Almost Lorenz Dominance

Transcription

1 Almost Lorenz Dominance Buhong heng Department of Economics University of Colorado Denver Denver, Colorado, USA August 6 Abstract: This paper extends Leshno and Levy s () approach of almost stochastic dominance to inequality measurement and inequality orderings. We de ne and characterize the notion of almost Lorenz dominance (ALD) and apply it to the US income data. An income distribution almost Lorenz dominates another distribution when the Lorenz curve of the distribution lies almost everywhere but not entirely above the other Lorenz curve. We show that this condition is equivalent to requiring that almost all Gini type inequality measures rank the former distribution to have less inequality than the latter distribution. We further de ne an almost composite transfer (ACT) and show that ALD is equivalent to a sequential application of such transfers. The empirical application to the US income data ( ) demonstrates the utility of this generalized notion of inequality ordering. JEL Classi cations: D63 Key Words: income inequality measurement, Gini coe cient, almost Lorenz dominance, almost stochastic dominance, almost composite transfer. Acknowledgement: I thank the session participants of The 3th Meeting of the Society for Social Choice and Welfare (Lund, Sweden) for valuable comments and suggestions.

2 Almost Lorenz Dominance. Introduction In measuring and ranking income inequality, Lorenz curve is a most intuitive and powerful tool. It is intuitive since it displays the income share of the cumulative population from the poorest to the richest and, as such, it has a clear and meaningful economic interpretation. It is powerful since a dominance relationship between any two Lorenz curves (i.e., one curve lies nowhere below and somewhere above the other curve) implies a unanimous agreement by all transfer-sensitive measures of inequality on the ranking of inequality between the underlying income distributions. Consequently, there is no need for any summary measure of inequality once a dominance relationship is established. Lorenz curves, however, do often cross. For example, in their year-to-year comparison of the US income distributions between 967 and 986, Bishop et al. (99) documented that Lorenz curves cross in seven of the nineteen comparisons. A corollary of the Lorenz dominance theorem is that in the presence of Lorenz-curve crossing, either distribution can be ranked as being more unequally distributed by some inequality measures. The standard protocol in handling crossing Lorenz curves is to declare that the underlying distributions are non-comparable and any judgement on their inequality rankings is then muted. But not all Lorenz-curve crossings are created equal; in some cases the crossings are apparent and unmistakable but in others they are barely noticeable. For example, in Bishop et al. s study, the Lorenz curves for 984 and 985 cross at the 5 percentile and the discrepancy between the two curves is clear; but the Lorenz curve of 985 for the most part falls below that of 986 and moves slightly above the 986 curve only at the 9 percentile. For the comparison between 984 and 985, one may be convinced that it is hard to draw a de nite conclusion regarding the ranking of inequality. But for 985 and 986, many would be inclined to believe that most moderate inequality measures should judge 986 as being more equal than 985 in income distribution. Here the issue arises naturally regarding the characterization of a moderate inequality measure that renders the judgement. The dilemma that a minor crossing, no matter how slight, might prevent a de nite judgement is not unique to inequality ordering. In fact it is common to all rankings with partial ordering tools such as Lorenz curve. Fortunately the issue has recently In theory, one may choose to further cumulate the Lorenz curves in the hope that crossings may disappear and dominance emerges. This practice amounts to imposing an additional restriction on the underlying inequality measures (oli, ; heng, ). It is useful to point out that the intuitive interpretation of Lorenz curve (ordinate) is lost when Lorenz curve is cumulated. In empirical applications (as in Bishop et al., 99), this approach is rarely employed and Lorenz curve remains the most popular tool. Bishop et al. (99) indeed reached the same conclusion albeit they attributed the slight crossing to the sampling variability.

3 received attention and been addressed in the nance literature on the use of stochastic dominance in ranking investment portfolios. In an important contribution, Leshno and Levy () characterized the conditions under which the ranking of portfolios is valid for almost all investors despite the failure of stochastic dominance (i.e., the curves cross). The theory of stochastic dominance states that when the dominance relationship holds at a certain degree (e.g., rst, second and third) between two portfolios, all investors (represented by di erent utility functions) of a certain type will unanimously prefer the dominating portfolio. For example, at the second degree, all more-is-better and risk-averse investors will select the dominating portfolio. Leshno and Levy () persuasively argued that the requirement of all investors is unnecessarily demanding as it includes some extremely conservative investors who may even prefer to receive $ for sure in lieu of a 99 percent chance of wining ve million dollars. This type of investors or utility functions is pathological and is practically irrelevant in real-world decision-making since these preferences assign a relatively high marginal utility to very low values or a relatively low marginal utility to large values (Leshno and Levy, ). By excluding these extreme investors, Leshno and Levy () demonstrated the possibility for a unanimous agreement among the remaining investors. Since the conditions they established are for almost all investors, they refer to them as almost stochastic dominance even though now there is no dominance relationship to verify. In this paper, we extend Leshno and Levy s approach to the inequality ordering of income distributions. Given the close tie between stochastic dominance and Lorenz dominance (second degree stochastic dominance is equivalent to Lorenz dominance when the means of the distributions are equal), it is natural to consider this extension. Although one could use normalized stochastic dominance (i.e., apply stochastic dominance to normalized income distributions which would have the same mean income) to rank income inequality as proposed in Foster and Sen (997) and heng et al. (), Lorenz curve is a very di erent ranking device than the stochastic dominance curve. Compared with a stochastic dominance curve, a Lorenz curve has the advantage that it has a much clear interpretation; it is conveniently de ned over a nite interval which is more manageable (between and in population space instead of and in income space); and it leads naturally to a familiar class of rank-dependent inequality measures (Donaldson and Weymark, 98; Shorrocks and Slottje, ). Besides, Leshno and Levy s results do not seem to be readily generalizable to Lorenz dominance and to rank-dependent inequality measures. All these point to the need to treat Lorenz dominance with independent interest and characterize the condition under which an agreement can still be achieved for a set of moderate inequality measures even when Lorenz dominance fails. Following Leshno and Levy (), we also refer to the ranking condition characterized as almost Lorenz dominance or ALD for short. The class of inequality measures we consider is the familiar class of rank-dependent Gini type inequality measures. A Gini type inequality measure is de ned as a 3

4 weighted average of the distance between the Lorenz curve of a distribution and the diagonal line (representing the distribution of perfect equality). The advantage in focusing on this speci c class is that the condition we derive provides a clear and intuitive interpretation. For a Gini type measure, the condition of ALD is to impose a restriction on the weights. Although a weight close to zero is allowed in theory, it is considered extreme and pathological in practice. The exact restriction depends on the ratio of the areas of each Lorenz curve above the other. After this characterization, the paper goes on to de ne an almost composite transfer (ACT) that consists of a progressive transfer and a regressive transfer. Di erent from the familiar Shorrocks and Foster s FACT (favorable composite transfer) or oli s FCPT (favorable composite positional transfer), our composite transfer keeps neither variance nor Gini coe cient constant, rather it keeps the ratio between the two crossing areas constant. We show that a distribution almost Lorenz dominates another distribution if and only if the former can be obtained from the latter by a sequence of such composite transfers. Finally, the paper illustrates the utility of ALD by applying the ALD condition to the crossing Lorenz curves in Bishop et al. s study. The rest of the paper is organized as follows. Section is the main body of the paper which is divided into four subsections. In the rst subsection, we brie y review the almost stochastic dominance literature and the main results therein. We then de ne the class of the Gini type inequality measures in the second subsection. In the same subsection, the notion of ALD is formulated and the ALD condition is characterized in terms of the Gini type inequality measures. In the third subsection, the concept of almost composite transfer (ACT) which is coherent with ALD is de ned and the characterization theorem on the link between ALD and ACT is proved. The empirical illustration of ALD to the US income data is presented in the last subsection. Finally, Section 3 concludes the paper.. Almost Lorenz Dominance Let x be individual income with x [; ) and F be a cumulative distribution function of x. De ne F (k) (x) = R x F (k ) (t)dt for an integer k with F () (x) = F (x). Let p [; ] be population proportion and de ne the income of an individual at the pth percentile in distribution R F as F (p) = infft : F (t) pg which is left continuous. Also de ne L F (p) = p F (t)dt where F F = R xdf (x) is the mean income of F... Almost stochastic dominance (ASD): a brief review Suppose G is another cumulative distribution function to be compared with F. Stochastic dominance (SD) is de ned by comparing F (k) (x) and G (k) (x) for k = ; ; ::: Here we consider only rst-degree stochastic dominance (FSD) and second-degree stochastic dominance (SSD). Speci cally, F rst-degree stochastic dominates G if F (x) G(x) for all x [; ) with the strict inequality holding for some x; F seconddegree stochastic dominates G if F () (x) G () (x) for all x [; ) with the strict 4

5 inequality holding for some x. The fundamental theorem of stochastic dominance is that, at a given degree, F stochastic dominates G if and only if the social welfare under F is greater than that under G for all utilitarian welfare functions in a certain class. For FSD, the corresponding utility functions are continuous and non-decreasing in income; for SSD, the corresponding utility functions are continuous, non-decreasing and concave in income. Leshno and Levy s ASD is de ned when stochastic dominance fails. That is, when the stochastic dominance curves cross. The crossing areas (the areas of either SD curve above the other SD curve) are important for ASD. To that end, de ne S (F; G) = fx [; ) : G(x) < F (x)g (a) and 3 S (F; G) = fx [; ) : G () (x) < F () (x)g: (b) Leshno and Levy () argued that if the area of S i (F; G); i = and ; is small when compared with the total area of crossing (i.e., R jf (x) G(x)jdx which is the sum of the areas in S i (F; G) and in S i (F; G) which covers G (i) (x) > F (i) (x)), then an almost stochastic dominance relationship can be established for F to dominate G. Since the ASD relationship critically depends on a parameter which cannot be larger than, the ASD at the rst two degrees are denoted as -AFSD and -ASSD, respectively. De nition (Leshno and Levy, ; Tzeng et al., 3). For, F dominates G by -AFSD if [F (x) G(x)]dx jf (x) G(x)jdx; (a) S (F;G) F dominates G by -ASSD if 4 S (F;G) [F () (x) G () (x)]dx jf () (x) G () (x)jdx: (b) 3 Originally, Leshno and Levy de ned S (F; G) as fx S (F; G) : R G(x)dx < R F (x)dxg. Tzeng et al. (3) showed that the ASD condition de ned over Leshno and Levy s original S (F; G) is not valid: it is neither su cient nor necessary. Tzeng et al. (3) replaced Leshno and Levy s S (F; G) with the one above and proved that the Leshno and Levy s ASD condition becomes valid. 4 Guo et al. (3) pointed out that -ASSD de ned by Tzeng et al. (3) is not implied by -AFSD, unlike the relationship between SSD and FSD (the latter implies the former, that is, the stochastic dominance relationship is nested). To overcome this drawback, Tsetlin et al. (5) de ned a generalized second-degree almost stochastic dominance. The modi ed ASD condition is much more complicated than -ASSD. In this paper, we stick to -ASSD as de ned in Tzeng et al. (3) since we are interested in inequality ordering in which the mean incomes of all distributions are equal (or to be made equal through normalization); there is no counterpart of FSD in inequality ordering so we need not concern the relationship between AFSD and ASSD. For our purpose, our ranking at the second degree is the same as Ekern s (98) second-degree ranking of risk. 5

6 Clearly, the notion of ASD is weaker than SD (the latter always implies the former, or we can regard SD as a special case of ASD with = ). Consequently, the fundamental theorem that SD implies welfare ranking by all utility functions in a certain class has to be weakened for ASD. Let u be a utility function and U and U the two sets of utility functions corresponding to FSD and SSD, respectively, Leshno and Levy () and Tzeng et al. (3) de ned the following two subsets of utility functions U and U. De nition (Leshno and Levy, ; Tzeng et al., 3). 5 For any, subsets U i (); i = and, are de ned, respectively, as U () = uju > and sup[u (x)] inf[u (x)] ; (3a) and U () = uju > ; u < and sup[ u (x)] inf[ u (x)] : (3b) Con ning to these two subsets, Leshno and Levy () and Tzeng et al. (3), respectively, proved the rst and the second parts of the following theorem. Theorem (Leshno and Levy, ; Tzeng et al., 3). For, F dominates G by -AFSD if and only if u(x)df (x) and F dominates G by -ASSD if and only if u(x)df (x) u(x)dg(x) for all u U (); u(x)dg(x) for all u U (): (4a) (4b) Leshno and Levy () gave a general interpretation to the conditions imposed on the utility function u. For example, for -AFSD, the condition requires not to assign a relatively high marginal utility to very low values or a relatively low marginal utility to large values. Speci cally, a satisfactory utility function s ratio between its minimal marginal utility and its maximum marginal utility is limited to for. For example, for = :, the maximum marginal utility cannot be more than four times of the minimum marginal utility over the entire range of income. Besides this, few tangible implications on the functional form of u have been 5 Note that U does not contain U as a proper subset. This echoes the point made in the last footnote - -AFSD does not necessarily imply -ASSD. 6

7 outlined. 6 In the next section, we will see that for the Gini type inequality measures, this restriction can be made much more concrete... Almost Lorenz dominance (ALD): the de nition and the condition Distribution F Lorenz dominates G if the Lorenz curve of F lies nowhere below that of G and somewhere strictly above that of G. That is, L F (p) = F p F (t)dt G p G (t)dt = L G (p) for all p [; ] with the strict inequality holding for some p. An important theorem on Lorenz dominance is that if F Lorenz dominates G then all relative and transfer-sensitive inequality measures will unanimously rank F as being more equally distributed than G in income distribution. Note that the proof of this result is rather indirect; it is through a well-known characterization of Lorenz dominance (e.g., Fields and Fei, 978): a Lorenz dominating distribution can be obtained from a Lorenz dominated distribution - which also has the same mean income - through a sequence of progressive (i.e., from high to low) transfers of income. When Lorenz curves cross, we denote the area of L F (p) below L G (p) as S L (F; G) = fp [; ] : L F (p) < L G (p)g : (5) Similar to ASD, almost Lorenz dominance (ALD) with a parameter, denoted -ALD, can be de ned as follows. De nition 3. For, F dominates G by -ALD if [L G (p) L F (p)]dp jl F (p) L G (p)jdp: (6) S L (F;G) Next we examine the restrictions of ALD on the underlying summary inequality measures. Aiming to derive intuitively interpretable implications, we limit our scope to the important class of rank-dependent Gini type inequality measures. A Gini type measure is Lorenz curve based and it is a normalized weighted average of the distance between the Lorenz curve in question and the diagonal line (representing the equal distribution). Following Shorrocks and Slottje (), a Gini type measure is de ned as R [p L(p)](p)dp I(x) = R p(p)dp : (7) 6 The ASD restrictions have important implications for the Atkinson-Kolm-Sen measure of inequality which is where is the EDEI (equally distributed equivalent income). For example, for -AFSD, since the restriction on marginal utility means that the utility function will register relatively less for what happens at extreme low incomes and relatively more at extreme high incomes, it follows that would be greater than without the restriction. Thus inequality will become less severe when it is measured with a restricted AKS inequality measure. Besides this implication, it would be useful to examine other issues such as how the principle of transfers will be a ected and what would a satisfactory AKS measure look like. We leave these for future research. 7

8 Clearly the value of the de ned measure lies between and. We denote G the set of all such inequality measures I(x) for all possible weighting functions (p). Also as noted by Shorrocks and Slottje (), the set G contains all known rank-based inequality measures (such as Donaldson and Weymark s (98) single series Ginis) as special cases. Denote ~ (p) (p) = R p(p)dp; then the above measure can also be written as and can be further written as I(x) = I(x) = [p L(p)] ~ (p)dp [p L(p)]d^(p) (8) where ^(p) = R p ~ (q)dq is the cumulative weight for the pth percentile. Note that R ~ (p) is normalized in the sense that all weights add up to one, i.e., p~ (p)dp =. Further, through di erentiation by parts, we have I(x) = [p L(p)]^(p) = ^(p)dp + ^(p)dl(p) ^(p)dl(p) ^(p)dp: (9) Since ~ (p) >, ^ (p) = ~ (p) >. Ignoring the constant term R ^(p)dp, the measure I(x) is akin to the de nition of a decomposable inequality measure with F (x) being replaced by L(p). Since L(p) can be viewed as a cumulative distribution function de ned over a bounded interval (e.g., Kendall and Stuart, 958), Aaberge () de ned a speci c form (by letting ^(p) = p m for m = ; ; :::) of I(x) and refers them to as the Lorenz family of inequality measures. Clearly, Aaberge s family of inequality measures are also members of the Gini-type class that we just stated. With this speci cation, we can now de ne a restricted subset of G in a similar fashion as in restricting members of U with u being replaced by ^. De nition 4. For any, a subset G () of G is de ned as G () = I(x)j^ > and sup[^ (p)] inf[^ (p)] : () What is the meaning of this restriction? Since ^ (p) = ~ (p), the condition states that the largest weight used in weighting the distance [p L(p)] cannot be 8

9 times larger than the smallest weight. For example, for = :, the ratio between the maximum weight and the minimum weight cannot be more than 4. Note that if Lorenz dominance prevails (i.e., = ), there is no restriction on what the weights can be. We can now state the following ALD condition. Theorem. For, F dominates G by -ALD if and only if I F (x) I G (x) for all I(x) G (): () Proof. The proof is similar to Leshno and Levy (, part of Theorem ) and Tzeng et al. (3, Theorem ). For completeness, we sketch a proof below. Since R ^(p)dp is the same for all distributions, I G (x) I F (x) = = = ^(p)dlg (p) ^(p)dlf (p) ^ (p)[lf (p) L G (p)]dp () ^ (p)[lf (p) L G (p)]dp + ^ (p)[lf (p) L G (p)]dp S L S L where S L and S L stand, respectively, for S L (F; G) and S L (F; G) which is the complement of S L (F; G), i.e., S L (F; G) = [; ]ns L (F; G). To prove the su ciency, let ^ = inf p[;] [^ (p)] and = sup p[;] [^ (p)]. Since we can reasonably assume ^ >, then I G (x) I F (x) = ^ (p)[lf (p) L G (p)]dp + ^ (p)[lf (p) L G (p)]dp S L S L [L F (p) L G (p)]dp + ^ [L F (p) L G (p)]dp (3) S L S L = (^ + ) [L F (p) L G (p)]dp + ^ jl F (p) L G (p)jdp S L = (^ + ) ^ ^ + jl F (p) L G (p)jdp S L [L G (p) L F (p)]dp : By assumption, ^( ) which implies ^. Finally, by De nition 3, we ^+ obtain I G (x) I F (x) (^ + ) jl G (p) L F (p)jdp [L G (p) L F (p)]dp S L : (4) 9

10 To prove the necessity, suppose the proposition is not true and R S L [L G (p) L F (p)]dp > R jl F (p) L G (p)jdp. Let = ^ for some > ^ >. Denote ^+! = R S L pdp,! = R SL pdp and construct 7 ^ (p) = ~ (p) = (! +! ^ if p S L (F; G) ^! +! ^ if p = S L (F; G) : (5) Clearly the constructed inequality measure I(x) belongs to G (). But I G (x) I F (x) = ^ (p)[lf (p) L G (p)]dp + ^ (p)[lf (p) = = = S L (F;G)! +! ^ ^ +! +! ^ ^ +! +! ^ S L (F;G) [L F (p) L G (p)]dp + ^ [L F (p) S L S L [L F (p) L G (p)]dp + ^ jl F (p) S L ^ + jl G (p) L F (p)jdp [L G (p) S L which contradicts the assumption that I F (x) I G (x) for all I(x) G (): L G (p)]dp L G (p)]dp (6) L G (p)jdp L F (p)]dp < ; It is useful to clarify what the theorem actually says and what the theorem does not say. The theorem says that all inequality measures in G () together are su cient to characterize the almost Lorenz dominance de ned in (6). What it does not say is that they are the only measures in G that can do so. In other words, not all measures in GnG () will show the opposite direction in inequality ranking (i.e., I G (x) < I F (x)). In fact, it is fairly easy to construct an example of inequality measure I(x) with sup[^ (p)] > inf[^ (p)] - a violation of condition () - that entails I G (x) > I F (x). 8 Thus the set of inequality measures that will rank F as being more equal than G is larger than G () and the measures in G () can be regarded as a core for ALD. In a sense, this property is echoed in Lorenz dominance in that the set of all Gini type measures G is su cient to characterize the dominance relationship even though the set of all transfer-sensitive inequality measures - which includes the Gini set as a proper subset - will also characterize the same relationship. 7 Note that! and! are needed to satisfy the condition R p~ (p)dp =. 8 An example is discrete distributions ^F = (; 3; 5; 7; 9; ; 3; 5) and ^G = (; ; 7; 7; ; :5; :5, 5) with weights (:; :6; :; :; :; :; :; :). Using the method for discrete distributions (to be outlined in the following subsection), one can calculate that = 3=7 and = 3. Thus, in G (), the maximum weight cannot be 3 larger than the minimum weight. But the chosen weights do not conform to the above restriction (:6=: = 3 > 3 ), yet it is straightforward to verify that the weighted Gini measures yield I( ^G) = :439 > :4 = I( ^F ) which is in agreement with all members of G ().

11 .3. Almost Lorenz dominance (ALD): a characterization A celebrated result in the theory of Lorenz dominance is that with equal means the dominating income distribution can be obtained from the dominated distribution via a sequence of rank-preserving progressive transfers of income (e.g., Fields and Fei, 978). This result vividly illustrates the meaning of a reduction in inequality and completely characterizes the di erence between the two distributions in Lorenz dominance. For any dominance condition (even though it is almost ), to enhance our understanding, it is useful to provide a similar characterization. 9 For almost Lorenz dominance, as expected, these results will need to be modi ed since Lorenz curves may now cross. Figure goes about here. Consider a simple case of ALD where the two Lorenz curves cross only once as depicted in Figure. In the gure, Lorenz curve L F crosses Lorenz curve L G initially from above at point A. Thus before point A, F Lorenz dominates G, and after that point, G Lorenz dominates F. By the Fields-Fei theorem, over the population range [; p A ) that part of F can be obtained from the corresponding part of G via a sequence of rank-preserving progressive transfers; over the population range (p A ; ] the part of F can be obtained from the corresponding part of G through a sequence of rankpreserving regressive transfers (from low incomes to high incomes). Thus, in almost Lorenz dominance, the necessary transfers are composite in that they combine both progressive transfers and regressive transfers - akin to Shorrocks and Foster s (987) FACT or oli s () FCPT. This observation promotes us to de ne a new type of composite transfer for ALD - almost composite transfer (ACT). For the sake of simplicity in presentation and tractability in derivation, we now switch to working with discrete distributions. The cumulative distribution functions F and G are now replaced with their discrete counterparts ^F and ^G which both contain n individuals with positive incomes. We assume that the incomes are sorted in increasing order. We also assume that n is large (the population is dense) so that there is always an individual at a Lorenz curve crossing point (instead of falling into a no man s land between two adjacent individuals). For a discrete distributions, its Lorenz curve is de ned as the linear segment of all Lorenz ordinates with and as the two ends. The area underneath the Lorenz curve is the sum of one triangle and n quadrilaterals. When applying a progressive transfer between incomes i and j and a regressive transfer between incomes k and l with i < j < k < l, the resulting Lorenz curve will necessarily cross with the original Lorenz curve. The progressive transfer will reduce the inequality of the low part of the distribution while the regressive transfer will raise the inequality of the upper part of the distribution. The increase in the area underneath the Lorenz curve due 9 Tsetlin et al. (5) described a similar composite probability shift for ASD in a speci c setting but did not provide a characterization. Without this assumption, a kind messy adjustment needs to be made when the crossing happens between two individuals. For a dense population, the adjustment becomes negligible and vanishes.

12 to the transfer of from j to i is (j i) and the decrease in the area underneath the n Lorenz curve due to the transfer of from k to l is (l k). The total change in the n area underneath the Lorenz curve is the sum of the two areas [(j i) + (l k)]. n The area reduction due to the regressive transfer as a proportion of the total area change is (l k) (j i) + (l k) : (7) For any < <, to make the ratio in (7) to be no greater than, it is necessary l k and su cient that. Thus, for each, we can de ne an almost j i composite transfer, ACT(), as follows. De nition 5. For, an ACT() is a combination of a rank-preserving progressive transfer of income from the j th income to the ith income (i < j) and a rank-preserving regressive transfer of income from the kth income to the lth income (k < l) with l k : (8) j i Here either j < k or l < i, that is, no overlapping between the sets fi; jg and fk; lg. We can now present the characterization for ALD. Theorem 3. For, ^F dominates ^G by -ALD if and only if ^F can be obtained from ^G by a sequence of ACTs de ned in De nition 5. Proof. To prove the su ciency, consider a sequence of ACTs, ACT (), ACT (),..., ACT m (), that transfer ^G into ^F. Let ^G (j) be the distribution after the rst j ACTs have been sequentially applied to ^G with j = ; ; :::; m, ^G () ^G and ^G (m) ^F. By Theorem, ^G (j) dominates ^G (j ) by -ALD for j = ; :::; m, That is, for j = ; :::; m, I ^G(j)(x) I )(x) for all I() G ()). It follows that I ^G(j ^F (x) I ^G(x) for all I() G (). Again by Theorem, ^F dominates ^G by -ALD. For the necessity, we prove a more speci c result. We rst de ne a speci c almost composite transfer: in an ACT() if l = j k i ; (9) then it becomes a speci c almost composite transfer and we denote it SACT(). That is, a SACT() leads to the equality in De nition 3. We then wish to prove that if [L ^G(p) L ^F (p)]dp = jl ^F (p) L ^G(p)jdp; () S L ( ^F ; ^G) then ^F can be obtained from ^G by a sequence of SACT()s. We rst consider the case of a single crossing between two Lorenz curves. Suppose the Lorenz curve of ^F crosses with that of ^G once from above. This is the situation

13 depicted in Figure. For convenience, denote the area of L ^F (p) above L ^G(p) as and refer to it as the -area; denote the area of L ^G(p) above L ^F (p) as and refer to it as the -area. By assumption, = or = ( ). + By the Fields-Fei Lorenz domination characterization theorem, the -area can be eliminated through a sequence of progressive transfers applied to ^G and the -area can be eliminated through a sequence of regressive transfers applied to ^G. Recall that we have assumed that the population is dense (the population size is large) and there is at least one individual at the crossing point (region) and thus the two types of transfers can be independently applied. Also note that these transfers are not unique as any single transfer can be split into multiple transfers that move smaller amount of income between the same two individuals (for example, a transfer of $5 from person 5 to person can be split into ve transfers of $ from person 5 to person ). We call a transfer distinct if it transfers the largest possible amount of income between the two individuals (the transfer of $ in the above example is not a distinct transfer). Suppose there are M distinct progressive transfers involved in eliminating the -area and there are N distinct regressive transfers involved in eliminating the - area. Each of these transfers can be split into multiple smaller transfers (in the sense explained above). The required SACT()s are constructed from these two sets of transfers using the following 3-step procedure. Step. Compute the changes in the areas beneath the Lorenz curve of ^F generated by these transfers (using the formulae (j i) (l k) and described above) and rank n n them in increasing order. Denote these areas as i s and j s with < M and < N. Label the corresponding transfers as P T ; P T ; :::; P T M and RT ; RT ; :::; RT N, respectively. Note that the sequential application of the transfers is path independent, that is, it does not matter in which order the transfers are applied. That is, the values of i s and j s are not a ected by the order of application. Step. Consider rst to pair P T with RT. There are three possibilities between and : = ( ), < ( ), and > ( ). Case. If = ( ), then P T and RT together constitute a SACT(). Case. If < ( ), then the area generated by P T is smaller than what is required to pair with RT in forming a SACT. Now split RT into two smaller transfers RT and RT such that = ( ) where is the area reduction beneath the Lorenz curve as a result of RT. Note that since the area reduction is a continuous function of the amount of income transfer, such a split can always be performed. Now pair P T with RT to form the desired SACT. Case 3. If > ( ), then the area generated by P T is larger than what is needed to pair with RT in forming a SACT. Now split P T into two smaller transfers P T and P T such that = ( ) where is the area increase beneath the Lorenz curve as a result of P T. Then pair P T with RT to form the desired SACT. Step 3. For Case above, move to examine P T and RT ; for Case, move to examine P T and RT ; for Case 3, move to examine P T and RT. For each of them, 3

14 there are again three possible cases to distinguish. Steps and 3 are repeatedly applied to all P T i and RT j until they are perfectly match paired into a sequence of SACTs. This is feasible and necessary since = P M i= i and = P N j= j and any splitting of i and j keeps the total area unchanged. The conclusion of this single crossing case is: if the two areas ( type and type) are in the ratio of, then a sequence of SACT() can always be constructed to characterize the -ALD. Now we can use this result to prove the general case of multiple crossings. Suppose the Lorenz curve of ^F crosses that of ^G K times and initially from above. With a slight abuse of notation, denote the areas of L ^F above L ^G also as the -areas and the areas of L ^F below L ^G as the -areas, then there are K = K+ -areas and K = K K + -areas; K and K are either equal or di er by. Rank the -areas and -areas in increasing order and denote them ; ; ; K and ; ; ; K, respectively. By assumption, = P M i= PN i = j= j =. Utilizing the result from the single-crossing case, we want to match pair i s and j s so that the ratio between each pair is exactly. The 3-step procedure outlined above can also be modi ed to be applied to the general case. First, consider and, the two smallest areas in each category. If =, then the procedure employed in the special case can be directly applied to generate a sequence of SACTs. If <, then split the area into and (through creating smaller transfers) such that =. The transfers within and can then be combined into a sequence of SACTs. The remaining area is then moved forward to compare with. If >, then it is the -area needs to be split into and such that =. The transfers within and can then be combined to form a sequence of SACTs. The remaining area is then moved forward to compare with. This procedure is repeated until all areas are exhausted and the required sequence of SACTs are generated. It is useful to point out that the order of the progressive transfer and the regressive transfer does not matter in an ACT (or SACT). This is very di erent from Shorrocks and Foster s FACT and oli s FCPT where the progressive transfer must precede the regressive transfer. In ALD, what matters is the comparison between the two types of crossing areas, whether the -area is before the -area, or after the -area, or scatter For example, consider again the two discrete distributions ^F = (; 3; 5; 7; 9; ; 3; 5) and ^G = (; ; 7; 7; ; :5; :5; 5) used in Footnote 8. The two Lorenz curves cross once at the 4th income. ^F is obtained from ^G through one progressive transfer ($ from the 3rd income to the nd income) and two regressive transfers ($.5 each from the 5th income to the 6th and 7th incomes, respectively). The value is ( = = ), the value is.5 ( = :5 and = ), = :5=3:5 = 3=7 and = 3. Since = > :5 3 = ( ), it falls into Case 3 of Step. Split = into = 3 and = 3 we have = ( ) and = ( ). Thus the two SACTs are ( ; ) and ( ; ). 4

15 around the -area makes no di erence as long as condition (6) is maintained..4. Almost Lorenz dominance (ALD): an application In applying Lorenz dominance to income data, population is rst sorted by income in increasing order, and then typically it is divided into ten equal-size subpopulations. The Lorenz ordinates are computed for the rst nine subpopulations (the last Lorenz ordinate equals by de nition). The Lorenz curve for the population is then obtained by the linear segment of these ordinates and and (as the two end points). Using this algorithm, Bishop et al. (99) examined the year-to-year changes in US income inequality between 967 and 986. Of the nineteen comparisons, Lorenz curves cross in seven. Based on Bishop et al. s Table (p. 36), the table below (Table ) reports the di erences in the Lorenz ordinates for each pair of the crossing years. Table. Crossing US Lorenz Curves and the Di erences in Lorenz Ordinates Year Di erences in Lorenz Ordinates st nd 3rd 4th 5th 6th 7th 8th 9th Of the seven pairs of Lorenz curves, ve pairs cross once and two pairs cross twice (968 vs. 969 and 97 vs. 97). Bishop et al. (99), pointing out that the income data used are random samples drawn from the underlying populations, employed statistical inference to conclude that only the crossing between 973 and 974 is statistically signi cant. In this subsection, we apply the tool of almost Lorenz dominance to re-examine the issue. In doing so we ignore the statistical issue and treat the income data as population data. In applying the ALD approach, we rst compute the area of each year s Lorenz curve above the other year s Lorenz curve. These area values are reported in Table below: the -area is the area of the later year s (e.g., 969) Lorenz curve above the earlier year s (968) and the -area is the area of the earlier year s (968) Lorenz curve above the later year s (969). We then compute the -value which is the ratio between the smaller value of and and the sum of and. This calculation ensures but the direction of dominance needs to be carefully identi ed (the year for which the Lorenz curve is on top and, at the same time, has a larger crossing area below it ( or ) is the dominating distribution). A good example is the Gini coe cient which mostly likely belongs to G (). For Gini, it does not matter how the areas are located in the calculation - what matters is the sum of all areas. 5

16 Table. Almost Lorenz Dominance Year -area -area -value ALD dominates dominates dominates dominates dominates dominates 86 Note: -area is the area of the later year s Lorenz curve above the earlier year, -area is the area of the earlier year above the later year, = minf;g, and indicates no dominance. + In the table, for <, we identify the directions of dominance and report them in the last column. For = as in the comparison between 968 and 969, since it is a borderline case, we do not give the direction of dominance. In fact, when =, sup[^ (p)] = inf[^ (p)] and, thus, ^ (p) = ~ (p) = constant. Thus the only measure in G ( ) is the familiar Gini coe cient. Of course, according to Gini, the two distributions have the same level of inequality. For < and when becomes smaller, the set of weights enlarges (i.e., the gap between inf[^ (p)] and sup[^ (p)] increases). The largest set of weights in the table is the last comparison between 985 and 986, where the ratio between the largest weight and the smallest weight is = 37:46. Suppose the weights are distributed over [; ], this means that the :6 interval for ~ (p) could be something like [:; :3746] or [:5; :9365]. Since a Gini type measure is linear in weights, it does not matter which interval is selected for inequality comparison. Of course, for =, the interval becomes the default range [; ]. 3. Conclusion Lorenz curve is an intuitive and powerful tool in ranking income inequality when Lorenz dominance succeeds. It becomes much less useful, however, when Lorenz dominance fails (i.e., when Lorenz curves cross). In this paper, we have characterized the notion of almost Lorenz dominance and demonstrated that the new tool can be applied to rank income inequality when Lorenz dominance fails. Like Leshno and Levy s almost stochastic dominance (ASD), almost Lorenz dominance (ALD) is justi ed on the recognition that some pathological measures of inequality (determined by extreme weighting in the Gini type measures) should be excluded in measuring and ranking income inequality. The notion of ALD a ords researchers as well as policy makers a venue to assess the degree of the failure in Lorenz dominance and to decide whether the restrictions placed on inequality measures (the Gini type) in ALD are reasonable. Of course, 6

17 choosing an appropriate size for the acceptable -value is always not an easy task. Although a value such as.6 (as in 985 vs. 986) seems pretty non-controversial, a value at.4 (as in 976 vs. 977) may receive much less support in declaring dominance in inequality. In this sense, the -value can be regarded as the signi cance level in inequality rankings akin to that employed in statistical analysis. A decisionmaker picks such a level, researchers then can tell whether a dominance is reached at this signi cance level. This approach should be regarded as a progress in ranking inequality since otherwise many inequality comparisons would be unduly declared non-comparable even though the -value could be quite small (means that few inequality measures are excluded). As for the inequality measures considered for ALD, one of course wishes to extend the family to include all possible inequality measures, including decomposable measures. Since ASD and ALD critically depend on the sizes of the crossing areas between the two curves (SD and Lorenz), a direct connection between a decomposable inequality measure and Lorenz curve needs to be established in order to execute the extension. But so far we only know that Lorenz curve crossing implies and is implied by crossing in (normalized) second-degree SD curves, we do not know much about the relationship between the two types of crossing, e.g., the sizes of the crossing areas between the two sets of the curves. Without this knowledge, it is not feasible to establish the connection between ALD and decomposable inequality measures. We leave this topic for future research. Finally we point out that the results we established in the paper actually hold for a broader class of inequality measures than we de ned. In fact, any inequality measure that respects the principle of almost composite transfer in that an ACT() will reduce the level of inequality will be included in the broader class. It would be interesting to fully characterize this broader class. But unfortunately decomposable inequality measures will not be part of it since they are not rank-dependent. 7

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