𝑛𝑛

�
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

𝑛𝑛

ln �

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

(1) =

GE

(0) = −

1
𝑛𝑛

� ln �
𝑖𝑖=1

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

𝑛𝑛

1
1−𝜀𝜀 

(𝜀𝜀) = 1 −

1
𝑦𝑦�

�

1
𝑛𝑛

1−𝜀𝜀

� 𝑦𝑦𝑖𝑖
(𝑖𝑖=1)

Atk

𝑛𝑛

�

1
𝑛𝑛 

(1) = 1 −

1
𝑦𝑦�

�� 𝑦𝑦𝑖𝑖
𝑖𝑖=1

�

𝑛𝑛

𝑃𝑃 =

1
2𝑛𝑛

�
𝑖𝑖=1

|𝑦𝑦𝑖𝑖 − 𝑦𝑦�|
𝑦𝑦�

=

1
𝑛𝑛

𝑦𝑦𝑖𝑖 − 𝑦𝑦�
𝑦𝑦�

�
𝑦𝑦𝑖𝑖>𝑦𝑦�

Atk

𝑛𝑛

1
1−𝜀𝜀 

Atk

Decomposition of inequality changes by Pigou-Dalton 
transfers 

This is the proof of the theorem presented in Section 3.2.2. 

The algorithm can be written as follows: 

Initial distribution 

Final distribution 

1

𝑦𝑦

1
= (𝑦𝑦1

1
, … , 𝑦𝑦𝑛𝑛

)

𝑀𝑀

𝑦𝑦

𝑀𝑀
= (𝑦𝑦1

𝑀𝑀
, … , 𝑦𝑦𝑛𝑛

)

ECB Statistics Paper Series No 31 / December 2019 52 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Elements sorted ascendingly 

and 

; equal means 

1
𝑦𝑦𝑖𝑖

1
≤ 𝑦𝑦𝑖𝑖+1

𝑀𝑀
𝑦𝑦𝑖𝑖

𝑀𝑀
≤ 𝑦𝑦𝑖𝑖+1

 ∀𝑖𝑖 ∈ {1, … , 𝑛𝑛}

𝑀𝑀
����

1
��� = 𝑦𝑦
𝑦𝑦
Let 
𝑦𝑦
individual 
𝑐𝑐𝑡𝑡

(𝑦𝑦𝑘𝑘, 𝑦𝑦𝑗𝑗)
𝑦𝑦𝑘𝑘

, 

𝑗𝑗
𝑗𝑗
, … , 𝑦𝑦𝑛𝑛
�𝑦𝑦1
�
Then step 

  denote the transfer of amount 
. 
  to individual 

performed on distribution 

  from 

𝑡𝑡 

𝑦𝑦

Let the distribution 

𝑦𝑦𝑗𝑗

  resulting from step 

  of the algorithm be such that 

𝑗𝑗

𝑦𝑦
𝑗𝑗
∀𝑖𝑖 ∈ {1, … , 𝑛𝑛} 𝑦𝑦𝑖𝑖

𝑗𝑗
≤ 𝑦𝑦𝑖𝑖+1

  is obtained as follows: 

  and 

. 

𝑗𝑗

𝑦𝑦

=

𝑗𝑗
𝚥𝚥
��� = 𝑦𝑦
𝑦𝑦

1
��� = 𝑦𝑦

𝑀𝑀
����

  is the vector of 

𝑗𝑗 + 1

  elements 

  where 

𝑛𝑛

𝑗𝑗

𝑑𝑑

𝑗𝑗
= (𝑑𝑑1

𝑗𝑗
, … , 𝑑𝑑𝑛𝑛

)

𝑗𝑗
∀𝑖𝑖 ∈ {1, … , 𝑛𝑛} 𝑑𝑑𝑖𝑖

𝑗𝑗
= 𝑦𝑦𝑖𝑖

𝑀𝑀
− 𝑦𝑦𝑖𝑖

𝑗𝑗

𝑑𝑑
Set 

Set 

Set 

𝑗𝑗
𝑎𝑎𝑗𝑗 = min{𝑖𝑖 ∈ {1, … 𝑛𝑛} | 𝑑𝑑𝑖𝑖
𝑗𝑗
𝑏𝑏𝑗𝑗 = min{𝑖𝑖 ∈ {1, … 𝑛𝑛} | 𝑑𝑑𝑖𝑖

< 0}

> 0}

𝑗𝑗
𝑡𝑡𝑗𝑗 = min ��𝑑𝑑𝑎𝑎𝑗𝑗

𝑗𝑗
� , �𝑑𝑑𝑛𝑛𝑗𝑗

��

  is obtained from 

  by transfer 

𝑗𝑗+1

𝑦𝑦
𝑐𝑐𝑡𝑡𝑗𝑗
𝑦𝑦
the elements are sorted subsequently. 

𝑦𝑦

𝑗𝑗

𝑗𝑗

, i.e. 

𝑗𝑗
�𝑦𝑦𝑛𝑛𝑗𝑗

𝑗𝑗
, 𝑦𝑦𝑎𝑎𝑗𝑗

�

𝑗𝑗+1

𝑦𝑦𝑖𝑖

= �

  and 

𝑗𝑗
𝑦𝑦𝑖𝑖
𝑗𝑗
𝑦𝑦𝑖𝑖

+ 𝑡𝑡𝑗𝑗 𝑖𝑖𝑖𝑖 𝑖𝑖 = 𝑎𝑎𝑗𝑗
otherwise
− 𝑡𝑡𝑗𝑗 𝑖𝑖𝑖𝑖 𝑖𝑖 = 𝑏𝑏𝑗𝑗
𝑗𝑗
𝑦𝑦𝑖𝑖

The algorithm described above will, by construction, result in transfers that occur 
exclusively inside each of the groups defined by the intersections of the Lorenz curves 
of the initial and final distributions (LCG), as defined in Section 4.4. In each of these 
groups, the initial distribution can be transformed into the final distribution exclusively 
through either Pigou-Dalton or anti-Pigou-Dalton transfers among elements of the 
group. This procedure will result in a sequence of transfers that minimises the total 
change in the Gini coefficient and the magnitudes of the contributions of Pigou-Dalton 
and anti-Pigou-Dalton transfers to this change. 

  represent the sequence of distributions that result from performing the 

Let 
algorithm. Let 
(respectively, anti-Pigou-Dalton) transfers in 
𝑎𝑎𝑃𝑃𝑃𝑃(𝑆𝑆)
lighter reading. 

  (respectively, 

𝑃𝑃𝑃𝑃(𝑆𝑆)

𝑆𝑆

) denote the set of all Pigou-Dalton 

. 

  and 

  are used below for a 

𝑆𝑆

𝑃𝑃𝑃𝑃

𝑎𝑎𝑃𝑃𝑃𝑃

We recall the expression for the total change in the Gini coefficient: 

Gini

Gini

Gini

Gini 

We take the case of an LCG sub-group where only Pigou-Dalton transfers occur. 

𝑀𝑀

1

(𝑦𝑦

) −

(𝑦𝑦

) = 𝐶𝐶𝑃𝑃𝑃𝑃

+ 𝐶𝐶𝑎𝑎𝑃𝑃𝑃𝑃

Gini

𝐺𝐺𝑖𝑖𝑛𝑛𝑖𝑖
minS ��𝐶𝐶𝑃𝑃𝑃𝑃

�� = minS � � �Δ

𝑖𝑖
𝑐𝑐𝑡𝑡

∈𝑃𝑃𝑃𝑃

𝑖𝑖
�𝑐𝑐𝑡𝑡

��

� = minS

2𝑡𝑡|𝑗𝑗 − 𝑘𝑘|
𝑖𝑖
𝑛𝑛 ∑ 𝑦𝑦𝑖𝑖

�

��
𝑗𝑗<𝑘𝑘

Minimising the expression above can be seen as the problem of choosing the specific 
pairs 
particular case of the Earth Mover’s Distance (EMD) problem, of finding the flow 

. Furthermore, it can be written as a 

  and the corresponding transfer 

𝑗𝑗, 𝑘𝑘

𝑡𝑡

𝑖𝑖𝑖𝑖𝑗𝑗

ECB Statistics Paper Series No 31 / December 2019 53 
 
 
 
 
 
 
 
that minimises the EMD: 
and 

. In the case of only Pigou-Dalton transfers, 
𝑖𝑖,𝑗𝑗

∑ 𝑖𝑖𝑖𝑖𝑗𝑗𝑚𝑚𝑖𝑖𝑗𝑗

𝑚𝑚𝑖𝑖𝑗𝑗

  where 

  is the distance between elements 
. 
  and 

𝑖𝑖

𝑗𝑗

We then have: 

𝑚𝑚𝑖𝑖𝑗𝑗 = 𝑖𝑖 − 𝑗𝑗

∀𝑖𝑖 ≤ 𝑗𝑗, 𝑖𝑖𝑖𝑖𝑗𝑗 = 0 

𝑗𝑗

𝐺𝐺𝑀𝑀𝑃𝑃 = ∑ ∑ 𝑖𝑖𝑖𝑖𝑗𝑗(𝑖𝑖 − 𝑗𝑗)
given by individual 
∑ 𝑗𝑗  ∑ 𝑖𝑖𝑖𝑖𝑗𝑗
𝑖𝑖
= ∑ 𝑖𝑖 𝑃𝑃𝑖𝑖
implies that the EMD does not depend on the exact sequence of flows 
𝑖𝑖

− ∑ ∑ 𝑗𝑗 𝑖𝑖𝑖𝑖𝑗𝑗
𝑗𝑗<𝑖𝑖
𝑖𝑖
  is the total wealth received by individual 
𝑗𝑗<𝑖𝑖
𝑃𝑃𝑖𝑖 = ∑ 𝑖𝑖𝑖𝑖𝑗𝑗

𝑖𝑖
  and 
− ∑ 𝑗𝑗 𝑅𝑅𝑗𝑗
𝑗𝑗

= ∑ 𝑖𝑖 (𝑃𝑃𝑖𝑖 − 𝑅𝑅𝑖𝑖)

𝑅𝑅𝑗𝑗 = ∑ 𝑖𝑖𝑖𝑖𝑗𝑗

𝑖𝑖
= ∑ 𝑖𝑖 𝑃𝑃𝑖𝑖

. This 
  but rather 
𝑗𝑗

𝑖𝑖>𝑗𝑗

𝑖𝑖>𝑗𝑗

−

𝑗𝑗<𝑖𝑖

𝑖𝑖

𝑖𝑖

  where 
= ∑ 𝑖𝑖 ∑ 𝑖𝑖𝑖𝑖𝑗𝑗
𝑗𝑗<𝑖𝑖

  is the total wealth 

only on the total wealth to be received or given. 

𝑖𝑖𝑖𝑖𝑗𝑗

The EMD is thus independent from the specific transfers, as long as they are all 
Pigou-Dalton (symmetrically, this holds when all transfers inside the LCG are 
anti-Pigou-Dalton). This means that, inside an LCG, the choice of specific 
Pigou-Dalton transfers is irrelevant to the group’s contribution to the total change in 
the Gini, and the contribution of the Pigou-Dalton transfers is thus minimal. 

Expressions for decompositions of inequality indicators 

  represent the complete population formed by 

Population sub-groups: let 
households that can be partitioned into 
that 
  and 

𝑋𝑋

. 

  sub-groups of 

  households each, such 

𝑛𝑛

  is the sample average of sub-group 

. 

𝑛𝑛𝑘𝑘

𝐾𝐾
𝑘𝑘=1

𝐾𝐾

𝐾𝐾
𝑋𝑋 = ⋃ 𝑋𝑋𝑘𝑘
𝑘𝑘=1

𝑛𝑛𝑘𝑘

𝑛𝑛 = ∑

𝑋𝑋𝑘𝑘
𝑦𝑦𝑘𝑘���
Variable components: the variable net wealth can be expressed as a sum of 
components such that 
household 
  is the sample average of component 
𝑦𝑦𝑖𝑖𝑖𝑖
distribution of the wealth component 

  is the value of component 

𝑦𝑦𝑖𝑖 = ∑

  where 

𝑅𝑅
𝑖𝑖=1

𝑦𝑦𝑖𝑖𝑖𝑖

. 

. 

. 

𝑟𝑟

  for 
𝑅𝑅

  denotes the theoretical 

𝑖𝑖

𝑦𝑦𝑖𝑖�

𝑟𝑟

𝐹𝐹𝑖𝑖

Table A.2 
Decomposition by population sub-groups – Generalised Entropy 

𝑟𝑟

Indicator 

Within   

between 

within

𝑮𝑮𝑮𝑮𝜶𝜶(𝑿𝑿)
=  

+

where  GE

𝐾𝐾

𝑛𝑛𝑘𝑘
𝑛𝑛

𝑦𝑦�𝑘𝑘
𝑦𝑦�

�

�

�
𝑘𝑘=1

𝛼𝛼 GE

𝛼𝛼(𝑋𝑋𝑘𝑘)  

1

𝛼𝛼(𝑋𝑋𝑘𝑘)   =

𝛼𝛼(𝛼𝛼−1)

1
𝑛𝑛𝑘𝑘
𝑖𝑖=1
𝑛𝑛𝑘𝑘 ∑ ��

𝛼𝛼

𝑦𝑦𝑖𝑖𝑘𝑘
𝑦𝑦�𝑘𝑘 �

− 1�

where  GE

𝐾𝐾
𝛼𝛼(∪𝑘𝑘=1

∗
𝑋𝑋𝑘𝑘

1

) =

𝛼𝛼(𝛼𝛼−1)

𝛼𝛼

1
𝐾𝐾
𝑘𝑘=1
𝐾𝐾 ∑

𝑦𝑦�𝑘𝑘
𝑦𝑦� �

��

− 1�

Between 

GE

𝐾𝐾
𝛼𝛼(∪𝑘𝑘=1

∗
𝑋𝑋𝑘𝑘

)

Table A.3 
Decomposition by non-overlapping population sub-groups – Gini coefficient 

Indicator 

Within (

) 

Gini

within

between

=

+

= 𝑮𝑮𝒘𝒘
+ 𝑮𝑮𝒏𝒏𝒏𝒏

𝑮𝑮𝒘𝒘

𝑛𝑛𝑘𝑘𝑦𝑦𝑘𝑘���
𝑛𝑛𝑦𝑦�

𝐾𝐾

� 𝐺𝐺𝑘𝑘 𝑛𝑛𝑘𝑘

𝑘𝑘=1

𝑛𝑛

Between (

) 

𝑮𝑮𝒏𝒏𝒏𝒏

𝐾𝐾

𝑘𝑘−1

� � 𝐺𝐺𝑘𝑘ℎ �
𝑘𝑘=1
ℎ=1

𝑛𝑛𝑘𝑘
𝑛𝑛

𝑛𝑛ℎ𝑦𝑦ℎ���
𝑛𝑛𝑦𝑦�

+

𝑛𝑛ℎ
𝑛𝑛

𝑛𝑛𝑘𝑘𝑦𝑦𝑘𝑘���
𝑛𝑛𝑦𝑦�

�

where 

𝑛𝑛𝑘𝑘
𝑖𝑖=1

∑

𝐺𝐺𝑘𝑘 =

∑

𝑛𝑛𝑘𝑘
�𝑦𝑦𝑖𝑖𝑘𝑘−𝑦𝑦𝑗𝑗𝑘𝑘�
𝑗𝑗=1
2
2𝑛𝑛𝑘𝑘

𝑦𝑦𝑘𝑘����

where 

𝑛𝑛𝑘𝑘
𝑖𝑖=1

∑

𝑛𝑛ℎ
𝑗𝑗=1

∑

�𝑦𝑦𝑖𝑖𝑘𝑘−𝑦𝑦𝑗𝑗ℎ�

𝐺𝐺𝑘𝑘ℎ =

𝑛𝑛𝑘𝑘𝑛𝑛ℎ(𝑦𝑦𝑘𝑘����+𝑦𝑦ℎ����) =

𝑦𝑦𝑘𝑘����−𝑦𝑦ℎ����

𝑦𝑦𝑘𝑘����+𝑦𝑦ℎ����

Note: Case of non-overlapping sub-groups: 

  such that 

. 

𝑘𝑘, ℎ ∈ {1, … , 𝐾𝐾}

min𝑦𝑦𝑖𝑖(𝑋𝑋𝑘𝑘) > max𝑦𝑦𝑖𝑖(𝑋𝑋ℎ)

ECB Statistics Paper Series No 31 / December 2019 54 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Table A.4 
Alternative decomposition by population sub-groups – Gini coefficient 

Indicator 

Gini

𝑲𝑲

= � 𝑻𝑻𝒌𝒌
𝒌𝒌=𝟏𝟏

Contribution of sub-group 

𝒌𝒌

𝑇𝑇𝑘𝑘 =

𝑛𝑛𝑘𝑘
𝑛𝑛

𝑦𝑦𝑘𝑘���
𝑦𝑦�

𝐶𝐶𝑘𝑘

  is the concentration coefficient of the sub-group vector 

. The elements of the 

sub-group are 
𝐶𝐶𝑘𝑘
Gini coefficient of the curve that plots the cumulative proportions of vector 

  and the concentration coefficient is computed as the 

𝑥𝑥𝑘𝑘

  against 

𝑥𝑥𝑖𝑖𝑘𝑘 = 𝑦𝑦𝑖𝑖𝟙𝟙{𝑖𝑖∈𝑋𝑋𝑘𝑘}

the cumulative proportions of the total population ordered ascendingly according to 
their wealth. 

𝑥𝑥𝑘𝑘

Table A.5 
Decomposition by variable components – Gini coefficient 

Indicator 

Gini correlation 

Gini of component 

Share of component 

Gini

𝑹𝑹

𝑹𝑹

= � 𝑷𝑷𝒓𝒓𝑮𝑮𝒓𝒓𝑺𝑺𝒓𝒓

𝒓𝒓=𝟏𝟏

= � 𝑪𝑪𝒓𝒓𝑺𝑺𝒓𝒓
𝒓𝒓=𝟏𝟏

cov
cov

(𝑦𝑦𝑖𝑖, 𝐹𝐹)
(𝑦𝑦𝑖𝑖, 𝐹𝐹𝑖𝑖)
Concentration coefficient or pseudo-Gini: 

𝑃𝑃𝑖𝑖 =

𝐺𝐺𝑖𝑖 =

cov

𝒓𝒓

2

(𝑦𝑦𝑖𝑖, 𝐹𝐹𝑖𝑖)
𝑦𝑦𝑖𝑖�

𝒓𝒓

𝑆𝑆𝑖𝑖 =

𝑦𝑦𝑖𝑖�
𝑦𝑦�

𝐶𝐶𝑖𝑖 = 𝑃𝑃𝑖𝑖𝐺𝐺𝑖𝑖

  is the concentration coefficient of the wealth component 

. It is computed as the 

Gini coefficient of the curve that plots the cumulative proportions of wealth component 
𝐶𝐶𝑖𝑖
  against the cumulative proportions of the total population ordered ascendingly 
according to their total wealth. 
𝑟𝑟

𝑟𝑟

Table A.6 
Marginal effects of variable components 

Indicator 

Elasticity with respect to the mean   

Pseudo-indicator 

Gini 

Generalised Entropy 

GE

Theil (GE(

)) 

𝜶𝜶 = 𝟏𝟏

Mean log deviation (GE(

)) 

𝜶𝜶 = 𝟎𝟎

Pietra 

𝜂𝜂𝑖𝑖(𝐺𝐺) =

𝑆𝑆𝑖𝑖(𝐶𝐶𝑖𝑖 − 𝐺𝐺)
𝐺𝐺
GE

GE

𝜂𝜂𝑖𝑖(

(𝛼𝛼)) =

𝛼𝛼𝑆𝑆𝑖𝑖�

GE

𝜂𝜂𝑖𝑖(

(1)) =

GE

𝑆𝑆𝑖𝑖�

′
GE
(𝛼𝛼)𝑖𝑖

−
(𝛼𝛼)

GE

′
GE
(1)𝑖𝑖

−
(1)

(1)�

GE

𝜂𝜂𝑖𝑖(

(0)) =

GE
(𝑆𝑆𝑖𝑖 − 𝑆𝑆𝑖𝑖� )
(0)

𝜂𝜂𝑖𝑖(𝑃𝑃) =

′
𝑆𝑆𝑖𝑖(𝑃𝑃𝑖𝑖
𝑃𝑃

− 𝑃𝑃)

cov

2

𝐶𝐶𝑖𝑖 =

(𝑦𝑦𝑖𝑖, 𝐹𝐹)
𝑦𝑦𝑖𝑖�
𝑛𝑛

GE

(𝛼𝛼)�

′
(𝛼𝛼)𝑖𝑖

  =

1
𝛼𝛼(𝛼𝛼 − 1)

1
𝑛𝑛

� �
𝑖𝑖=1

𝑦𝑦𝑖𝑖𝑖𝑖
𝑦𝑦𝑖𝑖�

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

�

𝛼𝛼−1

− 1�

GE

𝑛𝑛

′
(1)𝑖𝑖

  =

1
𝑛𝑛

�
𝑖𝑖=1

𝑦𝑦𝑖𝑖𝑖𝑖
𝑦𝑦𝑖𝑖�

ln �

𝑦𝑦𝑖𝑖
𝑦𝑦�

�

𝑁𝑁/𝐴𝐴

′
𝑃𝑃𝑖𝑖

=

1
𝑛𝑛

�
𝑖𝑖,𝑦𝑦𝑖𝑖>𝑦𝑦�

𝑦𝑦𝑖𝑖𝑖𝑖 − 𝑦𝑦𝑖𝑖�
𝑦𝑦𝑖𝑖�

ECB Statistics Paper Series No 31 / December 2019 55 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Table A.7 
Decomposition of changes in time by variable components 

Indicator 

Change in share   

Change in 
inequality   

change in share

change in inequality 

+

change in share

change in inequality 

𝐶𝐶𝑖𝑖,𝑡𝑡 + 𝐶𝐶𝑖𝑖,𝑡𝑡−1
2

�𝐺𝐺𝑖𝑖,𝑡𝑡 + 𝐺𝐺𝑖𝑖,𝑡𝑡−1�
2

−

� �
𝑖𝑖

� Δ𝑆𝑆𝑖𝑖,𝑡𝑡

𝑆𝑆𝑖𝑖,𝑡𝑡 + 𝑆𝑆𝑖𝑖,𝑡𝑡−1
2

�
𝑖𝑖

Δ𝐶𝐶𝑖𝑖,𝑡𝑡

+

�(𝐶𝐶𝑖𝑖 − 𝐺𝐺)
𝑖𝑖

𝑆𝑆𝑖𝑖̇

� 𝑆𝑆𝑖𝑖𝐶𝐶𝑖𝑖̇
𝑖𝑖

𝚫𝚫𝑮𝑮𝒕𝒕 =

𝚫𝚫𝑮𝑮𝒕𝒕 =

Complementing charts 

Chart A.1 
Absolute change in Atkinson(1) with 95% confidence interval 

)
1
(
n
o
s
n
k
t
A
n

i

i

e
g
n
a
h
c
e

t

l

u
o
s
b
a

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

BE

DE

GR

ES

FR*

IT

CY

LU MT

NL

AT

PT

SK

FI

euro
area

Sources: HFCS and authors’ calculations. 

Chart A.2 
Absolute change in Theil with 95% confidence interval 

l
i

e
h
T
n

i

e
g
n
a
h
c
e

t

l

u
o
s
b
a

1.0

0.5

0.0

-0.5

-1.0

BE

DE

GR

ES

FR*

IT

CY

LU

MT

NL

AT

PT

SK

FI

euro
area

Sources: HFCS and authors’ calculations. 

ECB Statistics Paper Series No 31 / December 2019 56 
 
 
 
 
 
 
 
 
 
 
 
 
 
Acknowledgements 
We are grateful for discussions that took place at the meetings of the Household Finance and Consumption Network while the first draft of 
the paper was being prepared. We also acknowledge the insightful comments of the referees. Any errors are however entirely our own. At 
the time of writing, Rita was working at the European Central Bank. 

Rita Neves Costa 
European Central Bank, Frankfurt am Main, Germany; email: costa.ritaneves@gmail.com 

Sébastien Pérez-Duarte 
European Central Bank, Frankfurt am Main, Germany; email: sebastien.perez_duarte@ecb.europa.eu 

© European Central Bank, 2019 

Postal address 
Telephone 
Website 

60640 Frankfurt am Main, Germany 
+49 69 1344 0 
www.ecb.europa.eu 

All rights reserved. Any reproduction, publication and reprint in the form of a different publication, whether printed or produced 
electronically, in whole or in part, is permitted only with the explicit written authorisation of the ECB or the authors. 

This paper can be downloaded without charge from the ECB website or from RePEc: Research Papers in Economics. Information on all of 
the papers published in the ECB Statistics Paper Series can be found on the ECB’s website. 

PDF 

ISBN 978-92-899-3463-3, ISSN 2314-9248, doi:10.2866/957288, QB-BF-19-001-EN-N