Inequality Measures and the Median:

Transcription

1 Inequality Measures and the Median: Why inequality increased more than we thought by Frank A. Cowell STICERD London School of Economics Houghton Street London, W2CA 2AE, UK and Emmanuel Flachaire Aix-Marseille University AMSE, CNRS & EHESS 2 rue de la charité, Marseille, France, emmanuel.flachaire@univ-amu.fr February 2017

2 Abstract Many standard inequality measures, as Gini and Theil indices, can be written as ratios, where the denominator is the mean. When one income moves away from equality, both the numerator and the denominator vary in the same direction and, thus, such indices may decrease. This undesirable property is not shared by median-based inequality measures developped in this paper, where the mean at the denominator is replaced by the median. However, median-based inequality measures do not respect the principle of transfer. We show that the mean logarithmic deviation, or second Theil, index is the only measure that satisfies the principle of transfer and avoids the previous undesirable property. An application shows that the increase of inequality in the United Sates over the last decades is largely underestimated with the Gini index and, that the mean logarithmic deviation index should be preferred in practice. Keywords: inequality measures, median, axiomatic approach JEL codes: D63 1

3 1 Introduction Many standard inequality measures, as Gini and Theil indices, are expressed as ratios, where the denominator is the mean. Division by the mean ensures that they are scale invariant. However, when one income moves away from equality, both the numerator and the denominator vary in the same direction and, thus, such indices may decrease. This is an undesirable property. This feature can be characterized by considering the following principle of monotonicity in distance: if two distributions differ only in respect of one individual s income, then the distribution that registers greater distance from equality for this individual s income is the distribution that exhibits greater inequality. This principle is desirable for a measure of inequality. However, standard inequality measures don t respect this principle. To illustrate, let us consider the following distributions of income: x = {1, 2, 3, 4, 5, 9, 10} and x = {1, 2, 3, 4, 7, 9, 10}. (1) The two distributions differ only in respect with the the 5 th individual, for which income increases from 5 to 7. The income of this individual moves away from equality. 1 However, computing the Gini and the Theil indices, we find more inequality in x than in x : Gini(x) = > Gini(x ) = (2) Theil(x) = > Theil(x ) = (3) It is thus clear that the Gini and the Theil indices don t respect the principle of monotonicity in distance. In this paper, we develop a median-based class of inequality measures that respects the principle of monotonicity in distance, based on an axiomatic approach. We show that it is closely related to the Generalized Entropy class of inequality measures, where the mean at the denominator is replaced by the median. However, it does not respect the principle of transfer. Then, we show that the Mean Logarithmic Deviation (MLD) index, or second Theil index, which is a limiting case of the mean-based Generalized Entropy class of inequality measure, is also a limiting case of our medianbased class of measures. Thus, it shares properties of both mean- and median-based inequality measures. In other words, the MLD index is the 1 The mean of x is equal to and, the mean of x is equal to

4 only measure that respects both the principle of transfer and the principle of monotonicity in distance. Indeed, we find less inequality in x than in x with the MLD index: MLD(x) = < MLD(x ) = (4) The lack of the principle of monotonicity in distance may have strong implication in empirical studies. Studying inequality in Great Britain and in the United States over the last decades, we show that the Gini index underestimate variations of inequality and, the MLD index should be preferred in practice. 2 Inequality measurement 2.1 Axiomatic discussion Consider inequality as a weak ordering 2 on R n+1 + ; denote by the strict relation associated with and denote by the equivalence relation associated with. For any vector s denote by s (ς, i) the vector formed by replacing the ith component of s by ς R +. We can characterise the structure of the inequality relation using just five axioms: Axiom 1 [Continuity] is continuous on R n+1 +. Axiom 2 [Monotonicity in distance] If s, s R n + differ only in their ith component then (a) if s i e :s i > s i (s, e) (s, e); (b) if s i e: s i < s i (s, e) (s, e). Axiom 3 [Independence] If s (ς, i), s (ς, i) R n + satisfy (s (ς, i), e) (s (ς, i), e) for some ς then (s (ς, i), e) (s (ς, i), e) for all ς R +. Axiom 4 [Anonymity] For all s R n + and permutation matrix Π, (Πs, e) (s, e). Axiom 5 [Scale invariance] For all λ R + : if s, s, λs, λs R n + and e, e, λe, λe R + then (a) (s, e) (s, e ) (λs, e) (λs, e ); (b) (s, e) (s, e ) (λs, λe) (λs, λe ). 2 In other words the relation is complete and transitive. 3

5 Theorem 1 Given Axioms 1 to 5, is representable as [ ] 1 1 n I α (s; e) := s α i e α, (5) α [α 1] n where α R, or by some strictly increasing transformation of (5) involving e. [Proof: see Cowell and Flachaire (2014)] 2.2 Normalisation In the case of cardinal data, the mean of status is a natural reference point. Indeed, in case of perfect equality, everyone gets the same value - equal to the mean of status - and, inequality measure in (5) is equal to zero. In order to respect the scale invariance property, "income" cannot be used directly as status, it has to be normalised. Standard inequality measures are defined by using income shares as status, that is, income are divided by the mean. However, another normalisation could be used. Here, we propose to divide income by the median rather than by the mean. In this section, we first review the standard normalisation by the mean; then, we study the properties implied by a normalisation by the median Mean normalisation: principle of transfer To respect the property of scale invariance, incomes can be divided by the mean. It implies that inequality can be measured in terms of income shares, which gives the standard class of Generalized Entropy inequality measures: I α (x/µ; µ) := 1 α[α 1] [ 1 n n i=1 i=1 ( ) α xi 1], α 0, 1. (6) µ This normalisation ensures that I α (x/µ; µ) is well-defined and non-negative for all values of α and that, for any profile x, I α (x/µ; µ) is continuous in α. Normalisation by mean income in order to ensure scale invariance (Axiom 5) introduces a complication: the type of income variation considered in the statement of Axiom 2 will change the mean of x. This would mean, for example, that such an income variation will affect not only the numerator 4

6 of each fraction in (6), but also the denominator. Such normalised indices still have minimal-inequality property: starting from a situation of zero inequality, changing any one person s income will result in positive inequality. But to cover cases where inequality is not close to zero, for the normalised index we need to reconsider the monotonicity principle. This can be done by replacing Axiom 2 with the following principle: Axiom 2 [Principle of Transfers (PoT)]: If x, x R n + differ only in their i th and j th components and x i x i = x j x j then, if x i > x i and if i is below j in x then i remains below j in x, we have (x, e) (x, e). It guarantees that a transfer from a richer to a poorer individual does not increase inequality. Mean-based inequality measure in (6) respects this property, also known as the Pigou-Dalton transfer property. This is because this property consider cases where the mean remains unchanged and, for fixed mean income, (6) is increasing and convex in each x i so that an infinitesimal transfer of δ from j to i must reduce I α (x/µ; µ) if x i < x j. Normalisation by the mean implies that the class of inequality measures in (6) respects the principle of transfers (Axiom 2 ), not the principle of monotonicity in distance (Axiom 2) Median normalisation: principle of monotonicity To respect the property of scale invariance, we propose to divide incomes by the median rather than by the mean. It leads us to define a median-based class of Generalized Entropy inequality measures, as follows: I α (x/m; µ) := 1 α[α 1] [ 1 n n i=1 ] ( xi α ( µ ) α, α 0, 1. (7) m) m This normalisation ensures that I α (x/m; µ) is well-defined and non-negative for all values of α and that, for any profile x, I α (x/m; µ) is continuous in α (see section 3.1). Note that (7) can be rewritten I α (x/m; µ) := 1 µ α µ a (8) α[α 1] m α where µ α = 1 n n i=1 xα i is the α-th raw moment of the distribution. The main feature of this normalisation is that the new class of inequality measures in (7) respects the principle of monotonicity in distance (Axiom 5

7 2). As long as the median is fixed, (7) respected Axiom 2, since it has the form in (5). It is true for any x i and x i both striclty above or below the median. We need to see what happens when x i and x i are on either side of the median. For income distributions, skewed to the right, the median is less than the mean. Thus, incomes at each part of the median corresponds in Axiom 2 to the case where x i < m < x i < µ only. From x i to x i the mean decreases: it follows that (7) increases, which is consistent with Axiom 2. Figure 1 illustrates the principle of monotonicity in distance. It shows median-based (left) and mean-based (right) inequality measures for α = 1, computed from 1000 observations, drawn from a standard lognormal distribution, plus 1 additional observation x i, where x i ]0, 8]. First, let us consider median-based inequality measure. From Figure 1 (left), we can see that the index is minimum when x i is equal to the mean (symbol ) and, it increases when x i goes far away from the mean. The principle of monotonicity in distance is then respected. However, there is a particuliar behavior in the neighborhood of the median (symbol ): the index increases suddenly very much. This behavior is due to a change of the median, which affects every terms in (7), even if only one income varies slightly. Since the number of observations is odd, n = 1001, we have x 500 if x i x 500 m = x i if x i [x 500 ; x 501 ] (9) x 501 if x i x 501 where x k denotes the k th -order income. The median varies in a very narrow interval: its values are bounded by two adjacent s mid-rank incomes. In our example, we have m [ ; ]. Second, let us consider mean-based inequality measure. From Figure 1 (right), we can see that the index is not minimum when x i is equal to the mean (symbol ), but when x i = 2.9. It suggests that for any µ k 1 < k 2 < 2.9, the index will exhibit more inequality with x i = k 1 than with x i = k 2, which is not consistent with the principle of monotonicity in distance. Median-based inequality measures in (7) do not respect the principle of transfers (Axiom 2 ), but a weaker version of this principle. They respect Axiom 2 for any transfers strictly above/below the median, that is, as long as the median is unchanged. 3 However, we can easily find a counter example, 3 The median is unchanged and, for fixed median, (7) is increasing and convex in each x i when α > 1. 6

8 median based inequality x i '=median(x) x i '=mean(x) mean based inequality x i '=median(x) x i '=mean(x) x i ' x i ' Figure 1: Principle of monotonicity in distance: median-based (left) and mean-based (right) inequality measures for α = 1, computed from 1000 observations drawn from a lognormal distribution plus one additional observation x i, where x i ]0, 8]. where transfer from the median individual to a poorer individual increases I α (x/m; µ). For instance, let us consider the following distributions: 4 x = {1, 2, 3, 5, 10} and x = {1, 2.5, 2.5, 5, 10}. (10) From the principle of transfers, mean-based inequality measures always exhibit more inequality in x than in x. By contrast, a median-based inequality measure may exhibit less inequality in x than in x : for instance, we have I 1 (x /m; µ) < I 1 (x /m; µ). 5 4 This example has been given by Gastwirth (2014) for the Gini index, see section I 1 (x /m; µ) = and I 1 (x /m; µ) =

9 2.3 One measure satisfies both principles The limiting forms α = 0 and α = 1 of median-based Generalized Entropy inequality measures in (7) are, respectively, equal to: 6 I 1 (x/m; µ) := 1 n n i=1 I 0 (x/m; µ) := 1 n x i m log n log i=1 ( ) xi µ ( ) xi µ (11) (12) They are median-based versions of, respectively, the Theil index and the Mean Logarithmic Deviation (MLD) index. The median-based inequality measure with α = 0 doesn t depend on the median and, it is equal to the limiting form of the mean-based inequality measure in (6): 7 I 0 (x/m; µ) = I 0 (x/µ; µ) (13) This result is important: it shows that the MLD index in (12) is a particularly attractive measure, since it respects both the principle of monotonicity in distance (Axiom 2) and the principle of transfers (Axiom 2 ). Figure 2 (left) illustrates the principle of monotonicity, with similar samples as in Figure 1. We can see that the index is minimum when x i is equal to the mean (symbol ) and, it increases smoothly as x i goes far away from the mean. 3 Discussion 3.1 The sensitivity parameter α From (7), we can see that the parameter α puts more weights to different parts of the income distribution: α > 1 puts more weights to high incomes (when x i m), α < 1 puts more weights to low incomes (when x i m), 6 They are obtained from a simple application of l Hôpital s rule. 7 For the mean-based inequality measures defined in (6), the limiting forms α = 0, 1 are, respectively, I 1 (x/µ; µ) := 1 ( ) n x i n i=1 µ log x i µ and I 0 (x/µ; µ) := 1 ( ) n n i=1 log x i µ. 8

10 I x i '=median(x) x i '=mean(x) Gini median based x i '=median(x) x i '=mean(x) x i ' x i ' Figure 2: Principle of monotonicity in distance: MLD (left) and medianbased Gini (right) indices, computed from 1000 observations drawn from a lognormal distribution, plus 1 additional observation x i, where x i ]0, 8]. 1 α < 0 puts more weights to incomes close and above the median, 0 < α 1 puts more weights to incomes close and below the median. When α [ 1; 1], the index puts more weights to incomes in the middle of the distribution, rather than in the tails. The limiting case α = 0 is non-directional, in the sense that it does not put more weights to incomes above/below the median or above/below the mean. It can be seen by rewriting (12) as follows: I 0 (x/m; µ) = log µ log g = log (µ/g), (14) where g is the geometric mean of incomes. The MLD index is then the log differences between the arithmetic mean and the geometric mean. It is clear that in computing arithmetic and geometric means, every income has the same weight. 9

11 index mean based index median based index alpha Figure 3: Median- and mean-based inequality measures, for α [ 2; 5]. The relationship between median- and mean-based inequality measures is straigthforward. From (6), (7), (11) and (12), we have ( µ ) α I α (x/m; µ) = Iα (x/µ; µ) (15) m For income distributions, skewed to the right (m < µ), median-based inequality measures are always greater (less) than mean-based inequality measures for α > 0 (α < 0). To illustrate this feature, Figure 3 plots values of medianand mean-based inequality measures, as defined in (7) and (6), for different values of α, using a sample of 1000 observations drawn from the standard lognormal distribution. In this example, the ratio µ/m and, it is clear from Figure 3 that median-based indices are greater (less) than mean-based indices when α > 0 (α < 0). The two curves intersects in α = 0. Calculating median-based inequality measure doesn t require microdata: (15) suggests that knowing mean-based measure with the mean and the median income is enough. 10

12 3.2 Decomposability Let there be K groups and let the proportion of population falling in group k be p k, the class of inequality measures (7) can be expressed as ( K [ mk ] α I α (x/m; µ) = p k Iα (x k /m k ; µ k )+ 1 K ) [ µk ] α [ µ ] α p m α 2 k α m m k=1 k=1 (16) where m k and µ k are, respectively, the median and the mean in group k and, m and µ are the corresponding population median and mean. This means, for example, that we may partition a top income group T (for x i m) and a bottom income group B (for x i < m) and, using an obvious notation, express overall inequality as I α = w T I T α + w B I B α + I btw (17) where the weights w T, w B and the between-group mobility component I btw are functions of the income mean and median for each of the two groups and overall; comparing I T α and I B α enables one to say precisely where inequality has taken place. The MLD index, or limiting case α = 0, can be decomposed as follows: I 0 (x/m; µ) = K p k I 0 (x k /m k ; µ k ) k=1 K p k log k=1 ( ) µk µ (18) Shorrocks (1980) argues that the MLD index is the "most satisfactory of the decomposable measures", because it splits unambiguously overall inequality into the contribution due to inequality within subgroups and that due to inequality between subgroups. This property, called path independent decomposability by Foster and Shneyerov (2000), is not shared by (16), because the weights in the within subgroup terms are not independent of the between group term. Indeed, changing the income subgroup means, µ k, will also affect the within subgroup contribution in (16) - through m k /m, not in (18). It follows that the inequality that would result from removing differences between subgroups, the inequality within subgroups being unchanged, is given by the first term in (18), not in (16). 11

13 3.3 The Gini index The well-known Gini index respects the principle of transfer, not the principle of monotonicity in distance. The Gini index is defined as follows, G(x; µ) = 2µ = E( x 1 x 2 ) 2µ = n i=1 n j=1 x i x j 2µn 2 (19) where is the well-known Gini s mean difference measure. 8 It has the same drawback as mean-based inequality measures defined in (6). Indeed, when only one income increases, the mean increases and, thus, the contribution of each distance between all other incomes in (19) decreases. 9 For instance, the Gini index exhibits more inequality in distribution x than in x, defined in (1), which is not consistent with the principle of monotonicity in distance. Gastwirth (2014) proposed to replace the mean by the median in the standard definition of the Gini index, G(x; m) = 2m = E( y 1 y 2 ) 2m = µ G(x; µ) (20) m The median-based Gini index in (20) does not respect both, the principle of monotonicity in distance (Axiom 2) and the principle of transfer (Axiom 2 ), and it is not decomposable. It respects weaker versions of these two principles: for incomes or transfers strictly above/below the values taken by the median only. Figure 2 (right) illustrates the principle of monotonicity, with similar samples as in Figure 1. We can see that the index is not minimum when x i is equal to the mean (symbol ) or equal to the median (symbol ). The index is thus inconsistent with the principle of monotonicity in distance. However, if we exclude the few values taken by the median, [x 500 ; x 501 ] (see (9)), we can see that the index increases as x i goes far away from this interval, in both directions. The median-based Gini can be viewed as an inequality measure with the median as reference point. In addition, when we consider distributions x and x in (10), we have G(x, m) < G(x, m), 10 which is not consistent with the principle of transfers. We can make a link between the median-based Gini (20) and the medianbased Generalized Entropy measures in (7). Indeed, the median-based mea- 8 For more details on the standard Gini index, see Yitzhaki and Schechtman (2013). 9 When only x k, we have µ and, then, x i x j /µ for all i, j k. 10 G(x, m) = and G(x, m) =

14 sure in (7) with α = 2 is equal to I 2 (x/m; µ) = σ2 2m = E([y ȳ]2 ) = E([y 1 y 2 ] 2 ) (21) 2 2m 2 4m 2 We can see that G(x; m) and [I 2 (x/m; µ)] 1/2 are two very similar measures, both are ratios of a dispersion measure on twice the median. For the Gini, the dispersion measure is based on Manhattan L 1 -distance, while for the Generalised Entropy measure it is on based Euclidian L 2 -distance. Then, I 2 (x/m; µ) puts more weights to high incomes, compared to the Gini. 11 Using L 1 -distance, it is clear that the Gini index does not put more weights to high/low incomes. In the class of inequality measures defined in (7), it is the limiting case α = 0 which doesn t put more weights to high/low incomes (see section 3.1). Thus, Gini and MLD indices are both quite similar in terms of putting similar weights to every income. 4 Application In previous sections, we have seen that the Mean Logarithmic Deviation (MLD) index, defined in (12), is a particularly attractive measure, since it is the only one measure that shares properties of both median- and mean-based inequality measures and, it is decomposable (see sections 2.3 and 3.2). In this section, we study inequality in Great Britain and in the United States. We compare results obtained with the MLD and with the standard Gini indices, both giving similar weighting schemes to different parts of the distribution (see sections 3.1 and 3.3). 4.1 Inequality in Great Britain First, we study inequality in Great Britain, from 1961 to The values of Gini and MLD indices are given by the Institute for Fiscal Studies (Tools and resources: "Incomes in the UK"). 12 They are based on the Family Expenditure Survey up to and including 1992, and the Family Resources Survey thereafter. We use inequality indices before housing costs. 11 It is also true for mean-based inequality measures G(x; µ) and [I 2 (x/µ; µ)] 1/2, since both are ratios of a dispersion measure on twice the mean. 12 see 13

15 Figure 4 - top plot - shows values of the Gini and MLD indices, defined in (12) and (19). Both inequality measures describe quite similar patterns. We can see that inequality increased quite a lot during the 1980s and, it appears to have fallen slightly during the 1990s, as suggested by Jenkins (2000). However, we can see that inequality is relatively stable since the 1990s: if we estimate linear regressions of Gini and of MLD indices in log against time, over the period , we find slope coefficients not significantly different from zero. 13 Figure 4 - bottom plot - shows variations of inequality between two successive years, in percentage, for the Gini and MLD indices. We can see that variations from the MLD index are always greater than variations from the Gini index, with large differences. For instance, between 1962 and 1963, inequality increases by 9.15% with the Gini and by 21.44% with the MLD; between 1983 and 1984, inequality increases by 2.54% with the Gini and by 7.91% with the MLD; between 2009 and 2010, inequality decreases by 5.65% with the Gini and by 10.95% with the MLD. Such discrepancies are explained by the fact that a shift to the right (left) of income greater (less) than the mean, increases (decreases) both the numerator and the denominator of the Gini index. Since both the denominator and the numerator vary in the same direction, income variation is attenuated. This features is characterized by the principle of monotonicity in distance, not respected by the Gini index. It is also true for any mean-based inequality indices, as Generalized Entropy measures defined in (6). The MLD index is an exception: it is the same limiting case α = 0 of median-based indices (7) and of mean-based indices (6) and, thus, it respects the principle of monotonicity in distance. By contrast to the Gini index, the MLD index does not attenuate variation of income towards the top or the bottom incomes. 4.2 Inequality in the United States Second, we study inequality in the United States, from 1967 to The values of Gini and MLD indices are given by the U.S. Census Bureau, in 13 We obtain: log Gini t = (1.656) ( ) t and 14 log MLD t = (3.456) ( ) t.

16 the report Income and Poverty in the United States: 2015 (Table A-2, pp ). 14 The sample survey has been redesigned in 1994, it is then not fully comparable over time (McGuiness 1994). Figure 5 - top plot - shows values of the Gini and MLD indices. Both inequality measures describe quite different patterns. The increase of inequality, since the 1980s is much higher with the MLD index than with the Gini index. If we estimate linear regressions of Gini and of MLD indices in log against time, over the period , we find slope coefficients significantly different from zero and, equal to, respectively, and It means that, inequality increases at an average annual rate of 0.274% with the Gini index and, at an average annual rate of 1.362% with the MLD index. Thus, over the last 22 years, the rate of growth of inequality with the MLD index is approximately five times that of the Gini index. Figure 5 - bottom plot - shows variations of inequality between two successive years, in percentage, for the Gini and MLD indices. We can see that variations of the MLD index are mostly greater than variations of the Gini index, with a few opposite results. For instance, between 1979 and 1980, inequality decreases by 0.25% with the Gini, it increases by 1.62% with the MLD; between 1998 and 1999, inequality increases by 0.44% with the Gini, it decreases by 2.46% with the MLD; between 2013 and 2014, inequality decreases by 0.41% with the Gini, it increases by 0.82% with the MLD. Such opposite results could be explained by the fact that a shift to the right of income greater than the mean can lead to a decrease of mean-based inequality measures, as GE indices and the Gini. It is because these measures don t respect the principle of monotonicity in distance (see sections 1 and 2.2). By contrast, median-based inequality measures respect this principle and, thus, they would increase in such case. The shift of incomes towards top incomes in the United States over the last decades is well documented, among others see Atkinson and Piketty (2010), Krueger (2012) and Piketty (2014). 14 see 15 We obtain: log Gini t = t and log MLD t = t. (0.4451) ( ) (1.433) ( ) 15

17 4.3 United States vs. Great Britain Finally, we compare inequality in Great Britain and in the United States with Gini and MLD indices. Figures 6 - top plot - shows values of the Gini index. It is clear that inequality is always much higher in the United States compared to Great Britain. However, when we compare trends over the last 22 years, we can see that the increase of inequality is not very different. We have seen that, from 1994 and 2015, the annual rate of growth is not significantly different from zero in Great Britain and, it is equal to 0.274% in the United States (see footnotes 13 and 15). Figures 6 - bottom plot - shows values of the MLD index. It provides a quite different picture: inequality increases much higher in the United States than in Great Britain over the last 22 years. Indeed, we have seen that the annual rate of growth is not significantly different from zero in Great Britain and, it is equal to 1.362% in the United States (see footnotes 13 and 15) Overall, comparisons based on the Gini index suggests that the increase of inequality over the last 22 years is not very different between Great Britain and the United States, while it is found very different based on MLD index comparisons. 5 Conclusion The Gini index is the most commonly inequality index published by statistical agencies and used in empirical studies. This index puts the same weight to every income, it is scale invariant and, it respects the principle of transfer. The MLD index shares the same properties and, in addition, it respects the principle of monotonicity in distance and, it is decomposable with the path independent property. In this paper, we have seen that the lack of the principle of monotonicity in distance may have strong implications in empirical studies. Indeed, the Gini index may seriously underestimate variations of inequality. Our application suggests that the increase of inequality in the United States over the last 22 years is significantly underestimated with the Gini index. By contrast, the MLD index has more desirable properties, it estimates variations of inequality more accurately and, it should be preferred in practice. 16

18 References Atkinson, A. B. and T. Piketty (2010). Top incomes: A global perspective. Oxford University Press. Cowell, F. A. and E. Flachaire (2014). Inequality with ordinal data. Public Economics Programme Discussion Paper 16, London School of Economics, Foster, J. E. and A. A. Shneyerov (2000). Path independent inequality measures. Journal of Economic Theory 91, Gastwirth, J. L. (2014). Median-based measures of inequality: Reassessing the increase in income inequality in the U.S. and Sweden. Statistical Journal of the IAOS 30, Jenkins, S. (2000). Modelling household income dynamics. Journal of Population Economics 12, Krueger, A. B. (2012). The rise and consequences of inequality in the United States. Technical report, The White House. McGuiness, R. A. (1994). Redesign of the sample for the current population survey. Employment and Earnings 58, 7 8. Piketty, T. (2014). Capital in the 21st century. Harvard University Press. Shorrocks, A. F. (1980). The class of additively decomposable inequality measures. Econometrica 48, Yitzhaki, S. and E. Schechtman (2013). The Gini Methodology: A Statistical Primer. Springer Series in Statistics. Springer. 17

19 Great Britain United States Year Gini MLD Gini% MLD% Gini MLD Gini% mld%

20 Inequality in Great Britain inequality index Gini MLD variation in % (Gini t Gini t 1 )/Gini t 1 100(MLD t MLD t 1 )/Gini t Figure 4: Inequality in Great Britain from 1961 to

21 Inequality in the United States inequality index Gini MLD variation in % (Gini t Gini t 1 )/Gini t 1 100(MLD t MLD t 1 )/Gini t Figure 5: Inequality in the United States from 1967 to

22 United States vs. Great Britain Gini index Gini index in the United States Gini index in Great Britain MLD index MLD index in the United States MLD index in Great Britain Figure 6: Inequality in the United States and in Great Britain. 21