all could have been simultaneously, have a central role. This interpretation

Transcription

1 Chapter 7 Measures of inequality 7.1 Introduction In the previous chapter I argued for an individualistic interpretation of the ideal of equality in which the complaints of those who are worse o than all could have been simultaneously have a central role. This interpretation will of course have consequences for a measure of inequality which is meant to be an index for the moral seriousness of inequality. These consequences will be examined in this chapter and in the next chapter. These chapters make use of some mathematical reasoning. Although the details of the reasoning can be skipped by the reader it is nevertheless explained rather extensively in order to make the reasoning accessible for those who are not mathematical experts. Several measures have been suggested. They are formulated all under the supposition of one xed equalisandum for which there is a measure. 1 In this chapter I discuss some of them because the critical remarks that have been stated let appear the properties that an adequate measure of inequality should satisfy. An example is illustrative. The badness of inequality could be taken to be represented by the dierence between the best-o and the worst-o i.e. x max x min the range (x i is the amount ofx person i has max indicates the best-o and min indicates the worst-o). This measure although attractive because of its simplicity is inadequate. It leaves out of consideration all persons which are in between the worst-o and best-o. A distribution represented by A in gure 7.1 will be considered to be as bad regarding inequality as distribution B. 1 In the next chapter I will have some remarks on the measure of the equalisanda. 213

2 214 CHAPTER 7. MEASURES OF INEQUALITY A B Figure 7.1: Illustration of the range not being a proper measure x A B Figure 7.2: Illustration of the mean deviation not being a proper measure This example shows that a measure should take into account not merely how well o the best-o and the worst-o are but also those in between. A measure which takes into account also how well o those others are is for example the mean deviation the sum Pof the dierences of how well people fare with respect to the mean i.e. n i=1 jx x ij (x P is the mean i.e. 1 n n i=1 x i n is the number of persons). But this measure is not adequate either because now a distribution represented by A in gure 7.2 will be seen to be as unequal as distribution B. This critical remark exhibits the idea that transfers from a rich person to a poor person should make a dierence to the inequality. 2 2 Whether all transfers should do is however an open question. I return to it below

3 7.2. LORENZ DOMINANCE 215 Like the examples of measures mentioned above some other measures are discussed in this chapter in order to arrive at a list of properties a measure should satisfy. I start with those measures that follow the ordering of Lorenz dominance meaning roughly that the seriousness of the inequality of a distribution ~x 0 is greater than that of another ~x ~x I ~x 0 if the Lorenz curve of~x is above the Lorenz curve of~x 0 i.e. ~x LD ~x 0. The Lorenz curve which is explained in section 7.2 is a widely used way of representing the inequality of a distribution. In that section some further properties of Lorenz curves and Lorenz dominance are discussed. The relation of Lorenz dominance has a serious shortcoming: it is incomplete in its ordering of distributions. It does not indicate how toevaluate the inequality if the Lorenz curves do intersect each other. One response is to look at extensions. These are examined in section It appears that the measures considered there are not consistent with our moral ideas on equality. In that section some important properties arise out of the discussions. The lack of a satisfactory extension suggests to attend to another response. This other response is accepting incompleteness and look for explanations of incompleteness. These explanations can be seen as reasons against the possibility of a complete ordering. The explanations are discussed in section It is argued that they are not acceptable or are not leading to judgements parallel to Lorenz dominance. Therefore the characteristic properties of the Lorenz dominance have to be amended. This amendment is the subject of section 7.4. Finally in section 7.5 I end up with some properties which an adequate measure should satisfy. In the next chapter a simple measure is presented which satises these properties it is the nal argument against the view that there is no complete ordering regarding inequality in distribution problems. 7.2 Lorenz dominance In newspapers and economic literature it is usual to represent the inequality of distributions by citing how many of the persons possess a particular part of the wealth or income [Samuelson 1984 p. 80]. For example the income distribution in the Netherlands represented by a diagram as in gure 7.4 in this chapter. In the example here it is clear that a transfer should make a dierence. But is it also in case the transfer takes place in the upper part of the gure and a less rich person receives some of the good from an even more rich person? Does that make any dierence in the seriousness of inequality? I am not sure about that. Sen choose his examples carefully in order to be convincing in [Sen 1973 p. 26].

4 216 CHAPTER 7. MEASURES OF INEQUALITY 20% 15% 10% 5% 0%?? 0%-10% 90%-100% Figure 7.3: Income distribution in the Netherlands 1995 was presented in an article in the Volkskrant. 3 On the horizontal axis are put the groups ordered from poor to rich 1:::i:::n on the vertical axis xi nx. The Lorenz curve is the result of the cumulation these fractions. The income of the rst percentile will be similar to those in the former curve. The second percentile is added to the rst and the total amount they possess is represented on the vertical axis. On the horizontal axis we see the percentage of the total population ordered from poor P to rich represented i x by the percentiles and on the vertical axis we see i for the group up 1 nx to the ith percentile. Connecting the points will result in the Lorenz curve as in gure 7.4 The diagonal line is representing an equal distribution in which 10% of the poorest people possess 10% of the total amount of goods 30% possess 30% 50% possess 50% etc. A Lorenz curve of~x 0 below another curve of~x means that ~x 0 is worse regarding inequality than ~x it is Lorenz dominated ~x LD ~x 0. Apart from the fact it is a common practice to represent the inequality of distribution by a Lorenz curve there are some other reasons for Lorenz dominance being an attractive ordering regarding inequality. Lorenz dominance is implied by a social welfare function which isschur concave and monotone increasing in its arguments. The idea is that the judgements `all things considered' can be represented by awelfare function W (~x) of the distribution ~x the higher the value of this function the better. Schur concavity does mean that W (B~x) W (~x) in which B is a bistochastic matrix 3 Volkskrant 9 december 1995 an article concerning the inegalitarian eects of the reformation of welfare state by the Dutch governing coalition in 1995.

5 7.2. LORENZ DOMINANCE % 50% 0% q q q q!! " q q q q q q Figure 7.4: Lorenz curve of income distribution in the Netherlands 1995 in which all the elements b j i are :0 bj i 1 and the sum of the elements ofarow is 1 and the sum of the elements of a column is 1. Such a matrix can be considered to represent transfers from rich to poor. Monotonicity in its arguments does mean that W (~x) is non-decreasing if x i increases representing the Pareto principle or the principle of personal good or even better person aectingness. 4 It has been proved that: (x =x 0 & ~x LD ~x 0 )=) W (~x) W (~x 0 ) [Dasgupta & Sen & Starlett 1973] So Schur concave monotone welfare functions W are consistent with Lorenz dominance under the condition of an equal total amount of what is to be allocated. 5 Because those indices are representing the judgements on how well a society is `all things considered' Lorenz domination is seen to be an attractive ordering of inequality. It is common to many measures or indices of inequality and social welfare [Kolm 1968 p. 149]. 4 See chapter 6 p An example of suchawelfare function is utilitarianism W (~x) = P n i=1 w(x i) in which w is increasing (the more of xi the higher w(xi)) and concave meaning that the more of xi the less it will add to w i.e. the law of diminishing returns. Equalising will result in an increase in W the popular argument for equality within utilitarianism. See for some critical remarks p. 226 of this chapter.

6 218 CHAPTER 7. MEASURES OF INEQUALITY Beside the reason mentioned which relies heavily concave social welfare functions there is another reason for the attractiveness of Lorenz dominance as an ordering indicating the moral badness of inequality. It can be shown that Lorenz domination will result if symmetrical distributions distributions in which persons changed place with respect to the distribution in other words permutations of the distribution form an indierence class in combination with an amended version of the person aectingness of chapter 6 of which the best-o are excluded. This can be shown by following the line of argument of Kolm in [Kolm 1972 p ]. The echo of Suppes' grading principle and the non-envy condition can be heard. Furthermore the idea that the best-o are excluded from those to be reckoned with does reect the idea of equality asitwas developed in the previous chapter together with the idea that if the best o are equally well o the second best o should not count in determining how good a distribution is and so on. Let me follow Kolm's argument. If a distribution ~x has a similar total amount of goods as another distribution ~y and if the former is better than the latter then the Lorenz-curve of the former is everywhere above the latter. The explanation is quite simple and instructive. Take distributions ~x =(x 1 ;x 2 :::; x n ) and ~y =(y 1 ;y 2 :::; y P P n ) such that n i=1 x n i = i=1 y i. 6 Take apermutation of ~x say ~x 0 such that for all i : x 0 i x 0 i+1 similarly take for ~y; ~y 0 such that for all i : y 0 i yi+1. 0 Nowif for all i<m: x 0 i y0 i (m being the maximum number for which x m 6= y m ) then distribution ~x 0 is better or equal to distribution ~y 0 ~x 0 ~y 0 because of the amended version of person aectingness the best-o i = n are excluded from consideration or if these are equally well o in all distributions the second-best-o are excluded and so on. Because of symmetry we have ~x ' ~x 0 and ~y 0 ' ~y so~x ' ~x 0 ~y 0 ' ~y. But if ~x 0 ~y 0 then the amended version of person aectingness implies that the Lorenz curve of~x 0 or what comes to the same the Lorenz curve of ~x is nowhere below the Lorenz curve of ~y because: 8i <m: x 0 i y0 i =)8i : ix k=1 x 0 k in which m is the maximal number such that x 0 i 6= y 0 i. So far Kolm's argument. It is also valid that if the Lorenz curve of ~x is nowhere under the Lorenz curve of~y then ~x ~y. Consider x 0 m and y 0 m in which m is the maximal 6 It would be according the common rules to state ~x =(x1; x2; ::xn) T ; () T meaning the transpose of (). Because of typographical convenience and because it does not lead to great misunderstandings I sometimes use the vertical representation of vectors and sometimes the horizontal. ix k=1 y 0 k ;

7 7.2. LORENZ DOMINANCE 219 number such that x 0 i 6= y0 i. Make a transfer from y0 m to the m 1 person in such away that in the resulting distribution ~y 00 x 0 m = ym 00 without disturbing the ordering of the persons. P This is possible P because the Lorenz i curve of ~x is not under ~y i.e. 8i : k=1 x0 k i. It means that P n i=m x0 i P n k=1 y0 k i=m y0 i so y 0 m x 0 m.furthermore ~y 00 ~y 0 because y 00 m >y0 m and i m can be excluded from consideration. But also ~x 0 LD ~y 00. Repeat this procedure p times until m = 2. Now it can be concluded that if m = 2 the worst-o and the second-worst-o person can dier and all others are equally well o. The second-worst-o can be deleted from considerations so because the total amount of goods is not changed this means that x 1 y p 1. Consequently if ~x LD ~y then ~x ' ~x 0 ~y p ~y 0 ' ~y. So symmetry and the amended version of person aectingness excluding the best-o from consideration and if they are equally well o excluding the second-best-o etc. is equivalent to the ordering of Lorenz dominance provided the total amount of the goods to be distributed is constant. Actually as shown by [Foster 1985 p. 51] the class of indices I which are consistent with Lorenz dominance meaning that the index does not contradict the ordering of the Lozenz dominance in formula: ~x LD ~y =) ~x I ~y can be characterised by four properties: 1. Pigou Dalton principle meaning that a transfer from a poor person to a rich person will result in an increase of inequality. 2. Symmetry or anonymity principle based on the idea of impartiality meaning that taking a permutation of the distribution will make no dierence for the value of the index of inequality. 3. Population principle meaning that adding a population with the same distribution will result in a value of the index of inequality that is similar to the values of the index of the populations taken apart. 4. Homogeneity meaning that the index of inequality is invariant and will not change if the amount of goods received by all is multiplied by the same factor. 7 All measures that satisfy these four properties are consistent with Lorenz dominance and all measures consistent with Lorenz dominance have these properties. The latter is shown in a clear way by Foster. Lorenz dominance follows these properties. Take ~x LD ~y then from the construction of 7 I return to this property in the next chapter. This property can be argued for because there should be no dierence between measuring income in dollars or in dollarcents.

8 220 CHAPTER 7. MEASURES OF INEQUALITY the Lorenz curve it is clear that symmetry is satised. Changing the persons that receive a particular amount will not change the Lorenz curve the Lorenz curve is determined by the ordering from poor to rich. The Lorenz curve of~x and of ~x will be similar because is seen in both the numer- P x ator and the denominator of i n:x.furthermore the Lorenz curve of~x is similar to ~y if ~y is consisting of several replications of ~x because the percentiles do not change the fraction of the population possessing a particular fraction of the total amount does not change. Finally the Pigou Dalton principle can be shown by turning to the contraposition. If distribution ~x is the result of a transfer of a rich person j to a poor person i starting from a distribution ~y then for the m < iand m > jnothing changes P m1 k=1 y k x P x k = m1. For i m j we can note that the percentile to x k=1 which i belongs will receive more because some persons of the i + 1-the percentile will now belong to the i-th percentile so P m k=1 x k x P m k=1 resulting in ~x LD ~y. Now the other direction: if an index I satises the four properties it is consistent with Lorenz domination. It can be shown as follows [Foster 1985 p. 51]. Suppose an index I satises these properties and let ~x and ~y be two distributions with dimension n and m representing the numbers of persons. Dene ~x 0 as m replications of 1 x ~x and ~y 0 as n replications of 1 y ~y. Now because of the argument in the former paragraph ~x ' LD ~x 0 and also ~y ' LD ~y 0.If~x ' LD ~y then ~x 0 = ~y 0 because ~x 0 and ~y 0 have the same total amount of goods and the same number of persons. Consequently ~ x 0 ' I ~y 0 and so ~x ' I ~x 0 ' I ~y 0 ' I ~y because of symmetry homogeneity and the population principle of I accounting for the other outermost equivalencies. Suppose next that ~x LD ~y then because of the previous paragraph ~x 0 LD ~y 0. So also because of the previous paragraph ~x 0 can be obtained from ~y 0 by transfers from rich persons to poor persons. Because I satises the Pigou Dalton principle ~x 0 I ~y 0 and because of homogeneity and the population principle and symmetry we have ~x ' I ~x 0 I ~y 0 ' I ~y. So there are several reasons for using Lorenz dominance as basis for judgements on inequality. But there is a serious problem namely what to do if Lorenz curves do intersect? It is just a partial ordering. y k x 7.3 Incompleteness As mentioned the Lorenz dominance as an indicator of inequality does not lead to a complete ordering. There are two answers to this incompleteness. One is developing a complete extension of the ordering of the Lorenz dominance the other is accepting incompleteness as an inherent feature of judgements of inequality. These two answers are discussed respectively.

9 7.3. INCOMPLETENESS n B A C 1 n 1 n 1 n x 3 : 1 n:x x 2 : 1 n:x x 1 : 1 n:x Figure 7.5: Illustration of the GINI coecient Extensions of Lorenz dominance GINI coecient One natural way of extending the ordering of Lorenz dominance is by taking the areas under Lorenz curve as a representation of inequality. One commonly used index is known as the GINI coecient. It is the ratio of the area enclosed by the Lorenz curve and the diagonal A and the total area under or above the diagonal B G = areaa or what comes to the same areab one minus twice the area C under the Lorenz curve G =1 2C. The greater the area A the greater the inequality. The GINI coecient can be expressed as: G(~x) =1+ 1 n 2 n 2 x (nx 1 +(n 1)x x n1 + x n ) This can be made clear by the following reasoning. The area under the Lorenz curve C() is: C(~x) = 1 1 x 1 2 n nx + 1 x 1 n nx x 2 2 n nx + = n nx (x 1 + x 2 )+ 1 1 n nx ((n 1)x 1 +(n 2)x 2 + )= n nx nx n nx ((n 1)x 1 +(n 2)x 2 )=

10 222 CHAPTER 7. MEASURES OF INEQUALITY Finally: 1 2n n nx (nx 1 +(n 1)x 2 + ) 1 1 n nx (x 1 + x 2 + )= 1 2n + 1 n 2 x (nx 1 +(n 1)x 2 + ) 1 n = 1 2n + 1 n 2 x (nx 1 +(n 1)x 2 + ) G(~x) =1 2C(~x) =1+ 1 n 2 n 2 x (nx 1 +(n 1)x 2 + ) There is also another formulation of the GINI coecient which is cited more often: G(~x) = 1 jx i x j j 2n 2 x i=1 j=1 This formula indicates that the dierence between each pair is the central idea of a measure of inequality. 8 Although the GINI coecient is complete and of course consistent with Lorenz dominance it is not according to our moral intuitions. For example a transfer from a rich person to a poor person will lower the GINI coecient but the amount ofhowmuch is dependent onhowmuch groups there are between these rich and poor persons and not merely on the dierence of how well o they are. This can be seen from the rst formulation of the GINI coecient. It could happen that a transfer of a rich person to a poor person with a small dierence but with many groups between them is lowering the index more than a transfer from a rich person to a poor person where there are a few groups between them but where the dierence is much larger. For example a transfer T from group 2 to 1 with a dierence of 100 without changing the groups is lowering the GINI coecient with 2 n 2 (nt (n 1)T ). A similar transfer T from a group n to group n-100 x with a dierence of say 10between them will lower the GINI coecient by 2 n 2 x (100T T ). This indicates that the GINI coecient leads to some odd evaluations [Sen 1973 p. 32]. The argument against the GINI coecient makes clear that for an evaluation of an index of inequality the issue of how much a transfer from rich topoorchanges the index is considered to be relevant. To be more precise one idea about our evaluations of inequality is that a transfer from a rich to a poor person matters more if the transfer is in the lower part of the distribution. This principle is called the principle of diminishing 8 It is often said that it can be easily shown that both formulations are similar but it is seldom explained. The explanation is in appendix 1 of this chapter.

11 7.3. INCOMPLETENESS 223 transfers. As was shown above the GINI coecient can be in conict with this principle. The transfer T in the upper part of the distribution can lower the GINI coecient more than a similar transfer in the lower part. Confronted with these odd evaluations of the GINI coecient due to arbitrariness of some evaluations according to this index one wonders if there is a systematic way of extending the ordering of the Lorenz dominance. One way for example would be adding more restrictions and wonder whether the class of indices is restricted such that the orderings are complete. Let me turn to that possibility. Decomposable indices Decomposability is argued to be an attractive property for a measure because it enables to determine the inequality of the population on the bases of the inequality of parts of this population. If one knows the inequality of the subgroups I g the number of persons n g belonging to that group and how well o these subgroups are x g then it is possible to determine the inequality of the whole group. Decomposability means that the inequality of the whole group is the weighted sum of the inequality within the subgroups and the inequality between the subgroups. To be more formal. I(~x; n) = GX g=1 w g I(~x g ; n g )+I(x 1 ; x 2 x G ; G) Shorrocks demonstrated in a good accessible way that the class of indices that are i) symmetric ii) follow the population principle iii) are homogeneous and iv) have values of zero if there is equality and v) are greater than zero if there is inequality vi) are continuous and vii) have continuous rst order and second order derivatives is: [Shorrocks 1980]. I c (~x; n) = 1 1 n c(c 1) I c (~x; n) = 1 n I c (~x; n) = 1 n i=1 i=1 i=1 (( x i x )c 1); c6= 0; 1 log x x i ; c=0 x i x log x i x ; c=1

12 224 CHAPTER 7. MEASURES OF INEQUALITY The index with c=2 corresponds to the square of the coecient ofvari- ation the ones with c=1 and c=0 are suggested by Theil as measures of inequality. The coecient ofvariation is: rp (x xi ) 2 CV (~x) = 1 x This coecient is criticised by Sen because it is in conict with the principle of diminishing transfers [Sen 1973 p. 28]. Transfers in the upper part of the distribution count as much as transfers in the lower part. For example take an innitesimal transfer from x i to x it then the innitesimal change in the coecient ofvariation will be dcvx i dx i 1 it is a mean preserving transfer the change in for all 1 p nx n p nx 2 + dcvx it dx it. Because dcv is zero the is pp 1 (xx i) 2 2P i (x x i)(n 1). Hence it is equally for all [Atkinson 1972]. Apart from this issue of the principle of diminishing transfers meaning that transfers should have less eect on the inequality in the upper regions in case c=2 there remains still something arbitrary on these measures. 9 For example how to choose between the values of c. Furthermore it is not clear at all whether decomposability is consistent with the idea of equality which was argued for in the previous chapter. Following the meaning of the ideal of equality as it was developed in the previous chapter namely that inequality is bad because some persons are worse o than all could have been means that the inequality within the group above the reference does not inuence the inequality of the total population. 10 This means that lowering the inequality within a group which is better o than in the ideal reference will not aect the inequality of the whole group. For example decreasing inequality in the western prosperous world will probably have no consequences for the inequality on a global scale. So an improvement in the former will not have any eect on the inequality on the global scale. And indeed by considering the problems on a more global scale these local problems seem to evaporate. It is not said that inequality in local application is of no importance at all for the inequality. Political decisions leading to a decrease in inequality 9 The argument of Sen that the coecient is highly suspicious because only dierences with respect tot the mean are reckoned with can be rebutted by pointing to the lemma 5.12 stated by Kakwani in [Kakwani 1980 p. 86]. It reads: If in any pairwise comparison a person with lower income suers some depression proportional to the square of the dierence in incomes the average of all such depressions in all possible pairwise comparisons leads to the coecient of variation. 10 See chapter 6 p dx i

13 7.3. INCOMPLETENESS 225 on a local scale is not necessarily constitutive of a decrease in inequality on a global scale but it could in the end result in an improvementby starting a process of actions leading to less inequality for example by promoting and extending sympathy. This is a process in time and concerns the question: how to arrive at a more egalitarian world? This issue of how to promote equality on a global scale is not the subject of this study. Here I am concerned with the content of the ordering of inequality as part of an analysis of equality as an ideal. The ordering at a particular moment is at stake and its dependence on the orderings in and between groups. And within this analysis of properties of indices a reduction of inequality within the group above the ideal reference has no consequences for the moral badness of inequality. An intuition which is plausible and can be explained by the ideal of inequality developed in the previous chapter. Decomposability is not a property an index of inequality should have. There is still another method for arriving at extensions of Lorenz dominance circumventing the arbitrariness of measures considered so far namely by starting with the inuence of the distribution on the social evaluation `all things considered'. Measures which start from these social evaluations `all things considered' do not suer from the arbitrariness that the above extensions suered from. Let me look at these. Kolm-Atkinson measures The former extensions of the ordering of the Lorenz dominance suered from arbitrariness; there is no connection between the measure and the reason why inequality is wrong. The so-called normative measures start with the inuence of distributions on the evaluations of those distributions `all things considered'. The moral badness of the inequality is measured directly by the loss of welfare or what represents this welfare for example money as is suggested by Kolm [Kolm 1968]. Dalton takes the loss of welfare because of inequality directly as the measure of inequality. His index is corresponding to I D (~a) =W (a~u) W (~a) in which ~u =(u 1 ;u 2 ;:::)=(1; 1;:::). 11 This index represents how serious inequality is in terms of social welfare. Relevant indices should be invariant 11 Dalton's measure is mostly presented in a relativised form 1 W (~a) W (a~u) For W the utilitarian interpretation P w(ai) is commonly used.

14 226 CHAPTER 7. MEASURES OF INEQUALITY under certain transformations of W and not be inuenced by idiosyncratic properties of W. Usually such aw is only determined up to certain transformations. A utilitarian welfare function is determined up to linear transformations meaning that W and W 0 are equivalent ifw 0 = 1 W + 2 ( 1 and 2 are numbers and 1 ; 2 > 0). For the essential properties it should not make any dierence whether one should take W or W 0. However Dalton's measure is dependent on these particular idiosyncratic properties [Atkinson 1972]. A possibility which does not suer from the dependency on irrelevant characteristics is measuring the inequality in terms of the resources which are spoiled because of an unequal distribution. The measure is in terms of resources instead of in terms of welfare. The leading concept introduced by Kolm which is used later by Atkinson is the equal equivalent distribution ~a e [Kolm 1968] [Atkinson 1972]. This equal equivalent distribution is the distribution in which all receive the same amount of the resources a e such that W (a e :~u) =W (~a). The inequality is measured by a a e orbyits relativised form 1 ae. If the equal equivalent distribution is a:~u then the a index is 0. The welfare function chosen by Atkinson is of a utilitarian form in which all have the same utility function which is concave meaning it satises the law of diminishing returns. Although there is a diculty of all having the same utility function there is even a more serious problem pointed to by Hansson [Hansson 1977]. The measure would mean for example that if the utilitarian function becomes more concave then the inequality would in fact be less but the eect on the inequality index is dierent. It is illustrated by gure 7.6. Suppose we have a utility function w and a utility function w 0 which is in the lower regions equal to w but in the higher ones below w. Consider two distributions x p and a r in the lower and higher regions respectively then in case of w the dierence between w(a r ) and w(a p ) is greater than the dierence between w 0 (a r ) and w 0 (a p ) which is representing the dierence between what goods do to people. But 1 2 (w0 (a r )+w 0 (a p )) is lower than 1 2 (w(a r)+w(a p )) so a 0e a e. Consequently a a 0e a a e meaning the index of inequality is greater although the dierence between what the goods do to people is less. Apart from the suggestion of Hansson that one should measure inequality directly in terms of equalisanda and not in terms of distribuenda the example discloses even more. It shows that one cannot take the loss of social welfare to be an index of inequality [Sen 1973] [Sen 1978]. Inequality is just one aspect of social welfare and is not represented by it. Stating that they are similar is mixing up the meanings in an unclear way. Social welfare is one thing and inequality another.

15 7.3. INCOMPLETENESS 227 w(a r ) q w w(a r)+u(ap) 2 w 0 0 (a r)+w (ap) 2 w 0 (a r ) q q q w 0 w(a p )=w 0 (a p ) q a p a 0e a e a r Figure 7.6: Hansson's problem Stated dierently once one has a social welfare index representing judgements `all things considered' one does not need any longer an index of inequality [Osmani 1982]. An index of inequality is useful as a component of the social evaluation in order to arrive at a judgement `all things considered'. But if one wants to use these judgements `all things considered' in order to arrive at an index of inequality representing the way inequality affects the social evaluation function one has already in mind what is meant by inequality and one has already an idea of an index apart from the social evaluation. So although the social welfare approach in order to arrive at a normative measure of inequality seemed to be a promising way to avoid arbitrary measures it is not satisfactory because of an essential dierence between equality and social evaluations `all things considered'. Summarising extending the ordering of the Lorenz dominance is not successful because it led to measures with some odd consequences in case of the GINI coecient and to arbitrary judgements in the case of decomposable indices or in the case of the normative approaches they turn out to be not adequate at all. The discussion learned that an index should satisfy the principle of diminishing transfers i.e. it should be more inuenced by transfers in the lower region than in the higher one. Furthermore the argument of Hansson indicated that one should not turn to distribuenda or goods simpliciter but to the equalisanda what goods do to people in order to measure how serious inequality is. This failure to arrive at satisfying

16 228 CHAPTER 7. MEASURES OF INEQUALITY extensions of Lorenz dominance suggests the other answer to incompleteness of the ordering according to Lorenz dominance namely acceptance of incompleteness. Let me look at that answer to incompleteness Acceptance of incompleteness and its explanation One answer to the incompleteness of the Lorenz dominance is accepting the incompleteness of the ordering of inequality. Incompleteness could be considered to be expected in normative questions reecting complexity. Realising this complexity one could take for example as a measure of inequality what all the measures about which one is certain in a particular domain have in common. The nal ordering would be the intersection of these orderings. The ordering of Lorenz dominance could be interpreted in such a way. If all indices satisfy the principle of Pigou Dalton principle of population homogeneity and symmetry then the Lorenz dominance is something they all have in common. Consequently incompleteness is nothing mysterious and is just what was to be expected. Sen is suggesting this as part of an explanation of incompleteness [Sen 1973 p. 72] [Sen 1977]. This explanation of incompleteness is criticised by Temkin. He argues that if one is certain about the adequacy of a measure in a particular domain one can just rely on that measure; conrmation by other measurements is superuous [Temkin 1993 p. 141.]. If one of the measures is reliable in a particular kind of situations then the fact that other measures are unclear or unreliable in this domain is not a reason for withdrawing the judgement that one distribution is more unequal than another. Divergence of opinion of the dierent measures is no reason to abstain from a judgement. One trustworthy measurement is just enough. Consequently the intersection approach is not an appropriate explanatory reason for the incompleteness of the ordering regarding inequality. Although Temkin criticises the intersection approach suggested by Sen he accepts incompleteness and underlines the complexity of equality. 12 He argues that the idea of equality is exhibiting several principles. Temkin argues for these dierent principles underlying the idea of inequality because 12 Temkin writes: What we need it might be claimed is to arrive at a measure of inequality that accurately captures each of the aspects involved in that notion according them each their due weight. Such a measure would give us a way of accurately comparing many though perhaps not all situations regarding inequality. [Temkin 1993 p. 52]

17 7.3. INCOMPLETENESS 229!! A B C Figure 7.7: Temkin's series they explain according to him the dierent judgements in cases like the following series in gure 7.7. In the rst situation all except one have a reasonable amount of some good. In the next all except two have this same amount of good and the two exceptions have the lesser amount. Continuing we arrive at the situation in which all except one have the lesser amount of the good. The question is how to rank these situations according to the moral badness of inequality. Temkin argues several judgements to be reasonable. Some argue that the situations in the series become better and better because in the rst there is one person victimised exhibiting discrimination. As the series progresses there are more people worse o and consequently this victimisation is less. Others argue the situations become worse and worse because there are more and more worse o people consequently the total amount of complaints of the worse-o is increasing. Dierent principles can explain the dierent judgements. 13 This series used as illustration of dierences of judgement regarding inequality are situations with intersecting Lorenz curves. 14 Suppose the 13 In the next chapter I return to the judgements on this series again on p The other examples he presents in order to show that there are several dierent principles behind the idea of equality are also situations in which the Lorenz curves intersect. These situations described in chapter 3 of his book Inequality consist of four groups which from poor to rich receive abcd of some good respectively call e=a+b+c+d. By considering the Lorenz curves one could look at ve points (0; 0); ( 1 4 ; a ); ( 2 e 4 ; a+b); ( 3 e 4 ; a+b+c ); (1; a+b+c+d )Now it can be easily checked that the e e situations Temkin points to in order to argue for dierent principles are situations with intersecting Lorenz curves. See [Temkin 1993 p ].

18 230 CHAPTER 7. MEASURES OF INEQUALITY 6? Figure 7.8: Lorenz curves of Temkin's series level of the best-o is x max and the level of the worst-o is x min and the number of people is n. The amount ofavailable goods can be represented by (n m):x max + m:x min in which m is the numberofworse o people. The fraction of the goods available to the poorest 1 n fraction F x min; n 1 is x min (nm):x max+m:xmin. The fraction of the goods available to the best-o 1 n x fraction is max (nm)x max+m:xmin. Soapoint on the Lorenz curve linked to the all but the best-o fraction is the fraction of the goods owned by the people minus the richest fraction 1 n F x max; n 1 is 1 x max (nm)x max+m:xmin. In the series Temkin considers n is constant. By increasing m from 1 to n 1we can see that F xmin; n 1 = x min (nm):x max+m:xmin is increasing and F xmax; n 1 =1 x max (nm)x max+m:xmin is decreasing so the Lorenz curves of the situations in the series intersect illustrated in gure 7.8. The incompleteness of the ordering regarding inequality is explained by Temkin by the dierent principles behind the idea of equality. Unanimity of the principles in their evaluation of situations nds its counterpart in unambiguous judgements. It does not mean that if unanimity is lacking judgements are not possible; it might be that in some situations one principle might prevail above another; this is the dierence between this approach and the intersection approach. However it is not excluded that

19 7.3. INCOMPLETENESS 231 there might be conicts of evaluation which are not solved so easily. The situations pointed to by Temkin to clarify his arguments are examples of these conicts. The explanation based on the dierences of principles behind the idea of equality is attractive however it presupposes a view on ethics in which principles constitute the basis of moral judgements. In a particularistic framework argued for in this study it is not acceptable. Principles are not the basis of our moral judgements so they cannot be the basis for incompleteness of judgements. In order to arrive at an explanation we should turn to those situations again and see what precludes unambiguous judgements. Judgements regarding inequality are concerned with distributions and redistributions. It is likely that the main diculty is that of comparing the advantage of one person with the disadvantage of another. For example in the series cited above it is not clear how the disadvantage of the worse-o should be compared to the advantage of the besto vis-a-vis a reference situation. Or in other words one is not clear how achange of the position of one person is to be compared to the change of the position of another person. The compensation for losses is not clear. This diculty of comparing advantages and disadvantages explains in a more direct way why judgements regarding inequality are incomplete. After all incompleteness in judgements is not uncommon in normative matters. How is this diculty of comparison to be explained? An explanation due to the impossibility ofinterpersonal comparisons is dealt with in chapter 3 and is excluded. There is no a priori argument for holding that what goods do to people cannot be compared. What could be possible is uncertainty about how much disadvantage of one is set o by the advantage of another. This could be framed in terms of a moral conict like those of dilemmas but in the context of equality in which we assume a fundamental similarity and comparability of people this is excluded. The diculty is rather one of uncertainty than a dilemma. But uncertainty can be dealt with by invoking the vocabulary of probability and decision making under uncertainty. Can this uncertainty explain the incompleteness of the ordering of inequality represented by Lorenz dominance? Suppose intersecting Lorenz curves represent uncertainty about how much disadvantage can be set o against the advantage of someone else. Compare two distributions A and B with the Lorenz curves L A and L B which intersects L A. Following the above explanation of incompleteness the intersection of L A and L B means that there is uncertainty in judgements regarding inequality. B could be seen as distribution A followed by a transfer from a rich person to a poor person in the upper part T R r1!p1 and a transfer from

20 232 CHAPTER 7. MEASURES OF LB 0 - LA BN L B Figure 7.9: Illustration of uncertainty is not an explanation for the incompleteness of Lorenz dominance a poor person to a rich person in the lower part Tp2!r2. P The uncertainty is based on the uncertainty in comparing T R r1!p1 to Tp2!r2. P Suppose further there is a range 4T R p1!r1 in which the transfers T R p1!r1 result in uncertainty of judgements between A and B. If the transfer from a poor to a rich person is greater than those included in this range the resulting distribution is worse than A and transfers below those in this range will result in a distributions better than A. Bylowering the transfer T R p1!r1 and approaching zero there will be somewhere a distribution B 0 better than A. If T R p1!r1 = 0 this B 0 is certainly better than A because the L B 0 is above L A because the upper part is equal to L A but the lower part is above L A due to the transfer Tr2!p2. P But it is highly unlikely that there is no T R p1!r1 such that B 0 is as unequal as A. But if there is such a transfer then there is an intersection of Lorenz curves without uncertainty namely L B 0 and L A.So the intersections of Lorenz curves do not mean uncertainty on judgements regarding inequality. One has to forego the explanation of incompleteness of the ordering regarding inequality by uncertainty of comparing transfers of advantages and disadvantages or one has to turn away from Lorenz dominance as the nal expression of an unambiguous partial ordering. Because the rst is in line with the particularistic conception of moral judgements and dicult to refute I turn to Lorenz dominance and look more critically to its four dening characteristics.

21 7.4. PROPERTIES OF LORENZ DOMINANCE REVISITED 233 Summarising the Lorenz dominance which is a well-known ordering of situations regarding inequality suers from incompleteness. Extending the ordering to a complete one either appeared to introduce unacceptable parts of the ordering for example the GINI coecient or some arbitrary parts as in the decomposable ones or just inadequate measures such as the normative measures. Accepting incompleteness however failed an explanation why a partial ordering is the one according Lorenz dominance and why the situations with intersecting Lorenz curves pose problems. So it is natural to revisit Lorenz dominance in order to arrive at an adequate measure of inequality. 7.4 Properties of Lorenz dominance revisited As was mentioned above it was shown by Foster that indices consistent with Lorenz dominance can be characterised by four principles: 1. Pigou Dalton principle 2. Symmetry 3. Homogeneity 4. Population principle If Lorenz dominance is left then also these principles should be left. But also if these four principles are not satised the Lorenz dominance should be left. The Pigou Dalton principle seems to be indisputable a transfer from rich persons to poor persons will lessen the inequality. But as appeared in the discussion on the extensions of Lorenz dominance notably the GINI coecient and the coecient ofvariation a transfer should be more important in the lower region than in the higher. For an ordinal ordering in which one is only establishing whether or not a situation is worse than another regarding inequality and not how much better or how much worse the principle of diminishing transfers could mean only that there is a region in the upper part of the distribution in which transfers from rich topoor persons will not have any eect. 15 This is consistent with the idea developed in the previous chapter in which the complaints of those below the reference level should count. If the transfer is between those above the level there will be no inuence on how bad the inequality is. Although there is no reason to withdraw this principle categorically it has to be amended 15 Temkin also points to a limited importance of the Pigou Dalton principle in the higher regions [Temkin 1993 p. 79].

22 234 CHAPTER 7. MEASURES OF INEQUALITY by adding that not all transfers have an impact but only those aecting persons who are worse o than all could have been. Symmetry can be questioned if it is applied to distribuenda or resources and not to equalisanda because what goods do to people is not for all people similar. Sen points to the conict between the Pareto principle and Suppes' grading principle showing a permutation does make a dierence [Sen 1979a p. 149]. His example is as follows. There are two persons a Hindu and a Muslim and two situations: a) in which the Hindu is receives 0 units of pork and 2 units of beef and the Muslim 2 units of pork and 0 units of beef and a situation b) in which the Hindu receives 1 unit of pork and 0 units of beef and the Muslim 0 units of pork and 1 unit of beef. Now according to the Pareto principle situation b should be preferred to a because the Muslim is not interested in pork at all and the Hindu not in beef but according to the grading principle i.e. taking the permutation of the distribution the evaluation is reversed. A permutation does make a dierence. However if one turns to the equalisanda in which these dierence are accounted for the permutations will be equivalent. One should be clear what one understands by symmetry. By symmetry one can understand that in case there are two persons of which the characteristics are represented by an ordered pair where x i is the resource part and y i represents the other characteristics what goods do to people ((x 1 ;y 1 ); (x 2 ;y 2 )) ' (I) ((x 2 ;y 1 ); (x 1 ;y 2 )). Symmetry interpreted in this way as symmetry of resources is not acceptable as was shown by Sen's example. But symmetry regarding what goods do to people represented by ((x 1 ;y 1 ); (x 2 ;y 2 )) ' (I) ((x 2 ;y 2 ); (x 1 ;y 1 )) is dicult to deny. So symmetry is acceptable but again it should be applied carefully namely to the equalisanda and not to distribuenda. Homogeneity is a principle which is also dicult to deny if it is meant to express invariance to the units of measurement. But a proportionally increase in what all have for example an increase of 200% is not directly clear to have no inuence on the inequality. If all improve it could be that there should be a change of equalisandum. 16 If all can aord plenty of food the distribution problem is dierent and is no longer one concerning bare survival and it is transformed into the distribution problem of for example more luxury goods. This change of equalisanda from more urgent equalisanda to less urgent equalisanda could be an explanation for the idea presented by Temkin that the moral badness should be discounted the more auent the worst-o is [Temkin 1993 p. 185]. The principle of homogeneity should be used with care. If it is meant to account forachange of units it is undeniable valid but if it is used in situations in which all 16 See chapter 6 p. 210.

23 7.4. PROPERTIES OF LORENZ DOMINANCE REVISITED 235 improve proportionally it is not acceptable at all. Increasing the equalisandum for all could mean a call for a change in equalisandum because of the change of the distribution problem at hand. So its use is valid but dependent on a clear determination of the equalisandum. One should realise that one should compare primarily the equalisanda and not distribuenda. Homogeneity can be accepted but carefully. 17 The population principle is the most implausible of the four principles. If a distribution is morally wrong then two such distributions are even worse. One situation with an unequal distribution is already worse than what could be ideally the case but two is of course even worse. One could defend the principle by arguing that the badness of the inequality per person is not changed but then one turned to a relativised version of the badness of inequality. This way of looking might be useful if one is interested in the mechanism of (re)production of inequality. In that case proportional numbers might be more informative than absolute gures. Although the study and understanding of how inequality is (re)produced is not unimportant it is not the subject of this study here. Here I am concerned with the moral seriousness of inequality. It is plausible to hold that the more replicas of an unequal distribution the worse. This view however could be questioned by the following argument discussed by Temkin [Temkin 1993 p. 218]. Suppose the more replicas or the more people with the same unequal distribution the worse the inequality then the magnitude of the dierence of the best-o and the worst-o could be traded o against the number of persons in the distribution. It could be possible that a large gap between the worst-o and the best-o for only a few persons could be even at least as good regarding inequality as a situation with a small gap between the best-o and the worst-o while the worst-o are even better-o than the worst-o in the former situation as represented in gure So it seems to be possible that whatever the gap is there is always a number n whatever the gap is such that A and B are equally unequal. This consequence called by Temkin the repellant conclusion of the idea the more replicas of a distribution the worse is dicult to accept. The argument is rather convincing but not unavoidable. In the argument it is assumed that all dierences can be traded o by numbers independent ofhow small this gap is. But this is not clear to be necessarily so. One can argue that the gap between the worst-o and the best-o is of such an important nature or it concerns such an urgent equalisandum that it cannot be traded o by a lesser gap concerning a less urgent equalisandum distributed unequally among a greater number of people. This method of blocking the repellant conclusion is quite natural and ts in nicely within 17 In the next chapter I return to the use of homogeneity on p. 247.