The properties of the extended Gini measures of variability and ...

METRON - International Journal of Statistics 2005, vol. LXIII, n. 3, pp. 401-433

SHLOMO YITZHAKI – EDNA SCHECHTMAN

The properties of the extended Gini measures of variability and inequality

Summary - The aim of this paper is to survey and investigate the properties of the extended Gini family of inequality measures. The paper surveys the alternative ways of spelling the extended Gini for continuous distributions (such as via covariance, via Lorenz curves, etc.), and the metrics that motivate them. It also offers the adjustments needed for consistent estimation in the case of discrete distributions. The relationship between the family and welfare dominance is discussed. Then, the equivalent parameters to the covariance and correlations that are required for the decomposition of a sum of random variables into the contribution of components are defined, and the new frontiers it opens for estimating a regression based on those measures is illustrated. Finally, the implications of analyzing the effect of policies intended to change the income distribution are discussed. Key Words - Extended Gini; Lorenz curve; Relative deprivation; Stochastic Dominance; Decomposition.

1. Introduction Researchers in the area of income distribution agree that a higher mean income increases social welfare while higher inequality decreases it. The question is how to measure inequality. In his seminal paper, Atkinson (1970) proved that if one Lorenz (1905) curve is always not lower than the other, then the income inequality in the distribution with the upper Lorenz curve is smaller than the inequality in the lower curve for every additive concave social welfare function, provided that the two distributions have equal means. If, on the other hand, the Lorenz curves of two distributions intersect, then one can always find two additive concave social welfare functions that will rank inequality in the two distributions in a reversed order. Shorrocks (1983) extended the above result to comparisons of distributions with different means. He showed that having a distribution with an absolute Lorenz curve that is always not lower than the other forms a necessary and sufficient condition for the expected value Received September 2005.

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of every concave social welfare function to be greater than the expected welfare of the distribution with the lower absolute Lorenz curve. There are several implications of the Atkinson/Shorrocks results. The most important is the following: in some cases it is possible to perform welfare comparisons without assuming the exact shape of the social evaluation of the marginal utility of income. On the other hand, in some other cases, i.e., when the Lorenz curves intersect, it is impossible to evaluate whether one distribution should be preferred over the other without further specifications of the social evaluation of the marginal utility of income. The latter means that in many practical cases, measuring inequality cannot be done in an “objective” or “scientific” statistical way, and the inequality measure used should be compatible with our social preferences. In other words, an agreement about our social preferences should precede the measurement of inequality. Without an agreement about the social evaluation of the marginal utility of income, two researchers comparing the same distributions may reach contradicting conclusions. In this sense, economic analysis is back in the arena of political economics. The Atkinson/Shorrocks approach was originally developed in Finance, and was referred to as Second degree Stochastic Dominance (SSD). (Hadar and Russel, 1969; Hanoch and Levy, 1969). One major weakness of this approach is that it is not capable of providing a complete ordering of possibilities, leaving many relevant and interesting cases undecided. As a way out, Atkinson (1970) suggested a family of social welfare functions that can produce alternative competing inequality measures. The family is referred to as the Atkinson family of indices of inequality. Each index produces a complete ordering, and reflects the social attitude of the user. Each index is compatible with the preference of at least one social welfare function, namely: the one it is based upon. In this sense, each index produces a necessary condition for SSD. However, in most cases it is really hard to defend the choice of a specific index, and the use of the family is limited to sensitivity analysis. Several developments that are relevant to the topic of this paper followed Atkinson’s paper. The Gini coefficient(1 ) is twice the area between the diagonal and the Lorenz curve. This property led Yitzhaki (1982b) to construct necessary conditions for second-degree stochastic dominance using the Gini coefficient. This is so although the Gini is not based on an individualistic additive social welfare function (Newbery, 1970). Also, following Atkinson (1970) criticism, the Gini coefficient was extended (Donaldson and Weymark, 1980, 1983; Kakwani, 1980; Yitzhaki, 1983) into a family of inequality measures that reflect different social preferences. We will refer to it as the extended Gini (EG). It is also referred to in the mathematically oriented literature as the S-Gini. Those (1 ) Gini, C. (1912, 1914). The Gini Mean Difference was developed in 1912, the Gini coefficient, which is the GMD normalized by the mean income, was proposed in (1914). See Giorgi (1990, 1993) for a biographical portrait.


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indices are based on the area enclosed between the diagonal and the Lorenz curve. Like the family of Atkinson indices, they produce a family of inequality measures, with each inequality measure forming a complete ordering of distributions and providing necessary conditions for SSD(2 ). Yitzhaki (1979, 1982a) supplied theoretical justification of the family as being based on the theory of relative deprivation, which is an alternative to the social welfare function. In an unrelated development, several authors have discovered the possibility of expressing the Gini Mean Difference (GMD) as a covariance between the variate value and its cumulative distribution(3 ). (Stuart, 1954; Pyatt, Chen, and Fei, 1980; Lerman and Yitzhaki, 1984). The covariance is a powerful tool with many good properties that allow the decomposition of the Gini of a sum of random variables into the contributions of the Gini’s of the components (plus some extra terms), which is a basic requirement for regression analysis and Analysis of Variance (henceforth ANOVA). The implication of the discovery is that one can easily define the equivalents of covariances and correlations based on the GMD and the EG. The application of this finding to the extended Gini has been developed by Schechtman and Yitzhaki (2003). In some sense, it means that (almost) every econometric text-book can be replicated with the EG family of measures of variability. The connection with the Lorenz curve and concentration curves will allow testing for monotonicity of the relationship between variables. The covariance presentation can be helpful in turning the Gini and the EG from measures of variability into a powerful tool for statistical analysis. However, this is infant research and all we can do is review the stateof-the-art, and point out possible directions for future research. As a result of these developments, the analytical capabilities of Gini’s mean difference, the Gini coefficient, and the EG have been greatly improved, not only as statistical measures but as representing additional ways for social modeling. To attest that: (a) the EG is a popular measure of inequality that is compatible with several social theories (Runciman’s relative deprivation, 1966), Yaari’s dual approach to inequality measurement and to decision-making under risk, 1987, 1988). (b) It is a statistical measure of variability with many good properties, such as robustness and decomposability, and (c) it is a member of a large family of inequality measures, which enable the user to check the sensitivity of his/her conclusions with respect to the measure of variability used. In a recent paper published in Metron by one of the authors (Yitzhaki, 2003), the properties of Gini’s Mean Difference (hereafter, GMD) were surveyed and confronted with the properties of the variance. The survey has revealed the richness of presentations of the GMD, and the advantages of using it when (2 ) There are several variations of the EG, the most notable is Chakravarty (1988, 1990). Lambert (2001) offers an excellent survey of the literature. (3 ) Gini’s Mean Difference is the expected difference between two random draws from the distribution. The Gini coefficient is one half of the GMD, divided by the mean of the distribution.

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the underlying distributions are not normal. The aim of this paper is to survey another advantageous property of the GMD. It belongs to a family of variability and inequality measures, so that one can always have the ability to examine how sensitive the results are to the use of the GMD as a measure of variability. This paper is a follow up of the survey on the GMD and covers the same topics. Our main interest is to use its decomposability properties so that we will be mainly interested in using its covariance presentation. Also, the focus is on the descriptive properties. Therefore, several important topics such as estimation procedures, inference, variations of the extended Gini and many other topics are not covered in this survey. We assume throughout that the reader is familiar with those properties of the GMD surveyed in Yitzhaki (2003). However, since the research on the GMD is much more mature than the research on the EG, in many instances we will have to present half-baked ideas. The structure of the paper is the following: Section 2 presents alternative ways of “spelling” the extended Gini for continuous distributions, and presents the metric of “congested city block” which is the metric behind the EG. Section 3 comments on some divergence between the different presentations when dealing with discrete distributions. Section 4 presents the equivalents of covariances and correlations and their properties. Section 5 discusses the decomposition of the EG of a sum of random variables into the contributions of the variables involved, and Section 6 discusses the problems encountered in trying to replicate ANOVA with the EG. Section 7 concentrates on the relationship with stochastic dominance, Section 8 is concerned with applications to welfare economics, while Section 9 points out relationship with regression analysis. Section 10 concludes.

2. Alternative ways of spelling the extended Gini-continuous distributions 2.1. The basic presentations Yitzhaki (1998) lists more than a dozen alternative presentations of the GMD. Those presentations tend to have different properties and to behave differently when one tries to extend their use to issues that are beyond the simple issue of representing the GMD. Fortunately, not all the presentations of the GMD have counterparts in the extended Gini, hence the task of describing the different presentations seems to be simpler for the EG than in the case of GMD. On the other hand, the EG formula is more complicated than the GMD, so the presentations are more complicated. The discussion in this section is limited to continuous distributions with finite bounds. Some of the results hold for distributions with no bounds, but we will not elaborate on them, while the

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adjustment to discrete distributions will be handled in Section 3. The extended Gini family of measures of variability depends on one parameter, ν. In the four subsections below we will give four alternative ways to present the extended Gini. 2.1.1 – The area-based definition The extended Gini can be written as the weighted area enclosed between the absolute Lorenz curve and the diagonal. EG(X, ν) = ν(ν − 1) where A( p) =

Xp

−∞

0

1

(1 − p)ν−2 ( pµ X − A( p))dp

x f X (x)d x and X p is indirectly determined by p =

(2.1)

Xp

−∞

f X (x)d x.

A( p) is the absolute Lorenz curve (Shorrocks (1983) refers to it as Generalized Lorenz curve), pµ X is referred to (Yitzhaki and Olkin, 1991) as the Line of Independence (LOI) or egalitarian line and the parameter ν(> 0) is a parameter determined by the investigators(4 ). Special cases of interest are: — ν = 2 is Gini’s Mean Difference (actually one half, but constants will be ignored). — ν → ∞ reflects an attitude toward inequality that reflects the attitude of a Max-Min decision maker(5 ). — ν → 1 reflects an attitude of someone who does not care about inequality, that is, the implied assumption is a constant social evaluation of the marginal utility of income. —- 0 < ν < 1 reflects an attitude of an inequality lover. The variability (inequality) index in this case is negative, and using it may complicate the presentation. Hence in the rest of this paper we will assume that ν > 1 although many of the results we report can be applied without modification to an inequality loving case. In the literature of income distribution, the index used is normalized by dividing (2.1) by mean income. In Finance and Econometrics, the absolute version (2.1) is the one that is used. We will refer to the former as the EG coefficient and to the latter as the EG index of variability. The difference between them is identical to the difference between the coefficient of variation and the standard deviation. (4 ) See Aaberge (2000), and Kleiber and Kotz (2002) on additional connections between the Lorenz curve and extended Gini. (5 ) The terminology is borrowed from the income distribution field. For application in Finance, the term “inequality” should be substituted with the term “risk”.

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2.1.2 – The covariance-based presentation The extended Gini index can be written as a covariance between the variate and a power function of its cumulative distribution. Specifically: EG(X, ν) = ν COV(X, −[1 − FX (X )]ν−1 ) ,

(2.2)

where FX (X ) is the cumulative distribution. The advantage of the covariance formula is that one can extend it quite easily to define the Gini covariance and Gini correlation. The latter can be used to decompose the EG of a sum of random variables to the contributions of the EG of each variable and the correlations, while other properties of the covariance may enable the imitation of ANOVA-like analysis(6 ). 2.1.3 – The EG as the expected value of the minimum of several i.i.d. variables The third type of presentation that is useful is the presentation of the EG index as the difference between the expected value of a variable and the expected value of the minimum of several i.i.d. variables. (Note that risk loving is not covered under this sub-section.) That is EG(X, ν) = E{X } − E{Min(X 1 , . . . , X ν )} ,

(2.3)

with ν restricted to be an integer. The usefulness of (2.3) is that the theory of extreme value distribution can be used. It is easy to see that the larger the value of ν, the larger is the value of the EG, for a given distribution. In some cases in the area of stochastic dominance it is convenient to use the mean minus the EG as the certainty equivalent of the distribution. In this case, CEG(X, ν) = µ − EG(X, ν) = E{Min(X 1 , . . . , X ν )} .

(2.4)

CEG is the certainty equivalent of the distribution(7 ). 2.1.4 – The integral version of EG (Dorfman, 1979) Writing (2.3) as an integral over cumulative distributions, one gets: EG(X, ν) =

a

∞

{[1 − FX (t)] − {[1 − FX (t)]ν }dt .

(2.5)

(6 ) The reader may be puzzled by the use of −[1 − FX (X )] instead of FX (X ) in (2.2). The reason is that the index is required to imitate the social welfare function, which is increasing and concave. In pure statistical use, one may prefer the alternative extension. (7 ) In Atkinson’s (1970) terminology, CEG is identical to Equally Distributed Equivalent, EDE.


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Unlike (2.3), ν can be a continuous parameter here. (2.5) is also useful when one wants to present the EG as a weighted sum of the sections of the range of the random variable. (dt is the section. The term in the curly brackets is always positive and represents the weight.) This presentation will turn out to be useful whenever one is interested in checking the monotonicity of relationship between random variables (Section 7 and on). 2.1.5 – A peculiar property We will end this sub-section by pointing out a property that seems to be unique to the EG, which makes it in some sense counter-intuitive. Provided that the distribution of X is non-symmetric around its mean, then EG(X, ν) = EG(−X, ν) for ν = 2 .

(2.6)

The intuitive explanation to property (2.6) is that since the weighting scheme is not symmetric around the mean – the variability measure is sensitive to the direction of the a-symmetry of the distribution. This property will be important in understanding the differences between EG decompositions and variance decompositions, and in understanding the range of the EG correlations. 2.2. The norm (metric) 2.2.1 – Three observations The aim of this section is to show that each EG, like the variance and the GMD, is based on two kinds of principles: one is the metric (or the norm) that is behind the index, while the other is the averaging operator (see equation (2.3)). To demonstrate the norm, the easiest way is to start with three observations. Since all EG indices are not sensitive to adding or subtracting a constant from all observations, it is convenient to look at the differences between successive observations. Let x1 , x2 , x3 be three observations. Let δ1 = x[2] − x[1] ≥ 0 be the difference between order statistics 2 and 1, and δ2 = x[3] − x[2] ≥ 0, then one can plot the equal-variability curves for each EG. It is demonstrated in Yitzhaki (2003) that the shape of equal-variance is a (quarter of a) circle, while the equal-GMD curves portray a straight line whose slope is 45 degrees. That is, the variance relies on the Euclidean norm, while the GMD is based on “city block” norm. It is easy to see that all the equal-EG curves are based on straight lines, with the slope being dependent on ν. The higher ν, the steeper is the line. That is, the extended Gini places a lower weight, the higher is the

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rank of the observation. To see that: using (2.5), we can write the equal EG curve as: (2.7) c = [(2/3) − (2/3)ν ]δ1 + [(1/3) − (1/3)ν ]δ2 where c is a given constant. Then the slope of the equal-EG curve is: dδ2 1 − (2/3)ν−1 = −2 . dδ1 1 − (1/3)ν−1 For ν = 2 (the GMD), the slope is −1, while for ν = 3, the slope is −(5/4). That is, the trade-off between distances between observations for the GMD is equal, while for ν = 3, the trade-off between δ2 and δ1 is that an increase of one unit in the distance between x[2] and x[1] is equivalent in its effect on the measure of variability to an increase of 5/4 in the distance between x[3] and x[2] . In other words, the distance in the lower end of the distribution gets a higher weight than the distance in the upper end. The GMD is sometimes referred to as a “city block” norm. The reason is that its norm is equivalent to the distance between points when one is allowed to move vertically or horizontally, but not in a diagonal. Along the same line, the EG norm can be referred to as “hilly city block” or “congested city block”, where movements diagonally are not allowed, while movements vertically or horizontally have a constant ratio in terms of contribution to variability. Figure 1 illustrates “equal variability” curves for several choices of ν. Each line represents the locus of all points that make a given EG index. The EG is homogenous of degree one in δ1 and δ2 – therefore, changing the constant c, moves the equal EG lines in a parallel way(8 ). δ2

city block (EG):ν =3

D ·

city block (GMD):ν =2 Euclidean (variance)

1 B·

city block (EG):ν =0.5

·C A ·

1

δ1

Figure 1. Equal GMD, EG and Variance Curves

(8 ) The ν = 0.5 is intended to illustrate that one may apply the EG in cases where the interest is in convex functions. Note, however, that in this case the index of variability is negative and some of the presentations and arguments do not apply.


409

What norm should one use in the social sciences is beyond the scope of this paper(9 ). All we can say is that the answer is not a simple one. In the area of income distribution we can say that if one has a clear quantitative social view, then the task of quantifying the “social evaluation of the marginal utility of income” is easy and this is the parameter needed to determine the appropriate norm(10 ). The use of the norm (i.e., the ν selected) determines our social views, at least as it is reflected in the indices one uses. The other principle that is invoked is the principle of averaging. The variability index is a weighted sum of the distances between observations. The principle of averaging determines many of the properties of the estimators of the EG, which are beyond the scope of this paper. 2.2.2 – A general distribution When analyzing the weighting scheme, it is convenient to restrict the distribution of X to be in the range [a, b], and to approximate it by a discontinuous distribution. In this case, (2.5) becomes EG(X, ν) =

n

{[1 − F(xi )] − [1 − F(xi )]ν }δi

(2.8)

i=1

where δi = x[i] − x[i−1] and x[0] = a. Note that (2.8) is a weighted sum of δi , the sum of which is the range [a, b]. Equation (2.8) is an extension of (2.7) to n observations. It can be used to recover the ratio of substitution between the δi ’s along the “equal EG” curve. The weight attached to each δi is wi (x) = {[1 − FX (xi )] − [1 − FX (xi )]ν }. The overall index is a weighted sum of the distances between observations. It is interesting to note that the weight is determined by the rank of the variate in that section. To see how the weight changes along the distribution we investigate the weights as a function of F. ∂w(x) = ν[1 − Fx (x)]ν−1 − 1 ; ∂ FX (x) ∂ 2w = −ν(ν − 1)[1 − Fx (x)]ν−2 ≤ 0 ∂ 2 F(x)

(2.9) for ν > 1 .

For the GMD (ν = 2), the first derivative is [1 − 2F(x)] so that the weight increases till the median and then declines symmetrically. The larger the ν, (9 ) It is worth noting that the norm of the EG index is totally different from the norm of the EG coefficient. However, it can be easily discovered by applying the same approach. The reason for the difference in the norms is that along the equal EG curve, the expected value of the distribution may change, and the expected value is a parameter which affects the EG coefficient. (10 ) The equivalent term in finance is the notion of risk aversion.

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the lower is the quantile with the maximum weight. To see this, set the first derivative to zero and solve for F with the maximum weight. F ∗ = 1 − (1/ν)1/(ν−1) .

(2.10)

Table 1, taken from Yitzhaki (1983, p. 623) presents F ∗ as a function of ν(11 ). Table 1: The rank with the maximum weight as a function of ν. ν

0.001

0.5

1

1.5

2

5

1000

F∗

0.999

0.75

0.63

0.56

0.5

0.33

0.007

As can be seen, the rank with the maximum weight decreases with ν. The weight increases till it reaches the maximum and then declines(12 ). An additional view of the change in the weighting scheme is to look at the change in the weight when ν is changed. To see that, we investigate wi (x) of equation (2.8) as a function of ν. ∂w(x) = − ln[(1 − F(x)][1 − F(x)]ν > 0 . ∂ν

(2.11)

As can be seen, the weights increase with an increase in ν. The second derivative is: ∂ 2 w(x) = − ln2 [(1 − F(x)][1 − F(x)]ν < 0 . ∂ 2ν This means that the rate of increase in the weight declines with ν. As a result, µ − EG(X, ν), which can serve as the certainty equivalent of the distribution (defined in Section 2.1.3, and discussed in Section 7), declines with ν. In applications in welfare economics, finance and regression, higher ν can be interpreted as a higher stress on the lower part of the distribution. The importance of the weighting scheme is that it determines our implicit social views. (See Yitzhaki (1996) for elaborations). (11 ) Equation (2.10) is not defined for ν = 1. However, one can evaluate the number it converges to. (12 ) The property that the farther the δi from the middle the lower the weight may seem at a first glance as contradicting the sensitivity of the EG index to mean-preserving spread. To see that this observation is incorrect note that a mean preserving spread means an increase of the δ at the middle of the distribution while decreasing those that are far from the middle. Hence, a mean preserving spread will increase the EG.


411

Finally, it is worth investigating the cross derivative ∂ 2 w(x) = [1 − Fx (x)]ν−1 + ν[1 − Fx (x)]ν−1 ln(1 − F(x)) ∂ FX (x)∂ν = [1 − F(x)]ν−1 [1 + ν ln(1 − F(x))] , which indicates a turning point in the derivative. Increasing ν does not imply increasing the weights everywhere. We will return to the implication of this point in Section 7(13 ).

3. Adjustment required for discrete distributions When the distributions are discrete with different probabilities, then the discrete versions of equation (2.1) and equation (2.2) cease to be identical. In other words, the area enclosed between the Lorenz curve and the egalitarian line ceases to be equal to the covariance presentation of the Gini. Clearly, the extensions to EG’s cease to be identical. The intuitive explanation is the following: when the distribution is discrete, the usual practice is to present the cumulative distribution as a step function (continuous from the right). On the other hand, in terms of areas, the usual practice is to obtain a Lorenz curve of a discrete distribution as a piecewise linear curve by connecting the points by straight lines (Gastwirth, 1971, 1972). This, in turn, implies that one is treating the cumulative distribution as continuous. This is the source of the discrepancy. The aim of this section is to reconcile those differences by applying an adjusted covariance formula for achieving identity between the two presentations. That is, the aim of this section is to show that one can still define the EG as a weighted sum of the area between the 45-degree line and the Lorenz curve and apply a covariance formula, similar to (2.2), to calculate it or to use the properties of the covariance in manipulating it. To perform the adjustment, the cumulative distribution function F is replaced by another function of X , to be defined below. The discrete cases for Gini’s Mean Difference and Gini coefficient (i.e., for ν = 2) were discussed in Lerman and Yitzhaki (1989). Let X [1] , . . . , X [n] be the ordered observations, having weights p1 , . . . , pn , respectively. Let Fi∗ = F ∗ (X [i] ) =

i−1 j=1

pj +

pi . 2

(3.1)

(13 ) This property implies that an increase in ν cannot be interpreted as greater concavity of the social welfare function everywhere, but that increasing ν increases the concavity at the lower end of the distribution while decreasing it at the upper end. The family of Atkinson’s indices suffers from the same property.

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Then, a covariance-based Gini, that is: the discrete version of (2.2), with F ∗ replacing F, is given by EG(X, 2) = 2 cov(X,

FX∗ (X ))

=2

n

∗

pi (X [i] − X )(FX∗ (X [i] ) − F )

(3.2)

i=1

where F¯ ∗ =

n

pi Fi∗ .

i=1

The extension to a general ν is discussed in Chotikapanich and Griffiths (2001), and the connection to a covariance-based representation is given in Schechtman and Yitzhaki (2005, Lemma 3.1) as follows: EG(X, ν) = −ν cov F (X, (1 − F ∗ (X ))ν−1 ) ,

(3.3)

where F ∗ is defined implicitly by (1 − Fi∗ )ν−1 =

(1 − p1 − . . . − pi−1 )ν − (1 − p1 − . . . − pi )ν . νpi

(3.4)

It is interesting to note that EG(X, ν) can also be represented as the difference between two covariances, each involving the original (discrete) F, as follows; EG(X, ν) = covU (X, Z i ) − covU (X, Z i−1 ) ,

(3.5)

where covU (, ) means calculating the covariance as if the distribution is uniform (i.e., ignoring the probabilities when calculating the covariance), while Z i = (1 − p1 − . . . − pi )ν and Z i−1 = (1 − p1 − . . . − pi−1 )ν .

(3.6)

The proof of (3.5) can be easily grasped if one moves the denominator of (3.4) to the left hand side of the equation, and substitutes the left hand term by the right hand term in the covariance formula. Chotikapanich and Griffiths (2001) present similar solutions to the problem, and offer numerical evaluation of the differences. Note that if one starts by defining the EG using the covariance formula, then no adjustment is required for discrete distributions. In the rest of the paper we will consider continuous distributions, ignoring the adjustment needed if one wants to define the EG using the Lorenz curve. Finally, it should be emphasized that at this stage it is not clear to us which adjustment is the “correct” one. It may turn out that in different applications there may be an advantage to one formulation over the other. For example, further research is needed in order to determine which is the appropriate version for constructing necessary conditions for stochastic dominance rules, while for purposes of decompositions, yet another version is appropriate(14 ). (14 ) The same problem appears when one defines the EG equivalents of covariance, especially when one variable is discrete and the other is continuous.


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4. Extended Gini equivalents of covariances and correlations The covariance formula for the family of EG’s is given in (2.2). It can be extended to produce the equivalents of the Gini covariance, the Gini correlation, and the (simple) Gini regression coefficients. The importance of the Gini covariances and correlations arises when a decomposition of the extended Gini of a sum of random variables is desired and when one is interested in the association between random variables. As in the case of the variance of a sum, the terms involved are the individual variances, the covariances and/or the correlation coefficients. (See discussion in Section 5.) It is worth noting that (2.2) is not the only way to define the covariances and correlations, but it is the most convenient one. However, one could develop it using the other definitions as well. Define (4.1) EG(X, Y, ν) = ν COV(X, −[1 − FY (Y )]ν−1 ) as the family of EG covariances between X and Y , depending on one parameter ν. The family of Gini correlations between X and Y is defined as: ζ (X, Y, ν) =

COV(X, −[1 − FY (Y )]ν−1 ) . COV(X, −[1 − FX (X )]ν−1 )

(4.2)

ζ (Y, X, ν) =

COV(Y, −[1 − FX (X )]ν−1 ) . COV(Y, −[1 − FY (Y )]ν−1 )

(4.3)

Similarly,

Note that one variable is taken in its variate values, while the other is represented by its cumulative distribution function. Therefore, it is expected that the Gini correlation coefficient will be similar in its properties to Pearson correlation with respect to the variate-valued variable, and similar to Spearman rank correlation coefficient with respect to the variable which is represented by its cumulative distribution function only. The family of EG correlations has the following properties: 1. Let H (X, Y ) be the joint cumulative distribution, and FX (X ) and FY (Y ) be the marginal ones. Then for every H (X, Y ) and for every ν, ζ (X, Y, ν) ≤ 1, for all X , Y . Note that the common lower bound (−1) for correlation coefficients does not exist. (See discussion and example below). 2. If Y is a monotone increasing function of X , then ζ (X, Y, ν) = 1 for all ν. 3. If X and Y are independent, then ζ (X, Y, ν) = ζ (Y, X, ν) = 0 for all ν. 4. ζ (X, Y, ν) is invariant under all strict monotonic increasing transformations of Y . That is, with respect to Y , ζ (X, Y, ν) has properties that are similar to Spearman correlation coefficient.

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5. Let (X, Y ) have a bivariate Normal distribution with correlation coefficient ρ, then ζ (X, Y, ν) = ζ (Y, X, ν) = ρ, for all ν. The fact that the extended Gini correlation is not symmetric in X and Y appears (and causes some difficulties) in the decomposition of the extended Gini of a sum of random variables. Therefore, it is of importance to see under what conditions the two correlations are equal. 6. Invariance under exchangeability. Let (X, Y ) be exchangeable up to a linear transformation. Then, ζ (X, Y, ν) = ζ (Y, X, ν) for all ν. An alternative sufficient condition for the equality of the two EG correlation coefficients can be formed using concentration and Lorenz curves. It seems to be different than exchangeability up to a linear transformation and further research is needed for comparing the conditions. Since it is harder to interpret the latter condition than the one that is based on the covariance formula – we will only illustrate its existence by geometric arguments(15 ). Starting from equation (2.1), the Gini covariance can be defined as the area between the absolute concentration curve of Y with respect to X and the diagonal (Blitz and Brittain, 1964). Consequently, the EG covariance is a weighted sum of this area, with the weights similar to those of the EG. Hence, the EG correlation is the ratio of the weighted area enclosed by the concentration curve divided by the weighted area enclosed by the appropriate absolute Lorenz curve. Equality of the correlation coefficients (between X and Y and between Y and X ) can then be defined in terms of the ratios of areas enclosed between concentration curves to Lorenz curves. The only (non-trivial) problem is to interpret the properties of the distributions that will lead to this equality. Exchangeability up to a linear transformation should lead to that equality, but it may be that other properties can guarantee it as well. Although the extended Gini correlations seem to have properties which are a mixture of Pearson and Spearman correlations, there is one familiar property that is missing from properties (1)-(6): the EG correlation coefficients are NOT bounded from below by −1 (except for the special case with ν = 2). To see this, let Y = −X . For this case, a Gini correlation (between X and Y ) of (−1) implies that cov(X, F(X )) = cov(X, F 2 (X )), which is generally not true. For example, let F(x) = x 2 , 0 ≤ x ≤ 1, then cov(X, F(X )) = 1/15, while cov(X, F 2 (X )) = 4/63. (See details in Schechtman and Yitzhaki, 2003.) The lower bound depends on the underlying distribution of X (the variable which is taken in its variate values) and on ν – the larger the value of ν the lower is the bound. For example, when X has an exponential distribution with scale parameter of 1, then the bound is −5/4 for ν = 3, and smaller (= −13/9) for (15 ) See Schechtman and Yitzhaki (1999, 2003), Shalit and Yitzhaki (1994) and Mayshar and Yitzhaki (1995) for the exact derivation.


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ν = 4. The bound is given by several alternative ways, as follows:

ζ (X, Y, ν) ≥ =

− EG(−X, ν) F(x)(F ν−1 (x) − 1)d x = ν−1 ((1 − F(x))(1 − (1 − F(x)) )d x EG(X, ν)

µ − E max(X 1 , . . . , X ν ) . µ − E min(X 1 , . . . , X ν )

The expression on the far right-hand side applies only when ν is an integer. The lower bound is obtained when Y = −X . There are two special cases for which the lower bound of (−1) is obtained. The first case is when X comes from a symmetric distribution, and the second is when ν = 2 (i.e., the case of Gini’s mean difference). Further research is needed to understand and interpret the implication of this property.

5. The decomposition of EG of a sum of random variables An important property in any analysis is decomposability. In ANOVA, the basic concept lies in the partitioning of a sum of squares into several sums of squares. In regression, the total variability in the data is partitioned into the variability explained by the model, and the remainder (the error). In this section we deal with the decomposition of the variability measure of a linear combination of random variables into the individual contributions, plus some extra terms to be discussed below. In Section 6 we shall deal with the decomposition of the variability measure in a population into the contributions of the sub groups comprising the population, plus extra terms. In both sections, we start with the well-known and widely used measure of variability, namely the variance, and only then we introduce the decomposition of the extended Gini. n αi Yi be a linear combination of Y1 , . . . , Yn , where αi are Let Y0 = i=1 arbitrary constants. The variance of Y0 is given by: σ02

= Var

n

αi Yi =

i=1

n

αi2 σi2

+2

n

αi α j ρi j σi σ j .

(5.1)

i=1 j>i

i=1

As can be seen, the variance of the linear combination involves the individual variances and the Pearson correlations between each pair of the Yi ’s (not including Y0 ). The special case in which the Y ’s are uncorrelated provides a much simpler decomposition: σ02 = Var

n i=1

αi Yi =

n i=1

αi2 σi2 .

(5.2)

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The decomposition of the EG of a linear combination of random variables is more complicated than the above decomposition. However, it gives more information on the underlying distributions. It can be shown that it has the following representation (details can be found in Schechtman and Yitzhaki (2003)).

n αi Yi , where Y1 , . . . Yn are drawn from a multivariate Claim. Let Y0 = i=1 distribution with finite second moments. Then

EG20 = EG0

n

αi Di0 EGi +

i=1

+

n n

n

αi2 EGi2

i=1

(5.3)

αi α j [ξ(Yi , Y j , ν) + ξ(Y j , Yi , ν)] EGi EG j

i=1 i< j

for all ν, where ζ (Yi , Y j , ν) is the extended Gini correlation between Yi and Y j , and Di0 = ζ (Yi , Y0 , ν) − ζ (Y0 , Yi , ν). A closer look at this decomposition reveals that it involves the individual extended Gini’s and the extended Gini correlations. So basically, it does resemble the decomposition of the variance in structure. However, there is a very basic difference, which arises due to the fact that in contrast to Pearson correlation coefficient, which is symmetric in its arguments, the extended Gini correlation is asymmetric. As such, instead of simply having 2ρi j , we have ξ(X, Y, ν) + ξ(Y, X, ν). Also, an extra term appears, for each variable, which is Di0 = ζ (Yi , Y0 , ν) − ζ (Y0 , Yi , ν). The investigator is now faced with two issues. The first issue is – under what circumstances will the two decompositions have the exact same structure? In other words, what are the conditions for the extended Gini correlations to be symmetric? The second issue is – what can one learn from the fact that there are some additional elements in the extended Gini decomposition? A partial answer to the first question is given in Schechtman and Yitzhaki (2003): a sufficient condition for the extended Gini correlation to be symmetric in its arguments is that the two variables are exchangeable up to a linear transformation. Note that there are two types of Gini correlations involved in the above decomposition – correlations between Y0 and each of the individual Y ’s, and correlations between each pair of Yi ’s. Correlations between each pair of Yi ’s (i = 1, . . . , n) do not create a serious problem because one can define a new, symmetric EG correlation coefficient by averaging the two asymmetric EG correlation coefficients(16 ):

S(Y j , Yi , ν) = [ξ(Y j , Yi , ν) + ξ(Yi , Y j , ν)] 2 .

(5.4)

Inserting (5.4) into (5.3) changes the last two terms on the right hand side of (5.3) to be identical in structure to the right hand side of (5.1). The correlations between Y0 and each of the individual Y ’s create a much more complicated (16 ) See Yitzhaki and Olkin (1991) for various possible definitions.


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problem to resolve and at the same time may provide the clue as to why one may want to use the EG decomposition. The EG correlations between the sum and each variable give us information about whether the sum of the variables stays in the same family of distributions, or whether it moves toward a distribution with different properties. For example, if all the Yi ’s are coming from the multivariate normal, then the differences in EG correlations are zero and (5.3) will look like (5.1). (See property 5 of Section 4.) On the other hand, if all the Yi ’s are i.i.d. uniform variables, then although the variables are i.i.d. (5.3) will not look like (5.2) because of the extra term that will tell us that the distribution of the sum is not similar to the individual distributions. The importance of this property has to be explored in different applications. Following is an example, which illustrates one possible use of this property. Assume that the distribution of the linear combination of the Yi ’s is claimed to be converging to the normal distribution. However, whether it has converged for a specific n is yet to be determined. Let us divide the linear combination of the n variables to two linear combinations of equal size. Then, if the distribution of the linear combination of n/2 has already converged to normality we should expect (5.3) to converge to the structure of (5.1), with Y1 being the linear combination of the first n/2 observations, and Y2 – the linear combination of the last n/2. On the other hand, if convergence did not occur, (5.3) would have an extra term. As far as we can see, this is only one possible illustration. Further research is needed in order to determine the importance of this property. 6. Decomposition of EG of a composite population into the contributions of sub-groups Whether one can decompose the GMD and the Gini coefficient of an overall population into the contributions of the different sub-populations is the subject of a controversy in the income distribution literature. One side (see, for example, Shorrocks, 1980, 1984; Shorrocks and Wan, 2004; Cowell, 1995) forcefully argues that the GMD and the Gini coefficient are not additively decomposable and hence their decomposition, whenever the distributions overlap(17 ), is meaningless, while many others (Rao, 1967; Silber, 1989; Deutsch and Silber, 1999; Yitzhaki and Lerman, 1991; Yitzhaki, 1994; Milanovic and Yitzhaki, 2002) offered what they considered as meaningful, nonadditive decompositions. Frick et al. (2006) offer a decomposition of the GMD, (ANoGi- Analysis of Gini), which is comparable to the analysis of variance – ANOVA, and collapses into the structure of ANOVA whenever there is no overlapping. Essentially it has a similar structure to the decomposition of the sum of variables: given that the distributions of the sub-populations do not overlap – then the decomposition (17 ) Overlapping between two distributions occurs whenever the ranges of the distributions intersect.

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of the Gini is identical in structure to ANOVA. That is, one can decompose the overall GMD (or Gini coefficient) into two terms: one measures the intragroup variability, and the other measures the inter-group variability. On the other hand, if the distributions do overlap, then two additional terms are introduced into the decomposition: one indicates the effect of the overlapping on the intra-group variability and the other, on the inter-group variability. It is argued that these additional terms add important additional information about the underlying distributions. Also, as will be argued later, they play an important role in quantifying the theory of relative deprivation (Section 8). As far as we know, a straight-forward application of the Frick et al. (2006) ANoGi decomposition to the EG does not exist. To see this, note the following: Let Yi , Fi (y), f i (y), µi , and pi represent the income, the cumulative distribution, the density function, the expected value, and the share of sub-population i in the overall population, respectively. The overall population is composed of the union of the sub-populations. That is: Yu = Y1 ∪ Y2 ∪ . . . ∪ Yn . Note that Fu (y) =

i

pi [1 − Fi (y)] or, in other words

1 − Fu (y) = 1 −

pi [1 − Fi (y)] .

(6.1)

i

That is, the cumulative distribution (or ranks) of the overall population is the weighted average of the cumulative distributions of the sub-populations, weighted by the relative sizes of the populations. Applying (6.1) to the covariance formula of the GMD is the key starting point of the decompositions of Yitzhaki (1994) and Frick et al. (2006). However, since [1 − Fu (y)]ν =

n

pi [1 − Fi (y)]ν

i=1

the decomposition of the EG to the contributions of the sub-populations is not a straight-forward generalization of the GMD decomposition. Further research is needed to shed light on this issue.

7. Stochastic dominance Decision-making under risk requires the knowledge of both the utility function of the decision-maker and the distribution of the rewards. Given enough repetitions, the data generating process can be estimated, but the utility function


419

remains a black box. To be able to predict behavior without a specific knowledge of the utility function is an important step toward a meaningful theory. In general, one can either impose mild restrictions on the utility functions, or on the data generating process, or on both. Expected utility theory is a major theoretical paradigm in finance. (See Levy, 1992, for a review). Markowitz’ (1952, 1970) portfolio theory and the literature that follows it, which is the most practical innovation in finance, summarize the distribution of the portfolio by the mean and the variance. To make the two theories compatible one has to either restrict the distribution to the normal or other specific distributions, or, alternatively, to restrict the utility function to be a quadratic function(18 ). Stochastic dominance imposes some mild restrictions on the class of utility functions, and derives results that hold for all distribution functions (Hadar and Russel, 1969; Hanoch and Levy, 1969). Definition. Given two wealth (or income) distributions, X and Y , we say that X dominates Y according to Stochastic Dominance of degree n if E{U (X )} ≥ E{U(Y)} for all utility functions with the first n derivatives having a given sign(19 ). The basic SD rules are first-order stochastic dominance and second-order stochastic dominance for concave utility functions. Since our major interest is in the role of the GMD and EG, we restrict the discussion to SSD rules for concave functions. Proposition 7.1. A necessary and sufficient condition for X to dominate Y according to SSD rule is that the absolute Lorenz curve of X will be not lower than the absolute Lorenz curve of Y . On the other hand, if the absolute Lorenz curves intersect, then there exist two legitimate utility functions that will rank X and Y in an opposite order. (Shorrocks, 1983) (20 ). Variability measures are summary statistics. Hence, without restrictions on the distribution involved they are not capable of forming necessary and sufficient conditions. However, they are capable of forming necessary conditions. Proposition 7.2. A set of necessary conditions for X to dominate Y according to SSD is: µ X ≥ µY and µ X − ν cov(X, −[1 − F(X ))]ν−1 ) ≥ µY − ν cov(Y, −[1 − F(Y ))]ν−1 ). (Yitzhaki, 1982b, 1983). Proposition 7.2 implies that one can form necessary conditions for SSD with any EG. This is one of the advantages of the EG over the variance. The implication of Proposition 7.2 is that if one constructs portfolios according (18 ) There are extensions that allow compatibility with a two parameter distributions, but they do not change the nature of the problem of incompatibility between the two approaches. (19 ) The signs depend on whether the utility function is assumed to be convex or concave. See Yitzhaki (1999) for the role of GMD in forming necessary conditions for SSD for convex functions. (20 ) Shorrocks refers to the absolute Lorenz curve as Generalized Lorenz curve.

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to mean and EG then the efficient set of the Mean-Extended Gini (MEG) portfolio, adjusted according to 7.2, is included in the efficient set of SSD portfolios. Hence, it is impossible to prove that an advice given according to MEG and 7.2 is not compatible with the preference of at least one investor. (Shalit and Yitzhaki, 2003). Proposition 7.2 can be extended in two directions. The first is to limit the distribution functions involved so that necessary and sufficient conditions for SSD can be formed. The other direction is to develop necessary, and necessary and sufficient conditions for SSD on conditional distributions, using the EG covariances and correlations. The necessary and sufficient conditions for SSD are the following: Proposition 7.3. Provided that F(X ) and F(Y ) intersect at most once, then µ X ≥ µY

and

µ X − ν cov(X, −[1 − F(X ))]ν−1 ) ≥ µY − ν cov(Y, −[1 − F(Y ))]ν−1 )

(7.1)

for ν > 1 is a set of necessary and sufficient conditions for SSD. (Yitzhaki, 1983). It is worth pointing out that even if (7.1) holds for every ν > 1, it does not form a sufficient condition for SSD. (For an example – see Yitzhaki, 1983). Unfortunately, stochastic dominance rules can only be tractable among univariate distributions. In order to apply the stochastic dominance in a setting which may interest economists, the SSD approach should be redefined in terms of choices to be made among alternatives that are easily identified. Instead of trying to define it in a multi-variate environment, it turns out that it can be developed under limited conditional settings. To derive the additional extension one needs the following definitions: n n wi X i , where i=1 wi = 1 and the wi are given weights Let Y = i=1 (sometimes an additional constraint, that the weights are positive, is added). Assume now that a new distribution is created by reducing one weight and increasing another. That is, an additional constraint is added. dwk + dw j = 0 .

(7.2)

Definition. We say that variable k dominates variable j, according to the Marginal Conditional Stochastic Dominance criterion (MCSD), if the distribution produced by applying (7.2) SSD dominates the original distribution. (Shalit and Yitzhaki, 1994). Proposition 7.4. Variable k dominates variable j according to the MCSD criterion if and only if the absolute concentration curve of k is not lower than the absolute concentration curve of j. (Shalit and Yitzhaki, 1994).


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In this paper we are interested in the extended Gini. This family provides the following set of necessary and sufficient conditions for MCSD. Proposition 7.5. Any ν > 1 can provide a set of necessary and sufficient conditions for MCSD between X k and X j provided that: µk ≥ µ j

and

µk − ν cov(X k , −[1 − F(Y ))]ν−1 ) ≥ µ j − ν cov(X j , −[1 − F(Y ))]ν−1 ) , where Y =

n i=1

wi X i ,

n i=1

(7.3)

wi = 1.

Note that the right hand term in (7.3) can be rewritten as: µ j − ν cov(X j , −[1 − F(Y ))]ν−1 ) = µ j − β j (ν) cov(Y, −[1 − F(Y ))]ν−1 ) (7.4) cov(X j , −[1 − F(Y ))]ν−1 ) where β j (ν) = . cov(Y, −[1 − F(Y ))]ν−1 ) The term β j (ν) is defined in Olkin and Yitzhaki (1992) as Gini’s simple regression coefficient, to be described in Section 8. Using (7.4), Proposition 7.5 can be interpreted in terms of the EG regression coefficient. β(ν) can be interpreted as the “beta” in the Capital Asset Pricing Model (CAPM) literature (Shalit and Yitzhaki, 2002; Gregory-Allen and Shalit, 1999). It should be clear that propositions (7.1) to (7.5) represent the top of the iceberg. Restrictions on the Lorenz and concentration curves can supply necessary and sufficient conditions for higher degree dominance rules. Another theory of decision-making under risk, for which the EG can be used to form necessary conditions for dominance, is Yaari’s (1987) dual theory. To see this, it is sufficient to use equation (2.5) µ X − EG(X, ν) =

a

∞

[1 − FX (x)]ν d x ,

(7.5)

which is a special “utility” function according to Yaari. Hence, Proposition 7.3 can be reformulated to argue that portfolios constructed using the EG can form necessary conditions for SSD both according to expected utility and Yaari’s dual theory. This means that a portfolio that was constructed by minimizing the EG of the portfolio subject to a given expected return is in the efficient set of both the SSD and Yaari’s theories. The empirical literature, which uses MCSD rules and EG for inference, exists but it is beyond the scope of this paper. Another important area in which the EG is extensively used is hedging. It is beyond the scope of this paper. For a survey of the literature see Lien and Tse (2002).

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8. Applications in welfare economics Atkinson (1970) includes several contributions in the area of welfare economics, stochastic dominance and the combination of the two, and indirectly has also contributed to the development of the EG. The contribution in the area of stochastic dominance is the proof that SSD conditions can be presented in terms of Lorenz curves. Implicitly, this has led to the finding that the GMD can be used to develop necessary conditions for SSD. The contribution to welfare economics is by showing that whenever two Lorenz curves intersect, then it is impossible to rank distributions according to the expected social welfare without using a specific utility function(21 ). The extended Gini can be viewed as a reaction to (or imitation of) the Atkinson index. This section is divided into two subsections. The first part is actually a replication of Section 7, translated into the area of income distribution and evaluating tax reforms. Since this area requires specific knowledge and special terms in the field of taxation, we will not present the applications but refer the interested reader to articles in the field. The second subsection presents the relationship between the theory of relative deprivation and the EG. 8.1. Analyzing tax reforms The core of this section is based on propositions 7.1 to 7.5, adjusted to the field of income distribution. Also, it includes further developments that are specific to the field. 8.1.1 – Major adjustments: the extended Gini income elasticity As illustrated in the last section, one can analyze tax reforms using necessary conditions for stochastic dominance, or necessary and sufficient conditions. We start with the necessary conditions: Let Y = nk=1 X k , where Y is income while X k (k = 1, . . . , n) may represent sources of income (like wages, capital income, etc.) or alternatively outlays of expenditures such as bread, clothing, etc. The major adjustment needed between the application in finance to the applications in the field of income distribution is that while in finance the GMD and the absolute version of the EG are used, in the field of income distribution the Gini coefficient, and the extended Gini coefficient are used. The Gini coefficient is the absolute Gini of income normalized by its mean. The government controls taxes and prices. Consider the following exercise: assume that the government increases the price of commodity k by a small percentage e so that the new price changes from Pk0 to Pk (e) = Pk0 (1 + ek ), (21 ) See Zoli (1999) on the Gini and crossing Lorenz curves.


423

ek → 0, where Pk0 is the initial price of commodity k. Assume that we measure inequality in after tax real income by the EG, EG(ν). Then, one can ask what will be the impact of a small change in ek on inequality in real income. The answer to this question is given by the following formula (Lerman and Yitzhaki, 1985; Stark, Taylor and Yitzhaki, 1986, 1988). Formally, ∂ EG(y, ν) = EG(y, ν)Sk [η(ν, k) − 1] , ∂ek

(8.1)

where EG(y, ν) is the EG coefficient of after tax income, Sk = µk /µ y is the share of expenditure on k in after-tax income and η is the EG income elasticity. Specifically η(ν, k) =

β(xk , y, ν) cov(xk , −[1 − F(y)]ν−1 ) 1 = . ν−1 cov(y, −[1 − F(y)] ) Sk Sk

(8.2)

The numerator on the right hand of (8.2) is the Gini regression coefficient of commodity k on income, and as such, it estimates a weighted average of the marginal propensity to spend on commodity k. (The EG regression coefficient will be described in the next section). Hence, equation (8.2) can be interpreted as the marginal propensity to spend divided by the average propensity to spend, which is the definition of the income elasticity of spending on commodity k. It is referred to as the EG income elasticity because the weighting scheme used to estimate the marginal propensity to spend on k is derived from the EG coefficient (Yitzhaki, 1996). The EG income elasticity behaves like a regular income elasticity, and obeys all the rules governing the elasticity of a weighted sum. It enables one to determine whether a change in the tax on a commodity is regressive, neutral or progressive – where progressivity is determined according to the impact on EG in after tax real income. (i.e., whether the income elasticity is lower, equal or greater than one). As can be seen from equation (8.1), progressivity is determined by the EG income elasticity. Another crucial step in evaluating tax reforms is the impact on tax revenue. A first order approximation of the tax revenue gain or loss from the change in the tax rate is a function of ek and Sk . One can use this property to evaluate the change in several taxes that keep the tax revenue intact and increase/decrease inequality, or alternatively to find the change in the taxes that will keep inequality unaffected, while raising tax revenue. The analysis that uses the EG income elasticity includes the effect of targeting (i.e., when only a sub-group of the population is affected by the tax) and the effect of using grouped data. One can list over thirty papers that use the EG, including several country reports of the World Bank – hence the findings of those papers concerning income elasticity of many commodities in developing or developed countries can be the topic of

424


another survey. Wodon and Yitzhaki (2002) survey some of the papers and the approaches and the list mentioned there is a partial list of applications of the methodology. See also Duclos (2000) for an application with newly developed statistical inference procedures. 8.1.2 – Major adjustment: welfare dominance This subsection is devoted to the adjustment needed for the application of the necessary and sufficient conditions for SSD in welfare economics – referred to as welfare dominance. Propositions 7.1 to 7.5 can also be applied to welfare economics with two major adjustments. 8.1.2.1 – Introducing marginal efficiency cost of taxation In finance, the typical approach is to ignore transaction costs. A plausible explanation is that in a market economy, where transactions are done at the free will of the participants, the mere sign that a transaction is done indicates that the involved parties have benefited from the transaction. Hence we could write equation (7.2), which means a dollar withdrawn from asset k can be reinvested in asset j, ignoring transaction costs. In the area of taxation and welfare economics, the transactions of taxation are forced on the participants by the government, and therefore, transaction costs can be very high(22 ). A typical analysis cannot ignore the fact that as a result of changing the tax system, a dollar withdrawn from the tax of k may result in an amount that is larger or smaller than a dollar depending on the change in the excess burden of the tax system. The change in the excess burden is captured by the term “marginal efficiency cost of funds”. Mayshar and Yitzhaki (1995) introduce this term and use it to extend the analysis of Slemrod and Yitzhaki (1991), which was based on propositions (7.1) to (7.5) without adjustment. Lundin (2001) extended the analysis to include externalities. The marginal efficiency cost of taxation should have been used in forming of the necessary conditions too. However, as far as we know, it has not been done yet. 8.2. The theory of relative deprivation The concept of relative deprivation was put forward by Stouffer et al. (1949) in their monograph The American Soldier: Adjustment During Army Life, and the classic reference is Runciman (1966)(23 ). The main purpose is to explain (22 ) By transaction cost we include the cost to the society of imposing and collecting the tax, which includes the excess burden of taxation, and administrative costs. (23 ) Runciman (1966) wrote his book to explain another paradox – that according to his observations, in Britain, the lower the observed inequality, the higher the feelings of deprivation are.


425

when people feel deprived. Among economists, the concept has not been used extensively(24 ), and when it was used, there were several interpretations of it. We follow the interpretation developed in Yitzhaki (1979, 1982a) and Wodon and Yitzhaki (2005)(25 ). Unlike the social welfare function approach, which is concerned mainly with the decision-making process of the society, relative deprivation theory is interested in describing how people feel. There are two other major differences between the social welfare function approach and relative deprivation (henceforth RD): (a) the prevailing social welfare approach considers an individual who is maximizing an individualistic utility function, without externalities. Especially, the utilities of other individuals do not enter into the person’s utility function. In some sense, the individual in a social welfare framework is isolated from the society surrounding him, and the only influence of others is through the price system. Relative deprivation also assumes no externalities, and that the connection with others is through the price system, but the evaluation of the satisfaction of the individual from what he has is a function of the abundance of what he possesses in the society(26 ). This qualification introduces externalities in the feelings of the individual without affecting his consumption. (b) The social welfare function approach refers to the society as a whole, while RD allows for different reference groups. Handling of different reference groups is complicated mathematically and there are few underdeveloped examples (see Yitzhaki, 1982). Existence of different reference groups is the cause for the relativity of the concept, because changing the reference group may change the feeling of satisfaction or deprivation. The issue of introducing the reference groups into the analysis is mathematically complicated because of the large number of possible permutations and because there is no convincing theory about the formation of reference groups. In this survey, we ignore the formation of reference groups. Since we ignore reference (24 ) As noted by Pedersen (2004), a handful of economists have considered deprivation in applied work, including Layard (1980) on the role of status ranking in motivation, and members of the Leiden school for work on poverty and well-being (van Praag 1971; Kapteyn and Wansbeek 1982; van de Stadt et al. 1985). See also the axiomatic definitions of relative deprivation proposed by Podder (1996). There are two fields in which relative deprivation is extensively used: health and mortality (Deaton, 2001), and migration (Quinn, 2001; Stark, 1984; Stark and Taylor 1989; Stark and Wang, 2000; Stark and Yitzhaki, 1988). Bossert and D’Ambrosio (2005) offer dynamic extension to the theory. A newly interesting empirical approach is presented in D’Ambrosio and Frick (2004). (25 ) An alternative interpretation can be found in Hey and Lambert (1979); Berrebi and Silber, 1985; Bishop, Chakraborti and Thistle, 1991; Sen, 1973; Duclos, 1998). The major difference between the alternative approach and the one followed in this paper is that in the alternative approach individuals compare themselves to others, while in Yitzhaki’s approach each individual evaluates what he possesses. Duesenberry (1951), Frank (1985), and Kapteyn et al. (1980) present evidence on the dependency of preferences on others. For a survey of the issues involved, see Weiss and Fershtman (1998). (26 ) Each unit of income is evaluated according to its scarcity in the society, which can be interpreted as applying the principle of declining marginal utility at the social level. The relative “prices” are determined by the norm of the EG, as illustrated in Figure 1.

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groups – the theory is a theory of deprivation. Ebert and Moyes (2000) present axiomatic derivation for the relative deprivation function. Wodon and Yitzhaki (2005) present the following example to illustrate the difference between welfare economics and deprivation. Consider the decathlon, which is an athletic two-day event with ten different disciplines. A scoring table is used to award points for performance in each discipline, and the winner is the athlete with the highest total score after the ten events. To draw the analogy between the feelings of satisfaction from acheivements in the decathlon and market economy, the ten disciplines can be considered as commodities and the scoring system as prices. The utility function of each athlete may be defined over the physical units of achievements in every field. The first stage in constructing the analogy can be referred to as the micro-economic problem: each athlete allocates his/her practicing time in order to maximize his/her utility subject to a time constraint and the scoring structure (prices). A proper solution (in a competitive environment) is to allocate time so that for each athlete, the marginal cost of achieving each additional point is equal for the various events. The allocation of time and effort may also be interpreted as if each athlete was maximizing his/her points (income) subject to the time constraint. The second stage is the general equilibrium process, which determines the prices. To reach an “equilibrium”, the scoring structure is adjusted to avoid that any one discipline overshadows all others (in our analogy, this is akin to the demand and supply mechanisms in markets). The result of applying the prices (scores) to individual achievements to get the total score of the individual athlete is that we end up with points of equal value from the point of view of production (given the pricing system, each point requires an equal marginal cost or effort to produce). However, an athlete’s satisfaction depends not only on his/her total score, but also on the achievement of others; in other words, it depends on his/her ranking. Wodon and Yitzhaki (2005) show that given the distribution of abilities, and given Yitzhaki’s (1979) formalization of deprivation theory, the social evaluation for each athlete can be summarized by the rank of the athlete. Neither envy nor altruism is involved. The results are derived due to the need to be able to evaluate the achievement of the various athletes. Following those arguments Wodon and Yitzhaki (2005) show that the feelings of deprivation of each individual are a function of the level of income (scores) and her rank in the society, while average deprivation in the society can be summarized by µ∗ RG (ν), where RG(ν) is the extended Gini coefficient, i.e., the extended Gini divided by the mean. Average satisfaction in the society is summarized by µ∗ (1 − RG(ν)). As far as we can see, the decomposition of the extended Gini coefficient to the contributions of different groups can be used to analyze the implication of introducing reference groups into modeling relative deprivation.


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To understand the conceptual difference between the social welfare (SWF) approach and the deprivation approach – it is worth concentrating on the concept of income. Income, according to the SWF approach, is like a real physical quantity. It can be divided without affecting its size. On the other hand, the deprivation approach treats each unit of income as a different quantity whose value depends on its scarcity. A transfer of a dollar from one person to another may change the values attached to units of income.

9. Application to regression analysis The EG is a family of variability indices that is accompanied by the equivalents of the covariances and the correlations. Hence it should not be surprising that one can use it to derive regression coefficients. However, one has first to investigate the GMD based regressions that seem to be easier to handle. The properties of the simple Gini regression were investigated by Olkin and Yitzhaki (1992). It is found that since the GMD has two correlation coefficients associated with it, there are also two regression coefficients that can be related to it. One is identical in structure to the OLS regression coefficient, where each covariance is substituted by a Gini covariance and each variance is substituted by the appropriate GMD. The second is derived by minimization of the GMD of the error term. The former can be interpreted as a weighted average of slopes between adjacent points along the observations of the sample, while the latter can be interpreted as the weighted median of the same slopes. Also, the latter turned out to be what is known as R-regression (Jaeckel, 1972; McKean and Hettmansperger, 1978). However, R-regression was developed as a standalone regression, without association with the properties of the GMD. Further research is needed to see whether relating it to the Gini can add further insight into the regression. The extension of the OLS-type Gini regression to a multiple regression has been investigated by Schechtman and Yitzhaki (2005), while Schechtman and Yitzhaki (2004) extend the simple Gini regression to include an Instrumental Variable (IV) method. They show that the advantage of the IV Gini regression is in its connection to the Lorenz and concentration curves, that enables one to check whether the IV and the independent variable have a monotonic relationship or not. If they do not have a monotonic relationship, then a monotonic transformation of one of the variables involved may change the sign of the regression coefficient – both under OLS and Gini IV regressions. It also turns out that one can utilize the connection of the Gini covariance to the concentration curve to check whether a monotonic transformation can change the sign of the regression coefficient of both OLS and Gini regressions.

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Since the EG shares with the GMD the same basic properties that are required for regression, our conjecture is that there are also two simple regression coefficients associated with each EG. If all the pairs of EG correlations are equal, then it seems that the EG regressions are identical in structure to the OLS regression. The advantage of EG regression seems to be when the correlations are not equal. Also, one can use different EG’s to represent the variability of each independent variable, a property that can be used to non-parametrically investigate the curvature of the conditional regression curve (Schechtman, Yitzhaki and Artzev, 2005). The EG multiple regression is more complicated than the OLS. The reason is that on top of having two correlation coefficients that can differ in sign, one can use a different ν to represent the variability of each independent variable, and hence to stress different sections of the distributions of the independent variables in the regression curve. This will enable one to investigate the curvature of the regression curve conditional on keeping the weighting schemes of the regression with respect to other independent variables unchanged. As far as we know, no one has investigated the properties of the minimization approach to a regression while using the EG. It is clear to us that the EG family has the potential to contribute to regression analysis, but it is also clear that the possibilities of further research offered by the EG family are huge and the current research in this area is not mature enough to cover all the possibilities. Hence, we have chosen to point out to the reader some promising directions, but we do not feel we know enough to have definite conclusions.

10. Concluding remarks The EG family adds another dimension to the GMD – a family of measures of variability that enables the user of the GMD and the Gini coefficient to perform sensitivity analysis to check whether the results found are sensitive to the norm used by the GMD. The EG, like the GMD, adds new dimensions to the analysis whenever the underlying distribution is not multi-variate normal. In this paper, we have demonstrated that the EG can be useful in stochastic dominance, welfare economics, finance and tax reforms. Clearly, it is more complicated to work with than the variance. The decomposition properties of the family imply that (almost) every analysis that relies on the properties of the variance, can be replicated with additional insights, by the GMD and EG. In finance and welfare economics the ability to construct necessary conditions for stochastic dominance means that the method can be helpful when the variance fails to produce results that are compatible for economic theory. In the case of relative deprivation, the EG family produces an alternative approach to the social welfare function.


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Like the GMD, it has many properties that are under-investigated. We hope to have shown that although complicated, it is also promising.

Acknowledgments The authors thank Giovanni Giorgi and Peter Lambert for many helpful comments.

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SHLOMO YITZHAKI Central Bureau of Statistics Jerusalem (Israel) and Department of Economics The Hebrew University of Jerusalem 91905 Jerusalem (Israel) [email protected]

EDNA SCHECHTMAN Department of Industrial Engineering Ben-Gurion University Beer-Sheva (Israel) [email protected]