On the Measurement of Functional Income
Distribution∗
Marco Ranaldi†
January 30, 2017
Abstract
The present study seeks to investigate the issue of income composition from a theoretical perspective. In order to make this possible,
I introduce the concept of sources polarization among individuals (or
groups), together with a novel methodological framework whereby further examining this subject. Such a framework hinges on two major
elements. The first element is a new way to decompose individual and
aggregate income. The second is a specific measure, called Functional
Income Distribution Index, If , directly derived from the framework,
which summarizes with one number the polarization of income sources
across the population. I illustrate several mathematical properties of
the index, among which a singular connection with the Gini coefficient. In addition to that, I carry out a first empirical application
of the methodology, which focuses on the case of the US during the
20th century. I find strong evidence that income sources are no longer
polarized among rich and poor as in the past.
JEL-Classification: C430, E250
Keywords: Functional Income, Income Distribution, Inequality, Polarization of Sources, Statistical Methodology
∗
I would like to thank B. Amable, A. B. Atkinson, A. Bagnai, J. Clement, M. Corsi, C.
D’Ippoliti, T. Darcillon, M. De Rosa, M. Fana, E. Franceschi, G. Gabbuti, E. Guillaud,
R. Jump, M. Lavoi, A. Lochmann, M. C. Morandini, G. Moore, M. Morgan, M. Olckers,
T. Piketty, E. Stockhammer, D. Waldenström, M. Zemmour, G. Zezza as well as all
participants at the EAEPE Summer School 2016 (Rome), at the Applied Economics Lunch
Seminar at PSE (Paris), at the YSI Plenary 2016 at the Central European University
(Budapest), and at the EAEPE Conference 2016 (Manchester) for helpful comments and
suggestions.
†
Paris School of Economics and University Paris 1 Panthéon-Sorbonne (Centre
d’économie de la Sorbonne). [email protected]
1
1
Introduction
Every time a new research question is put forward, specific measurement
techniques are called upon to bridge the gap between theory and reality. In
current economic debate the issue of income distribution is gaining momentum, and after a long period of silence, the ”principal problem in political
economy” (Ricardo, 1817) has been brought ”in from the cold” (Atkinson,
2009) once again. However, if much has been said about the measurement of
income dispersion across people (Gini, 1912, Hoover, 1936, Atkinson, 1975,
Theil, 1979, Palma, 2011), little attention has been paid to the measurement
of income composition across people. Particularly, if the question ”How unequally is income distributed across the whole population” may sounds familiar to the reader, and relatively easy to tackle with today’s techniques, the
same cannot be said about the question ”How unequally is the composition of
income distributed across the whole population”. For this reason, the present
paper provides a novel methodological framework to answer this question.
Specifically, a new way to decompose personal and aggregate income in different sources is proposed, and a new measure, called Functional Income
Distribution Index (If ), is introduced. The If is, therefore, a synthetic
index that summarizes, with one number, the polarization of income sources
across the whole population. To better understand what it captures, let us
consider the following example. We suppose to analyze a society composed
by two people only, A and B. We call yA the income of A, and yB the income
of B. Total income y, which is the sum of yA and yB (y = yA + yB ), can be
decomposed in profits, π, and wages, w: y = π + w. Let us now suppose
yA = yB . In such a similar scenario, A and B have same income level, thus
income is equally distributed across the population (i.e. the Gini coefficient
is zero). However, what can we say about the distribution of sources among
the two people? If we suppose yA = π and yB = w as in Kaldor (1955),
the two sources are fully polarized between A and B. Therefore, even though
total income is equally split ”across population”, the two main actors play
completely different roles in the economy: A produces goods, and B consumes them. On the contrary, if we suppose yA = 12 π + 12 w = yB , the two
sources are no longer polarized, and the main actors’ economic role changes
considerably: both individuals are producers, and consumers at the same
time.
Even though the latter example strongly oversimplifies reality, it is helpful
to grasp the concept of income sources polarization. Indeed, if we knew that
income sources were fully polarized between two groups of people, we would
have a better understanding not only of its economic structure, but also of
each group’s political interest. If the richest 10% of the population owned
2
capital only, they would likely be unfavorable to a capital-tax increase. At
the same time, if the bottom 90% of the population owned wages only, they
would definitely prefer a capital-tax increase instead of a wage-tax increase.
Along the same line, Atkinson argues that factor shares are not only concerned with depicting dissimilarities among individuals (Atkinson, 2009). In
fact, the latter can also be seen as measures of similarities, in terms of interest groups. A specific group of people, whose income is mainly dependent on
wages, for instance, will have different economic, as well as political, interests from the group of entrepreneurs, whose income is strongly conditioned
by profit dynamics. Therefore, in a society where the income of an individual is likely to be jointly fostered by different sources, the research topic in
question is of utmost importance, and urges to be investigated further.
The paper is structured as follows. Section 2 introduces both the algebraic
framework and the index. Section 3 extends the framework up to the case
of n individuals. Section 4 empirically tests the index on the case of the US
during the 20th century.
2
The Methodology
Figure 1: Graphical representation of the If in which two people (or groups
of people) with different income (y1 < y2 ), and two sources of the same
amount (π = w) are compared.
Let us assume to have a population of size n, and a sequence of values yi , with
i = 1, . . . , n, indexed in non-decreasing order (yi ≤ yi+1 ∀i),
represent
Pwhich
n
individuals’ income. I equalize total income to one (y = i=1 yi = 1), and
3
allow for the following decomposition of personal income:
yi = αi π + βi w ∀i = 1, . . . , n
(1)
where π = YΠ and w = W
are the share of profits and wages to income
Y
wi
respectively, while αi = ππi and
relative shares of profits and
Pn βi = wPthe
n
wages owned by i, such that i=1 αi = i=i βi = 11 .
Let us now focus on the case of n = 2, in which
the population
P
P is divided
in two groups. Formally, we can write y1 = ki=1 yi ≤ y2 = ni=k+1 yi , with
k ∈ {1, . . . , n − 1}2 .
We say that income sources are fully polarized between the groups if at
least one group owns one source only. To better grasp the latter definition,
we can start considering the following situation. I suppose total income y
to be evenly divided between the two groups, thus y1 = y2 = 12 . I also
assume, for now, that y is composed by the same amount of profits and wages
(π = w = 12 ). As a consequence, we face total polarization of income sources
among the two groups if y1 = w and y2 = π (or, equivalently, y1 = π and
y2 = w)3 . However, if the assumption π = w = 21 is removed, and for instance
w > π, the concept of full polarization can no longer be characterized by
the one-to-one relationship between income group and income source. This
justify, therefore, the need for the definition previously proposed.
On the other hand, we say that income sources are not polarized between
groups if the relative shares of profits and wages owned by each group i are
the same (that is αi = βi ∀i = 1, 2), or if one of the two sources equals zero
(i.e. π = 0 or w = 0)4
2.1
An Index of Functional Income Distribution
In order to introduce the measure of income sources polarization, let us
consider figure 1. The latter displays the 1×1 box from which to create
the Lorentz Curve (Lorentz, 1905), a function that, for every individual (or
group) i indexed in non-decreasing order of yi , associates the cumulative level
of income assumed by the bottom ni % of the population. The bisector in the
box, therefore, represents two groups with equal income level (y1 = y2 ), while
1
It should be noticed that the formal decomposition of income in different sources dates
back to the pioneering work of Ricardo (1817), who was the first economist to tackle the
issue of distribution. As a consequence, the decision to split total income in profits and
wages is rather a conventional one. Indeed, in the empirical section I will show how to
make the analysis more tangible.
2
Obviously, k is such that the income of the first group is lower than that of the second.
3
Such a difference in ”who is owning what” will be analyzed after.
4
I will further explain the second case in the following section.
4
the black line right below represents two groups with different income level
(y1 < y2 ). Instead, the red line cuts the latter into two equal parts, displaying a zero-polarization of sources scenario. Any deviation from the red line
(see the blue line below) rises the polarization level. Indeed, the blue line
shows that individual 1 has α1 π level of profits, and therefore β1 w > α1 π
level of wages at its disposal. All the rest is in the hands of individual 2.
When the blue line coincides with the horizontal-axe along the interval (0, 1),
then we face full polarization of sources between the two individuals. In fact,
in such a configuration y1 = β1 w, and y2 = β1 w + π, which means that at
least one of the two classes own one single source only. I therefore define the
measure of income sources polarization, that I will call Functional Income
Distribution Index from now on, the area between the zero-polarization line
and the Lorenz Curve for profit, normalized between zero and one. According to figure 1, the Functional Income Distribution Index is thus the ratio
between green area and green plus blue area. Formally5 :
If =
2
wπ(β1 − α1 ) = µρIf
y1
(2)
where µ = y21 , ρ = wπ and If = β1 − α1 , and it is assumed y1 ∈]0, 21 [.
The index is thus composed by three components: µ, ρ and If . More precisely, they are the normalization, the level, and the distribution’s component
respectively. To better grasp their meaning, let us rewrite the If in the following way6 :
w 0 β1 α1 If = µ 0 π β2 α2 The Functional Income Distribution Index can be thus rewritten as the product between the determinants of two matrixes and a normalizing coefficient.
The first determinant (ρ) adjusts the degree of polarization for the level of
sources: if we registered a negative shock in the share of profit, it would be
meaningless to talk about polarization when one of the two sources is missing
in the economy. The second determinant (If ) is, instead, the channel through
which I address the distribution issue7 . Indeed, when If = 0 the percentage
5
See the appendix for further details.
w 0 β1 α1 is
It can be noticed that the product between the two determinants 0 π β2 α2 β 1 w α1 π
nothing more than the determinant of the following matrix: A =
.
β 2 w α2 π
β1 α1
7
The matrix A∗ =
(such that det A∗ = If ) is part of the relationship
β
2 α2
y1
w
y = A∗ x, where y =
and x =
, which is equivalent to the following system of
y2
π
6
5
of profits of each class is equal to its percentage of wages (thus βi = αi ∀i);
inversely, when If = 1 group 1 owns all the wages and group 2 all the profits
(identity matrix). Finally, when If = −1 the opposite happens: the first
group owns all the profits and the second all the wages. Even though it may
seems of little interest to consider negative values of the index, this has a
powerful meaning in terms of income composition dynamics. To understand
it properly, let us consider the following definition.
Definition 2.1. Let signt,t+1 be the sign of Ift · Ift+1 , where Ift is the Functional Income Distribution Index at time t, while Ift+1 the one at time t + 1.
We say that a change in the structure of income composition among groups
occurs at time t if signt,t+1 is negative.
Indeed, when a change in sign occurs at time t (i.e. signt,t+1 < 0), the
”wage-owners” become ”profit-owners” and vice versa.
In addition to what has been previously said, still other properties of the If
deserve to be illustrated. First of all, let me allow for the following way to
arrange the profit to wage ratio:
Π
π
=
=
w
W
1
1+ϕ
ϕ
1+ϕ
− β2
(3)
− α1
where ϕ = YY21 , with ϕ > 0. Such a way to express
the following result.
Result 2.1. A variation of ϕ has no effect on
Π
W
Π
W
enables us to introduce
iff If = 0. Formally8 :
Π
∂W
= 0 ⇐⇒ If = 0
∂ϕ
(4)
This sheds light on the relationship between income inequalities (ϕ) and
Π
factor shares ( W
) in a given economy. These two aspects are, in fact, intimately tied by the If , that is the way through which sources are differently
allocated among the two groups. Indeed, a variation of ϕ does not affect
Π
when sources are fully polarized between the groups. This is a rather
W
intuitive result.
Another aspect of the If I want to underline is its relationship with the
between-group Gini Coefficient 9 .
(
equations:
8
y1 = β1 w + α1 π
y2 = β2 w + α2 π
∂
, describing the evolution of incomes.
Π
W
> 0 when If > 0. In particular, when If = 1, an increase
It is easy to notice that ∂ϕ
Π
of ϕ raises the ratio W of the same amount. Indeed, when If = 1 then Y1 = Π and
Π
Y2 = W , thus W
= YY12 .
9
See the appendix for further details.
6
Result 2.2. Let us have x be the share of population belonging to y1 10 , then
a variation in the wage share of income, w, causes a change in the Gini
coefficient of the following amount:
∂G
= −If x
∂w
(5)
According to the previous result, an increase in the wage share (↑ w) reduces the between-groups inequality (↓ G) depending on the degree of sources
polarization (If ), and on the share of poor people (x). Indeed, when sources
are strongly polarized, and therefore one group’s income almost entirely depends on wages, it is straightforward to see how a wage increase would make
the poorest (i.e. the wage owners by construction) better-off, by consequently
reducing the overall level of income dispersion11 . The result sheds also new
light on the relationship between income inequality, and functional distribution of income from a purely theoretical point of view. After Piketty’s FCC
(Fundamental Contradiction of Capitalism), which empirically states that
the rate of return on capital has been higher than the growth rate of the
economy (Piketty, 2014), so that economies will tend to have ever-increasing
ratio of wealth to income, the impact of a rise in π (and thus a fall in w12 )
on income inequality can also be tackled in light of previous relationship.
2.1.1
Percentiles and Polarization Levels
0.0
Country A
0.74
0.5
Country B
−1.0
−0.5
Functional Income Distribution Index
0.5
1.0
λ−curve
0.0
0.2
0.4
0.6
0.8
1.0
Cut−off point i
Figure 2: λ-curves for countries A and B.
10
Thus the income class with lower income level.
The reader can find the mathematical proof in the appendix.
12
Such a trend in data has been largely shown in recent literature; for instance, see
Stockhammer (2013).
11
7
Until now I have been focusing the attention on the dynamics of income polarization, by holding constant the main groups in question. In this section, on
the contrary, I will allow for different groups to be scrutinized. Specifically,
the following question will be addressed: if we considered two population
groups, such as the bottom x% income earners (for instance, the bottom
90%), and the top (1 − x)% income earners (for instance, the top 10%),
which value of x would engender the highest, or the lowest level of sources
polarization?
The principal reason to further investigate this issue is to be able to simultaneously compare the income distribution structure of several countries. Let
us consider the following example. We have two countries, A and B, characterized by two different Lorentz Curves for income. It is assumed, for both
countries, that the income of the poor is mainly composed by salaries, while
the income of the rich by capital. However, a slight difference between the
two countries can be observed: country A sees the capital to be more condensed among those at the very top of the distribution of income (let us say
the top 10%), while country B sees the capital to be more spread along the
whole population. At this point, we might wonder how to precisely compare
the two countries in term of sources polarization, for all possible percentile
groups. A way forward on this issue is to study the evolution of If counterpoising two groups: the bottom x%, and the top (1 − x)% income earners,
for all values of x. Formally, let us define the Functional Income Distribution
Index If with cut-off point i as follows:
If (i) =
2
wπ(βi − αi ) = µi ρIf,i
F (i)
(6)
P
P
P
where F (i) = ij=1 yj (with F (i) < 1 − F (i)), βi = ij=1 βj , αi = ij=1 βj ,
Pi
Pi
whence µi = F 2(i) and If,i =
j=1 αj . When F (i) > 1 − F (i)
j=1 βj −
2
(i.e. y1 > y2 ), then If (i) = 1−F (i) wπ(αi − βi ). Therefore, the latter way to
express the index allows us to counterpose two groups, the bottom ni % and
the top n−i
%, for all values of i. For instance, if ni = 12 we do compare the
n
bottom 50% of the distribution with the top 50%.
At this point, by plotting all couples of values (i, If (i)) on a graph, we obtain
what I call the λ-curve:
λ : i 7−→ If (i)
Figure 2 shows two plausible λ-curves for countries A and B. As we can see
from the graph, in country A a zero polarization level of sources is reached
when the bottom 74% income earners is counterposed to the top 26%, while
in country B when the bottom half is counterposed to the top half. The
8
latter meaning that in country A we need more people from the bottom of
the distribution (74%) than in country B (50%) to display the same income
composition of the people from the top.
3
Extension of the Methodology
Figure 3: Graphical representation of the If in which ten people (or groups
of people) with different income (y1 < · · · < y10 ), and two sources of the
same amount (π = w) are compared. The red line displays zero polarization
of sources among all individuals, while the blue line represents the Lorentz
Curve for the relative shares of profits, Lα : i 7−→ A(i).
Section 2 has tackled the concept of sources polarization among two income groups. Instead, the following part of the work aims at extending
the methodology up to n individuals (or groups). However, if the intuition
behind the concept of polarization in a two-groups-two-sources scenario is
relatively easy to grasp, the latter happens to be no longer true when we
deal with a n-groups-two-sources configuration. Therefore, before moving
on to the formalization, a discussion on the definition of polarization when
multiple agents are considered is needed.
Similarly to the case where two groups interact, we say that income sources
are not polarized between n individuals (or groups) if the relative shares of
the two sources in question (profits and wages in the current framework) are
the same for every individual i, i.e. αi = βi ∀i = 1, . . . , n, or if one of the
9
two sources, at the aggregate level, equals zero (i.e. π = 0 or w = 0).
On the contrary, in order to adapt such an extended framework to the definition of full polarization provided in previous section, an additional aspect
needs to be taken into consideration. When we have n individuals, it may
happen that, for instance, both the income of the richest and of the poorest
are fueled by the same source, say the salary13 . However, by grouping these
two people in the same income source category we would implicitly attribute
same economic role to both of them; it shall be noticed that if the poorest
can solely use his wage to consume, the richest can save out of wages to
a considerable extent. Therefore, the definition of polarization of sources
among n individuals has to account also for the level of sources, and not only
for its nature, which, in fact, is no longer a static concept.
That having been said, we say that income sources are fully polarized among n
individuals if at least one group of them, either among the richest or the poorest, owns one source only. For instance, let us suppose person n, thus the richest, owns the whole profits present in the economy, i.e. yn = π.PAs a consen−1
yi = w.
quence, the remaining n−1 people own all the wages, i.e. y−n = i=1
In such a situation, society faces full polarization of sources among its citizens.
From a mathematical point of view, the construction of the functional income
distribution index If follows in a straightforward manner from the way in
which the simplest version has been already formulated in section P
2.1 (see
Figure 3 for a graphical representation). Indeed, if we call A(i) = ij=1 αi
the cumulative function for the relative shares of profit, and µ∗ the new coefficient of normalization14 , I define the functional income distribution index
with n groups and two sources of income as follows:
n
If =
h X
| (F (i) − A(i)) + (F (i − 1) − A(i − 1)) | µ∗
2n i=1
(7)
It can be noticed that, when n = 2, equation A.3 equals equation 2. Therefore, from a purely graphical point of view, the extended coefficient is nothing
more than the area between the Lorentz Curve for the zero polarization scenario, Le , and that for the relative shares of profits, Lα , ranged between 0
and 115
13
This example happens to hold true in a society where la montée des hautes salaires
is becoming a widely known phenomenon (see Piketty, 2014).
14
See the appendix for details.
15
It can be noticed that the general case does not allow for the three components decomposition (µρIf ). This is due to the fact that the n-groups-two-sources configuration
cannot exploit the same properties of the 2×2 scenario, such as the symmetry of matrix
A.
10
4
Empirical Application
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
Top 10% Income Share − US
1920
1940
1960
1980
2000
years
Figure 4: Evolution of top 10% income share throughout the 20th century.
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Top 10% Capital & Labor Shares − US
1920
1940
1960
1980
2000
years
Figure 5: Capital share (red line) and labor share (blue line) of top 10% in
the US.
11
A first empirical analysis, with the objective to study the dynamics of income
sources polarization, is undertaken in the current section16 . The data used
are taken from the WID17 , and I will focus on the case of the US. The data
are available from 1913 to 2014, which allows us to cover almost entirely the
20th century. I will employ the two-groups-two-sources framework, where the
capital-labor polarization dynamics among the top 10%, and the bottom 90%
of the income distribution will be firstly tackled. For the second round of
analysis, I will study the same dynamics but with different groups, notably
the top 1%, and the bottom 99%. Turning now to the variables employed,
I will allow for the same definition of capital proposed by Piketty (2014),
which is rather a financial capital. Indeed, the latter is composed by the
following sources: income from equity, net interest, housing rent and capital
component of mixed income. On the other side, the labor is here considered
as the sum of compensation of employees, labor component of mixed income
and property income paid to pensions. As it can be noticed, I do not allow
mixed income to be a different category for the moment18 .
0.4
0.6
0.8
Capital & Labor Shares − US
0.2
1990
1920
1940
1960
1980
2000
years
Figure 6: The capital share is the red line, while the labor share the blue
one.
16
A complete empirical investigation of this issue, both in the direction of a comparative
(across countries) and a long-run study will be presented after (forthcoming).
17
World Wealth & Income Database.
18
The principal reason for this choice is to keep the analysis as simple as possible, and
to allow for different configuration of sources in a separate work.
12
0.35
0.40
0.45
0.50
Functional Income Distribution Index − US
0.25
0.30
2000
1943
1966
0.20
1940
1920
1940
1960
1980
2000
years
Figure 7: Top 10% vs bottom 90%.
Now, before getting to the heart of the matter, I will separately analyze
the major elements that play a role in the dynamics at stake; this will therefore help me to further highlight the new information the If brings on the
table.
To start, let us focus the attention on the evolution of the top 10% income
share (and therefore of the bottom 90%, which has the opposite dynamics)
over the period considered (figure 4). This is rather a well-known stylized
fact after Piketty’s magnum opus, which displays a U-shaped pattern of the
top 10% income share, meaning that today’s income share at the top of
the income distribution is the same as it was in the Pre-World War II era.
Therefore, the last 35 years (i.e. from 1980s on) have seen a profound rise in
the level of income inequality between rich and poor. If we now decompose
the top 10% income share in capital and labor, we can notice that both factors have increased after the 1980s (see figure 5). This amounts to say that
both sources have contributed to rise income at the top of the distribution,
and not only the capital as we might have expected. Furthermore, it can be
even noticed that, regardless the positive trends of both shares over the last
three decades, the labor’s share is times more stable than that of capital,
which markedly fluctuates.
Let us now take a closer look at the evolution of both the capital and the
labor share in the economy as a whole (figure 6). The two shares appear to
diverge from 1913 to 1990, and to very slightly converge afterwords, with the
13
amount of labor being at least the double of the capital’s one throughout the
whole 20th century.
0.12
0.14
0.16
0.18
0.20
Top 1% Income Share − US
1920
1940
1960
1980
2000
years
Figure 8: Evolution of top 1% income share throughout the 20th century.
0.1
0.2
0.3
0.4
0.5
0.6
Capital & Labor Shares − US
1920
1940
1960
1980
2000
years
Figure 9: Capital share (red line) and labor share (blue line) of top 1% in
the US.
14
At this point, in light of all this information, what can we practically say
about the polarization of sources between rich and poor? Indeed, even if all
previous macroeconomic trends are embedded in the index, it is difficult to
track the dynamics of sources polarization by separately analyzing its components. Therefore, let us focus the attention on figure 7, which displays the
evolution of the If during the century.
The first thing to notice is that it fluctuates considerably within the period,
reaching a peak of 0, 5 in 1943, and a low of 0, 21 in 2000. This appears to
be an ample variation, considering that it ranges between 0 and 1.
The second aspect which requires special care is the vertical leap observed
between 1940 and 1943. The latter may be certainly explained by the occurrence of the Second-World-War, but given that neither during the FirstWorld-War, nor during the Great Depression such a violent jump is observed,
the likely scenario of a change in the data registers must be taken into account. If this hypothesis happened to hold true, and we therefore allowed
for a smooth transition, rather than a sudden shock over this five yearstimeframe, a decreasing trend of the If would be observed all during the
century. In other words, a decreasing trend in the polarization of sources
between rich and poor would be recorded in the US during the last 100
years. Such a plausible scenario would contribute to further enhance the
thesis that today’s individuals are more concerned with multiple, instead of
single sources of income. However, even without the change in the data registers’ scenario the latter conclusion stays relevant, at the only expense of a
smooth transition during this time lag.
Let us now focus on the top 1% vs bottom 99% dynamics. Figure 8 portraits
a similar evolution of the top 1% income share as that of the top 10% (still
well-know stylized fact in the literature). Also the dynamics of the different
components are remarkably similar (see figure 9). However, this does not
do enough to mark out completely similar alleurs of the two indexes. As it
can be noticed from figure 10, the previous positive shock observed in the
1940s (that of the top 10% vs bottom 90% dynamics) is now replaced by a
smoother transition occurring over a longer period of time, precisely from
1932 to 1964. In addition to that, it can be pointed out that in 1964 the
coefficient reaches a spike of almost 1. Indeed, in this very year the wage
share owned by the top 1% amounts to 5% only, while the capital share to
39%. Therefore, in line with the definition previously given of sources polarizations, this represents the case in which at least one group (i.e. the top
1%) owns one source only.
15
0.4
0.6
0.8
1.0
Functional Income Distribution Index − US
1964
2000
0.2
1932
1920
1940
1960
1980
2000
years
Figure 10: Top 1% vs bottom 99%.
Another relevant difference from previous dynamics is that highest levels
of polarization are registered between top 1% and bottom 99% all throughout the century. Furthermore, if polarization between top 10% and bottom
90% stabilizes today at lower levels than those in the past, thus suggesting
an ongoing process of structural change in terms of income composition between these two groups, the same cannot be said for the second dynamics.
Indeed, the latter registers almost same values (' 0, 47) in 1932 and 2000,
for instance. Finally, it can be observed that a more determined rise of polarization has been occurring over the last fifteen years (starting from the
dot-com bubble) between top 1% and bottom 99%, differently from what is
happening between top 10% and bottom 90%.
To conclude, I am perfectly aware that answering such a broad issue, as the
study of income sources polarization in the US, requires more accurate analysis and investigations than those previously presented. Needless to say that
this research cannot be disconnected neither from a sociological, nor historical overview of the evolution of such a society. This is the principal reason
why I do consider all these results more of a starting point on which to bring
out a more ample picture of the matter.
16
References
[1] Atkinson, A. B., On the Measurement of Inequality, Journal of Economic
Theory, 2, 244-263, 1970.
[2] ————, Bringing income distribution in from the cold, The Economic
Journal, 107, 297-321, 1997.
[3] ————, Factor shares: the principal problem of political economy?, Oxford Review of Economic Policy, 2009.
[4] Bernardo, J. L., Marinez, F. L., Stockhammer E., A Post-Keynesian
Response to Piketty’s ”Fundamental Contradiction of Capitalism”, Post
Keynesian Economics Study Group, Working Paper 1411, 2014.
[5] Gini, C., Variability and Mutability, 1912.
[6] Hoover jr, E. M., The Measurement of Industrial Localization, Review of
Economics and Statistics, 18, No. 162-171, 1936.
[7] Kaldor, N., Alternative Theories of Distribution, The Review of Economic
Studies, Vol. 23, No. 2, pp. 83-100, 1955.
[8] Lerman, R. I., Yitzhaki, S., Income Inequality Effects by Income Source:
New Approach and Applications to the United States, The Review of Economics and Statistics, Vol. 67, No. 1, pp. 151-156, 1985.
[9] Palma, J. G., Homogeneous Middles vs. Heterogeneous Tails, and the End
of the ”Inverted-U”: It’s All About the Share of the Rich, Development
and Change, 42(1): 87153, 2011.
[10] Piketty, T., Capital in the Twenty-First Century, Harvard University
Press, 2014.
[11] Ricardo, D., Principles of Political Economy, London, Dent, first published 1817.
[12] Stockhammer, E., Why have wage shares fallen? A panel analysis of the
determinants of functional income distribution, Conditions of Work and
Employment Series No. 35, 2013.
[13] Theil, H., The Measurement of Inequity by Components of Income, Economics Letters 2, 1979.
[14] Yitzhaki, S., Economic Distance and Overlapping Distributions, Journal
of Econometrics, 1994.
17
A
A.1
Appendix
Result 2.1
Provided that y1 = α1 π+β1 w, and y2 = α1 π+β2 w, where y1 +y2 = y = π+w,
we can write:
ϕ =
y1 (β2 w + α2 π)
π
w
π
w
π
w
=
=
=
=
π
=
w
β1 w + α1 π
β2 w + α2 π
y2 (β1 w + α1 π)
β1 y2 − β2 y1
α2 y1 − α1 y2
β1 − ϕβ2
−α1 + ϕα2
1 − (1 − ϕ)β2
ϕ − (1 − ϕ)α1
1
− β2
1−ϕ
ϕ
1+ϕ
⇐⇒
⇐⇒
⇐⇒
⇐⇒
− α1
If we now take the first derivative of
manipulate we obtain result 2.1.
A.2
⇐⇒
π
w
with respect to ϕ, and we further
Result 2.2
Let us rewrite y1 (from equation 1) as follows:
y1 = β1 w ± α1 w + α1 π
and after some algebraic manipulations we get:
y1 = If w + α1
where If is, indeed, the distribution’s component of If . Let us now recall
the formal expression of the Gini coefficient:
G=1−
n
X
(xk+1 − xk )(yk+1 + yk )
k=1
where the whole population is divided in n groups, and xk , yk represent the
bottom xk % of the population, and its cumulative income respectively. When
n = 2 we can write:
G = 1 − xy1 − (1 − x)y
18
where x is the share of population belonging to group 1, and (1−x) the share
belonging to group 2. By plugging the expression for y1 previously dervied,
we can write:
G = 1 − x(If w + α1 ) − (1 − x)
whence19
G = x(α2 − If w)
from which, by taking the derivative with respect to w, we obtain result 2.2.
A.3
Extension of the Methodology
The area covered by the Lorentz Curve for the zero polarization scenario, Le
(previous red lines in figures 1 and 3), is the following:
n
X
1 (F (i) + F (i − 1)
A =
h
n
2
i=1
where y0 = 0.
Instead, the area covered by the Lorentz Curve for profits Lα (blue lines in
figures 1 and 3):
n
X
1 (A(i) + A(i))
P=
h
n
2
i=1
where α0 = 0.
Finally, the surface below the line describing the highest polarization of income sources (which will be employed to range the index between 0 and 1),
can be derived as follows:
n
X
1
1 (F (i) + F (i − 1)) 1 (F (j) + h)
+
+
h
Q=
n
2
n
2
n − (j + 1)
i=0
where j is such that F (j) < h < F (j + 1).
At this point, the If can be easily generalized:
n
h X
sign(A − P)
If =
| (F (i) − A(i)) + (F (i − 1) − A(i − 1)) |
2n i=1
Q−A
19
It can be noticed that G = x(α2 − If w) = x(1 − y1 ) = xy2 , which is a different way to
express the between-groups Ginin coefficient. In fact, it clearly appears from the equation
that inequality rises when either the share of people belonging to the poor increases, or
when the income share of the rich augments.
19
−P)
where µ∗ = sign(A
is the normalization coefficient. It can be easily shown
Q−A
that the formula above, for n = 2, gives equation 2. Indeed, when n = 2 we
find Q − A = y41 , and by supposing sign(A − P) > 0, which simply means
the first group owns more wages than profits, we get: If = y21 wh(β1 − α1 ) =
µρIf .
20