Gini index and Robin Hood index

Rafał Kucharski

Notation

I Let N = {1, 2, . . . , n} be a set of individuals.I Incomes: x = (x1, . . . , xn) ∈ Rn

+\{0}.I Value xi denotes the income of the agent i .I For x ∈ Rn let x∗ denote the vector obtained form x by sorting its

element in non-decreasing way:

x∗1 ≤ · · · ≤ x∗n .

I The group of permutations on N: Sn = {σ : N → N : σ is bijective}.

Gini indexI Discussed in 1876 by F. R. Helmert (geodesist, discoverer of the

chi-squared distribution?),I „Rediscovered” by Corriado Gini in 1912.I For integrable random variable X :

G (X ) =E|X − Y |2E(X )

,

where Y is an independent copy of X .I Discrete equivalent:

G (x) =

∑ni ,j=1 |xi − xj |2n∑n

i=1 xi, x ∈ Rn.

I Another abstract formula:

G (X ) = 1− E(min(X ,Y ))

E(X ),

where Y is an independent copy of X .

I The attractiveness of the Gini index to many economists is that ithas an intuitive geometric interpretation: it can be defined as theratio of the area that lies between the line of perfect equality and theLorenz curve, over the total area under the line of equality.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Lorentz Curve

I Lorenz curve:{(F (x), L(F (x))) : x ∈ R},

where F is the cdf of X and

L(F (x)) =E(X · 1(X ≤ x))

E(X )=

∫ x−∞ t dF (t)

E(X ).

Gini index – Sen formula

I Sen [9] defined the Gini index as a function G : Rn+\{0} → R such

that

G (x) =1n

[n + 1− 2

∑ni=1(n + 1− i)x∗i∑n

i=1 x∗i

]. (1)

I For this discrete version, 0 ≤ G (x) ≤ n−1n for x ∈ Rn

+\{0}.I Equivalent formula for non-increasingly ordered x∗:

G (x) =1n

[n + 1− 2

∑ni=1 ix∗i∑ni=1 x∗i

]. (2)

Axioms

In Plata-Pérez et. al. [2015], the following properties were defined:

1. The inequality index I is said to be scale-independent if

I (λx) = I (x) for all x ∈ Rn+\{0} and λ > 0.

2. The inequality index I is symmetric if and only if

I (xσ) = I (x) for every σ ∈ Sn and x ∈ Rn+\{0}.

3. The inequality index I is comonotone separable if

I (βx + (1− β)y) = βI (x) + (1− β)I (y)

for every β ∈ [0, 1], every x , y ∈ Rn+\{0} that are comonotone and∑n

i=1 xi =∑n

i=1 yi . (We say that x and y are comonotone if(xi − xj)(yi − yj) ≥ 0, for all i , j ∈ N.)

Axioms cont.

4. The inequality index I is said to be standardized if I (vk+1) =k

n,

where

vk+1 =

(0, . . . , 0︸ ︷︷ ︸

k

,1

n − k, . . . ,

1n − k︸ ︷︷ ︸

n−k

), k = 0, . . . , n − 1,

The idea behind this axiom is such that:

I (v1) = I ( 1n , . . . ,

1n ) = 0, I (vn) = I (0, . . . , 0, 1) = n−1

n ,

and the differences

I (v1)− I (v2) = I( 1n , . . . ,

1n

)− I(0, 1

n−1 , . . . ,1

n−1

),

...

I (vn−1)− I (vn) = I(0, . . . , 0, 1

2 ,12

)− I(0, . . . , 0, 1

)are all equal.

Characterization

Theorem (Plata-Pérez et. al. 2015)Let I : Rn

+\{0} → R. Then I equals the Gini index given by

G (x) =1n

[n + 1− 2

∑ni=1(n + 1− i)x∗i∑n

i=1 x∗i

],

if and only if it satisfies the axioms of scale independence, symmetry,comonotone separability and standardization.

Proof idea

If I : Rn+\{0} → R is any index satisfying the four axioms, then using the

symmetry and scale independence axioms we have:

I (x) = I (x∗) = I

(x∗/

n∑j=1

xj

), x ∈ Rn

+\{0}.

The key idea is to show that Axioms 4 and 3 characterize the Gini indexon the convex subset

K =

{y ∈ Rn

+ :n∑

i=1

yi = 1 and y1 ≤ y2 · · · ≤ yn

}.

Robin Hood index

I also known as:I Hoover index (Edgar Malone Hoover jr., 1936),I Schutz (1951) [7]I Ricci index, (Umberto Ricci 1879–1946)I Pietra ratio, (G. Pietra)I Lindahl index (Erik Robert Lindahl 1891–1960),

I The share of total income which would have to be transferred fromhouseholds above the mean to those below the mean to achieve astate of perfect equality in the distribution of incomes:

H(x) =

∑{i :xi≥x} (xi − x)∑n

i=1 xi.

I Higher values indicate more inequality and that more redistribution isneeded to achieve income equality [6].

Robin Hood index – alternative formulas

H(x) =

∑{i :xi≥x} (xi − x)∑n

i=1 xi.

I Since∑{i : xi≥x} (xi − x) = −

∑{i : xi≤x} (xi − x):

H(x) =

∑ni=1 |xi − x |2∑n

i=1 xi. (∗)

I Dividing nominator and denominator by n:

H(x) =1n

∑ni=1 |xi − x |2x

=MAD(x)

2x. (∗∗)

I Dividing by the total income inside the absolute value in (∗):

H(x) =12

n∑i=1

∣∣∣∣ xi∑ni=1 xi

− 1n

∣∣∣∣ . (∗ ∗ ∗)

Robin Hood index – in L1 space

I Let us think about Rn as the normed space L1:

‖x‖1 =n∑

i=1

|xi |.

I For x ∈ Rn+\{0} we have xi ≥ 0, i = 1, . . . , n, hence

‖x‖1 =n∑

i=1

xi .

I Denoting by 1n = ( 1

n , . . . ,1n ) ∈ Rn, we can rewrite (∗ ∗ ∗) as:

H(x) =12

∥∥∥∥ x

‖x‖1− 1

n

∥∥∥∥1.

I We can exploit the properties of the L1 norm!

Robin Hood index and Lorenz curve

I For integrable random variables (referring to (∗∗)):

H(X ) =E|X − E(X )|

2E(X )=

12E∣∣∣∣ X

E(X )− 1∣∣∣∣ .

I We can easily show that (L is a Lorenz curve)

E|X − E(X )| = 2E(X )[F (EX )− L(F (EX ))] (?)

henceH(X ) = F (EX )− L(F (EX ))

(vertical distance between the Lorenz curve and the line of perfectequality at F (EX )).

Proof of (?)

Let us denote µ = E(X ).

E|X − µ| =

∫ ∞−∞|x − µ| dF (x) =

=

∫ µ

−∞(µ− x) dF (x) +

∫ ∞µ

(x − µ) dF (x) =

= µ

∫ µ

−∞dF (x)−

∫ µ

−∞x dF (x) +

∫ ∞µ

x dF (x)− µ∫ ∞µ

dF (x) =

= µF (µ)−∫ µ

−∞x dF (x) + (µ−

∫ µ

−∞x dF (x))− µ(1− F (µ)) =

= 2µF (µ)− 2∫ µ

−∞x dF (x) = 2µF (µ)− 2µL(F (µ)).

since L(F (t)) =∫ t−∞ x dF (x)/E(X ).

Robin Hood index and Lorenz curve

I Assuming F and L are differentiable respectively at t0 and F (t0),F ′(t0) = f (t0), the slope of tangent at (F (t0), L(F (t0))) is equal:

limt→t0

L(F (t))− L(F (t0))

F (t)− F (t0)=

= limt→t0

1E(X )

∫ t−∞ x dF (x)− 1

E(X )

∫ t0−∞ x dF (x)∫ t

−∞ dF (x)−∫ t0−∞ dF (x)

=

=1

E(X )limt→t0

∫ tt0x dF (x)∫ t

t0dF (x)

=1

E(X )· t0f (t0)

f (t0)=

t0E(X )

.

I The slope of the tangent of the Lorenz curve equals 1 (tangent isparallel to the line of perfect equality) for t0 = E(X ).

I Lorenz curve is convex ⇒ slope is increasing ⇒ vertical distancebetween the Lorenz curve and the line of perfect equality is greatestat F (E(X )).

Robin Hood index and Lorenz curve

Robin Hood index can be graphically represented as the maximum verticaldistance between the Lorenz curve and the 45-degree line that representsperfect equality of incomes.

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1

H(X )

F (E(X ))

Corollary: Robin Hood ≤ Gini

DA0.2 0.8

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0.4

0.6

0.8

1 C

F

B

E

I Let q = F (E(X )) = |AE |. We have H = |BF |, |BE | = q − H.I |AEB| = 1

2q(q − H) = 12(q2 − Hq),

I |BCDE | = (1− q)1+(q−H)2 = 1

2(1− q2 − H + Hq),I∫ 10 L(t) dt ≤ |AEB|+ |BCDE | = 1−H

2 ,

I G =12−

∫ 10 L(t) dt

12

= 1− 2∫ 10 L(t) dt ≥ 1− 2 · 1−H

2 = H.

I By-product: H = 1− 2(|AEB|+ |BCDE |) = 2|ABC |.

Robin Hood vs Gini: min and max difference

0 0.2 0.4 0.6 0.8 10

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0.4

0.6

0.8

1max: H = 0.5, G = 0.75

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1min: H = 0.5, G = 0.5

Attempt to characterize Robin Hood index

I Robin Hood index fulfils:I scale invariance,I symmetry,I standardization.

I Moreover (int-standardization):

H(x) =zeros(x)

n, for x ∈ Jn,

where zeros(x) = #{i : xi = 0} and

Jn = {(x1, . . . , xn) : xi = ki/n, ki ∈ N, i = 1, . . . , n,∑n

i=1 ki = n}.

Attempt to characterize Robin Hood index,cont.

I We say that x , y ∈ Rn+\{0} are co-mean if

(xi − x)(yi − y) ≥ 0, i = 1, . . . , n.

I Axiom 3b: Robin Hood index is co-mean separable:

I (βx + (1− β)y) = βI (x) + (1− β)I (y)

for every β ∈ [0, 1], every x , y ∈ Rn+\{0} that are co-mean and∑n

i=1 xi =∑n

i=1 yi .

HypothesisFunction I : Rn

+\{0} → R equals the Robin Hood index given by (∗ ∗ ∗) ifand only if it satisfies the axioms of scale independence, symmetry,co-mean separability and int-standardization.

Real dataBased on: OECD Income Distribution Database (IDD),http://www.oecd.org/social/OECD2016-Inequality-Update-Figures.xlsx

H = 0.745876G − 0.005591.

Estimation from decilesFrom [5]: The Robin Hood index may also be calculated by:

I summing the percentage of income for each tenth of an incomedistribution where the percentage exceeds 10%,

I and subtracting from this the product of the number of tenths thatmeet this criterion times 10%.

Example

Decile 1 2 3 4 5Percentage of total income 1.08 2.48 4.13 5.74 7.33

Decile 6 7 8 9 10Percentage of total income 8.97 10.83 13.09 16.41 29.93

In this case four deciles (7-10) exceed 10%, so the Robin Hood index

= (10.83% + 13.09% + 16.41% + 29.93%)− (4× 10%) =

= 70.26%− 40% = 30.26%.

Thank you for your attention.

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