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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 index I 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) =

∑n i ,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 it has an intuitive geometric interpretation: it can be defined as the ratio of the area that lies between the line of perfect equality and the Lorenz curve, over the total area under the line of equality.

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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) = 1 n

[ n + 1− 2

∑n i=1(n + 1− i)x∗i∑n

i=1 x ∗ i

] . (1)

I For this discrete version, 0 ≤ G (x) ≤ n−1n for x ∈ R n +\{0}.

I Equivalent formula for non-increasingly ordered x∗:

G (x) = 1 n

[ n + 1− 2

∑n i=1 ix∗i∑n i=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 , . . . ,

1 n − k︸ ︷︷ ︸

n−k

) , k = 0, . . . , n − 1,

The idea behind this axiom is such that:

I (v1) = I ( 1n , . . . , 1 n ) = 0, I (v

n) = I (0, . . . , 0, 1) = n−1n ,

and the differences

I (v1)− I (v2) = I ( 1 n , . . . ,

1 n

) − I ( 0, 1n−1 , . . . ,

1 n−1 ) ,

...

I (vn−1)− I (vn) = I ( 0, . . . , 0, 12 ,

1 2

) − 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) = 1 n

[ n + 1− 2

∑n i=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 index on 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 from households above the mean to those below the mean to achieve a state of perfect equality in the distribution of incomes:

H(x) =

∑ {i :xi≥x̄} (xi − x̄)∑n

i=1 xi .

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

Robin Hood index – alternative formulas

H(x) =

∑ {i :xi≥x̄} (xi − x̄)∑n

i=1 xi .

I Since ∑ {i : xi≥x̄} (xi − x̄) = −

∑ {i : xi≤x̄} (xi − x̄):

H(x) =

∑n i=1 |xi − x̄ | 2 ∑n

i=1 xi . (∗)

I Dividing nominator and denominator by n:

H(x) = 1 n

∑n i=1 |xi − x̄ | 2x̄

= MAD(x)

2x̄ . (∗∗)

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

H(x) = 1 2

n∑ i=1

∣∣∣∣ xi∑n i=1 xi

− 1 n

∣∣∣∣ . (∗ ∗ ∗)

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 , . . . ,

1 n ) ∈ R

n, we can rewrite (∗ ∗ ∗) as:

H(x) = 1 2

∥∥∥∥ x‖x‖1 − 1n ∥∥∥∥

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 ) =

1 2 E ∣∣∣∣ XE(X ) − 1

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

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

hence H(X ) = F (EX )− L(F (EX ))

(vertical distance between the Lorenz curve and the line of perfect equality 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:

lim t→t0

L(F (t))− L(F (t0)) F (t)− F (t0)

=

= lim t→t0

1 E(X )

∫ t −∞ x dF (x)−

1 E(X )

∫ t0 −∞ x dF (x)∫ t

−∞ dF (x)− ∫ t0 −∞ dF (x)

=

= 1

E(X ) lim t→t0

∫ t t0 x dF (x)∫ t

t0 dF (x)

= 1

E(X ) · t0f (t0)

f (t0) =

t0 E(X )

.

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

I Lorenz curve is convex ⇒ slope is increasing ⇒ vertical distance between the Lorenz curve and the line of perfect equality is greatest at F (E(X )).

Robin Hood index and Lorenz curve

Robin Hood index can be graphically represented as the maximum vertical distance between the Lorenz curve and the 45-degree line that represents perfect equality of incomes.

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H(X )

F (E(X ))

Corollary: Robin Hood ≤ Gini

DA 0.2 0.8

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1 C

F

B

E

I Let q = F (E(X )) = |AE |. We have H = |BF |, |BE | = q − H. I |AEB| = 12q(q − H) =

1 2(q

2 − Hq), I |BCDE | = (1− q)1+(q−H)2 =

1 2(1− q

2 − H + Hq), I ∫ 1 0 L(t) dt ≤ |AEB|+ |BCDE | =

1−H 2 ,

I G = 1 2−

∫ 1 0 L(t) dt

1 2

= 1− 2 ∫ 1 0 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

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1 max: H = 0.5, G = 0.75

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1 min: 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}.