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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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    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) = 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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    0.2

    0.4

    0.6

    0.8

    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}.