A Consistent Decomposition of the Redistributive Effect. An application to Taxes and Welfare Expenditures.

Transcription

1 A Consistent Decoposition of the Redistributive Effect. An application to Taxes and Welfare Expenditures. Abstract The ai of this work is to solve the proble of non-additivity revealed by the works that calculate the redistributive effect of the budget or public policies ade up of different instruents of incoe or public spending. We use Shapley s value to do this. This technique allows us to decopose the redistributive effect and the vertical and horizontal effects consistently and not arbitrarily. The ain result obtained for the case of taxes and social transfers in the US is that previous calculations undervalued the redistributive effects and their vertical and horizontal coponents for taxes and transfers. Undervaluation is ore iportant for taxes.

2 . Introduction Within public econoy studies there is an iportant tradition in the analysis of the redistributive effects of taxes. One of the ost used indicators to calculate redistributive effects is the difference in Gini indexes before and after taxes (Reynolds y Solensky, 977). Its versatility and siplicity has eant the extending of the easure to all kinds of public intervention instruents and even to the analysis of redistributive effect of the whole budget. An additional refineent is in decoposing the Reynolds- Solensky index to assess the effects on vertical equity and reranking, according to the ethod of Kakwani (984), or also on horizontal inequality, according to the proposal of Aronson, Johnson and Labert (994). The proble of works that calculate the redistributive ipacts of policies ade up of various instruents or the whole budget is that they tend to present inconsistent results. Additive inconsistence is produced when the su of the redistributive effects of each instruent does not coincide with the redistributive effect of the easures taken as a whole. This non-additivity is due to calculating the redistributive effects of each instruent only taking as a reference the original incoe and excluding the other instruents. That is, the interaction between instruents having an effect on the redistributive effect is not taken into account. The ost eleental solution to take this interaction into account is to sequentially aggregate the different instruents to calculate their redistributive ipacts. This was done by: Ferrarini and Nelson (003) in a copared study of the financing and spending of the social insurance in ten countries, Keselan and Cheung (004) to analyze the budget in Canada, Mahler and Jesuit (006) for the budget in developed countries via the Luxebourg Incoe Study (LIS) data set, or Wolff and Zacharias (007) for the budget in the US. The proble that this solution produces is that the redistributive effects calculated for the sae policy are different according to the sequence chosen. That is, a proble of arbitrariness and udgent values arises. The ai that we set out for this work is to solve the proble of inconsistency of the decoposition of the redistributive effects without incurring in arbitrariness. To achieve this we propose additively decoposing the Reynolds-Solensky index, and its disaggregation into vertical and horizontal effects in the anner of Kakwani (984) or Aronson, Johnson and Labert (994), applying the Shapley s value (953). In broad ters, this technique considers the arginal effect on budget of eliinating each of the contributory factors in sequence, and then assigns to each factor the average of its arginal contributions in all possible eliination sequences. Shorrocks (999), instigator of its use for the decoposing of inequality indicators, 3 shows that the su of these effects is consistent and syetrical. For spending policies in the US copared with other countries we can see Mahler and Jesuit (006), Marical, et. al (008), Tanzi (008), Warren (008) or Prasad (008). Aong the ost recent we find those of Atkitson (004), Seeding (005), Förster and Mira d Ercole (005) or Garfinkel, Rainwater and Seeding (006) in which the redistributive effects of the US Adinistration budget are copared with the results obtained in other countries. For other countries see Forteza and Rossi (009) for Uruguay, Bargain and Callan (00) for the EU, Clark and Leicester (004) for the UK or Fuchs and Lietz (007) for Austria. 3 Previous Works belong to Rongve (995) and Chantreuil and Trannoy (997). This ethodology has been eployed ore profusely in the decoposing of poverty indicators, see as exaples Kolenikov and Shorrocks (005), Sai and Duclos (008) or Deustsch and Silber (008).

3 To present analytical results we are going to take as a reference the recent work of Ki and Labert (009) in which the redistributive ipact is analyzed for taxes and social spending in the USA for the years 994, 999 and Our calculations will allow the consistent and syetrical decoposing of the total redistributive effect, the vertical effect, the horizontal inequality and the reranking that taxes and social benefits produce in the US and copare the results with those of the article cited. Fro this coparison we will be able to draw conclusions about the consequences of assessing redistributive effects that do not produce consistent results. This paper is organized as follows. The next section discuss the probles of traditional ethodologies of redistributive decoposition and the relative erits of the Shapley decoposition. In section 3 we apply our proposal. The last section suarizes the results obtained and offers soe concluding rearks.. Methodology One of the ost usual ways of easuring the redistributive effect is the coparison of the Gini index of the initial incoe (arket incoe) with the Gini index of the final incoe (post-tax incoe) (Reynolds and Solensky 977): = G G () X At the sae tie, the ost used ways of calculating the vertical and horizontal redistributive effects of the taxes are those devised by Kakwani (984) and Aronson, Johnson and Labert (994). 5 The first distinguishes between the vertical effect and reranking: = V RX T () And the second also identifies the horizontal inequality without an order change: 6 = V H RX T (3) 4 Other recent works that are concerned with the redistributive effect of this type of policies are those of Verbist (007) for taxes on work, pensions and uneployent benefits in the EU-5, Ferrario and Zanardi (009) for the National Health of Italy or Biggs et al (009) for the National Health in the USA. There is also a broad literature of health spending fro the work of Wagstaff and van Doorslaer (997)for the health spending and financing in Holland, which later Van Doorslaer et. al (998) extended to an iportant group of countries. 5 There is another decoposing alternative put forward by Leran and Yitzaki (995), in which the order change is assessed first, and then V is resolved as the incoe change, but with the final order. 6 In spite of the terinology being undistinguishable, it ust be pointed out that the original values of V and R are not coincident in both forulas. The work of Urban and Labert (008) recently put forward a reforulating of the values given by Aronson, Johnson and Labert that identifies the relationship between both equations.

4 The vertical effect is defined by the difference between pre-tax distribution inequality index and the inequality index of a fictitious incoe distribution obtained with the aid of taxation not generating either horizontal effects or reranking effects. This fictitious taxation leads to the pre-tax incoe of a household of rank in the distribution of pre-tax incoe being equal to its expected net incoe before taxation. Furtherore, the effect (V) easures the redistributive effect generated by the taxation syste. This difference is positive if the taxation syste is progressive and negative if it is regressive. The horizontal inequality (H), on the other hand, relates to unequal treatent of equals. A further concept of reranking (R) corresponds to the incoe scale rank-switching induced by the fiscal syste. 7 In theory, for each of the instruents a,(=,,) that ake up a policy M we can calculate its respective redistributive effects X +a and the proble of inconsistency or M asyetry is given because X +M X +a J. The sae takes place for V, H and R. J = As was cited above, the ost eleental solution to the proble is to establish a sequence. To see the proble that it produces, we take the exaple of the work of Wolff and Zacharias (007), in which the effects of taxes and spending for the years 989 and 000 in U.S. are analyzed. The authors first calculate the redistributive effects for the taxes (T), then for the transfers (Tr) and finally for public consuption (E) and the results fulfil the additive property 8. Now, why choose this sequence and not another (T+E+Tr, E+T+Tr, E+Tr+T, Tr+E+T o Tr+T+E)? If we did this, we would have obtained different redistributive ipacts for each instruent according to the sequence used. That is, the order influences the result. To su up, if we change the order, the redistributive effect values change, producing asyetrical results. If this is so, what value do we take? Why ust taxes precede public spending? It would not see reasonable to us to consider that the adinistration first receives payents and then spends. Then, this sequence does not have the guarantee of certainty. Spending can be financed fro the public deficit and in this case the spending is first and the taxes coe afterwards. What happens when taxes fall back on transfers received? The reality is that odern econoies are coplex and it is not possible to find unquestionable sequences in the budget procedure. This is why to choose a sequence iplies arbitrariness and the establishing of a value udgent that conditions the results. We can find the proble s solution calculating the decoposition via the value of Shapley (953). This ethodology coes fro gae theory and has been applied in the study of inequality since the work of Shorrocks (999). 9 The applying of Shapley s value allows an additive, syetrical and exact decoposition for any index. 7 See decopostion of taxes in Creedy and Van Ven (00), for Australia, Urban (008) for Croacia, Labert and Thoresen, T (009) for Norway. 8 This can be checked in their Table 6 (Wolff and Zacharias, 007, p.70). 9 See Chantreuil and Trannoy (999); Sastre and Trannoy (00), Deustch and Silber (008).

5 The syste consists of quantifying the ipact of a factor calculating the average of the arginal effects, deterined by the eliination of the said factor in all the possible sequences. Indeed, let M be the set of instruents a, M = {,,... }, that ake up a public policy. Let G be a subset of M ade up of g instruents. Let X + G be the value of when the instruents a, G M have been eliinated. If all the sizes are equally probable, a deterined size g will occur with a probability of / g. On the other hand, if we have a deterined instruent a that belongs to a subgroup G, the ( g ) instruents reaining of this subgroup can be chosen between the ( ) instruents reaining in a nuber of sequences equal to: ( )! [( ) (g ) ]!(g )! = ( )! (4) ( g)!(g )! The probability of one of these cobinations being chosen is the reciprocal. Therefore, the probability of a group G containing the eleent a is equal to the said reciprocal ultiplied by /. If we take into account the arginal contribution of a G to the inequality of the subgroup G it is: X +G = X +G X +(G a ) (5) The contribution of the instruent a to the total value of the index is equal to the suation of all the groupings that can contain it, pondered by the probability of the occurring: X +M = Such that: ( g)!(g )!! X +G X +(G a ) G M G X +M = X +M (7) = [ ] (6) This expression is generalizable to deterine the contribution of each V, X M H + and R such that: a to the indices V H R = = = = = = V H V (8) Therefore:

6 = X + M = VX + M H RX + M = V H = = = = With this ethodology the sequence ceases to be deterinant. All the possible cobinations are considered and the interaction of the instruents that ake up the policy is taken into account. To appreciate the consequences of its application, we are going to use the case of a policy with two instruents, for exaple taxes (T) and transfers (B). Such that for equation 7 ust be et: T = Being: T B =.( = ( B + X ) ) + ) + X (0) Taking these results, we can construct Table, in which are included the solution obtained via applying Shapley s value and the two possible sequences. These latter produce consistent results but they are not syetrical. () R (9) Table. Decoposition of the redistributive effect of a public policy ade up of two instruents, taxes and transfers. Taxes Transfers Total Sequence (-T+B) X T (I) (II) X T (VII) Source: own elaboration. Sequence (B-T) X (III) X (IV) X T (VII) Shapley Solution (V) T X T+ B = ( X T X ) ) + X T B X T+ B = ( X T X T ) + X (VI) X T (VII) The proble of inconsistency that tends to appear in works of this kind, such as happens in Ki and Labert (009), originates in calculating the total effect, VII and the effects I and IV. The three effects are estiated as the arginal effect on the initial incoe, but in VII the interaction of both instruents is taken into account, while in I and IV it is not. This inconsistency is also produced in the case of our calculating the partial effects III and II (calculating the redistributive effect of each policy, eliinating

7 it in the final incoe instead of aggregating it to the initial incoe), as can be observed in Inervoll et. al (006). For their part, in the works that adopt the sequential solution, as happens in the work of Wolff and Zacharias (007), sequence is opted for (values I and II), which fulfils the additive property, but incurs in arbitrariness. Sequence could be taken and the result would also be consistent, but with different values (values III and IV). 0 Shapley s solution consists of calculating the average of the arginal effects of each instruent in each sequence. It fulfills the properties of additivity and syetry (V+VI=VII) and takes into account any interaction between instruents. In the sae way in which we proceed with, the vertical, horizontal and reordering effects V, H and R can be decoposed. 3. Data and Results As was pointed out in the introduction, to contrast the efficiency of the ethodology proposed and its consequences, we eploy the data of the work of Ki and Labert (009), which is the ost recent study for the United States, and we will analyze the consequences that derive fro the new estiation. In this work, data of personal taxes and transfers for social benefits are taken in the United States for the years 994, 999 and 004 proceeding fro the U.S. Current Population Survey (CPS). The different indexes that we use are coputed with the relation specified in equation (3). In Table the data offered by the authors is reproduced and the new results with value s Shapley for the three years considered is included. As can be observed, taxes and transfers reduce the incoe inequality of the arket around 30%. The greatest inequality after taxes and transfers is in 004, as a consequence of the arket incoe being divided ore unequally. Taxes have their greatest redistributive effect and their greatest relative weight, close to 30%, in 999. Later they lose iportance in favour of transfers which are responsible for 76.9% of the redistribution in 004. Likewise, taxes and transfers lose their capacity to correct the vertical inequality of the 999 arket incoe. Nevertheless, the increase of the vertical equality of the transfers eans that in 004 taxes and transfers together generate the greatest horizontal inequality correction. In coparative ters we can see that the redistributive effects of taxes and transfers separately are greater than those previously calculated and explain the whole redistributive effect. To explain this, we take taxes as an exaple and with Shapley value we calculate the average of the two redistributive effects on the original incoe and on the incoe after social transfers (I and III of Table ). When taxes are applied after the social transfers, these have already reduced the inequality and consequently with the sae quantity of resources a greater equality can be attained. The ost relevant effect for the taxes is that their redistributive effect is broader: they explain between 0 As was pointed out above, in the work of Wolff and Zacharias the arginal effect on the initial incoe of the transfers is also calculated, which is why it can be seen that II and IV do not coincide. As we did not have at our disposal the interediate of the Gini indices of the different arrangeent sequences, we will liit ourselves to disaggregating the redistributive effect of transfers and taxes, without continuing the disaggregation of the into the different subeleents that they ake up.

8 9.7% (999) and 3.7% (004) of the respective total effects. The new calculations also show a greater weight in the vertical effects of the taxes. This situation is coherent with its greater total redistributive effect. It is also observed that, as Ki and Labert pointed out, there is a certain stability of equality loss due to reranking, between 30 and 35% of the redistributive effect. Now, with the new calculations we also oreover identify the contribution of each instruent to the reranking. Thus, in 994, 76% of the reranking was produced by transfers and 4% by taxes. In 004 the taxes hardly contributed 7.5% of the reranking, which eans that the transfers have a ore unequal distribution in ters of horizontal equality. Finally we have to highlight, also in the tax abit, the sign change of the H values. In the original calculations the taxes produced inequality as a consequence of unequal treatent of equals. The new calculations point out that taxes and transfers follow a siilar pattern. Inequality due to horizontal inequality is not produced, but rather this operates in favour of redistribution. Table. Redistributive Effect of taxes and transfers. Non -additive decoposition vs Shapley decoposition 994 Taxes (T) Non additive decoposition* Absolute value In percentage of Absolute value Shapley decoposition In percentage of In percentage of effect of taxes and public transfers 0,033 0, ,8% V K 0,0396 9,60% 0, ,4% 5,65% V 0,0396 9,64% 0, ,96% 5,73% H 0,0000 0,0% -0,0005-0,8% 0,65% R K 0,0065 9,6% 0,0685 3,4% 3,83% Public Transfers (B) 0,0638 0, ,7% V K 0,4075 0, ,35% V 0,399 3,5% 0, ,78% 74,7% H -0, ,80% -0, ,85% 89,35% R K 0, ,3% 0, ,6% 76,7% Total (N=T-B) 0,5445 0, ,00% V L 0,0769 0, ,00% T by V L decoposition 0,036 0,05375 B by V L decoposition 0,7643 0,5445 V 0,066 34% 0,066 33,78% 00,00% H -0,0007-0,69% -0,0008-0,70% 00,00% R K 0, ,47% 0, ,47% 00,00%

9 Taxes (T) Non additive decoposition* Absolute value In percentage of 999 Absolute value Shapley decoposition In percentage of In percentage of effect of taxes and public transfers 0,0349 0,0496 9,70% V K 0, ,7% 0, ,3% 7,38% V 0, ,7% 0,0563 0,8% 7,43% H 0,0000 0,0% -0, ,% 9,6% R K 0, ,5% 0, ,3% 9,79% Public Transfers (B) 0, ,07 70,30% V K 0,385 0, ,6% V 0,34 3,76% 0, ,8% 7,57% H -0, ,46% -0, ,48% 90,74% R K 0,0308 3,% 0, ,77% 80,% Total (N=T-B) 0,4467 0, ,00% V L 0,8875 0, ,00% T by V L decoposition 0,0344 0,05685 B by V L decoposition 0,573 0,37065 V 0,88 30% 0,88 30,0% 00,00% H -0, ,38% -0, ,37% 00,00% R K 0, ,48% 0, ,48% 00,00% Taxes (T) Non additive decoposition* Absolute value In percentage of Absolute value 004 Shapley decoposition In percentage of In percentage of effect of taxes and public transfers 0,0954 0, ,7% V K 0,0333,49% 0, ,3%,08% V 0,0336,57% 0, ,05%,4% H 0,0000 0,08% -0, ,0% 8,4% R K 0,00369,49% 0, ,3% 7,49% Public Transfers (B) 0,307 0,9 76,9% V K 0,535 0,678 77,9% V 0,576 35,0% 0, ,78% 77,86% H -0, ,66% -0, ,69% 9,76% R K 0, ,76% 0, ,47% 8,5% Total (N=T-B) 0,5885 0, ,00% V L 0,535 0,535 00,00% T by V L decoposition 0,063 0,04754 B by V L decoposition 0,89 0,678 V 0,444 35% 0,444 35,00% 00,00% H -0,0009-0,57% -0,0009-0,57% 00,00% R K 0, ,57% 0, ,57% 00,00% * Data obtained fro Table 3 fro the work of Ki, Labert (009). Source:own elaboration.

10 The previous inforation can be copleted with that which appears in Table 3. This includes the relative variation in the calculations for 004. We appreciate that the data of Ki and Labert (009) undervalued all the indices, but in coparative ters the indicators of the redistributive effect of the taxes were ore affected than those of the transfers. The undervaluing of the redistributive effect of the taxes triples that of the transfers. What is ore, we can also see that the previous calculations undervalued ore the inequality associated with the horizontal inequality and the reranking than the vertical redistributive effect. To su up, for the years analyzed the result we obtain is that the redistributive effects of taxes and social transfers are greater than those calculated on the initial incoe. At the sae tie, the calculations allow us to conclude that the undervaluing of the redistributive effect is greater in the case of the taxes. According to our estiation, the relative weight of the taxes would represent around 5% of the total redistribution, which contrasts with % attributed by Ki and Labert for 004. That is, the weight of the taxes in the redistribution in the US for the years considered is notably greater when it is assessed taking into account the rest of the policy it is part of or with which it is associated. Table 3. Non additive decoposition vs Shapley decoposition. % differences. 004 Taxes Public Transfers (B) Non additive Percentage Non additive Shapley Percentage Shapley ethodology decoposition change decoposition ethodology change (I) (II) ((I-II)/II) (I) (II) ((I-II)/II) R E 0,0954 0, ,56% 0,307 0,9-6,70% V K 0,0333 0, ,0% 0,535 0,678-8,53% V 0,0336 0, ,93% 0,576 0,6697-8,5% H 0,0000-0, ,67% -0, , ,38% R K 0, , ,65% 0, ,0466-3,8% Non additive decoposition Taxes and Public Transfers (C) Shapley ethodology Percentage change Non additive decoposition A+B-C Shapley ethodology Percentage change (I) (II) ((I-II)/II) (I) (II) ((I-II)/II) R E 0,5885 0,5885 0,00% 0,307 0,9-6,70% V K 0,535 0,535 0,00% 0,535 0,678-8,53% V 0,444 0,444 0,00% 0,576 0,6697-8,5% H -0,0009-0,0009 0,00% -0, , ,38% R K 0,0565 0,0565 0,00% 0, ,0466-3,8% Source: own elaboration whit table. 4. Repercussions for redistributive ipact studies As we have been able to check, applying Shapley value to the Reynolds-Solensky index, and its decoposition according to Aronson, Johnson and Labert (994), we have anaged to deterine the contribution relative to the redistributive effect and its decoposition into vertical and horizontal ipacts of the instruents considered in Ki and Labert (009). We have consistently calculated the contribution of taxes and transfers to the total redistributive effect of the social well-being policies of the United

11 States for the years 994, 999 and 004 proceeding fro the U.S. Current Population Survey (CPS). That is to say, we have achieved syetrical results without arbitrarily introducing criteria, which has at least two repercussions for the redistributive assessent of public policies. Firstly, that to value a policy we ust take into account the proble of udgent values and their repercussion on the results. We are going to use Table 4 to show our arguent. In this table, we have reproduced table, but with the values calculated in Table for the redistributive effect of 004. In it one can clearly appreciate the possibility of arbitrary influence in the results. Thus, if taxes and transfers are evaluated on initial rent (alternative of the boxes I and IV), exactly as we have seen in Table 3, the effects of taxes and transfers are undervalued. On the contrary, if taxes and transfers are evaluated on incoe post-transfers or incoe post-taxes, respectively, (take the opposite alternative, that defined by boxes II and III) then what we do is to overvalue the effect of both. On the other hand, if we choose sequential sequence (initial incoe-taxes+transfers) the redistributive effect of the taxes is undervalued and that of the transfers is overvalued. Finally, if we choose sequence (initial incoe+transferstaxes), the opposite occurs: the overvaluing is produced in the value of the taxes. That is to say, the choice of one alternative or the other gives rise to a specific bias. That is, selecting a ethodology we can arbitrarily influence the results and these results have a greater or lesser redistributive ipact on one instruent or another or both. Faced with this situation, Shapley s ethodology also produces consistent results and reduces the technical possibility of arbitrary anipulation. Table 4. Additive decopositions. Redistributive Effect of Taxes and Transfers 004 Sequence (-T+B) Taxes (I) Transfers 0.93 (II) Total (VII) Source: own elaboration fro Table. Sequence (B-T) (III) (IV) (VII) Shapley Solution (V) 0.9 (VI) (VII) The second repercussion is that for policy refors we can distinguish two types of redistributive effects: a arginal one and a redistributive effect of the instruent. Indeed, let us suppose that we wish to assess the redistributive effect of a policy refor. If the refor consists of introducing a new instruent, we could calculate its redistributive effect according to boxes II or III. Now, the ipact calculated in this anner easures the arginal incidence of its incorporation, but not the redistributive effect attributable to the new instruent within the whole policy. To easure the latter The possibility of there being arbitrary anipulation does not,evidently, iply that this is practised.

12 that is, the redistributive effect of the instruent- the appropriate procedure is decoposition via Shapley s value. The difference is iportant, as the redistributive effect contains two effects: the arginal effect of the instruent itself and that which derives fro the alteration that is produced in the effects of the reaining instruents. Let us iagine, for exaple, that social transfers do not exist and that they are created. The arginal effect of the creating of the transfers in 004 would be 0.93 (box II) but the redistributive effect of the transfers, as coponent of one policy, according to Shapley s decoposition is 0.9 (box VI). The difference, 0.008, is the increase of the redistributive effect attributable to the taxes and is because its redistributive effect is reinforced when it is created together with the transfer instruent. Finally, there exists an additional positive repercussion for the studies of redistributive ipact. When we apply Shapley s additive decoposition to the redistributive effect and its vertical and horizontal coponents, we can copare the relative weight of the effect of each instruent in percentage ters. This possibility iproves the inforation of any copared study, not only the interteporal coparison as carried out in this work, but also the policy coparison or that of tax systes of different countries and the coparison of alternatives of the coposition of spending or incoe. References Aronson, R., Johnson, P. and Labert, P., Redistributive Effect and Unequal Incoe Tax Treatent, Econoic Journal, 04 (March), 6 70, 994. Atkitson, A., Increased Incoe Inequality in OECD Countries and the Redistributive Ipact of the Governent Budget, in Cornia, G.A. (eds), Inequality, growth, and poverty in an era of liberalization and globalization, Oxford University Press,Oxford, -48, 004. Bargain, O. and Callan, T., Analysing the effects of tax-benefit refors on incoe distribution, a decoposition approach, Journal of Econoic Inequality, 8(), -, 00. Biggs, A., Sarney, M. and Taborini, C., A Progressivity Index for Social Security, U.S., Social Security Adinistration, Issue Paper No, 009-0, January, -5, 009. Creedy, J., Van de Ven, J., Decoposing redistributive effects of taxes and transfers in Australia, annual and lifetie easures, Australian Econoic Papers, une, 85-98, 00. Chantreuil, F. and Trannoy, A., Inequality Decoposition Values, Mieo, Université de, Cergy-Pontoise Deceber, 997. Chantreuil, F. and Trannoy, A., Inequality decoposition values, the trade-off between arginality and consistency, Discussion Paper DP 994, Universite de Cergy-Pontoise, France, 999. Clark, T. and Leicester, A., Inequality and two decades of British tax and benefit refor, Fiscal Studies, 5 (), 9-58, 004. Deustch, J. and Silber, J., On the Shapley value and de decoposition of inequality population subgroups with special ephasis on the Gini index, in Betty, G, Lei,A,

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