aa r X i v : . [ ec on . T H ] F e b Extended Gini index
Ram Sewak Dubey * Giorgio Laguzzi † February 11, 2021
Abstract
We propose an extended version of Gini index defined on the set of infinite utility streams, X = Y N where Y ⊂ R .For Y containing at most finitely many elements, the index satisfies the generalized Pigou-Dalton transfer principlesin addition to the anonymity axiom. Keywords:
Anonymity, Extended Gini index, Generalized Pigou-Dalton transfer principle, Socialwelfare function.
Journal of Economic Literature
Classification Numbers:
C65, D63, D71. * Department of Economics, Feliciano School of Business, Montclair State University, Montclair, NJ 07043; E-mail: [email protected] † University of Freiburg in the Mathematical Logic Group at Eckerstr. 1, 79104 Freiburg im Breisgau, Germany; Email: [email protected] Introduction
The Gini index (also referred to as Gini Coefficient, Gini (1997) is a widely used measure of inequality in incomeor wealth distribution in the society. It is a measure of dispersion of income distribution for a given population andis sensitive to redistribution of income from rich to poor. In this paper, we propose an extended version of the Giniindex as a real valued representation of the infinite utility streams. The new index satisfies a generalized version ofPigou-Dalton transfer principle and the anonymity axiom.A brief review of the two equity axioms is as follows. Anonymity axiom is an example of procedural equity .It applies to the situations where the changes involved in the infinite utility streams do not alter the distribution ofutilities. The anonymity axiom requires that the society should be indifferent between two streams of well-being, ifone is obtained from the other by interchanging the well-being of any pair of generations.The second equity concept is an example of the consequentialist equity . The particular version considered in thispaper is the well-known Pigou-Dalton transfer principle. Pigou-Dalton transfer principle compares two infinite utilitystreams ( x and y ) in which all generations except two have the same utility levels in both utility streams; regarding thetwo remaining generations (say, i and j ), if y i < x i < x j < y j and y i + y j = x i + x j , then utility stream x is sociallypreferred to y . This equity principle ranks utility sequence x superior to y as x is obtained from y by carrying out anon-leaky and non-rank-switching transfer of welfare from a rich to a poor generation.It is easy to infer that this definition would also help us in ranking utility sequence x superior to y if x is obtainedfrom y by carrying out non-leaky and non-rank-switching transfers of welfare among any finitely many pairs ofgenerations. To enable us to compare sequence x and y when x is obtained from y by carrying out arbitrarily many(possibly infinitely many) pairs of rich and poor generations, we have introduced a generalized version of the Pigou-Dalton transfer principle in Dubey and Laguzzi (2020). It has been shown that the generalized infinite Pigou-Daltontransfer principle admits real-valued representation if and only if Y does not contain more than seven distinct elements.In this short note we consider a weaker version of the generalized Pigou-Dalton principle and show (in Proposition1) that an index defined along the lines of the Gini index, which we call the extended Gini index , satisfies the Pigou-Dalton principle in addition to the anonymity axiom when Y contains finitely many distinct elements.The rest of the paper is organized as follows. Section 2 contains the relevant definitions and in section 3 we discussthe generalized Pigou-Dalton transfer principle which is the focus of our paper. Section 4 contains the two results withdetailed proofs. We conclude in section 5. Let R and N be the sets of real numbers and natural numbers respectively. For all y , z ∈ R N , we write y > z if y n > z n , for all n ∈ N ; we write y > z if y > z and y = z ; and we write y ≫ z if y n > z n for all n ∈ N .A partial function f : X → Y is a function from a subset S of X to Y . If S equals X , the partial function is said to betotal. Domain and range of function f are denoted by dom ( f ) and ran ( f ) respectively. Definition.
A partial function α : N → N is called a pairing function if and only if α satisfies ∀ n ∈ dom ( α ) , α ( α ( n )) = n . Note that dom ( α ) = ran ( α ) for every pairing function α . We denote the set of all pairing functions by Π . Let | S | denote the cardinality of the finite set S ⊂ N . The lower asymptotic density of S ⊂ N is defined as: d ( S ) = lim inf n → ∞ | S ∩ {
1, 2, · · · , n }| n . The idea of anonymity was introduced in a classic contribution, Ramsey (1928), who observed that discounting one generation’s utility relativeto another’s is “ethically indefensible”, and something that “arises merely from the weakness of the imagination”. Diamond (1965) formalized theconcept of “equal treatment” of all generations (present and future) in the form of an anonymity axiom on social preferences. The inequality reducing property was initially hinted at by Pigou (1912, p. 24) as “The Principle of Transfers”. Dalton (1920, p. 351) describedit as “If there are only two income-receivers and a transfer of income takes place from the richer to the poorer, inequality is diminished.”. Theversion of equity axiom described here for the infinite utility streams was introduced and discussed in Sakai (2006), Bossert et al. (2007), andHara et al. (2008). upper asymptotic density of S ⊂ N is defined as: d ( S ) = lim sup n → ∞ | S ∩ {
1, 2, · · · , n }| n .If the two coincide for a set S ⊂ N , it is called the asymptotic density of set S , d ( S ) . Let Y , a non-empty subset of R , be the set of all possible utilities that any generation can achieve. Then X ≡ Y N is theset of all possible utility streams. If h x n i ∈ X , then h x n i = ( x , x , · · · ) , where, for all n ∈ N , x n ∈ Y represents theamount of utility that the generation of period n earns. We consider transitive binary relations on X , denoted by % andcalled social welfare relation (SWR), with symmetric and asymmetric parts denoted by ∼ and ≻ respectively, definedin the usual way. A social welfare order (SWO) is a complete social welfare relation. A social welfare function (SWF)is a mapping W : X → R . Given a SWO % on X , we say that % can be represented by a real-valued function if thereis a mapping W : X → R such that for all x , y ∈ X , we have x % y if and only if W ( x ) > W ( y ) . The following axioms on social welfare orders are used in the analysis.
Definition. (Anonymity - AN): If x , y ∈ X , and if there exist i , j ∈ N such that x i = y j and x j = y i , and for every k ∈ N \ { i , j } , x k = y k , then x ∼ y . Definition. (Pigou-Dalton transfer principle - PD): If x , y ∈ X , and there exist i , j ∈ N and ε >
0, such that x i + ε = y i < y j = x j − ε , while y k = x k for all k ∈ N \ { i , j } , then y ≻ x .Anonymity is an example of procedural equity. Pigou-Dalton transfer principle is an example of consequentialistequity. The key observation and reason of the present paper for studying some extended versions of these consequentialistequity principles is motivated by the following observation. Given a set of utilities Y := {
1, 2, 3, 4, 5 } ⊂ R , consider thetwo infinite streams x := h
2, 3, 5, 2, 3, 5, 2, 3, · · ·i , and y := h
1, 4, 5, 1, 4, 5, 1, 4, · · ·i .Following the expected interpretation of a redistributive equity principle, we should be able to always rank y ≺ x . Inthe finite case, PD together with transitivity is sufficient to secure such a ranking, but in the infinite case transitivitycannot be extended to infinite chains. Hence PD even with transitivity is not a sufficient condition to secure the desiredranking y ≺ x , and so an extension is necessary. Definition. (Generalized Pigou-Dalton, GPD): Given x , y ∈ X if there is α ∈ Π such that for every j / ∈ dom ( α ) , x j = y j , and for every i ∈ dom ( α ) there is ε i > y i + ε i = x i < x α ( i ) = y α ( i ) − ε i or y α ( i ) + ε i < x α ( i ) < x i < y i − ε i , then y ≺ x .In Dubey and Laguzzi (2020), we have investigated the existence and representation of these generalized equityprinciples. The results show that when we do not put any further restriction to the combinatorial characteristics ofthe pairing function, representation of SWRs satisfying those principles is rather demanding and hard to obtain. Thisleads us to investigate more restricted and weaker forms of generalized Pigou-Dalton transfer. A first line of weakervariants of GPD is given by imposing some combinatorial restrictions on the pairing functions as explained in whatfollows. Consider two streams x , y ∈ Y N , with Y = {
1, 2, 3, 4 } : x ( n ) = (cid:12) ∃ k ∈ N ( n = k ) y ( n ) = (cid:12) x ( n ) =
13 else.3y GPD we get x ≺ y even if the welfare improving re-distributions, from (
1, 4 ) in x to (
2, 3 ) in y occurs via a pairingfunction α such that for every N ∈ N there exists n ∈ dom ( α ) such that | n − α ( n ) | > N . In other words, the distancesbetween the generations linked by the pairing function α grows in an unbounded manner. To avoid this feature, wecan require the pairing function α having some limitations in term of being bounded, in a similar fashion as anonymitycould require some restrictions on the family of permutations. In this paper we focus on the following types of pairingfunctions. Definition.
We say that α ∈ Π is a fixed-step pairing function ( α ∈ Π s ) if and only if there exists h ∈ N (called the step of α ) such that ∀ n ∈ N ∀ k ∈ (( n − ) h , nh ] , one has α ( k ) ∈ (( n − ) h , nh ] .One can then introduce the following weakening of GPD. Definition. (Fixed-step Generalized Pigou-Dalton, s-GPD): There exists h ∈ N such that for every x , y ∈ X if thereis α ∈ Π s with step h such that for every j / ∈ dom ( α ) , x j = y j , and for every i ∈ dom ( α ) there exists ε i > y i + ε i = x i < x α ( i ) = y α ( i ) − ε i or y α ( i ) + ε i = x α ( i ) < x i = y i − ε i ,then y ≺ x .Note that GPD ⇒ s-GPD. A second line works as follows. Since we are considering an infinite time horizon, theranking induced by ≺ should be sensitive not only to few changes, but to a number of changes as large as possible.For instance, in an infinite setting one could require that the number of individuals/generations linked via the pairingfunction (where we can appreciate a reduction of inequality) should be at least an infinite set, and possibly with somenon-zero density. This is in line with the weaker forms of Pareto principles extensively studied in the literature, such asinfinite Pareto, asymptotic Pareto and weak Pareto. The following definitions capture this relevant idea for our study. Definition. • (Infinite Pigou-Dalton, IPD): Given x , y ∈ X if there is α ∈ Π such that dom ( α ) is infinite, forevery j / ∈ dom ( α ) , x j = y j , and for every i ∈ dom ( α ) one haseither y i + ε i = x i < x α ( i ) = y α ( i ) − ε i or y α ( i ) + ε i = x α ( i ) < x i = y i − ε i , then y ≺ x .• (Asymptotic Pigou-Dalton, APD): Given x , y ∈ X if there is α ∈ Π such that d ( dom ( α )) >
0, for every j / ∈ dom ( α ) , x j = y j , and for every i ∈ dom ( α ) one haseither y i + ε i = x i < x α ( i ) = y α ( i ) − ε i or y α ( i ) + ε i = x α ( i ) < x i = y i − ε i , then y ≺ x .• (Weak Pigou-Dalton, WPD): Given x , y ∈ X if there is α ∈ Π such that dom ( α ) = N , for every j / ∈ dom ( α ) , x j = y j , and for every i ∈ dom ( α ) one haseither y i + ε i = x i < x α ( i ) = y α ( i ) − ε i or y α ( i ) + ε i = x α ( i ) < x i = y i − ε i , then y ≺ x .Note that GPD ⇒ IPD ⇒ APD ⇒ WPD. In Dubey and Laguzzi (2020) we focus on GPD, IPD and WPD, whereasin this paper we focus on versions of APD.
Remark 1.
Note that we can then easily combine the two types of weaker variants, and obtain principles like s-APD,where we require both that the pairing function α ∈ Π s and that d ( dom ( α )) > Representation of fixed-step asymptotic equitable social welfare relations
In Dubey and Laguzzi (2020) we have proven that any SWO satisfying IPD is not representable when the utility do-main Y has at least eight elements. A careful scrutiny of the proof shows that actually non-representability persistseven if we weaken IPD to APD, since the pairing functions defined in that proof actually has domain with strictlypositive density. But what is crucial regarding those pairing functions is that they do not satisfy any particular charac-teristics in line with Definition 3. The following result shows that if we put some restrictions on the structure of thepairing functions, namely fixed-step with asymptotic density >
0, then we obtain an elegant social welfare function,which recalls an extended infinite version of the well-known Gini index.
Proposition 1.
Let X = Y N where | Y | < ∞ and Y ⊆ [
0, 1 ] . Then there exists a social welfare function W : X → R satisfying fixed step asymptotic Pigou-Dalton (s-APD) and anonymity (AN) axioms.Proof. We present the result for Y := {
1, 2, · · · , M } , i.e., we get W : Y N → R satisfying s-APD and AN. Let h ∈ N bethe fixed step, I n := (( n − ) h , nh ] for n ∈ N and H n := sup I n . Define W N ( x ) := H N H N X k = H N X j = | x ( k ) − x ( j ) | , (1)and W ( x ) := − lim inf N → ∞ W N ( x ) , (2)We claim W satisfies AN and s-APD. Anonymity is trivial to show. We need to prove W satisfies s-APD. Let x , y ∈ Y N be such that there exists α ∈ Π s with d ( dom ( α )) > k ∈ dom ( α ) , either x k < y k < y α ( k ) < x α ( k ) or x α ( k ) < y α ( k ) < y k < x k ;• for all k / ∈ dom ( α ) , x k = y k .In order to show W ( x ) < W ( y ) , we first need to compare H N X k = H N X j = | x k − x j | and H N X k = H N X j = | y k − y j | .The choice of x and y reveals that, for every n ∈ N , H n X k = H n X j = | x ( k ) − x ( j ) | > H n X k = H n X j = | y ( k ) − y ( j ) | + | dom ( α ) ∩ [ H n ] | · | dom ( α ) ∩ [ H n ] | .To show that, we can proceed by an inductive argument on all pairs ( k , j ) ∈ H n × H n , by computing the values | x k − x j | ’s compared to | y k − y j | ’s. For every j , k ∈ H N we have four possible non-trivial cases:1) k = α ( j ) or j = α ( k ) : Then ε j = ε k and trivially | x j − x k | = | y j − y k | + ε j .2) x k = y k and x j = y j : Then | x k − x j | = | y k − y j | .3) x j = y j and y k < x k : For { j , k } and { j , α ( k ) } and note that | x j − x k | + | x j − x α ( k ) | = | y j − y k | + | y j − y α ( k ) | + ε k − ε k = | y j − y k | + | y j − y α ( k ) | .4) y k < x k and x j < y j : We compute the values given by { k , j } , { k , α ( j ) } , { j , α ( k ) } , { j , α ( j ) } , { α ( j ) , α ( k ) } . Combina-torial computations provide the following:• | x k − x j | > | y k − y j | + µ ε k + µ ε j , where µ takes a values in { −
1, 1 } depending on x k , y k , x j , y j using thefollowing criteria: µ = y k > y j , µ = − y j > y k .• | x k − x α ( j ) | > | y k − y α ( j ) | + µ ε k − µ ε j , where µ takes a values in { −
1, 1 } depending on x k , y k , x α ( j ) , y α ( j ) using the following criteria: µ = x α ( j ) < y k , µ = − x k < y α ( j ) , and µ ∈ { −
1, 1 } is similarly chosenwhen y α ( j ) x k or y k x α ( j ) , and depending on whether ε k ε j or not.5 | x j − x α ( k ) | > | y j − y α ( k ) | − µ ε k + µ ε j , where µ takes a values in { −
1, 1 } depending on x j , y j , x α ( k ) , y α ( k ) using the following criteria: µ = x α ( k ) > y j , µ = − x j > y α ( k ) , and µ ∈ { −
1, 1 } is similarly chosenwhen y α ( k ) > x j or y j > x α ( k ) and ε k ε j , and depending on whether ε k ε j or not.• | x α ( k ) − x α ( j ) | > | y α ( k ) − y α ( j ) | − µ ε k − µ ε j , where µ takes a values in { −
1, 1 } depending on x α ( k ) , y α ( k ) , x α ( j ) , y α ( j ) using the following criteria: µ = x α ( k ) > x α ( j ) , µ = − y α ( k ) < y α ( j ) , and µ ∈ { −
1, 1 } is similarly chosen when y α ( k ) > y α ( j ) or x α ( k ) x α ( j ) and ε k ε j , and depending on whether ε k ε j or not.• | x j − x α ( j ) | = | y j − y α ( j ) | + ε j .All together we obtain: | x k − x j | + | x k − x α ( j ) | + | x j − x α ( k ) | + | x α ( k ) − x α ( j ) | + | x j − x α ( j ) | >> | y k − y j | + | y k − y α ( j ) | + | y j − y α ( k ) | + | y α ( k ) − y α ( j ) | + | y j − y α ( j ) | + ε k ( µ + µ − µ − µ ) + ε j ( µ − µ + µ − µ ) + ε j . (3)By construction we have µ µ , µ µ , µ µ and µ µ . Hence the last line is > ε j .The other combinations, like j = k or y k < x k and y j < x j , are simply reducible to one of these four cases,or analogous proof-arguments. Proceeding inductively and comparing all pairs through the induction on j and k following the four cases, we can therefore observe two facts. Firstly, cases 2) and 3) show that whenever at leastone of the pairs involved does not belong to dom ( α ) , the sum of the absolute values of symmetric differences of theconsidered combinations of stream x and stream y coincide. Secondly, case 1) and 4) show that whenever the pairsboth belong to dom ( α ) , then the combinations considered always reveal that the sum involving the values of stream x is strictly larger than the ones referring to y . Hence, putting these two observations together, we obtain: H N X k = H N X j = | x k − x j | > H N X k = H N X j = | y k − y j | + ε · | dom ( α ) ∩ [ H N ] | · | dom ( α ) ∩ [ H N ] | ,where the fraction comes from the counting in (3), where it is shown that for a pair j , k ∈ dom ( α ) the sum of the 5combinations considered for x overcome the analog sum for y by a factor 2 ε := (cid:8) ε j , ε k (cid:9) , which gives that thetotal double sum for x is larger than the total double sum for y by 2 ε over 5. Therefore we obtain the desired property,as by the characteristics of Y it holds ε >
1. Hence we get: − W ( x ) = lim inf N → ∞ H N H N X k = H N X j = | x ( k ) − x ( j ) | > lim inf N → ∞ H N (cid:16) H N X k = H N X j = | y ( k ) − y ( j ) | + | dom ( α ) ∩ [ H N ] | · | dom ( α ) ∩ [ H N ] | (cid:17) > lim inf N → ∞ H N (cid:16) H N X k = H N X j = | y ( k ) − y ( j ) | (cid:17) +
25 lim inf N → ∞ H N (cid:16) | dom ( α ) ∩ [ H N ] | (cid:17) > − W ( y ) + d ( dom ( α )) .Since by assumption d ( dom ( α )) >
0, we therefore get W ( x ) < W ( y ) as desired.Note that in the inequalities we have used the property lim inf ( a + b ) > lim inf a + lim inf b , lim inf ( a · b ) > lim inf a · lim inf b and dom ( α ) is strictly positive. Therefore the proof cannot be adopted to work for s-IPD as well,and this is perfectly coherent with [Dubey and Laguzzi (2020, Theorem 1)] where it is shown that the combination ofs-IPD and AN is not representable for every non-trivial utility domain.We conclude by stating an impossibility result, whose proof is delegated to a future work. Proposition 2 belowshows that Proposition 1 cannot be extended if the utility domain becomes too complicated. Specifically, Proposition2 shows a limitation when Y ⊆ [
0, 1 ] contains some infinite subsets with particular order type. More specifically, if theset Y contains a pair of infinite sets one increasing and the other decreasing with well-defined minimum or maximumelements for each subset of Y , then s-APD and AN together are not representable.6 roposition 2. Let Y ⊆ [
0, 1 ] contain as a subset (cid:8) + k + : k ∈ N (cid:9) ∪ (cid:8) − k + : k ∈ N (cid:9) . Then any SWO defined on X = Y N satisfying fixed step asymptotic Pigou-Dalton (s-APD) and anonymity (AN) axioms is not representable. In this paper, we have proposed a new version of Gini Coefficient. This index represents social welfare orders satisfyinggeneralized Pigou-Dalton transfer principle and anonymity on the space of infinite utility streams when individualagents’ utility is assigned values from a finite set Y ⊂ R . Since an explicit formula for the index is described, it isuseful for policy formulation. We also show that when we consider more general set Y (i.e., Y having infinitely manyelements of the type considered in Proposition 2) real-valued representation is impossible. It is an open question forus to explore in future if social welfare function exists in case Y ( < ) is a well-ordered infinite subset of real numbers. References
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