Gini Indices and the Redistribution of Income. Jean-Yves Duclos * June 1998

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1 Gini Indices and the Redistribution of Income Jean-Yves Duclos * CRÉFA and Department of Economics Université Laval June 998 Abstract Just as the Gini inequality index captures people s relative deprivation [Yitzhaki (979)], so, we show in this paper, Gini-based progressivity and horizontal inequity indices capture individual perceptions of relative fiscal harshness and ill-fortune. In fact, we find that these links between individualistic perceptions and the measurement of the distribution and redistribution of income generalise to the family of indices based on the extended Ginis of Donaldson and Weymark (98) and Yitzhaki (983). Through "leaky bucket" experiments, we also suggest how we can parameterise the inequality aversion present in these indices. Analysis of the Canadian gross and net income distributions (conducted using recently developed statistical inference procedures) shows the distribution and the aggregation of these individual indicators of relative deprivation, fiscal harshness and ill-fortune in 98 and in 99. JEL Numbers: H23, D3, D63 Keywords: Inequality, Progressivity, Redistribution, Horizontal inequity Address: Département d économique, Pavillon De Sève, Université Laval, Québec, Canada, GK 7P4; tel.: (48) ; fax: (48) ; jduc@ecn.ulaval.ca * This research was financed by grants from the Social Sciences and Humanities Research Council of Canada and from the Fonds FCAR of the Province of Québec. I am grateful to Martin Tabi for his research assistance, to two anonymous referees and to an Editor of the journal, Michael Keen, and to Stephen Howes, Peter Lambert, Satya Paul, Michel Truchon, François Vaillancourt and participants at the Third Meeting of the Canadian Public Economics Study Group in St-John s, Newfoundland, for their very helpful comments.

2 I. Introduction Relative deprivation has long been an important topic of research for sociologists and social psychologists. It is also the basis of a well-known theory of social attitudes to inequality. Its interesting link with the economic measurement of inequality, and more precisely with the standard Gini coefficient, has already been identified by Yitzhaki (979) and Hey and Lambert (98) 2. In their use of relative deprivation, they draw from the classic work of Runciman (966), who defines it as follows: "The magnitude of a relative deprivation is the extent of the difference between the desired situation and that of the person desiring it (as he sees it)." (p.) We start this paper by following Yitzhaki s and Hey and Lambert s lead and defining for each individual an indicator of relative deprivation which measures the distance between his income (or some other cardinal value of individual welfare) and the income of all those relative to whom he feels deprived. The theory of relative deprivation also suggests, however, that people sometimes specifically compare their individual fortune with that of others in similar or close circumstances. The first to formalise the theory of relative deprivation, Davies (959), expressly allowed for this by suggesting how comparisons with similar vs dissimilar others lead to different kinds of emotional reactions; he used the expression "relative deprivation" for "in-group" comparisons, and "relative subordination" for "out-group" comparisons (p.283). In the words of Runciman (p.29), "people often choose reference groups closer to their actual circumstances than those which might be forced on them if their opportunities were better than they are." In a discussion of the post-war British welfare state, he notes, for instance, that "the reference groups of the recipients of welfare were virtually bound to See, for instance, Olson et al. (986) for its link with subjective well-being, discrimination, social protest or feelings of injustice. 2 See also Sen (973), and Clark and Oswald (996) for useful and extensive references to the economic literature on the dependence of individual utility on reference income levels, and for its relevance to economic theory and policy. Clark and Oswald s paper also finds that "workers reported satisfaction levels are [...] inversely related to their comparison wage rates" (p.359).

3 remain within the broadly delimited area of potential fellow-beneficiaries. It was anomalies within this area which were the focus of successive grievances, not the relative prosperity of people not obviously comparable" (p.7) 3. In an income redistribution context, it is plausible that reference groups be established on the basis of gross incomes, and that comparisons of net incomes may then be made among these groups. Here, we let individuals assess their relative redistributive ill-fortune in "broadly delimited" reference groups of comparables by monitoring whether they are overtaken by or overtake these comparables in income status. More specifically, we measure individual fiscal ill-fortune as the sum of the incomes of those individuals by whom they are overtaken minus the sum of the incomes of those they succeed in outranking. Finally, in a manner analogous to the relative deprivation indicator and to capture the perceived effectiveness of the redistribution of income, we also define an individual perception of relative fiscal harshness on the rich which compares one s tax and benefit treatment with that of the richer in the population. The paper mainly indicates how aggregating these individual indicators of the "perceived equity" of the distribution and redistribution of income yield popular aggregate indices of inequality, progressivity, and horizontal inequity which may be called S-indices 4. These indices indeed derive from the S-Ginis of Donaldson and Weymark (98) and Yitzhaki (983) [see also Kakwani (98, p.444)], which include the 3 On this, note Runciman s allusion to reference group theory where "some similarity in status attributes between the individual and the reference group must be perceived or imagined for the comparison to occur at all" [Merton and Rossi (957), p.242, quoted in Runciman (966), p.4]. In his theory of social comparison processes, Festinger (954) also argues that: "If some other person s ability is too far from his own, either above or below, it is not possible to evaluate his own ability accurately by comparison with this other person. There is then a tendency not to make the comparison." (p.2) Thus, "given a range of possible persons for comparison, someone close to one s own ability or opinion will be chosen for comparison". (p.2) 4 Recent uses of S-indices include Achdut (996), Aronson et al. (994), Barrett and Pendakur (995), Creedy (996) and Palme (994). A two-parameter generalisation of S- indices is presented in Duclos (997). 2

4 standard Gini coefficient and are weighted gaps between the line of perfect income equality and the Lorenz curve for a distribution of income 5. The S-Ginis are thus straightforward to understand graphically, and they differ simply in the weights which each Gini applies on the distance between the cumulative population shares and the cumulative income shares at various percentiles of an income distribution. We first show that these extended Ginis can be intuitively interpreted as ethical means of individual relative deprivation across a population. We moreover demonstrate that the associated S-indices of progressivity and horizontal inequity can be interpreted, respectively, as ethical means of the individual indicators of fiscal harshness on the rich and of the individual indicators of relative ill-fortune in the allocation of taxes and benefits. This interpretation as aggregates of individual welfare (or ill-fare) demarcates these indices from others that may not be so intuitively understood. It can thus provide a "microfoundation" for the use of aggregate normative indices, and can also help generate micro explanations for changes or differences in aggregate indices of the distribution and redistribution of income. We also apply these indices to the distribution and redistribution of income in Canada in 98 and in 99. For this, we use recent developments in statistical inference procedures which take into account the classical sampling variability of estimates of conditional means as well as the sampling variability of quantile estimates (both types of estimates being needed for the computation of Lorenz and concentration curves). To focus on a limited range of the ethical parameter, we suggest the use of two "leaky bucket" approaches that weigh the competing goals of equity and efficiency. Furthermore, because of their linearity in income, it is well-known that S-indices of inequality and progressivity indices can easily be decomposed as the sum of inequality indices for various income components (including taxes, benefits, evaded taxes, unearned income, income from the underground economy, etc.). We can thus indicate the contribution of separate components of the tax and benefit system to overall progressivity and fall in relative deprivation. We find that although inequality and 5 This is more generally true of the Mehran (976) class of linear inequality measures. 3

5 relative deprivation increased significantly between 98 and 99 for the distribution of gross incomes, inequality decreased between these two years for the distribution of net incomes for most values of the ethical parameter. Furthermore, the 99 tax and benefit system reduces relative deprivation unambiguously more in 99 than in 98. Old age transfers account for more than a third of the total redistribution exerted by the tax and benefit system. With social assistance and unemployment benefits, they also account for most of the fall in relative deprivation between 98 and 99. We start in Section II by outlining the measurement of the distribution and redistribution of income using S-Ginis. Section III makes specific the definitions of the individual indicators of the "perceived equity" of the distribution and redistribution of income. Section IV shows how aggregating the individual indicators of Section III leads to the indices of Section II. Sections V and VI suggest, respectively, how we may interpret and calibrate the ethical weights used to aggregate the individual indicators. In Section VII, we recall how total progressivity can be decomposed into the sum of progressivity indices for individual taxes and benefits. Section VIII then illustrates the analysis using Canadian data, and Section IX concludes. II. The Measurement of the Distribution and Redistribution of Income Let X be gross incomes, T(X) be taxes (which can be negative), and N(X)=X-T(X) be the random net incomes corresponding to a level of gross income X. Conditional on X, we thus allow T(X) and N(X) to be continuous 6 random variables and we denote their finite expected value by T(X) and N(X). Denote by X(p) and N * (p) the inverse distribution (or quantile) functions for gross and net incomes, respectively. N(X(p)) is then the random variable for net incomes conditional on the p-quantile value of gross incomes, X(p), and N(X(p)) is its expected value. Finally, denote by N(q X(p)) the q- quantile value of the distribution of net incomes conditional on X(p). Let the Lorenz curve for X be L X (p), such that: 6 The assumption of continuous distributions is made for expositional ease. Analogous results for the case of discrete distributions can be found in Duclos (995). 4

6 L X (p) p µ X X(s) ds () where µ X is the mean of X. L X (p) shows the proportion of total income held by the p% poorest individuals in the distribution of X. Define in an analogous way the Lorenz curve for N as L N (p). We refer to the concentration curve for net incomes as C N (p), with C N (p) p µ N(t X(s)) dt ds, N p µ N N(X(s)) ds (2) The concentration curve for taxes, C T (p), can be defined similarly by replacing N( ) and µ N in (2) by T( ) and µ T respectively. Note that, for concentration curves, net incomes and taxes appear in increasing order of associated gross incomes. The S-Gini (or Single-Parameter Gini) indices of inequality in the distribution of X is given by [see Donaldson and Weymark (98) and Yitzhaki (983)]: G X (v) p L X p kp,v dp (3) where k(p,v)=v(v-)(-p) v-2 and v>. Note that G(2) yields the standard Gini coefficient, with k(p,2)=2. We will return later to the interpretation of the ethical weight k(p,v) and of the parameter v. We can interpret the difference p-l X (p) as the difference between (-L X (p)) and (-p). (-L X (p)) shows the proportion of total income which the richer than X(p) hold in the distribution of income. (-p) indicates the share of the population which these richer individuals represent; it also measures the proportion of total income that they would have held if income had been distributed equally. p-l X (p) is thus the income share of the rich [richer than X(p)] in excess of their "more equitable" share in an equal distribution of income. G(v) weights with k(p,v) these "excess" shares at different points of the income distribution. 5

7 We can measure the progressivity of the tax T(X) along either of two views: the Tax Redistribution (TR) view, and the Income Redistribution (IR) view [see Pfähler (987)]. A non-negative tax is said to be TR progressive if C T (p) L X (p) for all p [,], with the strict inequality holding somewhere 7 ; T(X) is IR progressive if C N (p) L X (p) for all p [,], with the strict inequality holding for at least some p. With this, we can define [see Kakwani (984)] the following indices K(v) and P(v) of TR and IR progressivity: K(v) L X (p) C T (p) k(p,v) dp (4) P(v) C N (p) L X (p) k(p,v) dp (5) When v=2, K(2) and P(2) yield, respectively, the well-known Kakwani (977) and Reynolds-Smolensky (977) indices of progressivity and vertical equity. We can interpret K(v) and P(v) as ethically weighted averages of the differences between -C T (p) and -L X (p), and between -L X (p) and -C N (p), respectively. -L X (p) is the total gross income share of those individuals richer than X(p); it should also be their share of total taxes (-C T (p)) and total net income (-C N (p)) if the tax was proportional to income and thus distributionally neutral. K(v) and P(v) weight the departures from tax proportionality L X (p)-c T (p) and from distributional neutrality C N (p)-l X (p) Unlike the Lorenz curve L N (p), C N (p) ranks individuals by the size of their gross incomes; we then have that C N (p) L N (p) for all p between and, with strict inequality somewhere if and only if the tax system reranks individuals. The greater the extent of reranking, the farther is C N (p) from L N (p). The difference C N (p)-l N (p) can thus be used to assess the horizontal inequity exerted by the tax T(X) since a horizontally equitable tax, besides treating equals equally, should not change the rank of individuals in the income distribution [see, for instance, Feldstein (976)]. We can then define the following 7 A transfer (a negative tax) is TR progressive if C T (p) L X (p) for all p [,], with the strict inequality holding for at least some p in that interval. 6

8 indices of horizontal inequity [see Duclos (993)]: E(v) C N p L N p kp,v dp (6) With v=2, E(2) gives the Atkinson (979) and Plotnick (98) index of horizontal inequity 8. A more robust approach to testing changes in inequality or differences in progressivity and horizontal inequity is to test the dominance of Lorenz or concentration curves, as in Howes (993) or Davidson and Duclos (997). The use of scalar indices, besides being popular, can however help to settle matters partially when dominance cannot be inferred; even when dominance can be inferred, scalar indices can summarise its strength, an important issue for most applications. III. Relative Deprivation, Fiscal Harshness and Fiscal Fortune As proposed in Section I, let the relative deprivation of individual i with income X(p i ), when comparing himself with j with income X(p j ), be given by the difference X(p j )- X(p i ) if j is richer than i 9. Otherwise, i feels no relative deprivation. For i comparing himself with j, relative deprivation then equals: δ p i,p j max, X p j X p i (7) The average deprivation felt by i over the whole population of individuals j is then: d p i δ p i,p dp (8) Relative deprivation in the distribution of net incomes can be similarly defined. Somewhat analogously to the relative deprivation of (7), now denote individual 8 A criticism of the reranking indices and approach to measuring horizontal inequity can be found, inter alia, in Kaplow (989) and Musgrave (99). 9 Again, see Yitzhaki (979) and Hey and Lambert (98) on this. For recent work on the measurement and aggregation of relative deprivation, also see Podder (996) and Paul (99,994). 7

9 i s perception of relative fiscal harshness on a richer individual j as: η(p i,p j ) T[X(p j )] T[X(p i )], if p j >p i, otherwise (9) Fiscal harshness η(p i,p j ) is negative when a richer individual j pays less tax than i, and positive if he pays more. It is equal to zero for all couples (p i,p j ) only in the presence of a poll tax, in which case the tax can be considered to be equally harsh on everyone. Hence, the value of the indicator of fiscal harshness η(p i,p j ) can be understood in reference to a uniform tax or transfer. The expectation of η(p i,p j ) is given by: η(p i,p j ) T[X(p j )] T[X(p i )], if p j >p i, otherwise () Individual i s average perception of fiscal harshness across the whole distribution of individuals j then equals h(p i ): h(p i ) η(p i,p) dp () An alternative indicator of the fiscal treatment of the rich compares i s net income with the net incomes of those with greater gross incomes. This gives "fiscal loosenes" λ(p i,p j ): λ(p i,p j ) N[X(p j )] N[X(p i )], if p j >p i, otherwise (2) Here, the standard of reference for fiscal looseness is the egalitarian outcome by which all net incomes are equalised: in that case, λ(p i,p j ) is zero for all couples (p i,p j ). When λ(p i,p j )>, the fiscal system fails to eliminate the "looseness" of net incomes; when λ(p i,p j )<, it is tighter than an egalitarian tax. The expectation of λ(p i,p j ) is: 8

10 λ(p i,p j ) N[X(p j )] N[X(p i )], if p j >p i, otherwise (3) Individual i s average perception of fiscal looseness over the population of j is then given by l(p i ): l(p i ) λ(p i,p) dp (4) The greater the value of l(p i ), the looser the equalisation of net incomes effected by the tax system. We may thus refer to l(p i ) as an individual indicator of fiscal looseness. As proposed in Section I, let an individual i also assess his relative ill-fortune by noting whether in the distribution of net incomes he outranks (or is outranked by) an individual j because of a particularly favourable or unfavourable fiscal treatment. Let, moreover, the intensity of that ill-fortune be measured by the net income of j, N j. The ill-fortune of i, relative to j, is then given by ρ(i,j): ρ(i,j) N (q j ), if q j < q i < p j N (q j ), if p j < q i < q j, otherwise (5) With ρ(i,j), i is assumed to monitor whether an individual j jumped below or above his net income position. For a jump above i s net income position, ρ is positive, and for a jump below, ρ is negative. Thus, ρ(i,j) can also be interpreted as a resentment indicator for the presence of some "newly rich" in the richer individuals than i at post-tax rank q i. To compute the expected value of ρ(i,j), it is then useful to think in terms of the post-tax rank q i of i. The average feeling of resentment r(q i ) for individual i indicates by what Measuring this intensity as N j -N i might perhaps appear more appropriate, but this would not aggregate consistently to the definition of the S-indices of horizontal inequity to be defined in the previous section. 9

11 amount the income of the richer class exceeds what the income of the richer class would have been if no "new rich" (or "upstarts") had displaced "old rich". It is given by: r(q i ) N (p) dp q i N (p) dp q i q i N s X(p) ds dp N X(p) dp q i (6) The first integral on the right of the second equality indicates the total net income of the ultimately better-ranked (the "new rich") than q i ; the second integral shows what net incomes the initially better-ranked (the "old rich") than q i enjoy, and thus the incomes of the rich had there been no reranking. IV. Gini-Based Indices and the Aggregation of Individual Perceptions We can now relate the individual indicators of relative deprivation, fiscal harshness and ill-fortune to the measurement of the distribution and redistribution of income. The proofs of the propositions are shown in the appendix. Proposition : The S-Gini indices of inequality are (mean-normalised) ethically weighted averages of relative deprivation in the population: G X (v) v µ X d p kp,v dp (7) Thus, if we give equal ethical weight to the relative deprivation of all individuals, we find the standard Gini coefficient, as Yitzhaki (979) and Hey and Lambert (98) have already shown for v=2. Proposition generalises the result to the whole class of S-Ginis. Proposition 2: The K(v) indices of TR tax progressivity are (mean-normalised) differences between the (ethically weighted) average perception of fiscal harshness and average relative deprivation in the population: K(v) v h(p) µ T d(p) k(p,v) dp µ X (8)

12 To interpret this result, we can think of deprivation as being harsh on the poor, and of the fiscal system as being (potentially) harsh on the rich. Proposition 2 contrasts these two aspects of harshness as viewed by individuals according to their ranks in the income distribution. If, overall, the harshness of the fiscal system on the rich exceeds that of relative deprivation on the poor, the tax is considered TR progressive [π(v)>]; if the two are equal, the tax is deemed proportional; if fiscal harshness falls below the harshness of relative deprivation, the tax is judged regressive. For v=2, k(p,v) gives equal weight, across the population, to all perceptions of fiscal harshness and relative deprivation. Proposition 3: The P(v) indices of IR tax progressivity are (mean-normalised) differences between (ethically weighted) average relative deprivation and the average looseness of the tax system in equalising net incomes: P(v) v d(p) µ X l(p) k(p,v) dp µ N (9) To interpret this result, it helps to think of relative deprivation as reflecting the looseness of gross incomes. A tax is then IR regressive if it is looser in levelling net incomes than the looseness of gross incomes (as measured by relative deprivation); the reverse holds for IR progressivity. Proposition 4: The E(v) indices of horizontal inequity are (mean-normalised) ethically weighted averages of ill-fortune r(p) in the population: E(v) µ N r p kp,v dp (2) Alternatively, E(v) can be interpreted as the average feeling of resentment towards the arrival of "parvenus" among the rich. V. Role of k(p) and v The ethical weights k(p,v)=v(v-)(-p) v-2 are a function of the parameter v. For v<2, more ethical weight k(p,v) is put on higher p, for v=2, equal weight applies to all p, and for v>2 greater weight is applied to areas with low values of p. From Propositions,2,3 and 4, we thus note that the greater the value of v, the more emphasis

13 is placed on perceptions of relative deprivation, fiscal harshness, fiscal looseness and illfortune of the most deprived individuals. Choosing v and aggregating individual indicators of "perceived equity" then falls in the spirit of Plotnick s (98) observation that any "inequity measure would embody a weighting scheme, the validity of which would (...) depend ultimately upon a normative judgement" (p.285) 2. We can also interpret k(p,v) as v times the density of a minimum income X(p) in a sample of v- incomes randomly and independently drawn from F X [see Muliere and Scarsini (989) and Lambert (993), p.29]. To see this, note that k * (p,v)=v[-(-p) v- ] is v times the probability that the minimum income in a sample of v- incomes fall below X(p), and thus that k(p,v)=dk * (p,v)/dp is v times the density of such a minimum income. For integer values of v, we may therefore interpret the weights k(p,v)/v as the relative frequency with which an individual with income X(p) would find himself the most deprived in randomly and independently sampled groups of v- individuals. The greater the value of v, the greater the density of the poorer relative to the richer for those who find themselves the most deprived in these random groups of v- individuals. G X (v), for instance, can then be described as the expected relative deprivation (as a proportion of mean income) of the most deprived individual in a group of v- persons. We can show that lim v > G X (v) (2) at which limit we become ethically insensitive towards inequality since we only care A different avenue for the aggregation of relative deprivation and for the definition of indices of inequality and horizontal inequity would be to define increasing and convex functions of d(p) and r(p), as implicitly suggested by Chakravarty and Chakraborty (984). Berrebi and Silber (985) alternatively suggest how we may express a number of inequality indices as variously (and somewhat arbitrarily) defined functions of relative deprivation. 2 R(2) attaches equal weight to the ill-fortune of every individual in the population. This thus seems to contradict the claim of Plotnick (98,p.285) that his index "attaches greater weight to inequities occurring among units with high preredistribution ranks than to those affecting lower ranking units". 2

14 about the relative deprivation of the richest individual, which is zero. Similarly, K(v), P(v) and E(v) all tend to when v tends to. Conversely, when v tends to infinity, we can demonstrate for a discrete distribution of H individuals that lim v > G X (v) d µ X H (22) where the S-Gini coefficient of inequality equals the relative deprivation felt by the most deprived individual in population. Similarly, when v becomes very large, we find that: lim v > lim v > K(v) P(v) lim v > f d µ T H µ X H d h µ X H µ N H v R(v) r µ N H (23) When v becomes very large, we consider only the perception of equity by the most deprived individual in computing indices of progressivity and horizontal inequity. This corresponds to a Rawlsian aggregation of individual perceptions of equity. VI. Calibration of v When carrying out an applied analysis of income redistribution, tax progressivity or horizontal inequity, a focus on a limited range of v values is generally needed for tractability and space constraints. We suggest here two methods to help in this choice of v. Both methods rely on an easily understood compromise between a desire for greater redistribution (or equity) and a desire for greater economic efficiency. First, we note that to the index of relative inequality G(v) corresponds a class of 3

15 homothetic social evaluation functions 3 function Ξ(v) is given by: whose equally distributed equivalent income Ξ X (v) µ( G X (v)) v X(p) ( p) v dp (24) where the last equality is obtained by integration by parts 4. Using the leaking bucket experiment of Okun (975), it is then possible to assess which range of v values is ethically sensible. Suppose that, with no effect on individual ranking, a tax of $a is enforced onto an individual with rank p j in the income distribution, so that a transfer of $a(-α) can be made to a poorer individual of rank p i in the distribution, where α is the size of the bucket leak in making that transfer ( α ). Agreeing on a α value which is socially tolerable, at the limit, will also determine a value for the ethical parameter v. A marginal increase in a tax of $a is indeed just socially acceptable if dξ(v)/da=, that is [using (24)], when α=α * and when α p j p i v (25) Were we to reach an ethical agreement on α *, equation (25) would give us the v value reflecting this ethical stance. A criticism of this first approach is that one generally needs to know the income value of individuals i and j to know whether a transfer from j to i is acceptable. This is because one might tolerate very little wastage in a transfer between two individuals ranked very differently but with relatively close incomes, but accept much more wastage 3 For the link between relative inequality indices and social evaluation functions, see Blackorby and Donaldson (978). 4 Since Ξ(v)=µ-µG, and because µg is the (un-normalised) relative deprivation of Proposition, the interpretation suggested for the inequality indices G(v) can easily be extended to the canonical form Ξ(v) of the social evaluation functions. Simply put, social welfare is per capita income µ minus the per capita income of which an ethically weighted random individual feels deprived in the population. 4

16 in transfers between the same two individuals if their incomes were much farther apart. One way to ease partially this informational problem is to consider transfers between wide regions of an income distribution, in a manner analogous to Blackburn (989). To make this clearer, take for instance a transfer of $(-α)a to everyone below the median that costs $a to everyone above the median. Using the same homothetic social evaluation functions as above, we find that the transfer is just socially acceptable if α=α *, with ( α ).5 v.5 v (26) Table displays α * for values of v ranging from. to 5. for different tax paying p j and transfer receiving p i (the first three columns) and for a transfer from everyone above to everyone below the median (the last column). For p i =.2 and p j =.8, for instance, specifying v=2 amounts to a social tolerance of efficiency leaks of up to $.75 for each dollar of tax on individual j. The next two columns indicate that, for a given value of v, we become less tolerant of efficiency losses and "bucket leaks" if the p i of the transfer-receiving and the p j of the tax-paying individuals are closer to one another. For a transfer from everyone above to everyone below the median, the limit values of α are approximately the same as those for p i =.25 and p j =.75. For v=4, the limit to the tolerable efficiency loss rises everywhere to above 85%, and to 98.4% in the first column; these are rather large limit values by most ethical standards. Allowing for larger values of v would imply an ethical acceptance of redistributive transfers that would be almost completely wasteful. In our applications later, we will thus limit our discussion to the results for values of v ranging from to 4. VII. Decomposition into Tax and Benefits Components The linearity of the K(v) and P(v) measures makes it straightforward to decompose total TR and IR progressivity into the sum of K(v) and P(v) indices for individual taxes and benefits 5. Denote t=µ T /µ X as the average global rate of gross 5 This decomposition is not new; see, for instance, Lambert (993), pp

17 income taxation, t m, m=,...,m, as the average rate 6 of gross income taxation of tax or benefit T m, with t=σ M=t m m, and K m (v) and P m (v) as the S-indices of TR and IR progressivity for tax or benefit m. Also define a transformed IR index P * m (v) as P * m (v)=(-t m )/(-t) P m (v). We can verify that: K(v) M m t m t K m (v) (27) P(v) M m P m (v) (28) and P m (v) t m t K m (v) (29) Finally, G X (v)-g N (v) captures the redistributive change in inequality caused by the tax and benefit system, which is simply the difference between relative deprivation in net and gross incomes. It is accounted for by the sum of the progressivity and horizontal inequity indices: G X (v) G N (v) t t K(v) E(v) P(v) E(v) (3) As shown in equations (27) and (28), K(v) and E(v) can also themselves be decomposed into the sum of progressivity indices for separate taxes and benefits. VIII. Income Distribution and Redistribution in Canada The distribution of income has been subjected to important disturbances in the 98 s in many countries around the world. Canada was no exception, having witnessed a severe recession between 98 and 983, followed by a significant recovery with relatively high growth rates until the end of 988, and with the beginning of another recession thereafter. To this were combined important labour market, demographic and 6 Both t and t m can be negative, but concentration curves and TR progressivity indices for T and T m are not defined when t or t m equal zero. 6

18 technological changes. The last decade was also the decade of major tax reforms; in Canada, taxation was particularly altered by the 987 revision of personal income taxation, which decreased the number of tax brackets, trimmed the top marginal tax rates, replaced a number of tax allowances by tax credits, broadened the tax base, and aimed, generally, to improve the perceived "fairness" of the tax system. The social security system (including the unemployment insurance, public pension, and social assistance schemes) also evolved significantly with changes in public policy and in the socio-demographic environment. To illustrate the application of the S-Gini indices of the distribution and redistribution of income, we thus use the Canadian Surveys of Consumer Finances for 98 and 99. These surveys contain, respectively, 38, and 45, observations on the distribution of pre-tax and pre-benefit family income, on the amount of personal taxes paid, and on various cash transfers received from the provincial and federal governments 7. To adjust income as well as tax and benefit data for heterogeneity in the size and the composition of families, we use the OECD equivalence scale; all monetary variables are thus in an "equivalised" individualistic form. For convenience, we have removed those families who reported negative gross or net incomes. The definition of the monetary variables is as follows [the percentages of 98 and 99 gross (i.e., market) incomes to which these variables amount are shown in parentheses]: Gross income (pre-tax and pre-benefit): Includes wages and salaries, self-employment income, private pensions, and total investment income; TAX: Total federal and provincial income tax (6.7; 22.); FAAL: Federal and Québec family and youth allowances (.;.8); CHILD: Child Tax Credit (.5;.5); OLD: Old Age Security Pensions and Guaranteed Income Supplement (3.; 3.5); PEN: Canada/Québec Pension Plan Benefits (.; 2.); 7 An important limitation of these data is that they do not allow for the modelling of indirect taxes (GST and provincial sales taxes). It is also the case that lifetime or panel data would present an alternative (and potentially more useful) picture of the dynamic and lifetime incidence of fiscal equity and discrimination. 7

19 UNEMP: Unemployment Insurance Benefits (.9; 2.7); SOCASS: Social Assistance Benefits and provincial income supplements (.;.4); OTHER: Various tax credits and grants to individuals, veterans pensions, pensions to widows, workers compensation, etc. (.7;.3). Table 2 shows the (mean-normalised) "perceived equity" of the distribution and redistribution of income for individuals at deciles to 9 in 98 and in 99. For relative ill-fortune, asymptotic standard errors are also shown using the distribution-free results of Davidson and Duclos (997). Relative deprivation is naturally higher at the lower deciles and it decreases quite quickly with deciles. For instance, an individual at the end of the first decile of gross incomes in 98 is relatively deprived of 92.6% of per capita gross income, but the median individual is relatively deprived of only 34.8% of the same per capita income. For gross incomes, relative deprivation is everywhere higher in 99 than in 98; for net incomes, relative deprivation is lower in 99 than in 98 for individuals at the end of the first four deciles but larger afterwards. Relative fiscal harshness is everywhere about four times as large as gross income relative deprivation, both in 98 and in 99; this shows the progressivity of the tax and benefit system. For instance, an individual at the end of the third decile of 99 gross incomes finds that, relative to his fiscal treatment, richer individuals pay excess taxes amounting to 27.% of the per capita net revenue of the government, although that same individual is deprived of only 59.3% of per capita gross income. Fiscal harshness thus exceeds the harshness of relative deprivation by 57.7% of per capita income at that point. For p * such that f(p * )/µ T =, we have that the taxes which individuals richer than X(p * ) pay in excess of T(X(p * )) equal the net tax paid in the population; in other words, the government could raise its overall net revenue simply by collecting only these excess taxes. Between 98 and 99, p * has moved approximately from the 58th percentile to the 6st percentile. The IR progressivity of the tax and benefit system is revealed in Table 2 by the fact that individual indicators of the relative looseness of the fiscal system are everywhere considerably lower than the losseness of (relative deprivation in) gross incomes. We also note that relative ill-fortune is highest at the first deciles, and decreases 8

20 continuously. Although this is not a necessary trend, it is not unexpected since the density of incomes is higher around the lower deciles, and it is also at these lower deciles that the benefit system is most operative and can therefore be the most discriminating. Ill-fortune ranges between.3% and.% of per capita income in 98, and exceeds % of per capita income for the first four deciles of 99. This indicator is also significantly and statistically higher (particularly for end of deciles 4, 5 and 6) in 99 than in 98, except for individuals at the end of the ninth decile 8. Table 3 shows the Kakwani and Reynolds-Smolensky indices [K m (2) and P * m (2)] for various components of the 98 and 99 tax and benefit systems. These are obtained when equal ethical weight is granted to relative deprivation and fiscal harshness or looseness across the population, as shown above 9. The asymptotic standard errors of the indices are also shown in parentheses, using again the method of Davidson and Duclos (997). To compare the size of the Kakwani indices across negative (transfers) and positive taxes, we must consider the negative of their values for negative taxes. The highest Kakwani index is then found for SOCial ASSistance benefits, followed by OLD age pensions and CHILD tax credit. As can be checked, these comparisons are statistically significant. The Kakwani index value for OTHER benefits and PENsions cannot be distinguished, but they are significantly greater, statistically, than the index values for UNEMPloyment benefits, FAmily ALowances and income TAXation. The harshness of deprivation on the poor is on average below the harshness of TAXes by around % of per capita income. As equation (29) indicates, the P * m (v) indices of IR progressivity are simple products of the TR progressivity indices and of the average rate of taxation as a proportion of net income. Indeed, because IR progressivity takes into account the 8 Simultaneously testing a number of statistically dependent hypotheses raises important inference issues. Here, we reject equality in favour of dominance if all quantile tests of equality are individually rejected with a rejection error of size 5%. The issues and the procedure are discussed, for instance, in Howes (993). 9 The aggregation is carried over the whole sample of individuals, not just over the individuals shown in Table 2. 9

21 importance of the average rate of taxation in the redistribution of income, it is a better indicator of the impact of taxes and benefits in reducing inequality than TR progressivity. For expositional constraints, it is therefore on these IR indices that we focus below. The second panel of Table 3 reveals that, in 98, OLD age benefits have the highest P * m (2) index value, followed by income TAXes, SOCial ASSistance, UNEMPloyment benefits, and public PENsions. These rankings are all statistically significant. OTHER benefits and FAmily ALlowances have the same index value, and the least IR progressive is CHILD tax credits. Those with high Reynolds-Smolensky indices have either high Kakwani indices (such as OLD age pensions and SOCial ASSistance) or large taxation rates (income TAXes). The P * m (2) ranking is the same for 99, with the exception of public PENsions which become significantly more progressive than SOCial ASSistance, and FAmily ALlowances which fall at the bottom of the list, even below CHILD tax credits. In 99, relative deprivation exceeds fiscal looseness by up to 3.5% of per capita income. Figure shows how the 99 IR progressivity ranking of the various groups of taxes and benefits varies with values of v. Recall that a rise in v increases the ethical weights granted to the relative deprivation of the most deprived in the population; it also renders indices of redistribution and of IR progressivity more dependent upon the change in the deprivation of those poorer individuals. The most sensitive group is OLD age pensions, which quickly (once v is greater than.4) becomes statistically more IR progressive than income TAXes and all other groups of transfers. For v=4, deprivation in gross incomes exceeds the fiscal looseness of OLD age pensions by almost 8% of per capita income, which alternatively indicates by equations (28) and (3) that the "tightness" of OLD age transfers tends to reduce G(4) by around.8. Income TAXes start from being the most progressive to being significantly less progressive than OLD age pensions, SOCial ASSistance benefits, and public PENsions (as soon as v lies above.8). Hence, for v s that grant sufficient weight to the relative deprivation of the poorer, income taxation, notwithstanding its relatively large rate of taxation, is significantly less progressive and redistributive than some fairly well targeted groups of benefits. FAmily ALlowances are generally the least IR progressive of all groups, followed by CHILD tax 2

22 credits and OTHER benefits. These last results hold for a wide range of v and are generally statistically very significant. Figure 2 depicts the contribution of each group of taxes and benefits to the total 99 IR progressivity indices, according to the decomposition of equation (28). As a percentage of total IR progressivity, IR progressivity for income TAXation falls rapidly with increases in v. The percentage contribution of most other groups is relatively constant, except for SOCial ASSistance and OLD age pensions whose relative importance in the redistributive process rises steadily. For larger v, almost a third of the total IR progressivity is exerted by OLD age pensions. Table 4 indicates how the S-Gini coefficients for the distribution of gross and net incomes evolved between 98 and 99, for various values of v. These S-Ginis are ethically-weighted averages of relative deprivation in the population, and we note that they naturally increase with v. The most deprived of two random individuals (v=3) is expected to be relatively deprived of 54% of total gross income in 98; for three random individuals (v=4), the corresponding figure is 63.2%. Gross income inequality witnessed a statistically significant increase between 98 and 99, for all values of v; for v=2, for instance, gross income inequality increased by.25, from.384 to.49, implying an increase of 7% in average relative deprivation. Net income inequality fell significantly, however, between 98 and 99, for all values of v equal to or greater than 2. Table 4 also portrays the values of the indices of total redistribution G X -G N for the two years. These indices of redistribution show by what proportion of total income average relative deprivation is decreased by the tax and benefit system. In 98 and taking v=2, for instance, the population s average relative deprivation was decreased from 38.4% to 29.6% of total income by the tax and benefit system. As equations (28) and (3) show, total redistribution is a function of the sum of IR progressivity for all groups of taxes and transfers, minus the index of horizontal inequity. In the light of the previous results on the change in inequality between 98 and 99 and on the change in the progressivity indices between these two years, it is therefore not surprising to note that the overall redistributive impact of the tax and benefit system increased significantly between 98 and 99, regardless of the value of v. The redistributive fall in average 2

23 relative deprivation increases as our ethical focus on the poor rises; relative to 98, the fall in 99 is also larger as v increases. For v=2, only 77% of the relative deprivation in the 98 gross income distribution remains after the income redistribution operated by the tax and benefit system; for 99, the figure is 7%. These percentage falls do not vary much across the different values of v. We also find in Table 4 the estimates of the E(v) indices of reranking and horizontal inequity. These indices are ethically-weighted averages of relative ill-fortune; for v=2, for instance, twice the equally-weighted ill-fortune equals.% of mean income in 98, and.7% in 99. The indices rise, respectively, to 2.8% and 4.2% of per capita income when we estimate the expected ill-fortune of the most deprived individual in random groups of 3 individuals (v=4). It is clear from Table 4 that horizontal inequity has increased between 98 and 99 for all v equal to or greater that.6. Furthermore, as equation (3) indicates, horizontal inequity lessens the P(v) progressivity impact on the reduction in S-Gini coefficients. We can check that, as a proportion of P(v), horizontal inequity increases with v, and is also about 5% larger in 99 than in 98. This suggests that, in mitigating income redistribution, horizontal inequity is more costly in 99 than in 98, and more costly too if we increase our ethical focus on the poor. IX. Conclusion We have shown how popular indices of the distribution and redistribution of income can be interpreted as population aggregates of individual indicators of relative deprivation, fiscal harshness, and ill-fortune in the allocation of taxes and benefits. These indices include the standard Gini and the S-Ginis of Donaldson and Weymark (98) and Yitzhaki (983), the Kakwani (977) index of progressivity, the Reynolds-Smolensky (977) index of vertical equity, and the Atkinson (979)-Plotnick (98) index of horizontal inequity. The aggregation of the individual indicators of "perceived equity" is performed using ethical weights that are a function of a parameter v. For an integer v greater than, these weights can be seen to average the equity perceptions of the most deprived individual in random groups of v- individuals. To help parameterise the indices, we also propose "leaky bucket" experiments that suggest limiting the range of plausible v to between and 4. 22

24 We apply the approach to the Canadian distribution and redistribution of income in 98 and 99. We find that relative deprivation in gross incomes in higher in 98 than in 99, but it fell for net incomes between these two years for the first four deciles. Fiscal harshness on the rich is about four times as large as deprivation relative to the rich, a feature which connects closely with the measured progressivity of the tax and benefit system. Relative ill-fortune, which is linked to indices of horizontal inequity, is highest at the lowest deciles and averages between.5% and.8% of per capita income. It is also about 5% greater in 99 than in 98. S-Gini indices and average relative deprivation increased significantly between 98 and 99 for the distribution of gross incomes, but fell for the distribution of net incomes for the standard Gini and all other Ginis with a greater focus on the relative deprivation of the most deprived. Increasing v from 2 to 4 increases our ethical expectation of relative deprivation in the population from about 4% to 65% of total gross income, and from about 3% to 5% of total net income. Taxes and benefits reduce this average deprivation by about 25% in 98 and 3% in 99. Unemployment benefits and social assistance amount, together, to about 25% of total redistribution, and Old Age and pension benefits for at least 35%, whatever the value of v. The estimated contribution of income taxes to total redistribution decreases from more than 25% to about % as we become more sensitive to the relative welfare of the more deprived. All taxes and benefits are (except family allowances) significantly more redistributive in 99 than they were in 98; public pensions, social assistance benefits, unemployment benefits, and old age benefits witnessed the greatest increases in that decade. 23

25 Appendix Proof of Proposition : Inserting equation (7) into (8) and using the definition () of a Lorenz curve, we find that d(p i ) µ X L X p i X p i p i (3) This yields the following (mean-normalised) ethically weighted average of relative deprivation: vµ d(p) k(p) dp X (v ) L X p ( p) (v 2) X p ( p) (v ) µ X dp (32) Proceeding by integration by parts for the term X(p)(-p) (v-) by integrating X(p) to yield L X (p) and differentiating (-p) (v-), we find that vµ d(p) k(p) dp X L X (p) k(p) dp (33) Since pk(p)dp (34) we find that (33) is also the S-Gini defined in (3). Proof of Proposition 2: Insert () into () and using the definition (2) of a concentration curve, we find: h(p i ) µ T C T p i T X p i p i (35) As for the proof of Proposition, integration by parts of the mean-normalised ethically weighted h(p) yields vµ h(p) k(p) dp T 24 C T (p) k(p) dp (36)

26 such that v h(p) d(p) k(p) dp µ T µ X L X (p) C T (p) k(p) dp (37) This is the K(v) index defined in (4). Proof of Proposition 3: The proof simply proceeds as for Proposition 2 above, replacing T by N and h(p) by l(p). Proof of Proposition 4: From (6), r(p i ) can be expressed as r(p i ) µ N L N p i µ N C N p i (38) Aggregating r(p) then leads straightforwardly to the horizontal inequity indices E(v) defined in (6): µ N r(p) k(p) dp C N (p) L N (p) k(p) dp E(v) (39) 25

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