In search of an international standard of inequality measurement Gini-Lorenz Conference, May 23-26, University of Siena

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1 In search of an international standard of inequality measurement Gini-Lorenz Conference, May 23-26, University of Siena Juan Vicente-Perdiz and Maria Teresa Rubio Sanz. and Valladolid University (Spain) ABSTRACT: The System of National Accounts (SNA) is the international standard for the compilation of economic statistics, the last major revision of which was undertaken in The aggregates of the SNA are composite values which measure the result of economic activity using a norm one, this is without considering where the flows come from or where they go to. The lack of the necessary micro data was probably the main cause of this omission in the past. However, nowadays the greatest obstacle is perhaps the lack of any consensus on the inequality measure to be used as an international standard. As Capéau and Decoster (2004) have recently pointed out, the debate about which inequality measure to use for assessing true world inequality has not yet begun. The new revision of the SNA, targeted for publication in 2008, provides a good opportunity to include some norm of aggregation that takes into account distributive issues. In order to contribute to the achievement of that goal this paper compares the Gini and Euclidian norms (and their related families) using three dimensional representations. After discussing some of the most relevant theoretical issues, we display the income within-country-distributions to explore the empirical relevance of using different inequality metrics. 1. The System of National Accounts and its review Inequality in the 1993 SNA and related guidelines The object (what) The subject (whom) The measure (how) Standard inequality measures Are the classical measures the reference measures? What about in practice The poorest-richest triangle (PRT) The Wider dataset Conclusion...16 References

2 1. The System of National Accounts and its review. The System of National Accounts (SNA) is the international statistical standard for the measurement of economic activity and its conceptual framework has been designed for economic analysis and policy-making purposes. The evolution of the SNA throughout the 1953, 1968 and 1993 versions reflects the compromise between the theoretical developments and the statistical limitations as well as emerging economic realities. The current version of the SNA was edited jointly by the main international organizations (UN, EU, IMF, OECD and WB, 1993), and in 1999 the Statistical Commission of the United Nations (UNSC) approved a procedure for updating it to continuous economic changes. After the incorporation of the first updates in an electronic version of the 1993 SNA and with an increasing list of issues under review, it was felt that an incremental mechanism did not ensure consistency of the System as a whole and it became an additional challenge for its implementation. In response to this dissatisfaction, in 2004 the UNSC approved a work program to review the SNA93 starting from the premise that its framework is robust enough not to require fundamental changes and therefore it restricts the updates to specific issues widely demanded and consistent with the whole System. The Inter Secretariat Working Group on National Accounts (IAWGNA) assisted by an Advisory Expert Group (AEG) at its first two meetings (February and December 2004) approved the list of conceptual issues for review and left the process only open to clarifying suggestions. During the next two meetings (November 2005 and May 2006) it is expected that the final decisions will be adopted and the consistency of the updates and the whole system evaluated. A final AEG meeting to be held in 2007 will take into account countries comments before adopting the draft of the SNA93 Rev. 1 to be submitted to the UNSC for approval. So far none of the proposals accepted or under consideration for review are related to distributive concerns. Such an omission is all the more remarkable given the increasing contrast between the lack of any reference to inequality throughout the SNA and the proliferation of economic models and political programs where inequality plays a major role. 2

3 2. Inequality in the 1993 SNA and related guidelines The 1968 SNA introduced some major extensions such as balance sheets and matrix presentation, but left the issue of household income distribution, as well as other important topics for future discussion and research. In 1977 the UNSC approved Provisional Guidelines on the distribution of income Consumption and Accumulation of Households (United Nations, 1977) and some work was undertaken to adapt them for consideration by the revision process from the 1968 SNA to 1993 SNA (Norrlof, 1985). However, although the household sector was one of the few topics with its own expert group (United Nations, 1989) and the 1993 SNA included some new features dealing with household accounts, the inequality issue was once again ignored. More recently under the auspices of the UNSC, several experts on household income statistics constituted in 1996 the Canberra group whose final report (Canberra, 2001) may be considered the current international guidelines on income distribution statistics (Eurostat 2003 and ILO 2003) The object (what) To the Camberra report (Canberra, 2001: 3, 24) household income is the most objective proxy of economic well-being for policy purposes and disposable income the preferred practical measure for international comparisons of income distribution, whereas other relevant aspects, such as consumption or changes and levels of wealth, still require further discussion and work. However, whatever the result of further research, it should be clear that no aggregate by itself, disposable income included, will able to summarize the economic status of a unit. In fact, more than to its relation with welfare the relevance of disposable income is due to its central position within the system. The SNA records economic flows in a closed and consistent circuit and disposable income is, on the one hand, the final balancing item of the production and distribution of income accounts and, on the other, the first resource of the use of income and accumulation accounts. Thus, it can be decomposed into its sources (Primary or Secondary income) or its uses (Consumption or Wealth accumulation) depending of the topic of interest P + S = Y = C + W ' (1) 3

4 2.2. The subject (whom) There is no question about the use of the household as the unit of analysis because not all of its members are entitled to own assets in its own right or are able to take decisions about and engage in economic activities, so the trouble is how to aggregate the households. The 1993 SNA (chapter IV, section F) distinguishes four household sub-sectors according to the nature of their largest source of income (employers, own-account workers, employees, and recipients of property and transfer incomes) and suggests many other criteria for household sub-sectoring including the size of total or per capita household income, yet concludes that individual countries are obliged to make their own decisions about what they considerer to be the most relevant classification for which international guidelines may not be helpful. In a later detailed analysis of the household sector the 1993 SNA (chapter XIX, sect. B) recognised that one could think of subdividing the household sector, in the context of national accounts, by income size and of trying to establish a full set of accounts for each income bracket, but at the same time it considered improbable that available data would permit it in practice. In this respect the Canberra report recommends (2001:32) gathering information at the lowest level of disaggregation possible and taking into account size in addition to other household characteristics by using equivalence scales, but without pinpointing any standard subsectoring of the household sector. The 1993 SNA introduced a separate chapter devoted to social accounting matrices, traditionally applied to the analysis on causes and consequences of inequality among household groups. A SAM is in fact merely the SNA in matrix format, allowing different classifications of households in each account (20.73) and choosing the most suitable for each topic of interest (19.5). Thus to deal with inequality issues the household sector may be disaggregated into the largest source of income in the primary (labour and capital) and secondary (transfers) income distribution accounts and by a functional classification in the current use of income and wealth accumulation accounts, both linked by common household subsectoring to income size l h... lh k c 1 1 t1 y k h 1 + t = = h y h c k 1 H th yh c h H w + wh wh l 1 1 (2) 4

5 2.2. The measure (how) There is no mention within the whole 1993 SNA (or previous versions) of any summary measure of inequality. The 1977 UNSC provisional Guidelines devoted a separate chapter to distribution statistics with a section of just five paragraphs to discuss summary measures, and the 2001 Canberra report devotes a chapter to data presentation with a ten page section on measuring income dispersion. The comparison between both texts roughly reflects the development of the topic and at the same time provides a good starting point in the search for useful recommendations to be considered by further SNA reviews. Concisely and almost literally: A ranking of households or individuals according to the size of a selected aggregate provides the data needed to calculate simple summary measures of dispersion or concentration such as the inter quartile range or more refined summary measures such as the coefficient of variation. The Lorenz curve is an effective graphic presentation obtained by plotting the cumulative s of income accruing to ordered fractile groups of income receivers and the Gini ratio is a measure of concentration (inequality) perceived visually as the normalized area between the line of equal distribution and the Lorenz curve. Other ways of presenting distributions are in general more complex and therefore less frequently used. (ONU, 1977: 59) A common approach is the use of distances between means, medians or quantile points of most representative quantile groups and the most basic presentation of income dispersion is the frequency diagram that ranks the units according to their income and it is at the foundation of most measures of income dispersion. The Lorenz curve is closely related. One distribution Lorenz-dominates (is less unequal than) others if it is unambiguously closer to the diagonal, but when two curves intersect some inequality index or summary measure is needed. One of the most widely used is the Gini coefficient that is fairly easy to interpret but quite sensitive to changes near the mean. Other frequently used measures are the Atkinson index which positional sensitivity depends on a parameter of inequality aversion, the Squared coefficient of Variation that is additively decomposable and Theil s entropy. In most cases the results will be unaffected by the chosen measure, but sometimes the messages are contradictory and therefore more than one measure should be presented. (Canberra, 200: ) 3. Standard inequality measures The international standards of the SNA are widely used by public organizations, economists, journalists or other analysts and influence popular and political judgements about the performance of economies and the success of programmes (paragraph 1.37). Since it is not only possible but also recommendable to present more than one measure, then it is necessary to state which inequality measures should be used for official reports. 5

6 3.1. Are the classical measures When a sample of, for instance, more than household incomes (INE, 1991) is merely displayed on a plane (figure 1) it is not possible to visualize inequality, which is why data are arranged. As there are no standard brackets to tabulate frequency distributions, for comparisons it is more practical to use the cumulative distribution function or its inverse, the quantile function: a parade of households arranged per income (which to save time may be accommodated in, for instance, ten coaches or deciles). Inequality may be perceived as the areas between the quantile function and the mean income and so measured by the relative mean deviation: M = yi 1 / n, where i = 1,, n are the household groups and y i is the equivalent income of the i group. Since M is not sensitive to transfers that do not cross the mean, alternative ways to aggregate the deviations may be used. The two most classical are the Coefficient of variation and the Gini index proposed as early as 1920 by Dalton, who also expressed certain preference for the latter because of its relation with the area between the diagonal and the Lorenz curve that plots the cumulative population and income s in a standard 1x1 square. 0 CV 2 2 n 1 = n yi n 1 (3) i= 1 n i n G n = yi i= 1 n / 2 n n, i y i + 1 y (4) Figure 1. Two dimensional presentations 1991Spanish FBS household incomes frecuencies by 300 intervals Quantile function Lorenz curve Mean poorer households poorest househols 6

7 3.2. the reference measures? A distribution of income s among three equivalent groups of households can be represented by a point within an equilateral triangle (Kolm, 1969, Blackorby and Donalson, 1977, Davies and Hoy, 1994), like the black point of figure 2a, the coordinates of which were the income s of the richest, middle and poorest thirds of Japanese households in Since as early as Condorcet (1785) it has been well known that the social metric does not exist. However, in order to valuate the inequality of distributions some ad hoc social norms are necessary. Most proposals are originally founded on philosophical principles, although final success also depends on their analytical handiness. Figure 2. Three dimensional presentations (0,1,0) (1/3,1/3,1/3) (0,3/4,1/4) (0,1/2,1/2) 0,55 0,15 0,30 (15,30,55) (30,55,15) (15,30,55) (15,55,30) (30,15,55) (15,30,55) poorest Ly richest PAK JPN (y ) (17,40,43) SWE 90 (15,30,55) JPN 80 (19,23,58) PAK 71 SWE (12,14,74) MRC 65 L -1 y MRC (1,0,0) (0,0,1) (a) Space of s (b) Schur convexity (c) Lorenz dominance The most popular principle (attributed to Dalton, 1920:351) -any transfer from a richer to a poorer, not so large as to reverse their positions, will diminish inequality-, combined with anonymity (inequality is not changed by permutations) implies that inequality decreases when, ceteris paribus, the difference between two income s diminishes. The measures that satisfy the above principles are symmetric and convex (SC), or Schur (1923) convex (points within the inner of the polyhedron whose vertices are the permutations of a given point are less unequal than such a point, figure 2b). Since the difference between two s, i j, diminishes when the initial distribution y (column vector) is pre-multiplied by a matrix E obtained from the unit matrix by replacing the ii and jj elements with λ ]0,1[ and the ij and ji λ ij with 1-λ, and a sequence of such progressive transfers, L = λ E ij, also reduces inequality, then the SC inequality measures will satisfy: I sc ( Ly) I ( y) I ( L 1 y) (6) sc sc 7

8 Which can be read as Ly Lorenz dominates y, because the Lorenz curve of Ly is unambiguously closer to the diagonal than that of y (L -1 y is Lorenz dominated by y). A dominance which is a partial order because one of two distributions cannot always be obtained from the other by means of a pure sequence of progressive transfers, in which case it is possible to decide the less unequal distribution by choosing the appropriate SC inequality measure (fig. 2c) The two simplest SC inequality measures (regular polyhedrons with the least number of edges) only take into account the status either of the poorest (solidarity, risk aversion, ) or of the richest (envy, complaints, ), founded respectively on the maximin (Rawls, 1973) and the minimax (Temkin, 1993) criteria. According to Rawls (Temkin) the points of the (inverted) triangle that contain the permutations of a particular point have the same inequality (the bigger the triangle the greater the inequality). Thus the income distribution of Japan in 1980 is more Rawls-unequal than that of Pakistan in 1971 (figure 3b), but less Temkin-unequal (figure 3c). Moreover, although inequality in Sweden in 1990 was lower than in Morocco in 1965, whichever SC measure is used, the perception of such a difference depends on the group on which attention is focused. Figure 3. The maximin and minimax orders poorest Ry (y ) JPN Ty JPN (y ) PAK richest (17,40,43) SWE 90 (15,30,55) JPN 80 (19,23,58) PAK 71 SWE (12,14,74) MRC 65 R -1 y MRC poorest (17,40,43) SWE 90 (15,30,55) JPN 80 (19,23,58) PAK 71 SWE (12,14,74) MRC 65 richest PAK T -1 y MRC (a) Two simple SC norms (b) Rawls norm (c) Temkin norm There is no reason to focus on the status of just one group. Leaving aside the above extreme cases, the two next measures whose level surfaces draw the regular SC polyhedron with the least (six) number of edges are the relative mean deviation (M) and the Gini index (G). The former uses only a pair of weights -one positive to those above the mean and another negative to the rest- and so it ignores transfers between groups at the same side of the mean (figure 4a). The latter gives a different weight to every group ( = [ 1+ ( n 1) G] / 2 i iy i +, yi yi + 1, i = 1,..., n ), and so it orders the space of distributions of income s by symmetric and strictly convex (SSC) polyhedrons. The Gini index (figure 4b) is therefore the SSC ine- 8

9 quality measure with the simplest contours (regular polyhedrons with the least number of edges). Figure 4. From SC to SSC orders poorest My SWE BGR PAK richest (y ) M -1 y (17,40,43) SWE 90 (17,34,49) BGR 77 (22,29,49) PAK 85 poorest G y richest JPN (y ) PAK (15,30,55) JPN 80 (19,23,58) PAK 71 G -1 y (a) relative Mean deviation (b) Gini index At the other extreme, the SSC measure whose contours are the regular polyhedron with most edges (or without edges) is the Coefficient of Variation that measures inequality as a Euclidean distance, d = 1/ 2 2 ( y, y) = y y CV ( y) / n where 2 y α 1/ α = ( ) α i yi and y is the egalitarian distribution. The concentric circles (figure 5a) are also the level surfaces of other commonly used measures such as Welfare (Kolm, 1966 and Atkinson, 1970) or Entropy (Theil, 1967 and Cowell, 1977) indices to the particular case of α=2 and their respective families can be obtained by tensing the circles simultaneously along the axes of coordinates towards the vertices of the inverted (minimax) or uninverted (maximin) triangle, the Theil index and Logarithmic Mean Deviation being ordinally equivalent to the particular cases of α 1 and α 0 respectively (figure 5a) E α W ( y) 1 α ( y) = n α 1 α = ( y y ) α 2 [(1 W1 α ) 1]/( α α) α α (7) (8) Similar procedures have been proposed (Donalson and Weymark, 1980) to obtain families of linear measures. Thus, for example, equation (4) may be rewritten, raising i to the power β, as follows: ( y y ) (2/ n) ( yˆ y ) 2 0 Gβ ( y) = (9) β β β β n where y β n = i i= 1 β y i, y i y i+ 1, and ŷ denotes the most concentrated distribution. 9

10 Figure 5. Linear and non linear families of orders Entropía β >1 β <1 (0,, 0) (0,, 0) α < 2 α >2 (0, /2, /2) ( /2, /2, 0) (0, /2, /2) ( /2, /2, 0) (0, 0, ) (, 0, 0) (0, 0, ) (, 0, 0) ( /2, 0, /2) ( /2, 0, /2) Gini To sum up, from a geometrical point of view at least, the Gini index and the Coefficient of variation are not only the classical measures of inequality, but also serious candidates to become the central references of most commonly used measures, and therefore the standards for international comparisons. 4. What about in practice Testing the relevance of the choice between alternative inequality measures is an old issue (Champernowne, 1974) and also a topic in other fields (Kokko et al., 1999). Computing progress, availability of large datasets and the relevance of inequality in growth models, explain the recent abundance of such works, which use simulated distribution samples or secondary datasets. 4.1 The poorest-richest triangle (PRT) To perceive visually the road to equality as an up-hill climb, the poorest third is represented in figure 6a in the vertical axis (first coordinate) and that of the richest in the left axis (third coordinate). The set of distribution s of three groups ordered in such a way are the points of the rectangular triangle at the left bottom side of the simplex: a useful tool to teach and learn about inequality measurement. The richest is bounded between the average and one. It is invariant for points lying on a perpendicular line to the hypotenuse, and such lines become the minimax norm. The poorest third is bounded between zero and 10

11 one time the average, and is invariant for points lying on a perpendicular line to the minor side, such lines becoming the maximin or Rawlsian norm. The Gini index is halfway between the maximin and minimax norms while the Coefficient of Variation -or Euclidean distance to equality- is more Rawlsian than (the) Gini (norm) when dealing with left skewed distributions (close to the minor side of the PRT) and less Rawlsian than Gini when dealing with right skewed distributions (near the hypotenuse) As shown in figure 6b all members of the welfare and entropy families (equations 7 and 8) are, as with the CV (α=2), more or less Rawlsian than Gini depending on the kind of asymmetry of distributions. The lowest is α the more right skewed are the distributions to which the index is as Rawlsian as Gini. Thus, for example, the behaviour of the Gini index and the Mean Logarithmic Deviation (α 0) will be similar when the sample consists of very right skewed distributions. Gini behaves like the Theil index (α 1) when distributions are less right skewed, and the conduct of Gini will be similar to that of the entropy- welfare measures with α>2 when distributions are left skewed. Figure 6. The poorest-richest triangle (PRT) and inequality norms (a) Gini and Coefficient of Variation Inequality (times the maximun) Shares (times the mean) (1/n,1/n,1/n) x 1 1,5 3/4 x 3 x 2 2 1/4 1/2 x x x ,5 G=1/2 1/4 richest 3/4 CV=1/2 (0,0,1) (0,1/n-1,1/n-1) 11

12 (b) Welfare (1-α) and Entropy (α) families rigth skewed β=-2 β->0 β=1 β=2 α = -2 α > 0 α > 1 β=-2 β=-1 β->0 β=5 β=3 α =20 β = 1 α = 4 α = 2 β=3 α = 3 α = 2 α = 0,5 α = > 1 left skewed (c) as Rawlsian as Gini α > 1 CV α = 3 Therefore the simulation exercises testing the correlation between indices are not independent of the method chosen to generate the sample of distributions, and the correlations with real data will depend of the database used. Thus, for example, the results showed in the table 1 suggest that the Harvey (2003) simulated sample may be mainly composed of left skewed and not so unequal distributions (in fact uses a concrete distribution function to simulate randomly one thousand income vectors with Ginis within 0,24 and 0,32) because the Atkinson indices are more correlated with Gini as α (=1-e) goes from negative values (Rawlsians norms) to one (Theil norm). The Figini-LIS real sample (2000) produces lower correlations but similar tendencies because the countries are mainly OCDE with distributions like of the above sample. And in the Champernowne (1974) the Gini index is highly correlated with several indices because the simulated sample includes very different kinds of distributions. 12

13 Table 1. Correlation between indices. Previous evidence Simulations (a) LIS database (b) α = 1-e (Gini, α) W/E(α =) 2 1 0,5 ->0 W/E(α =) 2 1 0,5 ->0 0,75 0,999 CV 1 CV 1 0,5 0,998 Theil 0,985 1 Theil 0, >0 0,996 Atkinson (0,5) 0,950 0,982 1 Atkinson (0,5) 0,845 0, ,964 Atkinson (1) 0,802 0,866 0,940 1 Atkinson (1) 0,674 0,830 0, ,775 Gini 0,974 0,966 0,993 0,901 Gini 0,872 0,979 0,955 0, ,605 (a) Champernowne (1974) 4.2 The Wider dataset (b) Figini (2000) (c) Harvey (2003) Secondary datasets about household income and expenditure surveys must be used with caution (Atkinson and Brandolini 1999). The most popular is probably that of Deininger and Squire (1996) that includes almost one thousand distributions with quintile information, of which two thirds are of good quality value, and from which it is possible to generate samples with different biases. The World Bank indicators enable income distribution to be related to the most relevant economic and social variables from just one statistical source. Thus, for example, figure 7b shows the apparent relation between inequality and the level of development (measured by GDP, other things being summarized by a geographical dummy). Figure 7. The World (Bank) inequality OCDE countries (18) GDP pc >10.000$ 1000$<GDP pc <10000$ East of Europe and North Africa (20) Latin America (19) GDP pc <1000$ Africa (22) Asia (12) Several projects, like the above-mentioned Luxembourg Income Study, have attempted to improve comparability at the expense of reducing sample size. These and other sources have been included in the WIDER World Income Inequality Database (WIID) (Murkhopadhaya, 2004). The current version released in December 2004 covers 4507 observations of which only 1530 provide disaggregated information by deciles (657) or quintiles (873). The s of the popu- 13

14 lation thirds used below have been obtained dividing the s of the common quantiles proportionally to the populations transferred. Although the value of the indices decreases with the number of groups (Davies and Shorrocks, 1989), there is a strong correlation between the results produced by different groupings. In fact when the values are scaled down dividing the indices by their top bounds most of the gap disappears. Starting by the sample of distributions with the necessary quantile information (1530) and taking as the reference the point (1/6,1/6,1/3), it can be seen from figure 8 that half of the distributions cannot be considered unambiguously, more unequal or less unequal than such a point, whatever the SC measure chosen. Moreover, if the reference point were (0, 1/2, 1/2), only five per cent of distributions could be considered less unequal while none is unambiguously more unequal. Figure 8. The WIDER database Distributions with quantile information (1530) (1/6,1/6,2/3) (0,1/2,1/2) Focusing on data with full coverage of area, population and age (1301) it can be observed from figure 9 that samples using households or families as the statistical unit are less stylised and more ambiguous than those using an individual basis. Finally, the distributions with the household as statistical unit and the per capita income as the unit of analysis (552) are classified under eight sub-samples by the income or expenditure definition used. As expected, gross income is more unequally distributed than net income (after tax). However, the subsamples are country biased and thus, for example, it is not possible to conclude whether the lowest level of inequality of the last subsample is due to the use of net monetary income or because the countries included are OECD and central European. 14

15 Figure 9. The WIDER database Distributions with full area, population and age covering (1301) Household units Person units Figure 10. The WIDER database Household per capita basis (1301) and subsamples by income or expenditure definition used All area,population and age Household per capita 552 Comsumption 104 Expenditure 87 Earning gross 19 Income gross 121 Monetary income gross 43 Income 49 Income Net 36 Monetary income net 93 15

16 In any case, the above subsamples are representative of the real world and can be used to test the empirical correlation among indices. At the first sight all of the subsamples considered appear to be through to the area in which the Theil index (α 1) behaves as Rawlsian as Gini (RG) as can be seen when figure 6c is mixed with figures 8 to 10. In fact the correlations between the Gini index and the members of the Generalized Entropy family (figure 11) reach a peak when the positional sensitivity parameter is between zero and two (the measures that are ordinally equivalent to the Mean Deviation Logarithm and the Coefficient of Variation respectively) Figure 11. Correlations between Gini and Welfare/Entropy indices 1,00 0,75 0,50 0,25 Correlation Coefficien All with quantile information (1530) Household basis (551) Individual basis (789) Positional sensitivity parameter 0, ,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 Monetary Income Net (93) All Hpc (552) Income Gross (121) Comsumption (104) Conclusion This paper argues in favour of the establishment of an international standard measure of inequality to be introduced in the official System of National Accounts as an alternative norm of aggregation of the disposable household income and of its main sources and uses. Among the linear measures from both theoretical and empirical points of view the Gini index is the halfway between the maximin and minimax extreme norms and it seems to be the candidate to become the reference measure. Among the non linear measures from a theoretical point of view the Coefficient of Variation seems to be reference measure, but from an empirical point of view the Theil index may be also a serious candidate to become a reference measure. 16

17 Of course the call for joining efforts in order to agree about some inequality measure to be introduced in the SNA it does not mean that inequality should be reduced to a simple issue. The elaboration of a chapter devoted to the satellite accounts of inequality, poverty, polarization and so on, could be the next steep. References AEG (2004) Summary Conclusions of the First Meeting of the Advisory Expert Group on National Accounts ISWGNA February Washington, D.C. ( AEG (2004) Summary Conclusions Second Meeting of the Advisory Expert Group on National Accounts 8-16 December 2004 United Nations, New York. ( AEG (2005): Advisory Expert Group on National Accounts Meetings. ( Atkinson, A. and A. Brandolini (2001): Promise and pitfalls in the use of secondary datasets: Income inequality in OCDE countries as a case study, Journal of Economic Literature, Atkinson, A.B. (1970): On the Measurement of Inequality, Journal of Economic Theory, 2, Blackorby, Ch. and Donalson D. (1977): Measures of relative equality and their meaning in terms of social welfare, Journal of economic theory, 18: Camberra group (2001): Final Report and Recommendations of the Expert Group on Household Income Statistics. The Canberra group, Ottawa 2001 Capéau, Bart and André Decoster (2004): The Rise or Fall of World Inequality A Spurious Controversy? WIDER Discussion Paper nº 2004/02 Champernowne, D., (1974), A comparison of measures of inequality of income distribution, Economic Journal, 84: Condorcet, Marqués de (1785) : Essai sur l'application de l'analyse a la probabilité des décisions rendues à la pluralité des voix, L Imprimerie Royale, Paris. Cowell, F. A. (1977): Measuring Inequality, Phillip Alland, Oxford. Dagum, C. (1990): On the Relationship Between Income Inequality Measures and Social Welfare Functions, Journal of Econometrics, 43, Dalton, H. (1920): The Measurement of the Inequality of Incomes. The Economic Journal, 30, 119, Davies, J. and Hoy, M. (1994): The normative significance of using third degree stochastic dominance in comparing income distributions, Journal of economic theory, 64: Davies, J.B. and A. F. Shorrocks (1989): Optimal grouping of income and wealth data, Journal of econometrics, 42: Deininger, K. and L. Squire (1996): A new data set measuring income inequality, World Bank Economic Review, 10, , Donalson, D. and J. Weymark (1980): A single-parameter generalization of the Gini indices of inequality, Journal of Economic Theory, 22: Eurostat (2003): Income and living conditions (EU-SILC). ( Figini, Paolo (2000): Measuring Inequality: On the Correlation Between Indices, Luxembourg Income Study. Working Paper No

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