The Gini Coefficient: An Application to Greece

Transcription

1 International Journal of Economics and Finance; Vol. 10, No. 3; 2018 ISSN X E-ISSN Published by Canadian Center of Science and Education The Gini Coefficient: An Alication to Greece Augustine C. Arize 1, Paraskevas Bakarezos 2, Ioannis N. Kallianiotis 3, John Malindretos 4 & John Phelan 5 1 Deartment of Economics and Finance, College of Business, Texas A&M University, USA 2 Agricultural Bank of Greece, Reublic of Greece 3 Deartment of Economics and Finance, Arthur Kania School of Management, University of Scranton, USA 4 Deartment of Economics, Finance and Global Business, Cotsakos College of Business, William Paterson University, Wayne, New Jersey, USA 5 Deartment of Economics, College of Business, University of New Haven, USA Corresondence: John Malindretos, Deartment of Economics, Finance and Global Business, Cotsakos College of Business, William Paterson University, Wayne, New Jersey, USA. MALINDRETOSJ@wunj.edu Received: August 15, 2017 Acceted: December 9, 2017 Online Published: February 28, 2018 doi: /ijef.v10n3205 URL: htts://doi.org/ /ijef.v10n3205 Abstract The Gini coefficient is a measure of income inequality. In this study we show that it needs to be adjusted to be a correct measure of income inequality. The result is that decomosition is ossible even without the interaction effect. The requirement however, is that there are data on individual incomes. Secondly, the aroach is alied to Greece. Third, there is the last section indicating extensions. Keywords: Decomosition, Gini coefficient, overlaing comonent of Gini coefficient, Greece 1. Introduction The measurement of ersonal income inequality has been for long a matter of interest in economics. There exist several measures of inequality and Stark (1972) gives a comrehensive list. The most widely used measure of inequality is robably the Gini coefficient. The investigation of its various characteristics on both the theoretical and the emirical level has also been long with major contributions by Atkinson (1970) and Sen (1973). An interesting develoment occurred when Bhattacharya and Mahalanobis (1967) and later Rao (1969) introduced the decomosition aroach. According to it the oulation of income-receiving units is divided into two grous and consequently inequality is distributed to disarities in income between grous. Pyatt (1976) analyzed the same disaggregation in matrix form and his method corresonds directly to that used here. The urose of this aer is to resent the rocess of decomosition and its relation to the Gini coefficient with emhasis on the exlanation of the nature of the overlaing comonent (or interaction effect) which arises during decomosition and it is usually treated as an awkward by-roduct of this rocess. It is shown that exact decomosition without the resence of the overlaing effect is ossible even when grou income distributions are overlaing under condition that data on individual incomes are available. The roblem is urely mathematical and it arises when decomosition is accomanied by relacement individual income comarisons with differences in mean grou income between airs of grous. In this case, the overlaing comonent can be comletely searated from the other two, the within and between inequality comonents. Furthermore, the case of groued data is considered that does not allow exact searation of the comonent. Exact decomosition without the resence of the overlaing effect, in the case of exact data, is the subject matter of the first section. The next section deals with the same case of exact data, the condition under which the overlaing comonent arises and its mathematical isolation and estimation. The third section considers the more realistic case of non-availability of exact data that makes exact searation of the overlaing comonent imossible and a combination of estimating techniques is suggested to aroximate it. The forth section consists of an illustration of all these results that are alied to data on income in Greece The alication is methodologically similar is not to analyze the comonents and the trend of inequality. The fifth section outlines briefly the necessary extensions of this analysis. The sixth section is a short summary with a resentation of the conclusions of the aer. In conclusion, this aer was develoed and written in the hoe of contributing to further understanding of the 205

2 rocess and nature of decomosition thus facilitating analysis of income inequality situation. 2. Exact Decomosition of the Gini Coefficient The most suitable formulation of the Gini coefficient for the uroses of the aer is (Note 1): Where is total number of income-receiving units, X d is income from the d th unit, d,f = 1, and Xk is average income. P G = d=1 f=1 X d X f 2 2 Xk (1) This formulation is therefore based on inter-unit income comarisons and it requires data on individual incomes, called here exact data as oosed to groued data (Note 2). The numerator in (1) can be considered as a norm of the matrix of absolute differences A of dimension (*) (Note 3). k It is ossible, however, to consider instead of number of units k number of grous of oulations 1, i=1 1 =. In this case, A can be artitioned into k 2 sub-matrices A i,j, i,j= 1 k. The sub-matrices along the diagonal of A are symmetric and of dimension ( i * i ), i= 1 k. The rest of the sub-matrices are of dimension ( i * i ), i,j= 1 k. i j, and deending on the division into grous, none, some, or all can be non-square. All this means that A can be decomosed into k 2 sub-matrices each having the following norm (Note 4): A ij = i r=1 = j X ir X jw Where i,j = 1 k w=1 (2) k k i=1 = j=1 A i,j = d=1 = f=1 X d X f The result of this rocedure is the decomosition of A which can be interreted as follows : The sub-matrices A i,j, i=j, contain as elements comarisons between airs of incomes within each grou. The rest of the sub-matrices A i,j, i j, consist of elements which are the comarisons of airs of income between units of i th and j th grous. The decomosition of A can be used to obtain exressions analogous to (1). These, for the sub-matrices A i,j, i=j, can be interreted as within grou Gini coefficients: G ij = A i,j X 1 kkkk, i = j (3) For the sub-matrices A i,j, i j, the following exressions can be defined which are interreted as between grou Gini coefficients measuring inequality between airs of income grous (Note 5): kkk+ Xkkk j G ij = A i,j 2 1 j X 1j kkkkk, i j (4) Where Xkkkk ij = X i 2 Using the results so far, matrix G can be defined with elements G ij from exression (3) and (4) and of dimension (k*k). Weighting these G ij by grou income and oulation roortions results in an alternative additively decomosable exression for G in (1) as follows: G = π Gm (5) Where π is a row vector (1*k) of aggregate grou income roortions (Note 6) and m is a column vector (k*1) of grou oulation roortions. Exression (5) is an exact and symmetric decomosition of the Gini coefficient (Note 7, Note 8). 3. Decomosition and Overlaing (Note 9) The result of the revious section is an exosition of the well-known fact that exact decomosition of the Gini coefficient is ossible when exact data are available, and it is included to reserve continuity of the discussion. Exact decomosability, however, is not usually encountered in ractice or it might be comutationally awkward even when it does. It would be of advantage to relace as many A ij in (2) with a summary measure. One ossible relacement is to exress A ij, i j in terms of differences in grou mean incomes, i.e. A ij = i r=1 = j w=1 X ir X jw = i j Xkkk 1 Xl j + R, i j (6) Where R is a remainder and the exlanation of its nature is the subject matter of this section. 206

3 The investigation of the relationshi between the left hand (LHS) and right hand (RHS) side exressions in (6) requires again analysis based on exact data as in the last section but before taking absolute values. The ordering of individual incomes in ascending (or descending) order according to size of income results in skew symmetric sub-matrices A ij, i=j, that is elements above the diagonal are the negatives of the elements below it. The elements of these diagonal sub-matrices are not affected by the RHS relacement in (6). The elements of the sub-matrices A ij, i j, are also laced symmetrically along the diagonal of the artitioned A. The difference between A ij, i=j, and A ij, i j, is that the elements of the latter might or might not be neatly searated with resect to their signs and that deends on overlaing between airs of grou income distributions. In the case of non-overlaing distributions each A ij, i j, above the diagonal consists of elements that are all negative, while each A ij, i j, below it (from now on denoted by A ij ) consists of the same elements laced symmetrically but all ositive (Note 10). When the distribution are overlaing A ij and A ij, i j, contain both ositive and negative elements not at random but in the following order: The air-wise comarisons in A ij between members of grou i and members of grou j for which X ir < X jw, come out negative while the rest of the comarison turn out ositive since X ir > X jw (and this is the result of overlaing).the corresonding comarisons in airs in A ij yield the same elements but of oosite sign (Note 11). It is the existence of elements of both sign within each A ij, i j, that creates the overlaing effect when LHS exression in (6) is relaced by the summary measure on the RHS combined with the existence of absolute values. The addition during the comutation of the double sum on the LHS of (6) is erformed after talking absolute values and that means that the outcome of each comarison (or element of A ij ) counts ositively regardless of its sign. However, the RHS is comuted differently since calculation of Xkkk 1 and Xl j requires addition before taking absolute values and differences of oosite signs off-set against each other (Note 12). In other words, the between grou Gini coefficient can be comuted using as numerator in (3) either the LHS or the first term in the RHS in (6). Since these two exressions are always equal, because they are not comutationally equivalent, it is necessary to make them equal under all conditions. This can be done in 2 ways: firstly, the addition on the LHS exression can be erformed before talking absolute signs thus allowing offsetting due to oosite signs on the LHS also and in this case R need not be include in the RHS (Note 13). Secondly, the order of oerations during comutation on the double sum in (6) remains unchanged while the RHS exression in (6) is increased by the amount R = 2 L u, u = 1 n ij, n ji being the number of differences of oosite sign in A ij and i = 1 k. This amount is, therefore, comuted as the absolute value of the sum of all differences of oosite sign (Note 14) multilied by two (Note 15) thus making all the differences on the RHS exression in (6) of the same sign (negative in our case). This can become clear by focusing temorarily on one comarison only, X ir X jw = L, X ir > X jw, by adding to L the quantity of -2L. Performing this for all the n ij u=1 differences within each A ij and searating the sum 2 L u, we allow addition to be erformed in the RHS exression in (6) before taking absolute values, while at the same time, we comensate for offsetting, since that is the only way to estimate Xkkk, 1 Xl j (Note 16). In the case of non-overlaing distributions L=0. The first way of making the two exressions in (6) equal by taking absolute values after erforming the addition (the equivalent of subtracting the above quantity from the LHS exression) is not correct taking exression (3) into consideration since it underestimates the Gini coefficient by losing the quantity from off-setting due to ositive and negative elements. The second way, of adding searately the same quantity to the RHS exression in (6) is in accord with exression (3) since no difference is lost only they are searated. Consideration of all A ij in A n ij u=1 results in the following decomosition of the Gini coefficient: G = π [B + X + V]m (7) Where B is a (k*k) diagonal matrix, each diagonal element being a grou Gini coefficient calculated according to exression (3). X is a (k*k) symmetric matrix with all diagonal elements equal to zero and all other elements equal to Gini coefficients measuring inequality between all airs of the k number of grous due to differences in mean grou incomes its calculation based on the RHS of exression (6). That is: X ij = X li Xl j (Xl i Xl (8) j ) 207

4 Where V is again a symmetric matrix with all diagonal elements equal to zero while all other elements are Gini coefficients their comutation based on overlaing elements as follows: R = V ij = 2 n ij u=1 L u 2 i j Xkkkk ij (9) These coefficients have no secial meaning but they are used to make exression (7) equal to (5). In the case of non-overlaing distributions V contains zeros as its elements (Note 17). 4. Decomosition and Groued Data It was shown in the revious section that the Gini coefficient can be neatly decomosed into three comonents as given in (7). However, a number of ractical roblems may arise with this analysis. The basic difficulty has to do with the fact that exact decomosition is feasible only when data on individual incomes are available. The first roblem arises even when data of this kind are available, and it relates to the dimension of matrix A and the difficulty of handling it mathematically even with the aid of comuters. One way to avoid this roblem, on the emirical level, is to consider average sub-grou incomes in the lace of individual incomes. In this case the dimension of A is drastically reduced. This formulation creates another well-known roblem however, since it underestimates the Gini coefficient. The reason for this is twofold: Firstly, it relaces sub-matrices along the diagonal of A with zeros thus destroying a number of differences and therefore making the numerator in (1) smaller than it would be in case of individual differences for the same data. Secondly, it further underestimates the numerator in (1) since it relaces blocks of elements in each off-diagonal sub-matrices. A ij by their means thus creating an overlaing effect if the income distributions of the sub-grous, which are now being relaced by means multilied by sub-grou oulations, are overlaing. The second roblem relates to the search of methods to imrove estimation in the case of groued data. Although this is not the objective of this aer, an attemt is made to develo a method with this roblem in mind. (See next section for its alication). Kakwani and Podder (1973) suggest an efficient method that can be used to estimate Gini coefficients. (The elements of B of the revious section in our case). Then the elements of X can be estimated using mean grou incomes. Following this the Gini coefficient for the total oulation (denoted by RG) can be comuted indeendently using the method suggested by Kakwani and Podder. Then another aroximate Gini coefficient for the total oulation can be calculated using a modified form of (7) that does not include V, that is: MG = π [B + X]m (10) Subtracting MG from RG we obtain an estimate of the total overlaing comonent as a residual.the final ste is to allocate the overlaing comonent among elements of V in (7). The straightforward way to do this is to comute overlaing elements using mean sub-grou incomes in the lace of individual incomes and then normalize taking as base the overlaing comonent comuted as a residual (Note 18). This method results in an exact decomosition of the Gini coefficient but at the same time it dums all other estimation inefficiencies onto the overlaing comonent since it is estimated as a residual. The ractical roblem is, therefore, to obtain some idea of magnitude of these inefficiencies relative to the magnitude of the overlaing comonent. A simle way is to comare the value of the comonent before and after normalization. This is done with the alication in the sections (See Fig. 3) (Note 19). 5. Decomosition of Inequality of Greece All the results of the revious sections were alied to a data base (Note 20) on incomes in Greece for the years The data allowed estimation of mean incomes and oulations of sub-grous. The sub-grous were allocated to three grous: The grou of ensioner including seventeen sub-grous, the grou of wage-salary earners with twenty-three sub-grous and finally the grou of entrereneurs containing thirty sub-grous. Taxes and transfer ayments are accounted for in the estimation of incomes. Since these are groued data, they are subject to all the roblems described in the last section. The decomosition of the Gini coefficient is done following the two methods outlined in the revious section of this aer (Note 21). The first method, of using sub-grou mean differences, multilied by the corresonding subgrou oulations, in the lace of individual incomes, is alied for two reasons: firstly, because it is exact and it serves the urose of verifying emirically all the results of this aer and secondly, because it can serve as an emirical comarison to the more efficient method described in the last section. Table 1 shows the alication of exressions (3) and (4) to the data. This is called the method of comarison between sub-grou means. Table 2 shows the results from the alication of exression (7) using again the method of comarison between sub-grou means. The lower case letters denote the weight Gini coefficient i.e. g11, g22, g33. Columns (1), (2), (3) are the elements of = [B] m in (7) where B is a (3*3) matrix with its diagonal elements equal to G11, G22, G33, columns (1), (2), (3) of table 1. Columns 208

5 (5), (6), (7) are again the elements of X in (7) weighted by grou oulation roortion and income shares, [X] m, while columns (9), (10), (11) show the weighted overlaing elements [V] m. Column (4), (8), (12) are the sums of column (1), (2), (3) resectively (the within inequality comonent), (5), (6), (7) (the between comonent), and (9), (10), (11) (the overlaing comonent). The sum of column (4), (8), (12), in column (13) equals the total (not decomosed) oulation Gini coefficient, denoted by G, that was calculated using exression (1). The same data are used to decomose the Gini coefficient following the rocedure suggested in the last art of section III. This method is called combined method of estimation, regression and comarison between subgrou means, and all the coefficients relating to it are labeled by the refix R. Table 3 is analogous to table 2 only the second method is used now. Column (1), (2), (3) show the weighted elements of the within comonent where the elements of B are calculated using the method suggested by Kakwani and Podder. It should be noted here that all regression coefficients used to calculate the elements of B are statistically significant. Column (5), (6), (7) include the same entries with the corresonding columns of table 2. Total oulation Gini coefficient, RG, column (14) is comuted indeendently using the method suggested by Kakwani and Podder. Columns (12), (13) show that normalized and estimated overlaing comonents corresondingly. It is interesting that none of the normalized overlaing elements, column (9), (10), (11) comes out negative and that fact could be considered to be an emirical verification of the discussion in section III. Figure 1 is a grahic exosition of how the three comonents from the decomosition vary among one another and also between each other according to the two methods of estimation. Figure 2 shows how the total oulation Gini coefficients, estimated by two methods, vary together. It is also observed emirically that the first method of estimation generally underestimates inequality (as suggested in section III), that is G < RG, excet for year 1968, and it is more volatile. Figure 3 shows how the estimated and normalized overlaing elements vary together. Table 1. Exact decomosition of the Gini coefficient into within and among grou Gini Coefficients for Greece (method of comarison between sub-grou means) year G11 (l) G22 (2) G33 (3) Gl2 (4) Gl3 (5) G23 (6) l962 l l l Note. G11, G22, G33 are calculated using exression (3) Gl2=G21, Gl3=G31. G23=G32 are calculated using exression All these constitute the elements of G in exression Calculation includes rimary data that are estimates or subject to revision l3 0.5 l ,

6 Table 2. Decomosition of the Gini coefficient into weighted comonents of within between and overlaing elements for Greece (method of comarison between sub-grou means) Year G11 (1) Within Elements G22 (2) G33 (3) Within Comonent g (4) Between Elements x12 (5) x13 (6) x23 (7) Between Comonents x (8) Overlaing Elements v12 (9) v13 (10) v23 (11) Overlaing Comonent v (12) Sum of Comonent G (13) Figure 1. Gini coefficient comonents 210

7 Figure 2. Total oulation Gini coefficients Figure 3. Normolized and estimated overlaing 6. Possible Extension (Note 22) The urose of this work is, as already stated, the exlanation of the nature of the overlaing comonent. This required in-deth analysis from the basic level which is the framework used by Bhattacharya and Mahalanobis (1967) as well as that used by Pyatt. The analysis, however, must be brought to higher levels to be in accord with its current status. To this end it can be extended in at least two resects. The first asect relates to generalization of the Gini indices as resented in Donaldson and Weymark (1980). They resent a class of relative inequality indices characterized by a single arameter and the class of social evaluation functions defined by this class. They also resent a corresonding class of absolute indices. The framework of our analysis, in the case of exact decomosition, seems to corresond to their absolute inequality 211

8 indices (it deends on income differentials). Accordingly, the introduction of between inequality, in the case of groued data, corresonds to their relative indices (only income shares are considered). It seems, therefore, necessary to investigate how the rocess of generalization is affected by decomosition. This refers to the additional restrictions that need to be laced on the generalization form of the index, in conjunction with the restrictions necessary to satisfy usual conditions an inequality index must satisfy as the rincile of oulation, transfer, and income homogeneity. The second asect relates to the analysis introduced by Shorrocks (1980). The whole analysis there derives the class of additively decomosable inequality measures under the condition that oulation sub-grous are disjoint. Since our work exlains the nature of the overlaing comonent it would be advantageous to investigate how the restriction imosed by additive decomosability on the form of the index and on satisfaction of the related rinciles are affected. The analysis in this resect could be further advanced along the lines of the analysis in Shorrocks (1984) to generalization to just decomosable Gini indices as suosed to only additively decomosable ones. It is obvious now that the final ste could be a synthesis of the two lines of analysis referred to with the finding of this work, i.e. the derivation of a generalized additively or just decomosable Gini index allowing division of the oulations considered into sub-grous in any ossible way and not just disjoint. 7. Summary and Conclusion A number of results have been derived in the revious sections of this aer, according to its urose to analyze the nature of the overlaing comonent that arises during the decomosition of the Gini coefficient. The first ste is to clarify the meaning of decomosition and this is done is section I. The conclusion there is that an exact decomosition of the Gini coefficient without the resence of overlaing is ossible in the case of exact (non-grou) data. The next ste is to investigate the condition under which the overlaing comonent arises as well as its nature. The conclusion in section II is: firstly, the roblem of overlaing aears when the between (grous) inequality is comuted using mean grou incomes in the lace of interersonal comarisons. Secondly, the overlaing comonent aears exactly because of this relacement and it constitutes a mathematical roblem since the order of oerations using the method of summing-u absolute values of interersonal comarisons is not comatible with the order of oeration during comutation of means. The two methods can be made comatible by comensating for summing-u after taking absolute values in the first case and vise-versa for the second and this comensation is the overlaing comonent. The consideration of certain roblems arising from the use of groued data is the urose of section III. The use of groued data makes estimation of the overlaing comonent non-exact and two methods of decomosition are suggested in the view of this. The first method treats sub-grou income means as individual incomes and it is introduced due to its exactness although it generally underestimates the Gini coefficient. The second method uses more efficient techniques and the overlaing comonent is treated as a normalized residual. All these results are alied to a data base on incomes in Greece for the eriod The results of this alication are resented in section IV. In conclusion the Gini coefficient might well be the most widely used measure of disarities among incomes and furthermore its decomosition is an interesting and significant ste towards better analyzing in equality in incomes. The roblem of the non-well defined overlaing comonent is a disadvantage of this ositive develoment and clarification of its nature strengthens the ower of analysis that decomosition offers. References Atkinson, A. B. (1970). On the Measurement of Inequality. Journal of Economic Theory, 2, htts://doi.org/ / (70) Bakarezos, P. (1984). Personal non-wealth Income Inequality and Tax Evasion in Greece Soudai, 34, Bhattacharya, N., & Mahalanobis, B. (1967). Regional Disarity in Household Consumtion in India. Journal of the American Statistical Assiciation, 62, htts://doi.org/ / Donaldson, D., & Weymark, J. A. (1980). A Single-Parameter Generalization of the Gini Indices of Inequality. Journal of Economic Theory, 22, htts://doi.org/ / (80) Kakwani, N. C., & Podder, N. (1973). On the Estimation of Lorenz Curves from Groued Observations. International Economic Review, 14, htts://doi.org/ / Kendall, M. G., & Stuart, A. (1963). The Advanced Theory of Statistics (Vol. I, 2nd ed.). London: Griffin. 212

9 Mookherjee, D., & Shorrocks, A. (1982). A Decomosition Analysis of the Trend in U.K. Income Inequality. Economic Journal, 92, htts://doi.org/ / Pyatt, G. (1976). On the Interretation and Disaggregation of Gini Coefficients. Economic Journal, 86, htts://doi.org/ / Rao, V. M. (1969). Two Decomositions of Concentration Retio. Journal of the Royal Statistical Society, Series A, 132. htts://doi.org/ / Sen, A. K. (1973). On Economic Inequality. Oxford: Claredon Press. htts://doi.org/ / Shorrocks, A. F. (1980). Inequality Decomosition by Poulation Subgrous. Econometrica, 52(6), htts://doi.org/ / Shorrocks, A. F. (1980). The Class of Additively Decomosable Inequality Measures. Econometrica, 48(3), htts://doi.org/ / Stark, T. (1972). The Distribution of Personal Income in the United Kingdon Cambridge University Press. Notes Note 1. See Kendall and Stuart (1963), Note 2. Groued data convey information in the form of mean incomes for grous and sub-grous of the total income-receiving oulation along with the size of the oulation of these grous. Exact data refers to the situation that individual incomes that can be attributed to individual income-receiving units exist. Note 3. The elements of the first row of A consist of the absolute differences from the air-wise comarisons between income of the first unit and all units, including own income and so on, for the second and subsequent rows u to the th row. If the units are arranged in ascending (or descending) order according to size of income, A has zeros as diagonal elements and it is skewed symmetric. Note 4. A i,j (boldface) denotes a sub-matrix while A i,j its norm as defined in (2). Note 5. It is also ossible to use for the calculation of G ij and its symmetric counterart G ij, since A ij = A ji, i j, instead of Xkkkk, ij Xl i, and Xl j corresondingly but then G ij G ji although A ij = A ji. The end result, however, will be the same as in (5) irresective of whether Xl i and Xl j or Xkkkk ij are used. Only in the first case, it is ossible for some of the G ji, i j, to turn out greater than unity and that means that they cannot be interreted as Gini coefficients. Note 6. It should be noted that π m = 1 Note 7. In the context of this section exact decomosition allows additive recomosition of the Gini coefficient from the within and between comonents to the original value, i.e. exression (5) equal exression (1). In the next section this definition is exanded to include the overlaing comonent as well, i.e. exression (7) equals exression (1). Note 8. Pyatt decomoses the Gini coefficient in a non-symmetric way similar to that suggested in footnote 5, while his method is to relace interersonal comarisons with a statistical game. The end result is the same as in (5) only certain of his equivalent to between grou Gini coefficients are greater than unity. Note 9. Overlaing between two grou or sub-grou distributions is defined as follows: Given Xl i, Xl j, the highest observed income(s) in grou i is greater than the lowest income(s) observed in grou j. Two distributions are non-overlaing when the highest observed income(s) in grou i is less than the lowest income(s) in grou j. Besides these two cases, it is also ossible to have the highest income(s) in grou i equal the lowest income(s) in grou j. Then the result, in terms of exact decomosition, is the same as in the case of non-overlaing distributions. The final consideration refers to the extreme case which occurs when all individual incomes in grou i are equal and also equal to all incomes in grou j (see footnote 17). Note 10. The elements of each A ij can be interreted as the result of comarisons made by members in the i th grou looking at income of members in the j th grou, Xl i < Xl j. The elements of the corresonding A ij are the differences of the oosite comarison. Note 11. The satiation described here sheds light on the relationshi between the statistical game suggested by Pyatt and each air of A ij, A ji. He seems to consider only ositive elements of A ij and only ositive elements of A ji, 213

10 since it is ossible to work only with the elements of A above or below the diagonal correcting exression (1) by eliminating division by two. The same result can be obtained by considering the oosite statistical game (elements with negative signs would be considered). In the case of non-overlaing distributions the game results in A ij with all elements equal to zero and all ositive elements in A ji. In the case of the oosite statistical game situation reverses itself. Note 12. Comutation if the grou income means relates to calculation of the double sum in the way shown in (6). Note 13. That is: i r=1 = j w=1 (X ir X jw ) = i j X kkk 1 Xl j Note 14. In our case (of ascending ordering in incomes) differences termed as of oosite sign are ositive for each A ij and negative for each A ji. This definition would be reversed in the case of descending ordering. Note 15. In our case of ascending ordering of incomes the roblem is to have: i r=1 = j w=1 X ir X jw = i j X kkk 1 Xl j = j i r=1 X ir i j X jw w=1 This can be done by adding to RHS exression the quantity R = 2 L u u=1, where X ir X jw = L, X ir > E jw considering elements of A ij (that is a difference of oosite sign), and n ij is the number of differences of oosite signs in A ij (ositive in this case). In other words, absolute signs are taken after addition is erformed in X ir and X jw since that is the only order of oerations that allows estimation of kkk, X 1 Xl j. Note 16. It is the case, when we have to use mean grou income and not individual incomes, for one reason or another, that creates the whole roblem. If we were free to use individual incomes, we would use decomosition as given in (5). Note 17. When the distributions are overlaing X ij in exression (8) will be zero when Xkkk 1 = Xl j, i j i.e. within inequality is zero since grou income means are equal. In this case, this LHS if exression (6) equals R (if we allowed addition in the LHS to be erformed before taking signs then inclusion of R is not receive the same amount of income. Note 18. There is no warranty that the estimated value of the overlaing element will be ositive. Nevertheless the rocedure of normalization can still be carried out and negative overlaing elements have the same meaning as ositive ones, that is they correct the elements of X due to consideration of mean instead of individual incomes. Note 19. The main result of this section which should be emhasized is that it is not ossible to decomose the coefficient exactly in the case of groued data and the best that can be done is to use some kind of aroximation as done here. This roblem is not a consequence of the rocess of decomosition but of the use of groued data which affects estimation of the coefficient even in the form of exression (1). On the other hand, it is clear that there exists some relation, albeit aroximate, between the cases of exact and groued data since the elements of B in (7) are calculated using some efficient method (the one suggested by Kakwani and Podder in this case) and the elements of X in (7) are the same during aroximation, when kkk X 1 Xl j,. Note 20. This data base was originally comlied by author from a wide variety of sources for the year , (1984). The base was re-calculated using an even wider variety of sources an exanded u to 1988 for the uroses of this study. Note 21. A Personal Comuter was used for the calculations. No comuter rograms for the required work existed so they were develoed by the author using LOTUS 123 and its macro commands (LOTUS 123 is a trademark of Lotus Develoment Cororation). Note 22. This is the content of my current research. n ij Coyrights Coyright for this article is retained by the author(s), with first ublication rights granted to the journal. This is an oen-access article distributed under the terms and conditions of the Creative Commons Attribution license (htt://creativecommons.org/licenses/by/4.0/). 214