Inequality and poverty orderings

European Economic Review 32 (1988) 654-662 North-Holland

INEQUALITY

AND POVERTY

ORDERINGS

James E. FOSTER Purdue

Uniuersity, West Lafayette,

IN 47907, LISA

Anthony F. SHORROCKS University

of Essex, Colchester CO4 3SQ. UK

1. Introduction Many policy issues require us to make inequality comparisons between alternative income distributions. One way of tackling this problem is to choose a specific inequality measure, and to use this measure to generate a complete ordering of the set of feasible distributions. There are, however, a large number of indices that might be employed for this purpose, and no compelling reason for settling on one particular functional form. As a consequence, there appears little likelihood of obtaining widespread agreement on the appropriate choice of inequality index, and hence little prospect of constructing a complete inequality ordering that would be generally acceptable. A consensus of views is more likely to be achieved if we seek only to rank certain pairs of distributions, rather than all conceivable pairs. For we may then appeal to the ‘unambiguous’ judgements which arise when all the members of some set of reasonable indices agree on their verdict. This alternative approach to inequality comparisons has been gaining in popularity in recent years, and poses two central questions. First, we have to decide the properties which characterise the set of ‘reasonable indices’. Secondly, we need to evaluate the power of the associated ordering or, in other words, to assess the probability that any randomly chosen pair of distributions will be ranked. These two issues are not unrelated. As extra properties are introduced, the set of admissible indices shrinks and the power of the induced ordering will normally increase. But the additional properties also carry the risk of eliminating indices which some people regard as satisfactory, and hence the risk of producing an ordering which no longer commands widespread appeal. Poverty indices offer similar opportunities for generating partial orderings that are more likely to be acceptable than the complete orderings derived 0014-2921/88/f3.50

0

1988, Elsevier Science Publishers

B.V. (North-Holland)

J.E. Foster

and A.F. Shorrocks.

Income

dlsrribution

and wealth

inequaluies

“55

from a single index. Indeed the opportunities are greater, since the calculation of poverty values requires us to specify both the form of the index and the level of the poverty line, thus introducing two potential sources of disagreement. In this paper, we provide a brief review of both inequality and poverty orderings. We begin with a discussion of Lorenz dominance. and show how the power of this ordering may be improved by imposing additional constraints on inequality measures. We then turn attention to the different types of ordering that can be constructed from poverty indices, and examine some of the relationships that exist between inequality and poverty orderings.

2. Inequality orderings We restrict attention to discrete, finite-population income distributions defined over positive income values. These can be represented by a vector of the form X=(X, ,..., x,), where xi~lw + + : =(0, co) is the income received by person i. Thus X: = {XIXE R: + for some finite n} is the set of all distributions under consideration. We will denote the dimension, mean and variance of XGX by n(x), p(x) and a’(x), respectively, and use X”(U): = {SEX/~(X) =n; ,u(x)=~) to signify the set of n-person distributions with mean income ~1.The ordered cersion of x is written as .?‘, where _C=l7x for some permutation matrix I7 such that 2, s_?z=<.--6_?.(,,. We say that x is obtained from _r by a permurafion if _i=J; by an (m-)replication if x =(y. y,. . . ,v) and n(x) = mn(y) for some positive integer m; and by a progressioe transfer if there exist i and i such that .Ui-yi=yj-xi>O, xj>yi and xlr=yll for all k#i, j. Finally we will say that x Loren: dominates y, and write xLy, if the Lorenz curve for x is nowhere below, and somewhere above, that of y. An inequality measure is a real valued function I:X+lR. The best known of the inequality partial orderings concerns the set of inequality measures .P which are symmetric, in that 1(x)=1(y) whenever x is obtained from _I*by a permutation, and satisfy the Pigou-Dalton Principle of Transfers. in that f(.x)
For every n, ,u > 0 and any x, y E X,,(p): I(x) > I(y) for ali I E .P ifl

Symmetry ;and the Pigou-Dalton condition are fairly minimal properties to impose on inequality measures, and the power of the ranking criterion is correspondingly weak. Theorem 1 does not let us compare distributions with different population sizes or mean income levels. Nor does it enable us to pass judgement on distributions whose Lorenz curves intersect. This situation is illustrated Itn fig. 1, which portrays the simplex X1@) corresponding to

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AS. Shommks, Income disrriburion and wealth inequaliries

Fig. 1, The Lorcnz dominance ordering.

3-person dist~butions with mean income ~1. For any arbitrarily chosen XE!%+&}, the set of distributions in X,(p) which Lorenz dominate x form a hexagonal region A(x) defined by (but not in~~uding~ the six permutations of X, while the three remaining crosshatched regions (again excluding permutations) labelled B(x) together comprise those distributions which are Lorenz dominated by x. The light areas contain all the distributions whose Lorenz curves intersect the Lorenz curve of x, and hence are not agreed to be more (or less) equal than x by all indices in the set 9. The ordering specified in Theorem 1 is clearly too weak to have much practical value, but it provides a basis on which to explore further constraints on the set of admissible inequality measures, and a consequent strengthening of the ranking criterion. One natural way of proceeding is to extend comparisons to distributions with different population sires and mean incomes, by considering only those indices which satisfy the replication inwriunce condition

w =f(Y)

whenever x is a replication of y,

and the scale invariance property &Ixxf=I(x)

for all xeS

and all R>O.

J.E. Foster and A.F. Shorrocks, income distributron and wealrh inequalities

Defining 3, to be the subset of 9 which satisfies the additional (1) and (2) we obtain Theorem 2.

For any x, y E 2

657

constraints

I(x) > I(y) for all I E 3, ifj‘xLy.

Replacing the set 9 with the set of relative inequality measures 3, considerably increases our ability to rank distributions. although there still remain the inconclusive judgements (corresponding to intersecting Lorenz curves) indicated by the light areas in fig. 1. Fig. 1 gives the impression that the criterion described in Theorem 2 will resolve about two thirds of pairwise inequality comparisons. In practice, the success rate seems to be more like 50 per cent,’ although the precise figure will clearly depend on the particular context under consideration. A number of other properties suggest themselves as candidates for further constraints on the set of admissable inequality measures. For example, it is reasonable to suppose that I(.) is continuous; that I(.) is zero when all incomes are equal; and perhaps also that 1( .) takes the additive form

I(X) = C +Cxii n(x),Ax)). i=* a restriction which follows if the measure is to be ‘subgroup consistent’ [Shorrocks (1988)]. However it turns out that incorporating these additional properties does not lead to any increase in the power of the derived ordering. More promising is the property of transfer sensitivity, recently explored in Shorrocks and Foster (1987). This property effectively strengthens the PigouDalton condition, by ensuring that the index places more weight on transfers which occur lower down in the distribution. The formal definition of transfer sensitivity involves the notion of a composite transfer which combines a progressive transfer with a regressive transfer at a higher income level, while preserving both the mean and the variance of the distribution. Specifically, a distribution s is said to be obtained from y by a favourable composite transfer (or FACT) if there exist i, j,k and 1 such that .r-Y=d(ei-ej)+8(e,-e,),

d>O, 6>0,

a2(x)= a2(y), xi

s xi s .Yk

<

(44 (4b)

x,,

Yi
(4c)

where e,,, denotes a vector of the form (0 ,..., 0, 1,0 ,..., 0) whose only non‘See, for example, Shorrocks (1983, Section II),

658

J.E. Foster

and A.F. Shorrocks,

Income

distribution

and wealth

inequalities

zero element occurs in the mth position. An inequality measure I(*) is then said to be transfer sensitioe if I(x)
For any x, y E Z&):

(T3a)

1(x)=-Z(y) for all IEX* if and only if

0-W

2 can be obtained from j by a non-empty finite sequence of rank preserving progressive transfers andJor favourable composite transfers.

The ordering induced by f* is somewhat stronger than that derived from 9 and enables some cases of intersecting Lorenz curves to be ranked. If we consider 3-person distributions, as in fig. 1, the Lorenz curves of any pair of distributions intersect no more than once, and (T3b) simplifies to the rule: (T3b’)

Either

xLy

or

g1 >ji

and

a2(x) 4a2(y).

The second part of (T3b’) corresponds to the increase in power provided by the ordering obtained from 4* rather than 9, and is indicated by the dark areas in fig. 2. The reduced size of the light areas (which remain unranked by f*) suggests that 9* resolves about half the cases of intersecting Lorenz curves, and this appears to be a reasonable rule of thumb in practical applications [Davies and Hoy (1986)J It is worth mentioning that, for distributions with a common population size and mean income, the orderings induced by 9 and 9* respectively coincide with second- and third-order stochastic dominance. This suggests that the inequality orderings may be further strengthened by considering higher degrees of stochastic dominance. However, unlike the second and third-order cases, there are no obvious interpretations of these higher degrees of stochastic dominance which suggest that they are relevant for inequality comparisons. 3. Poverty orderings A poverty index is a function P: .!Zx R + + -+R whose typical image P(x; z) indicates the degree of poverty associated with the distribution x when the poverty line is set at z. One example is the widely used headcount ratio, signifying the proportion of the population lying at or below the poverty line:

PI(x; 4: =4(x; 4/n(x), where q(x;r)

is the number

(5) of incomes in x that do not exceed z. Other

f.E. Foster and A.F. Shemxks.

Income distribution and

wealth

inequalities

659

Fig. 2. The ‘transfer sensitive’ ordering.

examples are provided by the Sen (1976) index and the family of indices proposed by Foster et al. (1984): 1

PA-~;4:

=n(x)

P(X--)

& t

Z-2.

a-1

( ) ---2 z

,

a2 1,

for which the headcount ratio corresponds to the case a= 1. The selection of a poverty index requires two major decisions to be made. First we have to choose the index form, or, in other words, the way in which the numerical value of the index is computed from data on individual incomes. Secondly, we have to specify the poverty line value 2. It is this second degree of freedom that distinguishes the construction of a poverty index from that of an inequality measure, and enables a wider variety of poverty orderings to be derived. One approach, discussed by Foster (1984) and Atkinson (1987), is the direct analogue of the method employed with inequality measures: we consider a class of poverty indices satisfying a number of reasonable properties, and examine when this class of indices agree on their ranking of any pair of distributions. A simpfe example is

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J.E. Foster and A.F. Shorrocks. Income distribution

and wealth inequalities

provided by the set of poverty indices which are symmetric, and which do not increase when a progressive transfer is made. Then, in the case of 3person distributions with a common mean p, we obtain a similar diagram to fig. 1, except that the regions ,4(x) and B(x) now include distributions that are ranked the same as x, as well as those ranked strictly above or below x. Another approach, considered by Foster and Shorrocks (1988a, b), focuses on the ambiguity associated with the level of the poverty line, and suggests that any given pair of distributions are not conclusively ranked if the ranking obtained at one poverty line is reversed at some other poverty standard. Formally, we say that x has unambiguously less poverty than y with respect to the poverty index P, and write xPy, if

P(x;4

s P(y;

4

for all z E R + + and < for some z.

(7)

In principle, this notion of a poverty ordering could be applied to any poverty index. However, the most interesting orderings appear to be those induced by the Foster et al. indices (6), and in particular those corresponding to a= 1,2 and 3. For example, when a = 2 we have the so-called ‘per-capita income gap’ index P,, and Foster and Shorrocks (1987b, Lemma 2) demonstrates that: Theorem 4.

For any x, y E 5?“(p): xP,y ifJ_xLy.

For distributions with a common population size and mean income, therefore, it follows from Theorem 1 that x has unambiguously less poverty than y with respect to the poverty index P, if and only if all inequality measures in the set 9 agree that x has less inequality than y. Similarly, for the poverty index P, we may deduce using Theorem 3 above and Foster and Shorrocks (1987b, Lemma 3) that: Theorem 5.

For any x, y E S”(p): xP,y ifl Z(x)
So, for distributions with a common population size and mean income, the unambiguous poverty ordering obtained with P, coincides with the inequality ordering based on the set of transfer sensitive indices 9*. For the 3-person case, it follows that the ordering P, generates a picture identical to that of fig. 2. Theorems 4 and 5 both refer to pairs of distributions with the same population size and mean income, and in this respect are the analogues of Theorem 1 rather than Theorem 2, which places no such constraint on the distributions under consideration. This leads us to ask whether we can construct a poverty ordering which coincides with Lorenz dominance on the complete set of distributions Z, and hence corresponds to the ordering derived from the set of relative inequality indices 9,. An affirmative answer

J.E.

Foster

and A.F. Shorrocks.

Income distribution

and wealth

inequaliGes

MI

can be given to this question, but the poverty ordering has to be defined in a slightly different way to {7), Instead of assuming that the poverty line is determined independently of the income distribution, we suppose that z is some fixed multiple i.>O of mean income. Then P(x;i.p(x)) becomes the poverty value corresponding to the distribution x and the ‘relative povrrty standard’ 2. Any uncertainty surrounding the choice of poverty line is now captured in an ambiguous value for 2, and, by analogy with (7). we may define the ‘relative’ poverty ordering P’ by xP’,v if and only if P(x; i.&I) s P(v; MY))

for all 2.E:R, + and < for some 2.

(8)

Now consider any distributions x, y ~5’. Define x’: = x/~(x~ and y’: = _v~‘&-), and let x“ and y” be any replications of x’ and y’ which satisfy n(Y) =n(y”). Using (6), we observe that P&x; ip(x))=P,(x”;& and hence xP~“,Yif and only if x”PJ’. Furthermore, x” and y” have the same mean p = 1, and a common population size. From Theorem 4 we may therefore deduce that xP:g is equivalent to x”Ly”. Since xLy if and only if x”Ly”, this immediately yields: Theorem 6.

For any x, YET:

xP;y iff xLy.

Thus the unambiguous ‘relative’ poverty ordering Pi is identical to Lorenz dominance on the complete set of distributions J.’ *A similar argument using Theorem 5 would allow us to deduce that xP;y is equivalent to the statement that f(x”)
References Atkinson. A.B. 1970, On the measurement of inequality, Journal of Economic Theory 2, 244263. Atkinson. A.B.. 1987. On the measurement of poverty, Econometrica 55, 749-764. Dasgupta. P.. A. Sen and D. Starrett, 1973, Notes on the measurement of inequality, Journal of Economic Theory 6. 180-l 87. Davies. J. and M. Hoy, 1986, Comparing income distributions under aversion to downside mequahty (Umversity of Western Ontario, London). Foster. I.E. 1984. On economic poverty: A survey of aggregate measures, in: R.L. Basmann and G.F. Rhodes. ed.. Advances in Econometrics. Vol. 3 (JAI Press, Greenwich, CT). Foster. J E.. J. Greer and E. Thorbecke, t984, A class of decomposable poverty measures, Econometruza 52. 761-766. Foster, J E. and A.F. Shorrocks. 1988a. Poverty orderings, Econometrica. Foster. J.E. and A.F. Shorrocks, 1988b. Poverty orderings and welfare dominance, Social Choice and Welfare. Sen, A.K., 1973. On Economic Inequality (Oxford University Press, Oxford). Sen. A.K., 1976, Poverty: An ordinal approach to measurement, Econometrica 44, 219-231. Shorrocks. A.F., 1983. Ranking income distributions, Economica 50.3-17. Shorrocks. AX.. 1988. Aggregatton issues in mequality measurement, in: W. Eichhorn, ed., Measurement in Economics (Sprmger-Verlag, New York). Shorrocks, A.F. and J.E. Foster, 1987, Transfer sensitive inequality measures, Review of Economic Studies 54,485--197.