Relation Between Two Income Inequality Measures: The Gini coefficient
and the Robin Hood Index
EDWARD ALLEN
Department of Mathematics and Statistics
Texas Tech University
Lubbock, Texas 79409-1042
UNITED STATES
Abstract: The objective of this investigation is to study the relation between two common measures of income
inequality, the Gini coefficient and the Robin Hood index. An approximate formula for the Robin Hood index
in terms of the Gini coefficient is developed from 100,000 Lorenz curves that are randomly generated based on
100 twenty-parameter families of income distributions. The approximate formula is tested against Robin Hood
indexes of commonly-used one-parameter Lorenz curves, income data of several countries, and reported results
of Robin Hood indexes. The approximate formula is also tested against results of a stochastic income-wealth
model that is introduced in the present investigation. The formula is useful conceptually in understanding why
Gini coefficients and Robin Hood indexes are correlated in distribution data and is useful practically in providing
accurate estimates of Robin Hood indexes when Gini coefficients are known. The continuous piecewise-linear
approximation is generally within 5% of standard one-parameter Lorenz curves and income distribution data and
has the form: R≈0.74Gfor 0 ≤G≤0.5, R≈0.37 +0.90(G−0.5)for 0.5≤G≤0.8, and R≈0.64 +1.26(G−
0.8)for 0.8≤G≤0.95, where Ris the Robin Hood index and Gis the Gini coefficient.
Key-Words: Robin Hood index, Gini coefficient, income inequality, Lorenz curve, Pietra index, Hoover index,
Schutz index
Received: June 19, 2021. Revised: February 6, 2022. Accepted: February 21, 2022. Published: March 8, 2022.
1 Introduction
The Gini coefficient was introduced by Corrado Gini
in 1912. Gini proposed that the area between two
curves describing equal and actual incomes be used
as a measure of inequality [1, 2]. The Gini coeffi-
cient ranges from 0 to 1 and is a measure of how far
a given income distribution differs from a distribu-
tion of complete equality. The Robin Hood index, a
second measure of income inequality, is known un-
der many names, in particular, the Pietra index, the
Hoover index, and the Schutz index. The Pietra in-
dex was introduced as an inequality measure in 1915
[3, 4]. Inequality measures equivalent to the Pietra
index were proposed several times in later investi-
gations, in particular, by Hoover [5, 6] in 1936, by
Schutz [7, 8] in 1951, and as the Robin Hood index
by Atkinson and Micklewright [9] in 1992.
The Gini coefficient is perhaps the most
commonly-used measure of income inequality
[10, 11]. The Robin Hood index is useful conceptu-
ally and is equal to the proportion of the population’s
total income that, if redistributed, would give perfect
income equality. An approximate formula for the
Robin Hood index in terms of the Gini coefficient
is useful in achieving additional understanding of
an income distribution when the Gini coefficient is
known. In the present investigation, a continuous
piecewise-linear approximation of the Robin Hood
index in terms of the Gini coefficient is determined
for Gini coefficients between 0.0 and 0.95. The
form of the approximate formula is indicated from
mathematical examination, improved through param-
eter value estimation of randomly generated income
distributions, and tested against standard Lorenz
curves and a stochastic income model. The resulting
approximation is generally accurate to within 5%
of Robin Hood indexes of standard one-parameter
Lorenz curves and income distribution data.
2 Problem Formulation
Let 0 ≤x≤1 be the cumulative proportion of the
population ranked by income level. Let 0 ≤I(x)<∞
be the income distribution, i.e., the proportion of the
population with income less than or equal to I(x)is
equal to x. It follows that Rx
0I(u)du/R1
0I(u)du is
the proportion of the total income earned by the frac-
tion xof the population with the lowest income and
R1
0I(u)du is the average income of the population.
As an example, if income is distributed exponentially
in a population with density p(I) = γexp(−γ(I−I1))
for I1≤I<∞, then I(x)has the form I(x) = I1−
1
γlog(1−x)for 0 ≤x<1.
The Lorenz function, L(x), of the population is de-
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fined as [12, 13, 14]:
L(x) = Rx
0I(u)du
R1
0I(u)du.(1)
The Lorenz function, L(x), is a continuous non-
decreasing function for 0 <x<1 and satisfies, for
example,
L(0) = 0,L(1) = 1,and L0(x)≥0,for 0 <x<1.
Conceptually, L(x)is the fraction of the total income
for the proportion xof the population having the low-
est incomes. For a population with complete in-
come equality, I(x)is constant for 0 ≤x≤1 and so
L(x) = x. Otherwise, L(x)≤xfor 0 ≤x≤1. The area
between the curves y=xand y=L(x)is defined to be
G/2 where 0 ≤G≤1 is the Gini coefficient [13, 15].
The Gini coefficient Gis a measure of the income in-
equality in the population with complete equality if
G=0.
The Robin Hood index is the proportion of the to-
tal income that can be redistributed to achieve income
equality. The Robin Hood index, R, is equal to the
maximum difference between the curves y=xand
y=L(x). That is, for x−L(x)achieving its maximum
value at x∗∈(0,1), then
R=x∗−L(x∗)where L0(x∗) = I(x∗)
R1
0I(x)dx =1.(2)
By Eq. (2), x∗satisfies I(x∗) = R1
0I(u)du and so,
R1
x∗I(x)dx −(1−x∗)I(x∗)/R1
0I(x)dx is the pro-
portion of total income to distribute so that the entire
population has income I(x∗). But R1
x∗I(x)dx −(1−
x∗)I(x∗) = RR1
0I(x)dx, so the proportion of the total
income to distribute from incomes above I(x∗)to in-
comes below I(x∗)is equal to the Robin Hood index1
R=x∗−L(x∗).
As G/2=R1
0(x−L(x))dx ≤R1
0(x∗−L(x∗))dx =
R, it is clear that Gand Rare closely related. Indeed,
by considering the area of the triangle defined by the
vertices (0,0),(1,1), and (x∗,L(x∗)), it can also be
shown [13] that R≤G. Thus, the Robin Hood index
satisfies the rough bounds
G/2≤R≤G.
Given a value of G, there are many different
Lorenz curves and, correspondingly, there are many
1As Robin Hood took from the rich and gave to the poor, a second
possible Robin Hood index would be ˆ
R= (x∗−L(x∗))/(1−L(x∗)) =
R/(1−L(x∗)) which is the proportion of all income above the average
income that, if redistributed to individuals with income below the average
income, would result in complete income equality. For example, for the
Pareto Lorenz function, ˆ
Rhas the form ˆ
R=2G/(1+G).
possible values of R. In the remainder of this inves-
tigation, given a value of G, it is shown that the val-
ues of Rare generally within 10% of each other. To
study this, many families of Lorenz curves are ex-
amined. One-parameter families of Lorenz functions
considered in the present investigation are assumed
to have Gas the specified parameter. That is, each
curve of a one-parameter family of Lorenz functions
is uniquely defined by the value of the Gini coeffi-
cient Gon some interval of [0,1]. In addition, any
n-parameter family of Lorenz functions considered
yields a unique Lorenz curve when npermissible pa-
rameter values are given. A Lorenz function from a
one-parameter family is written in the present inves-
tigation as L(x,G)and the Robin Hood coefficient for
the Lorenz function is written R(G).
Consider R(G)for a one-parameter Lorenz curve
L(x,G). From Eq. (2), the Robin Hood index is a
function of Gand satisfies
R(G) = x∗−L(x∗,G),where ∂L(x∗,G)
∂x=1.(3)
Of interest in the present investigation is how R(G)
is approximately related to G. Approximations for
R(G)can be inferred by Taylor series expansions of
R(G)about the Gvariable. In particular, for small
values of G, say 0 ≤G≤G1,
R(G)≈R(0) + RG(0)G,for 0 ≤G≤G1,
and for larger G, say G1≤G≤G2and G2≤G≤G3,
R(G)≈R(G1) + RG(G1)(G−G1),G1≤G≤G2,
R(G2) + RG(G2)(G−G2),G2≤G≤G3.
Equating by continuity the two expressions at G=G1
and G=G2and setting R(0) = 0, the following con-
tinuous piecewise-linear approximation2is obtained:
R(G)≈
αG,for 0 ≤G≤G1,
αG1+β1(G−G1),for G1≤G≤G2,
αG1+β1(G2−G1) + β2(G−G2),
for G2≤G≤G3,
(4)
where α,β1, and β2are constants.
An initial estimate of αcan be made by consider-
ing the Lorenz function L(x,G). The values of β1,β2
depend on the income distribution, i.e., the family of
Lorenz curves, and estimation of β1and β2is con-
sidered in the next section. To estimate α, it is as-
sumed that L(x,G)is approximately equal to xfor
Gsmall and so, L(x,G) = x+ε(x,G)for 0 ≤x≤1
where ε(x,G)is small. Assuming additionally that
2Continuous piecewise-linear approximations are common in applica-
tions [16].
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ε(x,G)is approximately a quadratic function of xfor
Gsmall, then
L(x,G)≈x+a1(G) + a2(G)x+b(G)x2
= (1−b(G))x+b(G)x2,(5)
as L(0,G) = 0 and L(1,G) = 1, where the coefficient
b(G)depends on G. Furthermore, as G/2=R1
0(x−
L(x,G))dx,R(G) = x∗−L(x∗,G), and Lx(x∗,G) = 1,
then
b(G)≈3G,x∗=0.5,and R(G)≈0.25b(G).
It follows that
R(G)≈0.75Gfor Gsmall
and, by Eq. (4), α≈0.75. In the next section, based
on randomly-generated Lorenz curves, β1,β2,G1,G2,
and G3are estimated and the value of αis modified.
The approximate formula is then tested against re-
sults from standard and derived Lorenz curves.
An example of a two-parameter family of Lorenz
functions, similar to the multi-parameter families
studied in the next section, is derived from a piece-
wise constant income probability density with param-
eters p1,p2, specifically,
p(I) = (p1for I1≤I<I2,
p2for I2≤I<I3,
p3for I3≤I<I4,
where I1=0, I2=1, I3=2, I4=3. As R1
0p(I)dI =1,
then p1+p2+p3=1 and so, p3=1−p1−p2is
fixed given p1and p2. The two positive parameters
p1and p2satisfy p1+p2≤1. The function I(x)for
this income probability density is
I(x) = Ii+ (x−xi)/pifor xi≤x≤xi+1,
for i=1,2,3 where x1=0, x2=p1,x3=p1+p2,
and x4=1. The Lorenz function L(x,p1,p2)and the
Gini coefficient depend on the values of the two pa-
rameters p1,p2, e.g.,
G=G(p1,p2) = 1−2Z1
0
L(x,p1,p2)dx.
For example, if p1=1/2,p2=1/4, then p3=1/4
and G=2/5. For this family of income densities,
more than one set of parameters can result in the same
value of G. For example, p1=1,p2=0,p3=0 and
p1=1/3,p2=1/3,p3=1/3 both give G=1/3. The
value of Gfor this family is exactly given by the for-
mula
G=1−2p2
1+6p1p2+8p2
2+6p1p3+18p2p3+14p2
3
3p1+9p2+15p3
where p3=1−p1−p2.The Robin Hood index can
also be explicitly found for this two-parameter family
of Lorenz functions and has the form:
R=
1
2p1φ,when 1/2≤φ≤1,
p1(φ−1
2) + 1
2p2(φ−1)2/φ,1≤φ≤2,
p1(φ−1
2) + p2(φ−3
2) + 1
2p3(φ−2)2/φ,
when 2 ≤φ≤5/2,
where φ=p1/2+3p2/2+5p3/2. For the three ex-
amples above, R=1/4=0.75Gwhen p1=1/3,p2=
1/3 or when p1=1,p2=0, and R=49/160 =
0.766Gwhen p1=1/2,p2=1/4.
3 Approximate formula for 100
families of income distributions
To estimate α,β1,β2,G1,G2,and G3in Eq. (4), 100
families of income distributions are considered where
each family has 20 parameters. (That is, twenty dif-
ferent parameter values must be specified to define
a unique Lorenz curve for each of the 100 different
families.) Then, α,β1,β2,G1,G2,and G3are esti-
mated by comparing how the Robin Hood indexes are
related to the Gini coefficients for the 100 families of
Lorenz curves. For each family, the income probabil-
ity densities are piecewise constant functions defined
for I≥0 where the number of income intervals, m, is
equal to twenty-one. First, a general but simple fam-
ily of income densities is initially studied. For this
family, the income density is piecewise constant and
has the form:
p(I) =
m
∑
i=1
piχi(I)for i=1,...,m=21,
where Ii+1=Ii+∆ifor i=1,2,...,m, with I1=0,
and
χi(I) = 1 for Ii≤I<Ii+1and χi(I) = 0 otherwise.
As R1
0p(I)dI =1, then ∑m
i=1pi∆i=1 defines one
value of piin terms of the others. The function I(x)
where 0 ≤x≤1 can readily be calculated for this
probability density and has the form:
I(x) = Ii+(x−xi)/pi,for xi≤x≤xi+1,i=1,2, ..., m,
where x0=0,and xi=
i−1
∑
j=1
pj∆jfor i=2,...,m+1.
For this initial density, the income intervals have
equal widths and the probabilities for each interval
are randomly chosen, i.e., ∆iand piare, respectively,
∆i=1 and pi=γi/Dfor i=1,2,...,m,where
γiare uniformly distributed random numbers on [0,1]
and D=∑m
i=1γi∆i. This represents a large number
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Figure 1: 100,000 points generated for the ini-
tial twenty-parameter family of income distributions
plotted along with the line R=0.75G.
of income probability densities, i.e., over the selec-
tions of {γi}i=1,m. The Lorenz functions can be found
for each income density and the Gini coefficients and
Robin Hood indexes rapidly calculated. For 100,000
randomly-selected distributions from this family, the
values of Gini coefficient, however, range only from
about 0.2 to 0.45. A plot of the Robin Hood index
with respect to the Gini coefficient is given in Figure
1 for 100,000 realizations of this family. Figure 1 also
illustrates that R(G)≈0.75Gfor this family.
To extend values of the Gini coefficient from
about 0.0 to 0.95, this original initial twenty-
parameter family of income distributions is expanded
to 100 twenty-parameter families. Each family is a
modification of the simple family above by generaliz-
ing the income interval widths and the interval prob-
abilities. There are undoubtedly many ways to gen-
eralize these interval widths and probabilities. In the
present investigation, the (j,k)th family is defined for
j,k=1,2,...,10 by
∆i,j= (i)j−3,pi,j,k=γi(k/5)i/Dj,k,for i=1,2,...,m,
with
Dj,k=
m
∑
i=1
γi(k/7)i∆i,j.
For example, the original family of probability den-
sities corresponds to j=3 and k=5. The forms for
the interval widths ∆i,jand the probabilities pi,j,kare
motivated by considering exponential and power-law
income probability distributions [17]. (In particular,
if income is distributed as p(I) = αe−αI, and income
interval widths are uniformly selected, then the aver-
age probability in the ith interval is proportional to ci
for a constant c. In addition, if income is distributed
such as p(I)∝Iαover an interval in Iand the proba-
bilities are uniformly selected, then interval widths
are approximately proportional to icfor a constant
Figure 2: 100,000 Robin Hood index/Gini coeffi-
cient points generated from the 100 twenty-parameter
families of income distributions.
c.) As the interval probabilities and widths can be
increasing or decreasing as iincreases depending on
the family (j,k)and the interval probabilities are as-
signed random magnitudes, a diverse and large num-
ber of income probability densities are represented by
these 100 families. A plot of the Robin Hood indexes
and Gini coefficients for 100,000 income densities
from these 100 families is given in Figure 2. Each
of the calculated 100,000 points is selected randomly
from the 100 families, i.e., each family is sampled
approximately 1000 times. Least squares fits to the
100,000 points give the following approximation for
the Robin Hood index Rin terms of the Gini coeffi-
cient G:
R≈(0.74G,for 0 ≤G≤0.5,
0.37 +0.90(G−0.5),for 0.5≤G≤0.8,
0.64 +1.26(G−0.8),for 0.8≤G≤0.95.
(6)
In comparing the approximate formula (6) with the
100,000 Robin Hood index/Gini coefficient points il-
lustrated in Figure 2, 52% of the points are within
2.5% of curve (6), 81% are within 5% of the curve,
and 97% of the points are within 10% of the curve.
Also, as illustrated in Figure 3, good agreement is
seen in a plot of the approximate formula (6) with
1000 Robin Hood index points obtained from sam-
pling the 100 families of income distributions.
4 Comparison with several
one-parameter Lorenz functions
For the one-parameter families of income distribu-
tions considered in this section, each Lorenz func-
tion, L(x,G), in the family is unique given a value
of G∈[0,1]. There are many one-parameter Lorenz
functions commonly used for studying income distri-
butions. One-parameter Lorenz functions are gener-
ally considered sufficiently accurate approximations
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0 0.2 0.4 0.6 0.8 1
Gini Coefficient
0
0.2
0.4
0.6
0.8
1
Robin Hood Index
Figure 3: 1000 points generated from the 100 families
of income distributions plotted with the approximate
formula (6) and with curves 10% above and below
formula (6).
for most income distributions [18]. Four commonly-
used Lorenz functions are compared in this section.
It is shown that the Robin Hood index in terms of the
Gini coefficient Gfor the four one-parameter Lorenz
functions is well-approximated by the proposed for-
mula (6). The four Lorenz functions considered are
the Pareto, Log-normal, Weibull, and Lam´
e. The
Pareto Lorenz function has the form:
LP(x,G) = 1−(1−x)1−G
1+G.
The Log-normal Lorenz function is given by:
LLN (x,G) = ΦΦ−1(x)−√2Φ−1G+1
2,
where Φ(x) = 1
2(1+erf(x/√2)is the standard normal
cumulative distribution. The Weibull function is:
LW(x,G) = 1−Γ1−log(1−G)/log(2),−log(1−x)
Γ1−log(1−G)/log(2),
where Γ(·,·)is the incomplete (upper) gamma func-
tion. (For example, in MATLAB, LW(x,G) =
gammainc−log(1−x),1−log(1−G)/log(2).)
The fourth Lorenz function considered is the Lam´
e
function given by:
LL(x,G) = (1−(1−x)α)1/α,
where G=1−(Γ(1/α))2
αΓ(2/α)(See, e.g, [14, 15, 18,
19, 20, 21] for more information about these Lorenz
functions.) Each of these Lorenz functions is
uniquely defined by the value of the Gini coefficient
and the Robin Hood index depends on the Gini co-
efficient. Generally, given the Lorenz function, find-
ing the Robin Hood index requires numerical solu-
tion. For the Pareto curve, however, the Robin Hood
0 0.2 0.4 0.6 0.8 1
Gini Coefficient G
0
0.2
0.4
0.6
0.8
1
Robin Hood Index R
Figure 4: Pareto (dotted) and Log-normal (dashed)
Robin Hood indexes compared with Eq. (6) (solid)
and with curves 10% above and below Eq. (6) (solid).
index, RP(G), is given explicitly by
RP(G) = 1−G
1+G1−G
2G2G
1+G.
The graph of the Robin Hood index R(G)for each
one-parameter Lorenz function has certain similari-
ties. In particular, for each Lorenz function, the graph
of the Robin Hood index with respect to the Gini co-
efficient increases from the origin, has initial slope
approximately equal to 0.75, and terminates at the
point (1,1). As Lorenz functions approach a dis-
continuous curve as Gapproaches unity and because
most income densities in applications have G<0.95,
Robin Hood index approximation is examined in the
present investigation for Gini coefficients between
0.0 and 0.95.
Plots of the Robin Hood index with respect to
the Gini coefficient for these four Lorenz curves are
shown in Figures 4 and 5 along with the approximate
formula (6). The approximate formula (6) is gener-
ally within 5% of the Robin Hood index curves for
all these Lorenz functions. In addition, the maximum
differences between approximate formula (6) and
these one-parameter Lorenz-function Robin Hood in-
dexes is: 6.7% for the Pareto function, 4.4% for the
Log-normal function, 4.1% for the Weibull function,
and 6.2% for the Lam´
e function.
5 Comparison with results from a
stochastic income-wealth model
Another check on the approximate Robin Hood index
formula (6) is made in this section by comparing the
approximate formula with Robin Hood indexes ob-
tained for two additional income distributions. The
distributions are derived from a stochastic income-
wealth model which is similar to many models of in-
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0 0.2 0.4 0.6 0.8 1
Gini Coefficient G
0
0.2
0.4
0.6
0.8
1
Robin Hood Index R
Figure 5: Lam´
e (dotted) and Weibull (dashed) Robin
Hood indexes compared with Eq. 6 (solid) and curves
10% above and below Eq. (6) (solid).
come and wealth such as those developed and investi-
gated in [22, 23, 24, 25]. In this model, the economy
is assumed to consist of a large population of individ-
uals having differing individual salaries and invest-
ment proficiencies. The stochastic model considered
has the form
I(t,X) = s(X) + r(X)K(t,X) + fD¯
I(t)−I(t,X),
∂W(t,X)
∂t=I(t,X)−γ2W(t,X)−γ3I(t,X)−c(t,X),
(7)
where ¯
I(t) = R1
0I(t,x)dx,K(t,X) = γ1W(t,X), and
where X∼U(0,1)is a uniformly distributed random
variable on [0,1]. In model (7), the stochastic process
I(t,X)is the rate of income, W(t,X)is the wealth,
and K(t,X)is the capital investment at time t. The
value of Xranges from 0 to 1 and identifies indi-
viduals in the population with a certain salary level
and investment proficiency. The capital K(t,X)is as-
sumed to be proportional to the wealth W(t,X)for
all individuals and all time. The salary rate for each
individual is assumed to be s(X). Income obtained
by investment depends on how the individual’s assets
are invested and is equal to r(X)K(t,X)where r(X)
is rate of capital growth. For example, I(t,xi)is the
income rate for the ith individual in the population,
s(xi)is the salary rate, and r(xi)is the individual’s
rate of capital investment where 0 ≤xi≤1. The pa-
rameter fD≥0 is a linear income redistribution pa-
rameter with no redistribution if fD=0 and increas-
ing redistribution as fD>0 increases. The average
income of the population is denoted ¯
I(t)at time t.
Taxes and costs on wealth and income are calculated
using flat-rate constants 0 ≤γ2,γ3<1. The param-
eter c(t,X)is a rate of consumption for individuals
in the economy. It is assumed, as in [26], that indi-
viduals consume a constant fraction of their wealth,
i.e., c(t,X) = γ4W(t,X). A sample of Nindividu-
als in the economy are described by Nrealizations of
model (7) where the individuals’ wealth and rate of
income, Wi(t)and Ii(t)for i=1,2,...,N, satisfy the
differential equation system:
Ii(t) = si+riKi(t) + fD¯
I(t)−Ii(t),
dWi(t)/dt =Ii(t)−γ2Wi(t)−γ3Ii(t)−ci(t),
where ¯
I(t) = ∑N
i=1Ii(t)/N,Ki(t) = γ1Wi(t), and where
xifor i=1,2,...,Nare uniformly distributed random
numbers on [0,1]and, for example, Ii(t) = I(t,xi).
To determine Lorenz functions and study how
Robin Hood indexes vary with Gini coefficient, the
equilibrium income distribution is considered where
wealth W(t,X), rate of capital K(t,X), and rate of in-
come I(t,X)are not changing with time, specifically,
W(t,X) = W(X),K(t,X) = K(X), and I(t,X) = I(X)
in (7). At equilibrium, model (7) has the form:
I(X) = s(X) + γ1γ5r(X)I(X) + fD(¯
I−I(X)),
W(X) = γ5I(X),K(X) = γ1W(X) = γ1γ5I(X),(8)
where ¯
I=R1
0I(x)dx and γ5= (1−γ3)/(γ2+γ4). As
K(X)and W(X)are proportional to income I(X),
only I(X)needs to be explicitly examined. Average
wealth satisfies ¯
W=r1¯
Iand so, the total wealth of
the individuals is proportional to the total income for
model (8).
For model (8), I(X)satisfies:
I(X) = s(X) + fD¯
I
1+fD−γ1γ5r(X).(9)
Two model cases are considered in the present in-
vestigation. First, it is assumed that rate of invest-
ment, r(X), is the same for all individuals in which
case r(X) = rand Eq. (9) reduces to
I(X) = a1s(X) + a2for constants a1,a2.(10)
In the second case, it is assumed that salary rate, s(X),
is the same for all individuals in which case s(X) = s
and Eq. (9) reduces to
I(X) = b1
1−b2r(X)for constants b1,b2.(11)
For the first model case, it is assumed that
the salary is distributed exponentially. Specifically,
p(s) = βexp(−β(s−s1)for s1≤s<∞is the prob-
ability density of salary s. It follows that for xuni-
formly distributed on [0,1], then s(x)has the form
s(x) = s1−log(1−x)/β. Solving (10) for I(x),
I(x) = I1−a1log(1−x)/βwhere I1=a1s1+a2, in-
dicating that income is also exponentially distributed.
This case’s type of income probability density there-
fore agrees with the type of income density observed
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by several investigators (e.g., [17, 27, 28]). As I(x)is
increasing in xfor 0 ≤x<1, xis the fraction of the
population with rate of income less than or equal to
I(x). Thus, the fraction of total income for the low-
est income fraction xof the population is equal to the
Lorenz function L(x,G) = Rx
0I(u)du/R1
0I(u)du and
is given by:
L(x,G) = x+c(1−x)log(1−x),for 0 ≤x≤1 (12)
for a constant c. The value of cis restricted to 0 ≤c≤
1 to ensure non-negativity of the Lorenz function. As
1−G=2R1
0L(x,G)dx, then c=2G. Thus, the Gini
coefficient Gis restricted to values between 0 and 0.5
for this case. The maximum of x−L(x,G)on [0,1]
occurs at x∗=1−e−1and the Robin Hood coeffi-
cient, R(G), for this case is equal to R(G) = 2G/e1≈
0.736Gfor 0 ≤G≤0.5. Hence, this model case with
exponentially distributed salaries agrees closely with
the approximate formula (6) for Gini coefficients less
than 0.5.
For the second model case, it is assumed that the
rate of investment probability density is a linear func-
tion. That is, p(r) = br for 0 ≤r≤r1=p2/bfor a
positive constant b>0. It follows that r(x) = r1x1/2
for xuniformly distributed on [0,1]. Solving for I(x),
I(x) = a
1−cx1/2
where a,care positive constants. As I(x)is increasing
in xfor 0 ≤x<1, xis the fraction of the population
with rate of income less than or equal to I(x). Thus,
the fraction of total income for the lowest income
fraction xof the population is equal to the Lorenz
function L(x,G) = Rx
0I(u)du/R1
0I(u)du and is given
by:
L(x,G) = cx1/2+log(1−cx1/2)
c+log(1−c),for 0 ≤x≤1.(13)
The Gini coefficient, G=1−2R1
0L(x,G)dx, for the
Lorenz curve in (13) is equal to:
G=2c+c2−1
3c3+ (2−c2)log(1−c)
c3+c2log(1−c),(14)
where 0 <c<1. When the value of the Gini coeffi-
cient Gis specified, then the value of ccan be calcu-
lated using equation (14) which in turn determines the
Lorenz curve in (13). The maximum of x−L(x,G)
occurs at x∗=1/c+c/2c+2log(1−c)2
and
the Robin Hood index is given by x∗−L(x∗,G). The
graph of the Robin Hood index for this case is com-
pared in Figure 6 to the approximate formula (6).
0 0.2 0.4 0.6 0.8 1
Gini Coefficient G
0
0.2
0.4
0.6
0.8
1
Robin Hood Index R
Figure 6: Model case 2 Robin Hood indexes
(dashed) compared with approximate formula (6)
(solid) and curves 10% above and below formula (6)
(solid).
Good agreement is seen between the two curves with
a maximum difference of 5.5%.
In addition, the income probability density for
this case can be derived, for example, from I(x) =
a/(1−cx1/2). Let z=f(x) = a/(1−cx1/2)and, as
f(x)is increasing for 0 ≤x≤1, then x=f−1(z) =
(1−a/z)2/c2. Let g(z) = f−1(z). Finally, let p(I)
be the probability density of income rate I. Then, for
ˆ
I∈[Imin,Imax]and ∆Ismall,
p(ˆ
I)∆I=Pˆ
I<I<ˆ
I+∆I
=Pg(ˆ
I)<X<g(ˆ
I+∆I)
=g0(ˆ
I)∆I=2
c2a
ˆ
I2−a2
ˆ
I3∆I.
Thus, the income probability density p(I)for the sec-
ond case is equal to:
p(I) = 2
c2a
I2−a2
I3for Imin ≤I≤Imax,(15)
where Imin =a, and Imax =a/(1−c).
6 Comparison with income data for
four different nations
To test the approximate formula (6), the formula is
compared with Robin Hood indexes calculated from
the income distribution data available from each of
four countries, Finland, United States, Brazil, and
South Africa. These four countries exhibit a wide
range of Gini coefficients. In 2016, Finland, United
States, and Brazil had Gini coefficients of 0.271,
0.411, and 0.533, respectively. Over the period from
1991 to 2014, South Africa reached a maximum Gini
coefficient of 0.648 in 2005. Income distribution data
for these four countries are listed in Table 1.
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Table 1: Cumulative income fractions of four coun-
tries for eight cumulative population fractions with
Gini coefficients 0.648 for South Africa, 0.533 for
Brazil, 0.411 for United States, and 0.271 for Finland.
(The income distribution data and the Gini coefficient
values are from [29].)
Cumulative South Brazil United Finland
Population Africa States
Fraction 2005 2016 2016 2016
0.00 0.000 0.000 0.000 0.000
0.10 0.010 0.011 0.018 0.039
0.20 0.026 0.033 0.052 0.094
0.40 0.073 0.107 0.155 0.234
0.60 0.148 0.228 0.308 0.409
0.80 0.290 0.420 0.533 0.634
0.90 0.458 0.579 0.696 0.776
1.00 1.000 1.000 1.000 1.000
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Cumulative Population Fraction
Cumulative Income Fraction
South Africa 2005
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Cumulative Population Fraction
Cumulative Income Fraction
Brazil 2016
Figure 7: Income data of South Africa and Brazil
with Gini coefficients 0.648, and 0.533, respectively.
Lorenz curves (13) shown have cvalues of 0.99598
(South Africa) and 0.98276 (Brazil) corresponding to
the respective Gini coefficients.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Cumulative Population Fraction
Cumulative Income Fraction
United States 2016
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Cumulative Population Fraction
Cumulative Income Fraction
Finland 2016
Figure 8: Income data of United States and Finland
with Gini coefficients 0.411 and 0.271, respectively.
Lorenz curves (13) shown have cvalues of 0.94719
(United States), and 0.84894 (Finland) corresponding
to the respective Gini coefficients.
First, the Lorenz function (13), derived in the pre-
vious section, is compared in Figures 7 and 8 with
the income distribution data for these four nations.
The values of cneeded for the Lorenz function (13)
are calculated using equation (14) for the four val-
ues of the Gini coefficients. For Finland, United
States, Brazil, and South Africa, the Gini coefficients
0.271, 0.411, 0.533, and 0.648 result in values of c
equal to 0.84894, 0.94719, 0.98276, and 0.99598,
respectively. The Lorenz function (13) consistently
provides a good fit to the income data for the four
countries. For all four countries, the maximum root
mean square error between the data points in Ta-
ble 1 and the Lorenz curves of Eq. (13) is 0.0103.
In comparison, the maximum root mean square er-
rors between the data and the standard one-parameter
Lorenz curves of Pareto, Log-normal, Weibull, and
Lam´
e [14, 15, 19, 20, 21] are 0.0380, 0.0155, 0.0343,
and 0.0177, respectively, for all four countries.
The Robin Hood index values for the four coun-
tries are: 0.5103, 0.3839, 0.2929, and 0.1910, cal-
culated using polynomial interpolants of the Lorenz
curve income data. The Robin Hood indexes using
formula (6) for the four Gini coefficients are: 0.5032,
0.3997, 0.3041, and 0.2005 which are all within 5%
of the Robin Hood indexes calculated from the in-
come data. A graph of approximate formula (6) is
illustrated in Figure 9 along with the Robin Hood in-
dex values for the four countries.
7 Comparison with several reported
results
Generally, Gini coefficients and Robin Hood indexes
are strongly correlated in reported investigations in
the literature [11, 30, 31, 32]. For example, the anal-
ysis of [30] indicates that income inequality measures
(such as the Gini coefficient and the Robin Hood in-
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0 0.2 0.4 0.6 0.8
Gini Coefficient
0
0.2
0.4
0.6
0.8
Robin Hood Index
South Africa 2005
Brazil 2016
United States 2016
Finland 2016
Figure 9: Robin Hood indexes for South Africa,
Brazil, United States, and Finland graphed with the
approximate Robin Hood index formula (6).
dex) behave very similarly and are highly correlated
[11]. Indeed, it has been observed for certain Lorenz
functions, such as the Lam´
e function, that the Robin
Hood index is approximately linear with respect to
the Gini coefficient (see, e.g., [18]).
In this section, the results of three previous inves-
tigations are compared with the approximate Robin
Hood index formula (6). The first study [33] exam-
ines the population density with distance from the
center of Chicago in 1900. The second study [34]
compares distributions of maternal and child health-
related workforces in provinces of Iran during 2010-
2012. The third study [35] examines income distri-
butions of users of three community forests in Nepal
with pre-community forestry usage occurring in 1995
and post-community forestry usage in 2009. The
Gini coefficients and Robin Hood indexes reported
for these studies are illustrated in Figure 10 along
with the approximate formula (6). The approximate
formula agrees well with the reported Robin Hood
index values for these three investigations. Formula
(6) is within 5% of the Robin Hood index values for
twenty-four of the twenty-six points and within 6%
and 7% for the two remaining points.
8 Summary
The Gini coefficient is a widely-used measure of
income inequality but is defined in a mathematical
sense. The Robin Hood index, which is not as widely
used as the Gini coefficient, is equal to the proportion
of the population’s total income that, if redistributed,
would give perfect income equality. An approximate
formula for the Robin Hood index in terms of the Gini
coefficient is useful in obtaining a better understand-
ing of income distributions.
An approximate formula for the Robin Hood in-
dex, R, in terms of the Gini coefficient, G, is de-
termined for values of Gbetween 0 and 0.95. The
0 0.2 0.4 0.6 0.8 1
Gini Coefficient
0
0.2
0.4
0.6
0.8
1
Robin Hood Index
* Population density from center of Chicago in 1900
x Health workforce distribution in Iran in 2011
o Income distribution of forestry users in Nepal in 1995 and 2009
Figure 10: Robin Hood indexes for three different
investigations plotted with the approximate Robin
Hood index formula (6).
approximation is developed from 100,000 Lorenz
curves that are randomly generated based on 100
twenty-parameter families of income distributions.
The formula is within 5% (within 10%) of the Robin
Hood indexes for 81% (for 97%) of the randomly-
generated income distributions examined.
The approximate formula is tested against Robin
Hood indexes of commonly-used one-parameter
Lorenz curves, income data of several countries, and
reported results of Robin Hood indexes. The ap-
proximate formula is also tested against results of
a stochastic income-wealth model which is intro-
duced in the present investigation. The continuous
piecewise-linear approximation is generally within
5% of Robin Hood indexes of standard one-parameter
Lorenz curves, income distribution data, and the
stochastic income-wealth model.
Acknowledgement
The author acknowledges the reviewers’ helpful com-
ments on improving the paper.
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