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STONE CENTER ON SOCIO-ECONOMIC INEQUALITY
WORKING PAPER SERIES
No. 87
Compositional Inequality
Measurement, Stylized Facts, and Theoretical Aspects
Marco Ranaldi
May 2024
REVISED
February 2025
COMPOSITIONAL INEQUALITY
Measurement, Stylized Facts, and Theoretical Aspects
Marco Ranaldi *
University College London
February 3, 2025
Abstract
This paper explores the concept, measurement, principal stylized facts, and
theoretical aspects of compositional inequality. Compositional inequality refers
to how the shares of capital and labor income vary along the income distribution.
This analysis is valuable for several reasons. From a macroeconomic perspective,
it elucidates the link between functional and personal distributions of income,
which is crucial for addressing the drivers of income inequality in a context ris-
ing capital share. From a comparative economic perspective, it locates economic
systems on the continuum between two extremes: classical capitalism, where the
rich earn predominantly from capital and the poor from labor, and new capitalism,
where the composition of capital and labor is uniform across the distribution. We
refer to the entire range of systems along this continuum as the distributional vari-
eties of capitalism. Recent empirical studies indicate that, in most countries, we are
far from classical capitalism, though with notable exceptions, such as Latin Ameri-
can countries. This underscores the need to evaluate the benefits of compositional
equality. The paper concludes that compositional equality is desirable for at least
two reasons: it promotes fairness and supports an inclusive, profit-driven regime
of accumulation and growth.
JEL-Classification: C46, D31, D33, D63, F63, H27, P10
Keywords: Capital and Labor, Inter-Personal Income Inequality, Functional Income
Distribution, Inequality Measurement.
*I wish to thank Laura Acevedo Schoenbohm, Ludovica Ardente, Stefano Di Bucchianico, Jan Fidr-
muc, Stefano Filauro, Roberto Iacono, all the seminar participants at Sant’Anna (Pisa) and at the Uni-
versity of los Andes (Bogotá) and two anonymous reviewers for the helpful feedback. All mistakes
remain my own.
1
1 Introduction
One of the central findings in Piketty’s work is the positive association between
the rise in the capital share of income and income inequality (Piketty,2014). Piketty
argues that capital income tends to be concentrated at the upper end of the income
distribution, leading to an increase in inter-personal income inequality as the overall
share of capital relative to output grows. While Piketty posits a positive association
between the functional — the composition of GDP in terms of profits and labor com-
pensation — and personal distributions of income in his analysis, the strength of this
relationship can vary significantly across countries and over time. Building upon a
recent body of research on this topic, this paper argues that a precise assessment of
the relationship between functional and personal income distributions necessitates
the introduction of the concept of compositional inequality, as introduced in Ranaldi
(2022). Moreover, examining this relationship offers valuable insights into different
types of capitalist systems, contributing to and enriching the comparative political
economy debate, as highlighted in the work of Milanovic (2017,2019) and Ranaldi
and Milanovic (2022).
Compositional inequality refers to how the shares of capital and labor income
vary along the income distribution. When compositional inequality is high, income
sources are distinct: the rich primarily derive their income from wealth, while the
poor rely on income from labor. Conversely, under low compositional inequality,
both the rich and the poor individuals earn income from capital and labor in sim-
ilar proportions. In simpler terms, high compositional inequality results in a clear
divide where the rich are capitalists, and the poor are workers, whereas low compo-
sitional inequality implies that both rich and poor individuals simultaneously hold
the roles of capitalists and workers. In cases of high (low) compositional inequal-
ity, an increase in the capital share automatically leads to greater (or lesser) inter-
personal income inequality. Thus, compositional inequality serves as a measure of
the strength of the link between the functional and personal distributions of income.
Furthermore, compositional inequality can shed light on different types of capitalist
2
systems. High compositional inequality characterizes what we refer to as classical cap-
italism, where capital income is concentrated among a few affluent individuals, while
the labor share of income is primarily associated with the impoverished majority. In
contrast, low compositional inequality describes a new capitalism in which everyone
benefits from capital accumulation while also suffering from capital destruction. We
refer to the entire range of systems along this continuum as the distributional varieties
of capitalism.
This article provides an overview of the key methodological tools and empirical
findings related to compositional inequality, laying the foundation for future research
in this area. It argues that the study of compositional inequality allows us to go be-
yond the unidimensional nature of income inequality (which informs us whether and
to what extent there are rich and poor individuals in society) and explore who these
individuals are, based on their income composition. Rather than directly address-
ing economic inequality, the concept of compositional inequality addresses political
economy instead, insofar as it describes the existence of groups of individuals earn-
ing income from different sources. These groups of people are likely to have different
policy preferences, as well as different political views. This paper concludes by ad-
dressing the desirability of achieving a state of compositional equality.
This paper is organized as follows. Section 2provides a brief review of the liter-
ature on the relationship between factor shares and income inequality. Section 3in-
troduces the concept of compositional inequality, the main definitions of capital and
labor incomes, and its measurement. Section 4describes the principal stylized facts
regarding the dynamics of compositional inequality both between and within coun-
tries. Section 5discusses the desirability of compositional equality in light of recent
stylized facts affecting modern capitalist economies. Section 6concludes the paper
and discusses potential avenues for future research on the topic.
3
2 Literature
The study of the relationship between the functional and personal distribution of
income can be traced back to the work of classical political economists, especially
David Ricardo. Classical political economists implicitly assumed that the dynamics
of aggregate factor shares determined income inequality, based on the simple obser-
vation that different income sources were earned by distinct groups of individuals oc-
cupying different segments of the total income distribution.1In his Principles, Ricardo
stated that the functional distribution of national income between wages, profits, and
rents was the central problem of political economy. However, it was only in the sec-
ond half of the twentieth century that this topic took a prominent role in economic
theory, particularly with the work of Kaldor (1961).
Kaldor, in the process of constructing a theoretical model of capital accumulation
and economic growth, proposed a set of stylized facts that any model describing the
development of capitalist societies should satisfy.2Among these, he emphasized the
stability of the capital share of income, highlighting the strong correlation between
the share of profits in income and the share of investment in output. This stylized
fact was contextualized within an economic system where the investment-to-output
ratio remained constant, with Kaldor using the United Kingdom as an example.
After a long period of silence, in his Presidential Address to the Royal Economic
Society, Atkinson (1997) argued that income distribution should be brought back from
the cold. He stressed that the study of the functional distribution of income is cru-
cial for understanding the broader dynamics of income distribution among individ-
uals.3In a similar vein, Brandolini (1992) contended that economic systems shape
“entitlement rules", which in turn influence the composition of income distribution.
These entitlement rules can also generate political tensions between different income
groups, particularly between capital-abundant and labor-abundant individuals.
1See Milanovic (2023) for a historical perspective on this matter.
2A similar attempt to summarize recent stylized facts on the distribution of income and wealth was
made by Stiglitz (2016).
3For a historical discussion of the reasons for the decline in the study of income distribution during
this period, see Milanovic (2023).
4
Later, Atkinson (2009), building on Glyn (2011), stressed the importance of factor
shares as a central issue in modern political economy. He identified three key reasons
for their relevance: (i) linking macroeconomic-level incomes (national accounts) to
individual incomes; (ii) improving our understanding of inequality in the personal
distribution of income; (iii) addressing concerns of social justice by evaluating the
fairness of different income sources.
Glyn (2011) also underscored that the study of the relationship between factor
shares and inequality remains relevant today for at least two reasons: wealth (particu-
larly high-yielding) remains highly concentrated, and employees’ sense of fairness is
undermined when corporate profits rise significantly faster than wages, leading to de-
mands for fairer income distribution. He further noted an ideological shift away from
a class-based view of the economy, reflected in the expansion of capital-funded occu-
pational pensions and homeownership. These arguments will be further explored in
the final theoretical section of this paper, following the discussion on compositional
equality.
In his important effort to identify a novel set of distributional stylized facts across
the world, Piketty (2014) challenged Kaldor’s assumption of a stable profit share,
showing that since the 1980s, the share of profits in many modern economies has
increased. This shift implies a gradual change in the composition of GDP. Addition-
ally, Piketty highlighted a strong rise in top-income shares, suggesting a historical
connection between functional and personal income distributions. As will be further
discussed later in this article, this connection has only been assumed, highlighting the
need for a thorough assessment of the relationship between these two distributions.
A series of empirical studies has since been devoted to examining this relation-
ship, using econometric techniques and assembling databases across countries and
years. However, the results have been conflicting. Bengtsson and Waldenstrom
(2018), using a novel historical cross-country dataset, provided further evidence of
a strong and growing link between functional and personal income distribution over
the past century. However, they acknowledged that this relationship has likely evolved
across different historical periods due to structural factors, including the role of in-
5
stitutions. Contrasting these findings, Francese and Mulas-Granados (2015) analyzed
data from up to 93 countries between 1970 and 2013 and concluded that labor and
capital income shares have not been a major factor in explaining income inequality.
Only recently, Milanovic (2017,2019) proposed a framework for understanding
this link using a Gini-like approach, arguing that the correlation between capital and
total income provides valuable insights into the structure of economic systems. Mi-
lanovic argues that the positive link between functional and personal income inequal-
ity holds if and only if two conditions are satisfied: i) the distribution of capital in-
come is more unequal than the distribution of labor income, and ii) there is a high
rank-correlation between capital and total income. As will be discussed in the fol-
lowing section, compositional inequality offers an alternative, more general frame-
work for analyzing the relationship between functional and personal distributions,
carrying a strong political economy message.
3 Framework
3.1 Concept
Compositional inequality describes differences between rich and poor in terms
of their income typology. Two income sources are considered in this study: labor
income, stemming from work, and capital income, arising from wealth.4When com-
positional inequality is high, the rich and the poor are earning their income from dif-
ferent sources: the rich mainly from wealth, the poor from labor.5When, by contrast,
compositional inequality is minimal, the rich and poor are earning income from cap-
ital and labor in the same proportion: for example, the twenty percent from wealth
and the remaining eighty percent from labor. The following definition formally de-
scribes the conditions underlying minimal compositional inequality:
4Although this study concentrates on compositional inequality in terms of capital and labor income,
the framework can be readily adapted to examine the composition of wealth in real and financial
assets, or the division of income into savings and consumption, among other decompositions.
5In this paper, we adopt the convention where maximal compositional inequality is characterized
by capital income at the top of the distribution and labor income at the bottom. A generalization of
this definition, both conceptually and analytically, is discussed in Ranaldi (2022).
6
Definition 3.1. Compositional inequality is minimal across a population when each individ-
ual has the same population shares of capital, π, and labor, w. Formally, this happens when
Wi
Πi
=w
π∀i, with Wiand Πirepresenting individual i’s labor and capital income, respectively.
Under minimal compositional inequality, individuals’ income shares mirror the
aggregate composition of GDP into capital and wages, assuming that the definitions
of capital and labor are comparable (further discussion on this follows in subsequent
paragraphs). The next definition formally describes instead the conditions underly-
ing maximal compositional inequality:
Definition 3.2. Compositional inequality is maximal when the bottom p%of the income
distribution has an income consisting only of the source z=π, wand the top (1 −p)% of the
income distribution has an income consisting only of the source z−.
Different compositional inequality scenarios – high and low compositional in-
equality – can occur even at identical levels of income inequality. Such different sce-
narios can therefore both describe societies with rich and poor individuals. What
makes them different lies in the income composition of these two groups of individ-
uals. Compositional inequality can thus characterize various typologies of capitalist
systems and inform discussions on comparative economic systems. High composi-
tional inequality describes classical capitalism (Milanovic,2017,2019), where capital
income is concentrated in the hands of a few rich individuals, while the labor share
of income is associated almost entirely with the poor majority. Low compositional in-
equality, by contrast, describes a new capitalism Milanovic (2017,2019). While classical
capitalism, characterized by the stark distinction between affluent capital owners and
poor workers as depicted by classical political economists like Marx and Ricardo,
dominated economies of past centuries, new capitalism represents a contemporary
form of capitalist configuration. This recent trend will be further explored in the em-
pirical section, where we will show how countries around the world range from high
to low compositional inequality. We refer to this type of analysis as the study of dis-
tributional varieties of capitalism.
From a macroeconomic standpoint, compositional inequality in capital and labor
income links the functional and personal distributions of income. The functional dis-
tribution of income describes how a nation’s output is divided into capital and labor
7
compensation. If individuals at the top of the total income distribution solely receive
all the capital income (as characterized by a maximal level of compositional inequal-
ity), then any rise in the share of capital income will inevitably augment the incomes
of those at the top, provided all other factors remain constant. This automatically
implies an increase in inter-personal income inequality. Section 3.3.1 describes this
relationship from a methodological standpoint.
3.2 Definitions of Capital and Labor Incomes
One of the key challenges in empirically studying the relationship between the
functional and personal distribution of income lies in harmonizing the definitions of
capital and labor at the macro and micro levels. As Flores (2021) points out, house-
hold surveys tend to underestimate the total stock of capital income reported in na-
tional accounts. This discrepancy arises not only from the limitations of survey data
in capturing capital income due to under-reporting or missing units,6but also from
the fact that while survey data reflect household-sector income, national accounts in-
clude additional components that do not directly appear in household balance sheets,
such as undistributed corporate profits. While one could assume that undistributed
corporate profits are distributed proportionally to individuals based on their capital
income, this is a strong assumption, given that such profits are typically reinvested in
the company rather than directly accruing to individuals.
Given these challenges, studies of compositional inequality have so far relied on
definitions of capital and labor income confined to household-sector income. This
means that, within the context of the studies discussed in this paper, the relationship
between functional and personal income distribution must be understood through
the household-sector capital and labor shares. These shares tend to systematically
underestimate both the overall capital share of income and the labor share, as shown
by Flores (2021).
The primary objective of compositional inequality studies have been so far to con-
struct measures of market income to better capture the dynamics between capital and
6See Lustig (2020) for a taxonomy of issues associated with survey data.
8
labor before accounting for taxes and transfers. Specifically, capital income is typically
defined as the sum of interest,dividends, and rental income,7while labor income con-
sists of wages and salaries. However, the classification of two additional sources of
income—self-employment income and pensions—requires further discussion.
Self-employment income embodies both labor and capital components. However,
there is no precise approach for decomposing this source into capital and labor. For
this reason, some studies have treated self-employment income as purely labor in-
come, while others have split it under specific assumptions. A common approach
is the 30-70 split, where 30% of self-employment income is attributed to capital and
70% to labor. However, other studies have opted to allocate this income based on the
aggregate share of capital and labor in output.8
Turning to pensions, most studies consider them as deferred labor income, given
that they result from past labor. However, in many countries, pensions also stem from
past investments. For this reason, they may need to be divided into capital and labor
components. As will be shown in the following section, splitting pensions into capital
and labor components can significantly impact the level of compositional inequality,
especially in the context of Scandinavian countries.
Finally, two additional remarks should be made. First, it is important to note that
additional sources of income, such as capital gains, could potentially be included in
the definition of capital. However, identifying this source can be challenging, partic-
ularly when relying on survey data and conducting comparative analyses. Overall,
as will become evident in the stylized facts section, different definitions of capital and
labor have significant implications for the level of compositional inequality, as well
as for its trends, albeit to a lesser extent. Second, the definition of capital and labor
income could be informed by a more thorough assessment of economic classes. In a
recent article, Villani and Giangregorio (2024) propose a new estimation of the labor-
ers’ and capitalists’ share of income on the basis of a class analysis that hinges on both
7Rental income can also include imputed rent, which represents the income an individual would
earn if they rented out their owned property.
8See Iacono and Ranaldi (2023) for an evaluation of the impact of different self-employment splits
on compositional inequality in Italy.
9
economic and sociological considerations. The authors employ a separate treatment
of managers, who are considered more as capital than labor earners, in contrast with
the recent literature that has treated CEO compensations as wages.
3.3 Measurement
This section introduces the main statistical measures used to study compositional
inequality. It differentiates between measures of this inequality dimension across the
entire distribution (Section 3.3.1), and within specific sections of the distribution, such
as the top 10% (Section 3.3.2). Finally, it presents some methodological results that
underpin the primary mechanisms connecting compositional and income inequality,
from a technical perspective (Section 3.3.3).
3.3.1 Compositional Inequality Across the Distribution
To measure compositional inequality across the entire distribution, we utilize the
Income-Factor Concentration (IFC) index as proposed by Ranaldi (2022). Before in-
troducing the IFC indicator, let us recall the notion of concentration curve. If we rank
individuals according to their total income, Y, and define the cumulative share of in-
come source Sj, for each j=1,...,n, such that Y=S1+· · · +Sn, of the bottom p%of the
total income distribution as L(Sj,p)=Pp=i
n
i=1
Sj,i
Sj, then the pairs (p,L(Sj,p)) describe the
concentration curve for income source Sj(Kakwani,1977). Concentration curves are
therefore statistical tools useful for determining whether and to what extent income
sources are positively, or negatively, correlated with total income.
The construction of the IFC index involves three distinct concentration curves: (i)
the actual-concentration curve, which illustrates the actual distribution of capital in-
come across different income levels; (ii) the zero-concentration curve, representing a
hypothetical scenario where all individuals have an identical composition of capi-
tal and labor income; and (iii) the maximum-concentration curve, depicting a situation
where the poorest individuals earn exclusively labor income, and the wealthiest earn
solely capital income (Figure 1displays the three concentration curves for Italy in
1989). Drawing an analogy to the Gini coefficient, the actual-concentration curve cor-
10
responds to the Lorenz curve, the zero-concentration curve to the equality line (i.e.,
the bisector), and the maximum-concentration curve to the axes xand y.9
The IFC index is defined by the following equation:
I=A
B,(1)
where Arepresents the area between the zero-concentration and the actual-concentration
curves, and Bdenotes the area between the zero-concentration and the maximum-
concentration curves.10 The IFC index values range from 1to −1, where I=1implies
that capital income is concentrated at the top and labor income at the bottom of the
total income distribution (classical capitalism), while I=0represents a situation in
which all individuals have the same shares of capital and labor in their total income
(new capitalism).11 It can be easily shown that the elasticity of the personal income Gini
coefficient (G) to changes in the factor share z=π, wis a function of the difference be-
tween the areas of the two concentration curves, which are the main components of
the numerator of the IFC index. Formally:12
∂G
∂z≈A.(2)
Equation 2shows that the IFC index, whose sign is determined by the indicator
A, can be regarded as a measure linking the functional with the personal income dis-
tributions. Other methodological approaches focusing on the measurement of such a
link can, for instance, be found in Atkinson and Bourguignon (2000).13
9See Appendix Afor a formal description of the three concentration curves used to measure com-
positional inequality via the IFC index.
10For an alternative formulation of the IFC index that involves pseudo-Gini coefficients, specifically
the areas under concentration curves, refer to Ranaldi (2022).
11A negative value of the indicator implies a distribution where individuals at the bottom of the total
income distribution predominantly earn income from capital, while individuals at the top primarily
earn income from labor. Although theoretically possible, this scenario is far removed from real-world
distributions.
12More details on this result can be found in Ranaldi (2022).
13The authors approach the measurement of this link by decomposing the squared coefficient of
variation of income, where there are two types of income: wage income and capital income. Specifi-
cally, the coefficient of variation of income V2can be written as a function of the capital share of income
π, the inequality of wage income Vw, the inequality of capital income Vk, and the correlation ρbetween
wage income and capital income: V2=(1 −π)2V2
w+π2V2
k+2π(1 −π)ρVwVk. Now, if we define λas the
relationship between wage income dispersion and capital income dispersion, then a rise in the capital
11
Among the properties of the IFC, it is worth mentioning that it can be decomposed
into between and within components, as described by Monroy-Gómez-Franco and
Ranaldi (2024):
If=κ
L
|{z}
between
+M
|{z}
within
+C
|{z}
residual
,(3)
where L=RπGB
π−RwGB
w,M=RπGW
π−RwGW
w,C=ϵπ−ϵwand κ=πw
2B. The superscripts
W and B respectively refer to the within and between Gini coefficients of the specific
income source. This decomposition elucidates the extent to which the dynamics of
compositional inequality at global or regional scales are influenced by movements in
the functional distribution across countries (between-compositional inequality) or within
countries and across individuals (within-compositional inequality).
While the IFC index will serve as the principal indicator in this paper, an alter-
native measure of compositional inequality across the entire distribution can also be
considered: the rank-correlation coefficient between capital and total income. This
metric is employed by Milanovic (2017) in his analysis of various typologies of capi-
talism. Formally, this is defined as follows:
Rπ=cov(r(y), π)
cov(r(π), π),(4)
where r(y)and r(π)represent the rankings according to y(total income) and π(cap-
ital), respectively.14 Figure 2illustrates the correlation between the IFC index and
Milanovic’s metric across a selection of EU-SILC countries (0.88). The reason for such
a high correlation between these two metrics is related to the fact that, within the
context of the IFC, it is constructed through concentration curves. These curves relate
the cumulative share of a given income source, such as capital income, to the total
income ranking. This exercise is therefore similar to studying the correlation between
the ranking of yand capital income, as captured by Milanovic’s metric.
share of income is transmitted into personal income inequality only when the following condition is
satisfied: π > 1−λρ
1+λ2−2λρ . In this way, they manage to show the condition under which an increase in the
capital income share is transmitted into an increase in overall income inequality, as measured by the
standard deviation of income.
14For further development of this measure, especially in relation to demand regimes, consult Tonni
(2023).
12
3.3.2 Compositional Inequality at the Top
Similar to the analogy drawn between the Gini coefficient, which measures in-
come inequality across the entire distribution, and top income shares, which solely
address the upper part of the distribution, compositional inequality can also be anal-
ysed in specific sections of the distribution. Several indicators have for instance been
developed to study the composition of income at the top of the distributions.
Berman and Milanovic (2023) construct an indicator to measure the phenomenon
of homoploutia. Homoploutia, as defined by the authors, represents the proportion
of individuals who simultaneously belong to the top 10% in both labor and capital
income distributions. Formally, this can be expressed as follows:
H10,10 ≡10
n
N
X
i=1
1
top10%π
(i)×
1
top10%w
(i)=10 ×Pr(top10%w∩top10%π),(5)
which is equivalent to the following expression:15
H10,10 =Pr(top10%π|top10%w)=Pr(top10%w|top10%π).(6)
Equation 6describes the probability of being in the top decile of capital (labor)-
income recipients conditional on being in the top decile of labor (capital)-income
earners.
Another measure suitable for examining compositional inequality at the top is the
alignment coefficient, introduced by Atkinson (2007) and further extended by Lakner
and Atkinson (2021). This metric assesses the degree of concordance between rank-
ings based on income from factor mand total income. It is defined as follows:
Ai,m=
˜
Si,m
Si,m
,(7)
where ˜
Si,m=˜
Xi,m
Xmrepresents the share of total income from factor mreceived by the top
quantile iof total income recipients.
15This result immediately follows from the properties of conditional probability.
13
3.3.3 Income and Compositional Inequality
In this section, we describe the technical relationship between income and com-
positional inequality by illustrating several results on this matter. Let us start from
the decomposition of the Gini coefficient proposed by Lerman and Yitzhaki (1985) for
two income sources, notably capital (π) and labor (w):
G=πRπGπ+wRwGw,(8)
where Gis the Gini coefficient for total income, Gπis the Gini coefficient for capital,
Gwis the Gini coefficient for labor and, finally, Rπand Rware the correlation ratios
between the respective source and total income. As described in the previous section,
these two correlation ratios can be considered as elasticities of personal income Gini
to changes in the functional income distribution (Milanovic,2017).
Through the same decomposition, it is possible to write the Gini coefficient for
total income, G, similarly to a Cobb-Douglas production function, as follows (Ranaldi
and Milanovic,2022):
G=Gα
πGβ
w,(9)
where α=GπRππ
Gand β=1−α=GwRww
Gare the relative contributions of capital and
labor inequality to overall income inequality, respectively. This reformulation allows
us to establish an analytical link between income and compositional inequality, as
measured by the IFC index, as follows:
I
G=α−π
2B.(10)
When therefore α=πthe ratio tends to zero.
As discussed by Ranaldi (2019), another approach to analyzing the joint dynamics
of income and compositional inequality is to decompose the variations in the level
of income inequality, as measured by the Gini coefficient, into three main compo-
nents: movements in the functional income distribution, movements in the income-
factor concentration, and movements in the income-factor inequality. For each of
these three movements to affect income inequality, a specific transmission condition
14
can be derived. This decomposition follows from the Gini decomposition previously
considered Lerman and Yitzhaki (1985). Equation 8can in fact be further rearranged
to obtain:
G=w(RwGw−RπGπ)+RπGπ,(11)
and by further development, one can write:
2(˜µπ−˜µw)=RwGw−RπGπ,(12)
where ˜µπand ˜µware the areas of the concentration curves for capital and labor, re-
spectively. We can then write:
G=2w( ˜µπ−˜µw)+RπGπ.(13)
Let us now consider the income Gini coefficient as a function of time. As a conse-
quence, all the elements of the decomposition are also functions of time; hence we
can write Equation 8as:
G(t)=G(π(t),Rπ(t),Gπ(t),w(t),Rw(t),Gw(t)).
If we now take the total derivative of G(t)with respect to time, from Equation 13, we
obtain:
∂G
∂t
=2( ˜µπ−˜µw)∂w
∂t
+2w∂( ˜µπ−˜µw)
∂t
+Rπ
∂Gπ
∂t
+Gπ
∂Rπ
∂t.(14)
Let us now assume as follows:
∂Rπ
∂t
=−ϵ∂( ˜µπ−˜µw)
∂t,(15)
with ϵ∈R. At this point of the analysis, we can observe that the term ( ˜µπ−˜µw)is one
of the components of the numerator of the IFC index. As the term ( ˜µπ−˜µw)is the key
driver of changes in income-factor concentration, we can further rearrange Equation
14 to get the following relationship:
˙
G≈2˜
If(w)×˙w+(2w−Gπϵ)×˙
If(w)+Rπ×˙
Gπ,(16)
where ˜
If(w)=˜µπ−˜µw=If(w)
B(w)πwis the non-normalized income-factor concentration
index, and the dots indicate derivatives with respect to time. The expression If(w)
15
denotes the specification of the IFC index that measures the extent to which labor
income is distributed at the top of the total income distribution, and capital income
at the bottom, distinct from the previous specification. Note that, by construction, we
can write If(w)=−If(π)(and, therefore, ˙
If(w)=−˙
If(π)). In other words, the higher
the concentration of labor at the top (and capital at the bottom), the lower the concen-
tration of capital at the top (and labor at the bottom). Equation 16 thus illustrates the
relationship between variations in income and compositional inequality over time.
Finally, one can study the relationship between income and compositional in-
equality by decomposing the sources of growth across the distribution. Specifically,
we can decompose growth rates as a function of changes in capital and labor income
inequality, as follows. If we assume that the overall growth rates of total, capital,
and labor income are equal to zero, and therefore focus exclusively on distributional
changes, the following can be shown (Ranaldi,2025):
gi=˙
˜
Gπ Πi−¯
Π
Yi!+˙
˜
Gw Wi−¯
W
Yi!,(17)
where ˙
˜
Gπand ˙
˜
Gware the pseudo-Gini coefficients of capital and labor income changes.16
According to Equation 17, the two terms Πi−¯
Π
Yiand Wi−¯
W
Yidetermine the differential
growth rates, gi, across the income distribution under two specific tax and transfer
schemes for capital and labor income, as measured by the two pseudo-Gini coeffi-
cients.17 For example, a 1% reduction in the pseudo-Gini coefficient of capital income
implies a positive income growth rate for the percentiles of the income distribution
where capital income is below the average.
4 Stylized Facts
This section documents the main stylized facts regarding the level and dynam-
ics of compositional inequality. It begins with a comparative analysis of countries
16Pseudo-Gini coefficients are Gini-like indicators, which are constructed from concentration curves,
rather than Lorenz curves.
17This approach follows from the decomposition proposed by Lakner et al. (2020), which relies on
the tax and transfer scheme firstly introduced by Kakwani (1993), and then further extended by Fer-
reira and Leite (2003).
16
worldwide before delving into specific case studies. In particular, it examines Italy,
where compositional inequality has exhibited a decreasing trend over the past three
decades. It then explores the Czech Republic and Slovakia, highlighting their diver-
gent trajectories of compositional inequality following their split. Next, it discusses
the exceptional case of Scandinavian countries, which combine low levels of income
inequality with high levels of compositional inequality. Finally, it presents the first es-
timates of global compositional inequality. While this selection of countries does not
provide a comprehensive global picture, it aims to guide future research in this area,
encouraging further studies to complement the analysis and address key unresolved
puzzles. Moreover, this section aims to highlight how different definitions of capital
and labor income affect both the level and trends of compositional inequality.
4.1 Compositional Inequality Across the World
Let us start with an overview of compositional inequality across several coun-
tries around the world, reporting the main results from Ranaldi and Milanovic (2022).
The authors construct a comprehensive database for the study of compositional in-
equality from a comparative perspective. Data is taken from the Luxembourg In-
come Study (LIS) Database, and has a coverage of 47 countries from Europe, North
America, Oceania, Asia and Latin America (≈80% world output), including 302
country-representative household surveys between 1995 and 2018. The income con-
cept adopted is market income plus pensions. Specifically, capital income is defined
as the sum of interest incomes, dividends, and rental incomes, while labor income
is the sum of wages, self-employment income and pensions, here considered as de-
ferred labor income. A second definition of income splits pensions into a private
(capital) and public (labor) components. Finally, the unit considered is that of the
individual.
Figure 3plots the level of income inequality against compositional inequality for
all countries in the sample. Three main results emerge from this graph. First, the
higher the compositional inequality, the higher the income inequality. In other words,
17
classical capitalism displays higher interpersonal income inequality than new capital-
ism. Second, three world clusters emerge from this analysis. The first cluster includes
Latin American countries, which are, on average, characterized by high levels of both
compositional and income inequality. These countries can, therefore, be considered
classical capitalist economies with extremely high interpersonal inequality. This re-
sult is further confirmed by Monroy-Gómez-Franco and Ranaldi (2024), who focus
on compositional inequality in Latin America (Figure 4). Their confirmation comes
through the use of a more comprehensive database that combines fiscal data, survey
data, and national accounts recently produced by De Rosa et al. (2023). The sec-
ond cluster is composed of western countries including the US, Canada and the UK.
This cluster is characterized by mild levels of both inequality dimensions. Finally,
the third cluster is the Nordic countries, which surprisingly combine high levels of
compositional inequality with low levels of income inequality. Third, the final result
emerging from this graph is actually a non-result: there seems not to be evidence
of any countries combining both low compositional inequality and extremely high
income inequality. In other words, new capitalism seems to be characterized by mod-
erate levels of interpersonal inequality. This result sheds light on the desirability of
achieving compositional equality, which will be further discussed in Section 5.
4.2 Italy
Italy experienced a steady decline in compositional inequality over the last three
decades, as evidenced by (Iacono and Ranaldi,2023) exploiting data from the Italian
Survey of Household Income and Wealth (SHIW) (see Figure 5). In other words,
Italy is moving from classical to new capitalism, with labor income flowing towards
individuals at the top of the distribution, given the surge surge in high-skilled jobs,
and capital income, especially in the form of imputed rent, which increases the capital
share of the middle class.18 Actual rent has, however, led to higher concentration of
18Focusing exclusively on wealth, housing constitutes the primary component of wealth for the
middle classes, while financial assets are more concentrated at the top of the wealth distribution, as
evidenced by Acciari et al. (2024).
18
capital incomes at the top in the decade preceding the outbreak of the financial crisis.
Villani and Giangregorio (2024) estimate the IFC index for Italy based on their
own class-based classification of laborers and capitalists. As previously mentioned,
this classification aims to better incorporate sociological considerations of class into
the composition of individual incomes. For example, managers are directly treated
as capitalists instead of wage earners, as is typically done in the literature. Their
results show that while IFC trends remain unaffected by the different categorizations
of capital and labor across individuals, the level is significantly higher in their class-
based formulation of the IFC. This also shows how different definitions of capital and
labor can impact the level of compositional inequality as measured by the IFC.
Lastly, the capitalization process in Italy does not seem to have kept pace with
global developments. As recently shown by Ranaldi (2024), which examines the as-
sociation between national and global distributions of capital and labor income in
Italy over the last three decades, Italians have lost ground in global rankings for both
capital and labor income up until 2016 (Figure 13). However, there have been signs
of recovery since then, though not to the levels seen three decades ago. This decline
in global standing has been uneven across Italian regions; for example, the South
and Centre of Italy have experienced a drop of up to 40 positions in the global labor
income distribution. Finally, while transfer incomes elevated the bottom half of the
distribution to global middle-class standards in the early 1990s, this is no longer the
case today.
4.3 Czech Republic and Slovakia
We now examine the dynamics of compositional inequality in the Czech Republic
and Slovakia from 1996 to 2015, reporting a novel stylized fact based on data from
Ranaldi and Milanovic (2022). The analysis of these two countries is highly rele-
vant, considering that during the period in question, they were initially part of the
same state, Czechoslovakia, before splitting in 1993 following the collapse of the for-
mer Soviet Union. Subsequently, the two countries pursued relatively different eco-
19
nomic transitions: while the Czech Republic rapidly implemented neoliberal market
reforms, Slovakia adopted a more gradual approach. This situation changed in the
late 1990s when Slovakia accelerated its economic reforms under a new administra-
tion, aiming to secure admission to the EU.19
Compositional inequality was the same in the two countries between 1996 and
2003 and, then, substantially diverged following their entrance into the European
Union: Czech Republic settled at a high compositional inequality regime, whilst Slo-
vakia at a low compositional inequality regime (see Figure 6). Currently, their differ-
ence is striking: Czech Republic displays a level of compositional inequality similar
to those of Latina American countries, which represent the most compositionally un-
equal region of the world, whilst Slovakia one of the lowest level of compositional
inequality in the world (Figure 3). The level and trend of income inequality in the
two countries is, instead, almost the same (Figure 7). Hence, while displaying the
same level and trend of total income inequality, Czech Republic and Slovakia are
moving towards two, distinct forms of capitalist systems.
While exploring the determinants of these two diverging trends is beyond the
scope of this study, one can argue that the differing paces of privatization processes—slow
in Slovakia and rapid in the Czech Republic—may have contributed to this diver-
gence. A similar dynamic is observable in the analysis of capital concentration in
China and Russia as these countries transitioned to market economies at the begin-
ning of the 1990s. With a gradual strategy, China slowed the pace of capital concen-
tration at the top, unlike Russia, which adopted a shock therapy approach (Novok-
met et al.,2018,Yang et al.,2019). Labor income dynamics across the distribution may
also have been a significant factor in explaining the diverging trends between the two
countries, as observed in many countries in transition (Estrin et al.,2009). However,
when analyzing the dynamics of the IFC index by separating movements in capital
and labor income, as measured by the areas of the concentration curves for each,20 it
19For a comprehensive overview of the economic development of the two countries, refer to Stolarik
(2016).
20More on the determinants of compositional inequality in Section 5.
20
is evident that capital income dynamics have played the largest role in shaping the di-
vergence in compositional inequality between the two countries. Figure 8illustrates
that although labor income dynamics have been similar across the two countries,
capital income dynamics have differed significantly: it shifted towards the top of the
income distribution in the Czech Republic, while moving towards the bottom in Slo-
vakia, particularly starting from 2003. As previously mentioned, while this result
does not shed any light on the causal mechanisms behind the divergence in compo-
sitional inequality between the two countries, it sets the stage for future research on
the topic.
4.4 Scandinavian Countries
Scandinavian countries, as already mentioned in Section 4.1, are exceptional as
they combine high compositional inequality with low interpersonal income inequal-
ity (Figure 9). The Scandinavian model of economic governance, notably established
through a significant compromise between capital and labor in the 1930s, has facil-
itated a unique form of capital accumulation and income distribution. According
to Rojas (1991) and Esping-Andersen (1990), this model preserves the rights to capi-
talist accumulation while implementing centralized bargaining of wages, creating a
balance that supports economic stability and growth. Moene and Wallerstein (2003)
and Moene (2003) further explain that this Nordic model harmoniously combines
wage compression with a "socially acceptable" high return on capital. This distinc-
tive arrangement results in a notably low inequality of earnings, as highlighted by
Fochesato and Bowles (2015), yet it paradoxically coexists with high wealth inequal-
ity, as documented by Davies et al. (2012). Furthermore, the region’s high social
mobility contributes to what is often referred to as Nordic exceptionalism in socio-
economic terms. Recent shifts in policy, such as the implementation of Dual Income
Tax reforms and the abolition of inheritance taxes in Sweden and Norway in 2004
and 2014, respectively, have however begun to alter the traditional composition of
income.
21
As noted by Iacono and Palagi (2022), the dual income taxation system signifi-
cantly influences the dynamics of compositional inequality in these countries. The
authors document a recent increase in compositional inequality in the Nordic coun-
tries, attributing this rise to the introduction of Dual Income Tax reforms in the early
1990s, characterized by a flat tax on capital incomes. Moreover, as shown by Ranaldi
and Milanovic (2022), the distinction between public and private pensions also plays
a critical role in shaping the level of compositional inequality. The authors show that,
when pensions are categorized into private and public, the cluster of Nordic countries
shifts towards Western countries in the income versus compositional inequality dia-
gram. This shift is explained by the compositionally equalizing effect of private pen-
sions, which distribute capital incomes towards the ends of the middle class. These
developments indicate a dynamic evolution in the Scandinavian economic model,
reflecting changes in global economic conditions and domestic policy choices.
4.5 Global Compositional Inequality
Let us now turn our focus to global compositional inequality. Similar to global in-
equality, global compositional inequality describes variations in the composition of
income in terms of capital and labor among individuals worldwide. This involves
considering a world social welfare function (Atkinson and Brandolini,2010), which is
symmetric, meaning each individual is counted as one unit of analysis (after adjust-
ments for different purchasing powers and inflation). As observed by Ranaldi (2025)
through the construction of the Global Capital and Labor (GCL) Database, this mea-
sure has decreased significantly in the 21st century. The world has transitioned from
levels of compositional inequality akin to those in Latin America to levels compara-
ble to Western countries. This reduction is attributed to two major factors. First, there
has been a significant individual-level capitalization process: the percentage of indi-
viduals worldwide with positive capital income increased from 20% to 32%. Second,
the global middle class, particularly in China, has benefited most from this process.
China, in particular, has experienced an exceptionally high average growth rate of
22
capital income across its entire distribution at the start of the 21st century, in con-
trast to mature economies, which have seen stagnation in this income source due to
global financial crises (see Figures 10 and 11). Having said that, labor income remains
a more critical determinant of global income status than capital income (Figure 12).
These results nicely complement our understanding of global inequality dynamics.
The extensive literature on this topic has consistently shown that global income in-
equality has decreased over the past three decades. A significant factor in this trend is
the reduction in between-country inequality components, as highlighted in studies by
Lakner and Milanovic (2015), Chancel and Piketty (2021), and Milanovic (2021a). This
reduction is largely attributed to the significant growth of the global middle class, pri-
marily consisting of Chinese and Indian populations, compared to the growth rates
of the upper middle class. Our finding of a decreasing trend in global compositional
inequality therefore aligns well with previous studies, further emphasizing the role
that different sources of income growth play in shaping global economic imbalances.
5 Theoretical Aspects of Compositional Equality
In discussions about income inequality, it is generally accepted that extremely
high inequality is worse than extremely low inequality. Although there is no con-
sensus on the ideal level of income inequality in society—whether it should be zero
or slightly more—we can agree that a society where a single individual owns all avail-
able resources is undesirable. However, with compositional inequality, the implica-
tions are less clear. Is a high level of compositional inequality detrimental or benefi-
cial for society? Is compositional equality inherently desirable to achieve inclusive and
sustainable growth? While these questions would require the construction of a nor-
mative framework of analysis, which is beyond the scope of this article, this section
begins discussing the desirability of compositional equality in light of recent stylized
facts affecting modern capitalist economies.
We begin with some technical considerations. It can be demonstrated that through
a straightforward tax and transfer scheme—wherein everyone’s income is taxed at a
23
rate τand everyone receives an equal absolute transfer—the following result emerges
(Ranaldi,2022):
ˆ
I≈ −τ, (18)
where the hat denotes percentage changes. Thus, a one percent increase in τresults
in a one percent decrease in the IFC index, a behavior similar to that of the Gini co-
efficient under the same tax and transfer scheme. This outcome, which arises from
the assumption that income components are redistributed proportional to the popu-
lation’s share of capital and labor income, suggests a movement toward a steady state
where income sources are equally composed across the population. Having said that,
it should be noted that taxation of capital and labor income is typically asymmetri-
cal, with a more pronounced burden on labor income than on capital. This disparity
implies that current taxation policies tend to increase, rather than decrease, composi-
tional inequality, as evidenced by several studies, including Iacono and Palagi (2022).
When discussing the dynamics of compositional inequality—whether it is dimin-
ishing or increasing—it is important to address the direction of change. Theoretically,
compositional inequality can be reduced in four distinct ways: (i) deconcentrating
capital at the top of the income distribution, (ii) increasing labor income at the top,
(iii) decreasing labor income at the bottom, and (iv) increasing capital income at the
bottom. Let us explore each of these scenarios in detail, highlighting the pros and
cons of each.
Scenario (i) offers the benefit of reducing the concentration of capital income across
the total income distribution, which is particularly desirable given the extremely high
levels of capital income inequality compared to labor income inequality (Piketty,
2014,Milanovic,2017, among others), a feature that characterizes both the devel-
oped and developing world.21 This reduction could be achieved by increasing capital
income taxation at the top, or through wealth taxation, such as inheritance taxes.22
21The situation differs, however, when viewing the world as a single nation (global inequality),
where the levels of capital and labor income inequality are relatively similar, as shown by Ranaldi
(2025).
22See Bastani and Waldenström (2023) for an analysis of the relative merits of wealth and capital
income taxes as instruments for taxing individuals at the top of the distribution.
24
Scenarios (ii) and (iii) typically result from structural changes in the labor market.
The increase in earning inequalities over recent decades is primarily attributed to
skill-biased technological change, which has widened the gap between high and low
earnings across the total income distribution. While this likely reduces compositional
inequality initially, it increases total income inequality which, in turn, can fuel com-
positional inequality in the long run by enlarging a high-skill elite who not only earn
from high wages but can also accumulate significant wealth (Milanovic,2021b). This
could reduce the mobility of income sources. Evidence for Scenario (ii) can already
be seen in the US with the rise in homoploutia (Berman and Milanovic,2023), which
refers, as previously discussed, to the increasing concentration of individuals at the
top of both the labor and capital income distributions (see Figure 14). Lastly, Scenario
(iv) would imply a process of capitalization occurring at the bottom of the distribu-
tion. An example of a policy aiming at going in that direction is the idea of a capital
endowment, also called minimum inheritance, paid to all at adulthood, as proposed by
Atkinson (2015), with the assumption that this would then generate capital incomes.
Let us now turn to some macroeconomic considerations of compositional equal-
ity. As previously discussed, compositional inequality links functional and personal
income distributions. This suggests that if the capital share of income continues to
rise, personal income inequality would also be impacted—a rise in the capital share
has been a well-established stylized fact over the past four decades. To better under-
stand this aspect, let us relate compositional inequality to Piketty’s simple model of
capitalism.
Piketty constructs a simple “machine” that encapsulates the essential features of a
capitalist economy (Milanovic,2014). According to Milanovic, this machine consists
of four key components. First, the definition K
Y=β, where Krepresents the capital
stock and Ydenotes the total income of the economy. Second, the first law of capitalism,
expressed as the identity Π
Y=α=r×β, where Πis capital income and rrepresents the
rate of return on wealth. Third, the second law of capitalism, which establishes an equi-
librium condition given by ¯
β=s
g, where sis the savings rate, gis the economic growth
rate, and the bar indicates a steady-state variable. This law relies on the assumption
25
that, in a closed economy, investment equals savings. Finally, Piketty highlights a
fundamental inequality relationship: r>g, which suggests that when the rate of re-
turn on capital exceeds the economy’s growth rate, wealth concentration increases
over time.
Formally, Piketty’s model in steady state can be summarized by the following
equation:
α=¯
βr=sr
gwith r
g>1.(19)
An additional equation can be included to relate the functional and personal dis-
tributions of income through compositional inequality, as follows:23
∂G
∂α ≈I,(20)
where Gis a measure of income inequality, and Iis a measure of compositional
inequality. Equation 19 captures the drivers of the capital share, and Equation 20
explores how this translates into an increase or decrease in income inequality within
society.
There are compelling arguments to suggest that the capital share will continue
to rise, one of which is the role of technological and particularly AI development.
The relationship between factor shares and AI technological development is already
generating a rich scholarly debate.24 A reduction in compositional inequality under
Scenarios (i) and (iv) would, therefore, imply not only a more equitable distribution of
assets but also a more favorable view towards the technologically-driven increase in
the capital share of income.
To conclude this section, I would argue that compositional equality is desirable for
at least two reasons. First, from a fairness perspective, given the extreme concentra-
tion of wealth—and thus capital income relative to labor income—a reduction in such
23Equation 20 implicitly assumes that the macroeconomic drivers of economic inequality, G, can be
expressed as a set of variables X1,...,Xn, which include the capital share. These variables are multi-
plied by a set of elasticities, ϵ1, . . . , ϵn, and are assumed to be orthogonal to each other.
24See Acemoglu (2010) for a discussion on an economy transitioning from labor- to capital-
augmenting technical change, Acemoglu and Restrepo (2018) for the implication of capital-
augmenting technological development on the factor shares, Acemoglu (2024) for a study on the rela-
tionship between capital accumulation and demand for labor in a context of automation, and Bastani
and Waldenström (2024) for a study on the implications of automation for the taxation of labor and
capital in advanced economies.
26
concentration would lead to a more equitable distribution of asset-derived rewards.
Second, from a macroeconomic standpoint, achieving compositional equality would
mean that, should the capital share continue to rise due to, for instance, new waves
of technological development (among other factors), individuals would not perceive
this trend as a threat to their economic well-being, but rather as an opportunity for
growth. In fact, under compositional equality (or even negative compositional in-
equality), an increase in the share of capital relative to overall GDP would imply
either equal or pro-poor (under negative compositional inequality) growth along the
income distribution.
6 Conclusion
This paper has provided an overview of recent research on compositional inequal-
ity. This research is positioned within recent scholarly advances in inequality stud-
ies, particularly highlighting the recent contributions by Thomas Piketty in unveiling
novel stylized facts on the dynamics of factor shares and inter-personal income in-
equality. It begins by reviewing the literature on the link between factor shares and
income inequality. While the study of this relationship can be traced back to classi-
cal political economists like David Ricardo, a general framework for analyzing this
relationship has been missing. The paper then introduced the concept and the princi-
pal methodological tools for its measurement. This discussion included indicators of
compositional inequality across and at the top of the distribution, drawing an analogy
between measures like the Gini coefficient and top factorial income shares within the
context of income inequality analysis. The paper presented major stylized facts re-
lated to compositional inequality, both between and within countries. It showed that
compositional inequality is currently far from maximal, unlike in the past when there
was a stark distinction between income-rich capital earners and income-poor labor-
ers. Countries like Slovakia and Taiwan, for instance, display extremely low levels
of compositional inequality, depicting alternative types of capitalist systems where
both rich and poor individuals have similar compositions of their incomes in terms
27
of capital and labor. These findings are important for contextualizing the long-run
stylized facts unveiled by Thomas Piketty regarding the joint evolution of the rising
capital share and increasing income inequality. While the empirical evidence pre-
sented in this article does not establish causality, it suggests that the strength of this
link is not fixed and depends on the extent of compositional inequality, which varies
across countries and over time. Finally, the paper discussed the desirability of com-
positional equality, arguing that it should be pursued for at least two reasons: fairness
and enhancing an inclusive, profit-driven regime of accumulation and growth.
Notwithstanding recent advances in studies of compositional inequality in terms
of capital and labor incomes, much more research is needed to enrich our understand-
ing of world dynamics in this dimension and its normative implications. Several
critical elements should be highlighted. First, data on compositional inequality need
improvement. While much emphasis has been placed on perfecting income distribu-
tion measures, especially at the top, more research is needed to refine our estimates
of income composition in terms of capital and labor across the entire distribution,
and particularly at the top. Second, the geographical coverage of compositional in-
equality studies is predominantly focused on the developed world, with a notable
lack of comprehensive data for African countries. This issue stems from the absence
of reliable data on income composition across the distribution in this region of the
world.25 Third, a better understanding of the determinants of compositional inequal-
ity is necessary. Petrova and Ranaldi (2024) recently showed that left-wing parties
tend to reduce, rather than increase, compositional inequality across the distribution
within the context of Europe. However, evidence for the rest of the world is scarce,
where different dynamics are certainly at play. Fourth, a normative economic theory
of compositional equality is needed to better grasp the extent to which it is desirable
and, if so, under which conditions. Fifth, the overall compositional inequality frame-
25This is particularly problematic when studying global compositional inequality, as it results in
overlooking a significant portion of a rapidly growing population (see Milanovic (2024) for a discus-
sion on this matter). While global inequality studies may use consumption data to supplement missing
income distribution information, this approach is not feasible for studies on the distributions of capital
and labor income across the population.
28
work can be adapted to study other types of compositions, such as the composition of
wealth in terms of real and financial assets, or the composition of income into savings
and consumption, or into taxes and market income.
29
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APPENDIX
A Concentration Curves
In this Section, we formally describe the concentration curves used to measure
compositional inequality via the IFC index. We primarily focus on the concentration
curve for capital, although the formalization could account for additional sources of
income.26 The actual concentration curve is defined as follows:
L(π,p)=π
p=i
n
X
j=1
αj,(21)
where α=Πi
Πrepresents the relative share of capital income for individual i. The
zero-concentration curve for capital income, denoted as Le(π, p), is defined by:
Le(π, p)=π
i
X
j=1
yj∀i=1,...,n.(22)
This curve is analogous to the Lorenz curve for total income, multiplied by the
capital share, π. It is a function of both the Lorenz curve for income and the capital
share. Unlike the egalitarian line used to construct the Gini coefficient, which is iden-
tical across all distributions, this curve is specific to each distribution. The maximum-
concentration curve for capital income, Lmax (π, p), varies depending on whether the
concentration curve for capital income lies below or above the zero-concentration
curve. In the former case, it is defined as:
Lmax (π, p)=Lm(π, p)=
0for p≤p′′
L(y,p)−z−for p>p′′ ,
(23)
and in the latter case, as follows:
Lmax (π, p)=LM(π, p)=
L(y,p)for p≤p′
zfor p>p′,
(24)
26For more information on the general framework, consult ?.
35
where p′and p′′ are such that L(y,p′)=πand L(y,p′′)=1−π, respectively. Under
this scenario, the maximum-concentration curve remains zero up to a certain income
percentile p′′ , after which it follows the shape of the Lorenz curve. Conversely, in the
second case, the maximum-concentration curve follows the Lorenz curve up to the
percentile p′, after which it becomes constant. The choice of p′and p′′ depends on
the shape of the Lorenz curve and the capital share. A graphical illustration of these
three curves for the case of Italy in 1989 can be found in Figure 1.
36
B Figures
Figure 1: Concentration Curves - Italy 1989
0
.2
.4
.6
.8
1
0 .2 .4 .6 .8 1
Population Deciles
Lorenz Curve Conc. Curve Capital Zero-Conc. Curve
Max-Conc. Curve
Note: The concentration curve for capital (red line), the zero-concentration curve (green line), the
Lorenz curve for income (blue line) and the maximum-concentration curve (black line) for Italy in
1989 are presented using data from the 1989 Survey on Household Income and Wealth (SHIW) carried out
by the Bank of Italy. Capital income is defined as the sum of property income and the capital compo-
nent of net self-employment income. Labor income is defined as the sum of payroll income and the
labor component of mixed income. Both the capital and labor components of self-employment income
are imputed following Glyn (2011). Pensioners are excluded from the analysis. Source: Iacono and
Ranaldi (2023).
37
Figure 3
Note: Relationship between compositional and inter-personal inequality. The graph shows on the
horizontal axis compositional inequality (as measured by the IFC Index) and on the vertical axis the
standard measure of inter-personal income inequality (Gini coefficient). Each dot is the unweighted
average for a country. Source: Ranaldi and Milanovic (2022).
39
Figure 4
Note: Relationship between compositional and inter-personal inequality in Latin American countries.
The graph shows on the horizontal axis compositional inequality (as measured by the IFC Index) and
on the vertical axis the standard measure of inter-personal income inequality (Gini coefficient). Source:
Monroy-Gómez-Franco and Ranaldi (2024) based on data from De Rosa et al. (2023).
40
Figure 5: Income and Compositional Inequality in Italy
.3
.4
.5
.6
.7
1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013 2015
Years
IFC Gini coefficient
Note: Series of the IFC index and the Gini coefficient constructed using the SHIW data. Capital income
is defined as the sum of property income and the capital component of net self-employment income.
Labor income is defined as the sum of payroll income and the labor component of mixed income.
Total income is the sum of capital and labor income. Both the capital and labor components of self-
employment income are imputed following Glyn (2011). Pensioners are excluded from the analysis.
Source: Iacono and Ranaldi (2023).
41
Figure 6: Compositional Inequality in Czech Republic and Slovakia
Note: The IFC (income-factor concentration) index measures the degree of compositional inequality in
terms of capital and labor income. Capital and labor incomes are defined as in Ranaldi and Milanovic
(2022): capital income is the sum of rent, interests and dividends, whilst labor income the sum of
wages, self-employment income and pensions, considered as deferred labor income. Source: Ranaldi
and Milanovic Database.
42
Figure 7: Income Inequality in Czech Republic and Slovakia
Note: The income definition adopted is market income (before-tax capital plus labor income) plus
pensions, considered as deferred labor income. Source: Ranaldi and Milanovic Database.
43
Figure 8
(a) Area of the Concentration Curves for Capital
(b) Area of the Concentration Curves for Labor
Note: The areas under the concentration curves for capital (above) and labor (below) describe the
distribution of capital and labor incomes across the total income distribution, respectively. A higher
(lower) value of these areas implies that the income source is predominantly distributed at the bottom
(top) of the total income distributions. Source: Ranaldi and Milanovic Database.
44
Figure 9
Note: Capital share and capital contribution to Gini in Nordic and other advanced economies. Note:
The horizontal axis shows the share of capital in total income and the vertical axis the contribution of
capital to total inequality. If capital income were distributed like overall income, the two shares would
be the same. The fact that capital income tends to be more important among the rich explains that the
capital’s inequality contribution share is always greater than capital income share. Source: Ranaldi
and Milanovic Database.
45
Figure 10: Pseudo-Growth Incidence Curves for Capital and Labor
Note: Y-axis displays the growth rate of the decile average income source, weighted by population.
Growth rates are quantified as the multiple by which the corresponding income increases. For exam-
ple, a value of 1 signifies a doubling, or 100% increase, of the income in the relevant decile. Growth
incidence is evaluated at decile groups of total income. Capital income is the sum of interests, divi-
dends and rental income. Labor income includes wage income and self-employment income. Total
income is, hence, the sum of labor and capital income. Source: Ranaldi (2025).
46
Figure 11: Regional Pseudo-Growth Incidence Curves for Capital
(a) China
(b) US
Note: Y-axis displays the growth rate of the decile average capital (red) and labor (blue) income.
Growth rates are quantified as the multiple by which the corresponding income increases. For ex-
ample, a value of 1 signifies a doubling, or 100% increase, of the income in the relevant decile. Growth
incidence is evaluated at decile groups of total income. Capital income is the sum of interests, divi-
dends and rental income. Labor income includes wage income and self-employment income. Source:
Ranaldi (2025).
47
