Normative comparisons of distributions of one attribute

Basic framework We assume here that social states that are visible by the evaluator (economist, philosopher, policy maker) are described by a lists of numbers (one such number for every individual) The number is supposed to represent the quantity of some attribute that is cardinally meaningful Best example: income, Other examples (not as good): life expectancy, schooling Notice the distinction between what is visible by the analysis and what may be normatively important

Formally : We compare objects such as: x = ( x 1,…, x m ) a distribution of the attribute between individuals 1,… m and y = ( y 1,…, y n ) a distribution of the attribute between individuals 1,..., n x and y differ by: 1) the number and identity of the individuals 2) The size of the cake that is distributed (efficiency) 3) The sharing of the cake given its size (equity, equality) 3a) The amount of the cake held by those who gets the least (tighly connected to 3) We want to compare income distributions in a way which captures each of these features

Differing number of individuals The handling of this difference is easy It is done by the so-called Dalton principle of replication ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m …, x m )

Differing number of individuals The handling of this difference is easy It is done by the so-called Dalton principle of replication ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m …, x m ) k times

Differing number of individuals The handling of this difference is easy It is done by the so-called Dalton principle of replication ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m …, x m ) k times

Differing number of individuals The handling of this difference is easy It is done by the so-called Dalton principle of replication ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m …, x m ) k times

Differing number of individuals The handling of this difference is easy It is done by the so-called Dalton principle of replication ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m …, x m ) k times “Nothing important is changed from an ethical point of view if a given distribution is replicated k times”

Differing number of individuals The handling of this difference is easy It is done by the so-called Dalton principle of replication ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m …, x m ) k times “Nothing important is changed from an ethical point of view if a given distribution is replicated k times” This principle enables one to compare distributions with differing numbers of individuals

Differing number of individuals The handling of this difference is easy It is done by the so-called Dalton principle of replication ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m …, x m ) k times “Nothing important is changed from an ethical point of view if a given distribution is replicated k times” This principle enables one to compare distributions with differing numbers of individuals Indeed suppose I want to compare distribution x containing m individuals with distribution y containing n individuals

Differing number of individuals

( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m,…, x m ) n times

Differing number of individuals ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m,…, x m ) ( y 1,…, y n )  ( y 1,…, y 1, y 2,…, y 2,…, y n …, y n ) n times

Differing number of individuals ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m,…, x m ) ( y 1,…, y n )  ( y 1,…, y 1, y 2,…, y 2,…, y n …, y n ) n times m times

Differing number of individuals ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m,…, x m ) ( y 1,…, y n )  ( y 1,…, y 1, y 2,…, y 2,…, y n …, y n ) n times m times nm individuals

Differing number of individuals ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m,…, x m ) ( y 1,…, y n )  ( y 1,…, y 1, y 2,…, y 2,…, y n …, y n ) n times m times nm individuals

Differing number of individuals ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m,…, x m ) ( y 1,…, y n )  ( y 1,…, y 1, y 2,…, y 2,…, y n …, y n ) n times m times nm individuals If I can compare together these two replicates of x and y then, by transitivity, I can compare x and y

Differing number of individuals ( x 1,…, x m )  ( x 1,…, x 1, x 2,…, x 2,…, x m,…, x m ) ( y 1,…, y n )  ( y 1,…, y 1, y 2,…, y 2,…, y n …, y n ) n times m times nm individuals If I can compare together these two replicates of x and y then, by transitivity, I can compare x and y Hence, we shall assume from now on that distributions have the same number ( n ) of individuals

Important principle: anonymity The name (identity) of the individuals does not matter Underlying this principle is the assumption that individuals do not differ in other dimensions than income Formally, x  A x for every n  n permutation matrix Recall that an n  n square matrix A is a permutation matrix if it satisfies A ij  {0,1} for every i, j =1,…, n and A i1 +….+A in =A 1j +…+A nj = 1 for every i, j For every x, x (.) denotes its ordered permutation x (.) = A x for some permutation matrix such that x (i)  x (i+1) for all i =1,…, n -1

Efficiency Let  ( x ) = (  i x i )/ n  ( x ) (the mean of x ) is a natural (and widely used) measure of the “size of the cake”. What principles should a measure of “efficiency” satisfy ? 1: Weak (Pareto) efficiency x  y  x is weakly better than y 2) Strong (Pareto) Efficiency x > y  x is strictly better than y 1a) Anonymous (Suppes) weak efficiency x (.)  y (.)  x is weakly better than y 2a) Anonymous (Suppes) Strong efficiency x (.) > y (.)  x is strictly better than y

Pareto and Suppes efficiency ? x2x2x2x2 x1x1x1x1 45 o yy2yy2 yy1yy1 yy2yy2 y1y1y1y1 ( y (1), y (2) ) Pareto better (y1,y2)(y1,y2) Pareto & Suppes better

Efficiency  verifies both Pareto and Suppes efficiency The notions of efficiency captured by the Pareto and Suppes principles are strong in the sense that they are not required to hold “ceteris paribus” with respect to inequality For example (1,1000)  (1,100) for both principles Yet, in some sense (to be made precise soon), (1,100) is “more equal” than (1,1000) There are other (weaker) notions of efficiency that restrict the scope of the principle to situations that are “identical” in terms of equality

Efficiency 1) For all real numbers t > 1, tx (.)  x (increasing all incomes in a common proportion t does not affect inequality if inequality is taken to be a “relative” concept). 2) For all strictly positive real number a, x (.) + a  x (adding a euros to everyone does not affect inequality if inequality is an “absolute” concept) Both principles are obviously satisfied by   (tx (.) ) = t  ( x) >  ( x) for every number t > 1  (x (.) + a ) = a +  ( x) >  ( x) for every strictly positive number a

Equality What do we mean by “equalizing” ? Basic idea: Reducing pair-wise discrepancies in incomes without “throwing cake away”. Pigou-Dalton notion of transfer x (.) has been obtained from z (.) by a bilateral Pigou- Dalton transfer if there are i and j  {1,…,n} and a strictly positive real number  such that x ( h ) = z (h) for all h ≠ i, j and x ( i ) = z ( i ) +   z ( j ) -  = x ( j ) Illustration: z(i)z(i) z(j)z(j)   x(i)x(i) x(j)x(j)

Equality x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers ( x (.)  PD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a bilateral Pigou-Dalton transfer for t = 0,…, T -1 Example: x (.) = (2,4,8,11,16) z (.) = (1,5,7,10,18)

Equality x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers ( x (.)  PD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a bilateral Pigou-Dalton transfer for t = 0,…, T -1 Example: x (.) = (2,4,8,11,16) z 3 (.) = (1,5,7,10,18)

Equality x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers ( x (.)  PD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a bilateral Pigou-Dalton transfer for t = 0,…, T -1 Example: x (.) = (2,4,8,11,16) z 2 (.) = (2,4,7,10,18)

Equality x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers ( x (.)  PD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a bilateral Pigou-Dalton transfer for t = 0,…, T -1 Example: x (.) = (2,4,8,11,16) z 1 (.) = (2,4,8,10,17)

Equality x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers ( x (.)  PD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a bilateral Pigou-Dalton transfer for t = 0,…, T -1 Example: x (.) = (2,4,8,11,16) z 0 (.) = (2,4,8,11,16)

Equality x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers ( x (.)  PD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a bilateral Pigou-Dalton transfer for t = 0,…, T -1 Example: x (.) = (2,4,8,11,16) z 0 (.) = (2,4,8,11,16) If x (.)  PD z(.)), then distribution x is unambiguously more equal than the distribution z because it results from a finite sequence of clearly “equalizing” elementary operations

Pigou-Dalton transfer and T-transforms An n  n matrix A is a T-transform if there exist i and j and a number   [0,1] such that: A ii = A jj = 1-  A ij = A ji =  A hh = 1 for all h ≠ i, j A hk = 0 for all h, k ≠ i, j, h ≠ k Let us visualize what a T-transform is

An example of a T-transform 10… …0 0 0.…10…0 0…01-  …  ….0 ………01 0…0  01-  …0.……0…0… 00……………1 i j ij

A Pigou-Dalton transfer is a T- transform Indeed, if x and z are two distributions of income for which x (.) = A z (.) for some T -transform Then: x (.) = (z (1),…, z ( i -1),(1-  ) z ( i ) +  z ( j ), z ( i +1),…, z (j-1),  z ( i ) +(1-  ) z ( j ), z ( j +1),…, z ( n ) ) If x (.) is obtained from z (.) by a Pigou-Dalton transfer, then x (.) = (z (1),…, z ( i -1), z ( i ) + , z ( i +1),…, z (j-1), z ( j ) - , z ( j +1),…, z ( n ) ) for some   [0, [ z ( j ) - z ( i ) ]/2] Which is a T -transform if  =  /[ z ( j ) - z ( i ) ]

Bistochastic matrices An n  n matrix A is bistochastic if it satisfies: A ij  [0,1] for all i, j.  j A ij =  i A ij = 1 for all i and j A permutation matrix is a bistochastic matrix So is a T -Transform If x = Ay for some bistochastic matrix A, then x can be said to be “more equal” than y. Indeed, what the premultiplication of a vector of numbers by a bistochastic matrix does is that it reduces the discrepancies between numbers without changing the mean Indeed if x = Ay for some bistochastic matrix A, then

Some results Result 1: x (.)  PD y (.) if and only x = Ay for some bistochastic matrix A Result 2 : x = Ay for some bistochastic matrix A if and only if x (.) = T 1 T 2 …T K y (.) for some finite sequence of T- transforms.

Lorenz dominance and Pigou-Dalton There is a nice test to check if x (.)  PD y (.) For any distribution x and number k =1,…, n define L k ( x ) by: For any two distributions of income x and y with the same Means, define the Lorenz binary relation  L by: x  L y  L k ( x )  L k ( y ) for all k  L is a quasi-ordering (reflexive and transitive) Theorem (Hardy-Littlewood-Polya): Let x and y be two distributions such that  (x) =  (y). Then x  PD y  x  L y

Example has (2,6,6,15,16) been obtained from (1,5,7,14,18) by a finite sequence of Pigou- Dalton transfers ? Let us do the Lorenz test and draw the Lorenz curves This curve shows the points ( k,L k ( x )) for any distribution x

Lorenz Curves position Cumulated income Lorenz curve for (1,5,7,14,18) Lorenz curve for (2,6,6,15,16) Hence (2,6,6,15,16) has been obtained from (1,5,7,14,18) by a finite sequence of Pigou-Dalton transfers (find the sequence!!)

Principle of diminishing transfers A bilateral Pigou-Dalton transfer is a clearly equalizing operation But it remains silent about the ranking of some distributions Consider for example (2,4,6,10) and (1,5,7,9) To go from (1,5,7,9) to (2,4,6,10) we have made a progressive transfer of 1 from 2 to 1 and a regressive transfer of the same amount from 3 to 4 Can it be said that (2,4,6,12) is more equal than (1,5,7,11) ? Yes if one adheres to the Foster & Shorrocks (1987) principle of diminishing transfer The key transformation involved in this principle is the composite transfer

Composite transfer ? x (.) has been obtained from z (.) by a composite transfer if there are h, i, j and k  {1,…, n } and a strictly positive real number  such that 1) x ( g ) = z (g) for all g ≠ h, i, j and k 2) x ( h ) = z ( h ) +   z ( i ) -  = x ( i ) 3) z ( j ) -  = x ( j ) 4) x ( k ) = z ( k ) +  5) z ( i ) – z ( h ) = x ( k ) – x ( j ) > 0 6) z ( h ) < x ( j )

A Composite transfer Is a combination of a progressive transfer in the lower tail of the distribution with a regressive transfer of the same magnitude in the upper tail of the distribution. Does not affect the variance of the distribution of income

A Composite transfer does not affect the variance Suppose x has been obtained from z by a composite transfer ( x and z are assumed to have the same mean  ) Let us calculate the variance V ( x ):

A Composite transfer does not affect the variance Suppose x has been obtained from z by a composite transfer ( x and z are assumed to have the same mean  ) Let us calculate the variance V ( x ):

A Composite transfer does not affect the variance Suppose x has been obtained from z by a composite transfer ( x and z are assumed to have the same mean  ) Let us calculate the variance V ( x ):

A Composite transfer does not affect the variance Suppose x has been obtained from z by a composite transfer ( x and z are assumed to have the same mean  ) Let us calculate the variance V ( x ):

A Composite transfer does not affect the variance Suppose x has been obtained from z by a composite transfer ( x and z are assumed to have the same mean  ) Let us calculate the variance V ( x ):

A Composite transfer does not affect the variance Suppose x has been obtained from z by a composite transfer ( x and z are assumed to have the same mean  ) Let us calculate the variance V ( x ):

A Composite transfer does not affect the variance Suppose x has been obtained from z by a composite transfer ( x and z are assumed to have the same mean  ) Let us calculate the variance V ( x ): Because z ( i ) – z ( h ) = x ( k ) – x ( j )

A larger class of variance- preserving composite transfers In a composite transfer, the amount transferred (from the rich to the poor at the bottom, and from the poor to the rich at the top) is the same We may want to consider a broader class of transfers that avoid this constraint but that keep the requirement that the variance be unaffected.

A larger class of variance- preserving composite transfers x (.) has been obtained from z (.) by a variance preserving composite transfer if there are h, i, j and k  {1,…, n } and strictly positive real numbers  and  such that 1) x ( g ) = z (g) for all g ≠ h, i, j and k 2) x ( h ) = z ( h ) +   z ( i ) -  = x ( i )  x ( j ) = z ( j ) -  3) x ( k ) = z ( k ) +  5) z ( i ) – z ( h ) > 0 and x ( k ) – x ( j ) > 0 6) V ( x ) = V ( z )

A stronger notion of « equalization » x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers and/or variance preserving composite transfers ( x (.)  COMP z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by either a bilateral Pigou-Dalton transfer or a composite variance-preserving transfer for t = 0,…, T -1 Example: x (.) = (2,4,6,12,16) z (.) = (1,5,7,9,18)

A stronger notion of « equalization » x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers and/or variance preserving composite transfers ( x (.)  COMP z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by either a bilateral Pigou-Dalton transfer or a composite variance-preserving transfer for t = 0,…, T -1 Example: x (.) = (2,4,6,12,16) z 2 (.) = (1,5,7,9,18)

A stronger notion of « equalization » x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers and/or variance preserving composite transfers ( x (.)  COMP z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by either a bilateral Pigou-Dalton transfer or a composite variance-preserving transfer for t = 0,…, T -1 Example: x (.) = (2,4,6,12,16) z 1 (.) = (2,4,6,10,18) Composite variance preserving

A stronger notion of « equalization » x (.) has been obtained from z (.) by a finite sequence of Pigou-Dalton transfers and/or variance preserving composite transfers ( x (.)  COMP z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by either a bilateral Pigou-Dalton transfer or a composite variance-preserving transfer for t = 0,…, T -1 Example: x (.) = (2,4,6,12,16) z 0 (.) = (2,4,6,12,16) Pigou Dalton

Empirical test of this stronger notion of equalization There is a nice test to check if x (.)  COMP y (.) Theorem (Foster & Shorrocks (1987): Let x and y be two distributions such that  (x) =  (y). Then x  COMP y  there exists a k  {1,…, n } such that L j (x)  L j (y) for all j  k (with 1 inequality strict) and L l (x)  L l (y) for all l > k and V(x)  V(y) In words, if  (x) =  (y), x  COMP y if and only if the variance of x is weakly smaller than the variance of y and the Lorenz curve of x starts above that of y and crosses that of y at most once.

Measuring inequality Now that we have defined what it means for a distribution to be “more equal” than another, we may want to measure the inequality of a distribution x by a single number I ( x ) I ( x )  I ( y ) means “inequality is no smaller in x than in y ” We shall assume for the sake of this course that any income distribution lies in  n + I may verify I (  x )= I ( x ) for any strictly positive real number  and every x   n + (relative) I may verify I ( x +  )= I ( x ) for any strictly positive real number  and every x   n + (absolute) We will restrict attention to relative indexes

Properties of inequality indices Schur-convexity: I ( x )  I ( A. x ) for every n  n bistochastic matrix A and every x in  n + Strict Schur-convexity: I ( x ) > I ( A. x ) for every n  n bistochastic matrix A that is not a permutation matrix Schur-convexity garantees (Hardy-Littlewood-Polya) that I will be (weakly or strictly) sensitive to Pigou- Dalton transfers Hence, for any I Schur Convex, x (.)  PD y (.)  I( x )  I ( y ) (or I( x ) < I ( y ) if Schur-convexity is strict) It also entails the property of symmetry Symmetry: I ( x ) = I ( x (.) ) for every distribution x   n + We could also require I to be sensitive to variance- preserving composite transfers. x (.)  COMP y (.)  I( x )  I ( y ) (Variance-Preserving Composite Transfers (VPCT) sensitivity)

Examples of inequality indices: Interquartile ratio Inter quartile ratio I Iq ( x ) : For any fraction q  [0,1] I Iq ( x ) is defined by This index measures (negatively) the ratio of the fraction of total income held by the q th poorest fraction of the population over the total income held by the q th richest fraction of the population This index is weakly (but not strictly) Schur-convex. It does not satisfy VPCT sensitity

Examples of inequality indices: Coefficient of variation This index ranks income distributions with the same mean as per their variance; the absolute version this index is the standard deviation This index is strictly Schur-convex but it violates VPCT sensitity Coefficient of variation : I CV ( x )

Examples of inequality indices: Coefficient of variation of the logarithm Coefficient of variation of the logarithm I CVLOG ( x ) : Comes from applied work where the distribution of income is often assumed to be log normal (the distribution of the logarithm of income is normal) This index is not Schur-convex and violates therefore the Pigou-Dalton principle

Examples of inequality indices: Theil Theil index I T ( x ) : This relative index is very closely related to the entropy index of diversity discussed earlier It is strictly Schur convex and verifies VPCT sensitivity

Examples of inequality indices: Atkinson-Kolm Atkinson-Kolm index (for a parameter  > 0) I AK  ( x ) : This relative index is strictly Schur convex and verifies VPCT sensitivity

Examples of inequality indices: Generalized entropy family

Generalized entropy family of indices (for a parameter c ) I GEc ( x ) :

Examples of inequality indices: Generalized entropy family Generalized entropy family of indices (for a parameter c ) I GEc ( x ) :

Examples of inequality indices: Generalized entropy family Generalized entropy family of indices (for a parameter c ) I GEc ( x ) :

Examples of inequality indices: Generalized entropy family Generalized entropy family of indices (for a parameter c ) I GEc ( x ) : For c =2, we have I GEc ( x ) = I CV ( x ) 2 /2 This family of relative indices contains several indices seen so far

Examples of inequality indices: Generalized entropy family Generalized entropy family of indices (for a parameter c ) I GEc ( x ) : For c =1, we have I GEc ( x ) = I T ( x ) This family of relative indices contains several indices seen so far

Examples of inequality indices: Generalized entropy family Generalized entropy family of indices (for a parameter c ) I GEc ( x ) : For 0 < c <1, we have I GEc ( x ) = [1/ c ( c -1)][1-[ I AK(1-c) ] c - 1) This family of relative indices contains several indices seen so far

Examples of inequality indices: Generalized entropy family Generalized entropy family of indices (for a parameter c ) I GEc ( x ) : All members of this family are strictly Schur-convex

Examples of inequality indices: Generalized entropy family Generalized entropy family of indices (for a parameter c ) I GEc ( x ) : Only those members for which c < 2 satisfy VPCT sensitivity

Examples of inequality indices: Gini Gini coefficient I G ( x ) :

Example of inequality indices: Gini Relative position Cumulated fraction of income 1/ n 1 n / n = 1 2/ n Line of perfect equality area =1/2

Examples of inequality indices: Gini Hence, the Gini coefficient is twice the value of the area between the perfect equality line and the (relative) Lorenz curve But this coefficient can be written (and interpreted) differently

Examples of inequality indices: Gini Gini coefficient I G ( x ) : Gini coefficient is (one minus) the weighted average of the relative income of people, with the weight being inversely related to the ranking of people in the income distribution

Examples of inequality indices: Gini Gini coefficient I G ( x ) : But there is yet another interpretation of the Gini coefficient

Examples of inequality indices: Gini Gini coefficient I G ( x ) : But there is yet another interpretation of the Gini coefficient

Examples of inequality indices: Gini Gini coefficient I G ( x ) : It is the sum of all income differences between people (relative to total income)

Examples of inequality indices: The single parameter Gini family Single parameter  family I G  ( x ) :   1 is a parameter that reflects the inequality aversion of the index  = 2 gives the standard Gini coefficient  = 1 gives the constant index = (n-1)/n

Which (relative) inequality index should we use ? Let us try to answer this question axiomatically Let us propose desirable properties that an inequality index should satisfy and see whether we can identify precisely the class of inequality indices that satisfy these properties

Desirable properties of an inequality index We always assume Schur-convexity (and therefore symmetry) Continuity: For every number n > 0 of people, and every distributions of income x and y between n people such that I ( x ) < I ( y ), there exists a strictly positive real numbers  such that I ( x  ) < I ( y ) for all income distribution x    n + such that | x i  - x i | <  Relative invariance: For every number n > 0 of people, and every distributions of income x and y between n people I ( x )  I ( y )  I (  x )  I (  y ) for all strictly positive real number  (trivially satisfied by any relative index) Absolute invariance: For every number n > 0 of people, and distributions of income x and y between n people I ( x )  I ( y )  I ( x+  )  I ( y+  ) for all strictly positive real number  (trivially satisfied by any absolute index)

Desirable properties of an inequality index Shorrocks Group Decomposition: For every set N = {1,…, n } of n individuals, for every partition of N into subsets A and B of N such that A  B =  and A  B = N, I ( x ) = I ( x A ) + I ( x B ) + I (  ( x A ),  ( x B )) (where, for every C  N with #C = c, x C is the vector in  c + defined by x i c = x i for all i  C ) Convenient if the index is to be applied to subgroups of a given population Another important notion: the Equally Distributed Equivalent Income Take an inequality index I:  n  n   and consider any distribution of income x between n individuals EDEI I ( x ) is the number such that I (EDEI I ( x ),…,EDEI I ( x )) = I ( x ) EDEI I ( x ) is the amount of income which, if given to everyone, gives an income distribution that is just as good as x for the inequality index I EDEI I ( x ) may not exist (it will not if I is a relative index) Recursivity: For any n, and all x   n + I ( x ) = I ( x (1),EDEI I ( x (2),…,x ( n ) ),…, EDEI I ( x (2),…,x ( n ) )) Increasingness: For all positive real numbers a and b such that a > b I(a,…,a) > I(b,…,b) (violated by relative indices; efficiency notion)

Result 1: Theorem (Shorrocks (1984)): A relative and Schur convex index I:  n  n   is continuous and satisfies the Shorrocks Group Decomposition axiom if and only if, for every x   n  n, I ( x ) = I GEc ( x ) for some real number c  0 Hence generalized entropy indices are the only continuous, Schur convex and relative indices that satisfy the Shorrocks group decomposition axiom If one wants VPCT sensitivity as well, then one needs to restrict attention to the smaller class of such indices for which c < 2

Result 2: Theorem (Donaldson and Weymark (1980, Bossert 1990) A Schur convex index I :  n  n   is continuous and satisfies recursivity, increasingness, Dalton replication invariance as well as relative and absolute invariance if and only if there exists a real number   0 such that I ( x )= I G  ( x ) for every x   n  n

Poverty Poverty is an issue that is the source of intense feelings. A person is considered poor if the income of this person is considered inferior to a certain treshold t: the Poverty line Defining and measuring poverty amounts therefore to: Identifying the poor (and therefore fixing the poverty line) Measuring poverty, given the identification of the poor Let us consider each of these in turn.

Identifiying the poor It is admitedly difficult to fix a poverty line that separate poor and non-poor It is always difficult to « draw the line ». Two approaches exist: absolute and relative Absolute approach: the poverty line is an amount of income that is necessary (given prices) to someone to achieve a minimal nutritional, clothing, housing, educational, etc. objective Absolute approach: the poverty line is independent from the characteristic of the income distribution to which it applies Example: $1/day for the Word Bank millenium objective (based on the cheapest way in India to get 2400 calories per day). Relative approach: the poverty line depends upon the distribution to which it is applied (for instance one half the median income)

Measuring poverty The most widely used class of poverty measures (given a poverty line) is the Foster, Greek, Thorbeck FGT (1987) class. This class is parameterized by a positive real number a. Given such a number a and a poverty line t the FGT measure of poverty in a distribution of income x, denoted P a ( t, x ) is defined by: This class contains two widely used measured of poverty that correspond, respectively, to a = 0 and a = 1

Headcount poverty a = 0 corresponds to headcount poverty (counting the number (or the fraction) of poor in the population) Indeed, for a = 0, the formula for the FGT class of index writes : This measure of poverty is widely used under the convention that 0 0 = 0 It is not very sensitive to the intensity of poverty

Poverty gap a = 1 corresponds to poverty gap the poverty gap of an income distribution for a given poverty line is the minimal amount of money that is needed to eliminate totally poverty Indeed, for a = 1, the formula for the FGT class of index writes : This measure of poverty is also quite used It is more sensitive to the intensity of poverty one sometimes use the square of the poverty gap (by fixing a = 2)

Contrasting these poverty measures For a given poverty line, these poverty measures give very different evaluations of poverty For instance suppose the poverty line is 6 and consider the following three distributions x = (5,5,5,7), y = (3,3,7,7) and z = (1,7,7,7) x has more poor than y which has more poor than z Poverty gap is 6 in y, 5 in z and only 3 in x Square of poverty gap is 25 in z, 18 in y and 3 in x.

Dominance rankings of distributions of one attribute We would like now to put together the various notions that we have seen (efficiency, equality, poverty) so as to rank distributions of income in a way that commands wide support. We would like to connect our rankings of income distributions to firm ethical theories (welfarist or not) There are several beautiful dominance theorems that enable us to do this

1st order (efficiency) dominance The following four statements (that apply to two income distibutoins x and y ) are equivalent 1) #{ i : x i < t }  #{ i : y i < t } for all poverty lines t. 2) x Suppes dominates y 3) W ( x )  W ( y ) for all increasing and symmetric functions W :  n +   (ethical robustness for non-welfarist ethics) 4) F ( u ( x 1 ),…, u ( x n ))  F ( u ( y 1 ),…, u ( y n )) for all Pareto inclusive and symmetric welfarist functionals F and all individual utility functions u :  +   that are increasing in income (ethical robustness for welfarist ethics)

1st order (efficiency) dominance First order dominance of a distribution over another is particularly robust ethically Indeed, the class of ethical judgments (welfarist or not) who aggree with the Suppes dominance ranking is very large. It is also nice to know that a distribution Suppes dominate another if and only if the number of poor is lower in the dominating distribution than in the dominated one for all poverty lines Any objective of poverty reduction for a poverty line, with poverty measured by the headcount, will therefore be pleased to see a Suppes improvement Suppes dominance is often useful in practice to rank income distributions Consider the ordered vectors of income of the OECD countries discussed in the beginning of the course

Some empirical Suppes dominations

Suppes dominance chart SwitzerlandUS UK Australia Canada AustriaFranceGermany Sweden Italy Spain Portugal India

Inequality dominance Consider now two distributions x and y with the same mean The following 5 statements are equivalent: 1) x  PD y (one can go from y to x by a finite sequence of Pigou Dalton transfers) 2) P 1 ( t, x )  P 1 ( t, y ) for all poverty lines t (Poverty gap dominance) 3) L k (x)  L k (y) for all k = 1,…, n (Lorenz dominance) 4) W ( x )  W ( y ) for all Schur-concave functions W :  n +   (ethical robustness for non-welfarist ethics) 5) F ( u ( x 1 ),…, u ( x n ))  F ( u ( y 1 ),…, u ( y n )) for all Pareto inclusive and Schur concave welfarist functionals F and all individual utility functions u :  +   concave in income (ethical robustness for welfarist ethics)

Inequality dominance Concerns only distributions with the same mean income Notice that in the ethical statements 4 and 5, the property of concavity of individual utility function (welfarism) or of Schur concavity of the social evaluation function has replaced the property of increasingness. What if we combine inequality and efficiency dominance ? We can define robust rankings of distributions if we define the following (efficiency) elementary operation

Income increment x (.) has been obtained from z (.) by an income increment if there is an individual i and a strictly positive real number  such that x ( h ) = z (h) for all h ≠ i and x ( i ) = z ( i ) +  x (.) has been obtained from z (.) by an income increment if an individual has received some amount of money “from the sky” (mana ?) x (.) has been obtained from z (.) by a finite sequence of Pigou- Dalton transfers and/or increment ( x (.)  IPD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a either bilateral Pigou-Dalton transfer or an increment for t = 0,…, T -1

An example: Canada and Australia The distribution of income in Australia in 1998 could have been obtained from that of Canada in the same year by a finite sequence of increments and Pigou- Dalton (at least if the two income distribution are aggregated by deciles) Let us see this

Australia and Canada

A Pigou Dalton transfer between 3 and 1

Australia and Canada

a set of income increments

Second-order dominance The following 5 statements are equivalent: 1) x  IPD y (one can go from y to x by a finite sequence of Pigou Dalton transfers and/or increments) 2) P 1 ( t, x )  P 1 ( t, y ) for all poverty lines t (Poverty gap dominance) 3) L k (x)  L k (y) for all k = 1,…, n (Generalized Lorenz dominance) 4) W ( x )  W ( y ) for all Schur-concave and increasing functions W :  n +   (ethical robustness for non- welfarist ethics) 5) F ( u ( x 1 ),…, u ( x n ))  F ( u ( y 1 ),…, u ( y n )) for all Pareto inclusive and Schur concave welfarist functionals F and all individual utility functions u :  +   increasing and concave in income (ethical robustness for welfarist ethics)

Second-order dominance Except for statement 2), the equivalence looks very much like that established for two distributions with the same mean Mild difference for statements 4 and 5 (where the requirement that W and u (respectively) be increasing is added to that they be concave Hence the class of ethical functions over which unanimity is looked for is smaller dominance) We talk about Generalized Lorenz domination because the Lorenz curves that are appealed to in statement 3) do not have the same mean (and therefore do not “end” at the same point) as is the case with usual (non-generalized) Lorenz curves (with the ending point being often normalized at 1) Here is the second order ranking of our countries

Generalized Lorenz dominance chart SwitzerlandUS UK Australia Canada Austria FranceGermany Sweden Italy Spain Portugal India

Comments on 2nd order dominance The ranking is driven to a large extent from 1st order dominance According to some, this is symptomatic of an excessive weight given to efficiency as opposed to equality Indeed, the notion of efficiency which underlies 1st order dominance is very strong (it is not defined ceteris paribus with respect to equality) What happens if we weaken the notion of efficiency only to proportional increases in income ?

Proportional increase in income x (.) has been obtained from z (.) by a proportional increase in incomes of t if there is real number t greater than 1 such that x ( h ) = tz (h) for all h x (.) has been obtained from z (.) by an income increment if all individuals have seen their income increased in proportion t x (.) has been obtained from z (.) by a finite sequence of Pigou- Dalton transfers and/or proportional increase in income ( x (.)  PIPD z (.) ) if there exists a sequence of distributions { z t (.) }, with t =0,... T such that z 0 (.) = x (.), z T (.) = z (.) and z t (.) has been obtained from z t+1 (.) by a either bilateral Pigou-Dalton transfer or a proportional increase in all incomes for t = 0,…, T -1

A result by Shorrocks The following 3 statements are equivalent: 1) x  PIPD y (one can go from y to x by a finite sequence of Pigou Dalton transfers and/or increments) 2) l k (x)  l k (y) for all k = 1,…, n where, for every distribution z, l k (z) = L k (z) / n  ( z ) and  (x)   (y) (relative Lorenz and mean dominance) 3) W ( x )  W ( y ) for all Schur-concave functions W :  n +   satisfying W ( tx ) > W ( ty ) for all real numbers t > 1 ethical robustness for non- welfarist ethics)

Comments on this result No statement on poverty or on welfarist dominance Implementable criterion: requires both mean dominance and relative Lorenz dominance Relative Lorenz criterion: compare the share of total income held by the k poorest individuals (no matter what k is) This criterion is very incomplete Here are the relative Lorenz curves for some of our countries

Relative Lorenz curves

Mean income country ranking countryMean income US Switzerland Australia Canada UK Austria France Germany Sweden Italy Spain Portugal India1 885

Efficiency-equality dominance chart Sweden US India AustraliaCanada Austria GermanyFrance Switz. Italy SpainPortugal non comparable to any other country

TO BE COMPLETED