Document(s) 6 of 20

4.1 Lorenz curves

The Lorenz curve has been for several decades the most popular graphical tool for visualizing and comparing income inequality. As we will see, it provides complete information on the whole distribution of incomes relative to the mean. It therefore gives a more comprehensive description of relative incomes than any one of the traditional summary statistics of dispersion can give, and it is also a better starting point when looking at income inequality than the computation of the many inequality indices that have been proposed. As we will see, its popularity also comes from its usefulness in establishing orderings of distributions in terms of inequality, orderings that can then be said to be "ethically robust".

The Lorenz curve is defined as follows¹:

The numerator sums the incomes of the bottom p proportion (the poorest 100p%) of the population. The denominator sums the incomes of all. L(p) thus indicates the cumulative percentage of total income held by a cumulative proportion p of the population, when individuals are ordered in increasing income values. For instance, if L(0.5) = 0.3, then we know that the 50% poorest individuals hold 30% of the total income in the population.

¹ DAD: Curves|Lorenz.

A discrete formulation of the Lorenz curve is easily provided. Recall that the discrete income values yi are ordered such that y₁ ≤ y₂ ≤... ≤ y_n, with percentiles p_i = i/n such that Q(p_i) = y_i. For i = 1,...n, the discrete Lorenz curve is then defined as:

If needed, other values of L(p) in (4.2) can be obtained by interpolation.

The Lorenz curve has several interesting properties. As shown in Figure 4.1, it ranges from L(0) = 0 to L(1) = 1, since a proportion p = 0 of the population necessarily holds a proportion of 0% of total income, and since a proportion p = 1 of the population must hold 100% of aggregate income. L(p) is increasing as p increases, since more and more incomes are then added up. This is also seen by the fact that the derivative of L(p) equals Q(p)/μ:

This is positive if incomes are positive, as we are assuming throughout. Hence by observing the slope of the Lorenz curve at a particular value of p, we also know the p-quantile relative to the mean, or, in other words, the income of an individual at rank pas a proportion of mean income. An example of this can be seen on Figure 4.1 for p = 0.5. The slope of L(p) at that point is Q(0.5)/μ, the ratio of the median to the mean. The slope of L(p) thus portrays the whole distribution of mean-normalized incomes.

The Lorenz curve is also convex in p, since as p increases, the new incomes that are being added up are greater than those that have already been counted. This is clear from equation (4.3) since Q(p) is increasing in p. Mathematically, a curve is convex when its second derivative is positive, and the more positive that second derivative, the more convex is the curve. Formally, the second-order derivative of the Lorenz curve equals

Note that by definition that p ≡ F(Q(p)). Differentiating this identity with respect to p, we have that 1 ≡ f(Q(p)) d(Q(p))/dp. Thus,

and we therefore have that

Figure 4.1: Lorenz curve

The larger the density of income f(Q(p)) at a quantile Q(p), the less convex the Lorenz curve at L(p). The convexity of the Lorenz curve is thus revealing of the density of incomes at various percentiles. On Figure 4.1, this density is thus visibly larger for lower values of p since this is where the slope of the L(p) changes less rapidly as p increases.

Some measures of central tendency can also be identified by a look at the Lorenz curve. In particular, the median (as a proportion of the mean) is given by Q(0.5)/μ, and thus, as mentioned above, by the slope of the Lorenz curve at p = 0.5. Since many distributions of incomes are skewed to the right, the mean often exceeds the median and Q(p = 0.5)/μ will typically be less than one. The mean income in the population is found at that percentile at which the slope of L(p) equals 1, that is, where Q(p) = μ and thus at percentile F(μ) (as shown on Figure 4.1). Again, this percentile will often be larger than 0.5, the median income's percentile. The percentile of the mode (or modes) is where L(p) is least convex, since by equation (4.4) this is where the density f(Q(p)) is highest.

Simple summary measures of inequality can readily be obtained from the graph of a Lorenz curve. The share in total income of the bottom p proportion of the population is given by L(p); the greater that share, the more equal is the distribution of income. Analogously, the share in total income of the richest p proportion of the population is given by 1 – L(p); the greater that share, the more unequal is the distribution of income. These two simple indices of inequality are often used in the literature.

An interesting but less well-known index of inequality is given by the proportion of total income that would need to be reallocated across the population to achieve perfect equality in income. This proportion is given by the maximum value of p – L(p), which is attained where the slope of L(p) is 1 (i.e., at L(p = F(μ))). It is therefore equal to F(μ) – L(F(μ)). This index is usually called the Schutz coefficient.

Mean-preserving equalizing transfers of income are often called Pigou-Dalton transfers. In money-metric terms, they involve a marginal transfer of $1, say, from a richer person (of percentile r, say) to a poorer person (of percentile q < r)that keeps total income constant. All indices of inequality which do not increase (and sometimes fall) following any such equalizing transfers are said to obey the Pigou-Dalton principle of transfers. These equalizing transfers also have the consequence of moving the Lorenz curve unambiguously closer to the line of perfect equality. This is because such transfers do not affect the value of L(p) for all p up to q and for all p greater than r, but they increase L(p) for all p between q and r.

Hence, let the Lorenz curve L_B(P) of a distribution B be everywhere above the Lorenz curve L_A(P) of a distribution A. We can think of B as having been obtained A through a series of equalizing Pigou-Dalton transfers applied to an initial distribution A. Hence, inequality indices which obey the principle of transfers will unambiguously indicate more inequality in A than in B. We will come back to this important result in Chapter 11 when we discuss how to make ethically robust comparisons of inequality.

4.2 Gini indices

If all had the same income, the cumulative percentage of total income held by any bottom proportion p of the population would also be p. The Lorenz curve would then be L(p) = p: population shares and shares of total income would be identical. A useful informational content of a Lorenz curve is thus its distance, p – L(p), from the line of perfect equality in income. Compared to perfect equality, inequality removes a proportion p – L(p) of total income from the bottom 100 .p % of the population. The larger that "deficit", the larger the inequality of income.

If we were then to aggregate that deficit between population shares and income shares in income across all values of p between 0 and 1, we would get half the well-known Gini index²:

The Gini index implicitly assumes that all "share deficits" across p are equally important. It thus computes the average distance between cumulated population shares and cumulated income shares.

4.2.1 Linear inequality indices and S-Gini indices

One can, however, also think of other weights to aggregate the distance p–L(p). The class of linear inequality indices is given by applying percentile-dependent weights to those distances. Let those weights be defined by κ(P). A popular one-parameter functional specification for such weights is given by and depends on the value of a single "ethical" parameter ρ That parameter must be greater than 1 for the weights κ(P; p) to be positive everywhere. The shape of κ(p;ρ) is shown on Figure 4.2 for values of ρ equal to 1.5, 2 and 3. The larger the value of ρ the larger the value of κ(P;ρ) for small p.

² DAD: Inequality | Gini/S-Gini Index.

Figure 4.2: The weighting function κ(p;ρ)

Figure 4.3: The weighting function w(p;ρ)

Using (4.8) then gives what is called the class of S-Gini (or "Single-Parameter" Gini) inequality indices, I(ρ)³:

E:18.8.2

Note⁴ that I(2) is the standard Gini index. This is because κ(p;ρ = 2) ≡ 2, which then gives equal weight to all distances p – L(p). When 1 < ρ < 2, relatively more weight is given to the distances occurring at larger values of p, as shown by Figure 4.2. Conversely, when ρ > 2, relatively more weight is given to the distances found at lower values of p. Changing ρ thus changes the "ethical" concern which is felt for the "share deficits" at various cumulative proportions of the population.

Let ω(p;ρ) be defined as

The shape of ω(p; ρ) is shown on Figure 4.3 for ρ equal to 1.5, 2 and 3. Note that ω(p; ρ) > 0 and that dω(p;ρ)/dp < 0 when ρ > 1. Since for any value of ρ the area under each of the three curves on Figure 4.3 equals 1 too. Using (4.10) and integrating by parts equation (4.9), we can then show that⁵:

E:18.8.31

This says that I(p) weights deviations of incomes from the mean by weights which fall with the ranks of individuals in the population. Since, in equation (4.11), I(p) is a (piece-wise) linear function of the incomes Q(p), it is a member of the class of linear inequality measures, a feature which will prove useful later in measuring progressivity and vertical equity. The usual Gini index is then given simply by:

Yaari (1988) defines "an indicator for the policy maker's degree of equality mindedness at p" as –ω⁽¹⁾(p;ρ)/ω(p;ρ), where ω⁽¹⁾(p;ρ) is the first-order

³ DAD: Inequality|Gini/S-Gini Index.

⁴ DAD: Curves|Lorenz.

⁵ DAD: Inequality|Gini/S-Gini Index.

derivative of ω(p; ρ) with respect to p. This indicator thus captures the speed at which the weights ω(p;ρ) decrease with the ranks p. It gives:

Thus, the local degree of "equality mindedness" for ω(p;ρ) is a proportional function of the single parameter ρ. As definition (4.13) makes clear, this degree of inequality aversion is defined at a particular rank p in the distribution of income, independently of the precise value that income takes at that rank. The larger the value of ρ, the larger the local degree of equality mindedness, and the faster the fall of the weights ω(p;ρ) with an increase in the rank p. Therefore, the greater the value of ρ, the more sensitive is the social decision-maker to differences in ranks when it comes to granting ethical weights to individuals.

The functions κ(p;ρ) and ω(p;ρ) can also be given an interpretation in terms of densities of the poor. Assume that r individuals are randomly selected from the population. The probability that the income of all of these r individuals will exceed Q(p) is given by [1 - F(Q(p))] ^r. The probability of finding an income below Q(p) in such samples is then 1 - [1 - F(Q(p))] ^r = 1 - [1 - p] ^r. 1 - [1 - p] ^r is thus the distribution function of the lowest income in samples of r individuals. The density of the lowest income rank in a sample of r randomly selected incomes is the derivative of that distribution with respect to p, which is

This helps interpret the weights κ(p;ρ) and ω(p;ρ). By equation (4.8), κ(P; ρ) is ρ times the density of the lowest income in a sample of ρ- 1 randomly selected individuals; analogously, by equation (4.10), ω(p;ρ) is the density of the lowest income in a sample of ρ randomly selected individuals.

We might be interested in determining the impact of some inequality-changing process on the inequality indices of type (4.11). One such process that can be handled nicely spreads income away from the mean by a proportional factor λ, and thus corresponds to some form of bi-polarization of incomes away from the mean (loosely speaking). This bi-polarization process is equivalent to adding (λ - 1)(Q(p) - μ)to Q(p), since

does indeed spread income away from the mean by a proportional factor λ. As can be checked from equation (4.11), this changes I(p) proportionally by λ:

Equation (4.16) also says that the elasticity of I(p) with respect to λ, when λ equals 1 initially, is equal to 1 whatever the value of the parameter

Such bi-polarization away from the mean is also equivalent to a process that increases the distance p - L(p) by a factor λ. That this gives the same change in I(ρ) can be checked from equation (4.9). This bi-polarization process thus increases the deficit p - L(p) between population shares p and income shares L (p) by a constant factor λ across all p. We will see later in Chapter 12 how this distance-increasing process leads to a nice illustration of the possible impact of changes in inequality on poverty.

As shown on Figure 4.3 and in equation 4.11, the larger the value of ρ, the greater the weight given to the deviation of low incomes from the mean. When ρ becomes very large, the index I(ρ) equals the proportional deviation from the mean of the lowest income. When ρ = 1, the same weight ω(p; ρ = 1) ≡ 1 is given to all deviations from the mean, which then makes the inequality index I(ρ = 1) always equal to 0, regardless of the income distribution under consideration. Thus, S-Gini indices range between 0 (when all incomes are equal to the mean or when the ethical parameter ρ is set to 1) and 1 (when total income is concentrated in the hands of only one individual, or when ρ is large and the lowest income is close to 0). Since the Lorenz curve moves towards p when a Pigou-Dalton equalizing transfer is implemented, the value of the S-Gini indices also naturally decreases with such transfers.

Hence, ρ is a parameter of "inequality aversion" that captures our concern for the deviation of quantiles from the mean at various ranks in the population. In this sense, it is analogous to the parameter ε of relative inequality aversion which we will discuss below in the context of the Atkinson indices. For the standard Gini index of inequality, we have that ρ = 2 and thus that ω(p;ρ = 2) = 2(1-p); hence in assessing the standard Gini, the weight on the deviation of one's income from the mean decreases linearly with one's rank in the distribution of income. In a discrete formulation, the weights ω(p; ρ) take the form of:

4.2.2 Interpreting Gini indices

The S-Gini indices can also be shown to be equal to the covariance formula

a formula which can simplify their computation with common spreadsheet or statistical softwares. The traditional Gini is then simply:

and is just a proportion of the covariance between incomes and their ranks. Note here the interesting analogy of (4.19) with the variance, given by

A further useful interpretive property of the standard Gini index is that it equals half the mean-normalized average distance between all incomes:

Thus, if we find that the Gini index of an income distribution equals 0.4, then we know that the average distance between the incomes of that distribution is of the order of 80% of the mean. Again, note the interesting link of (4.21) with another definition of the variance, which is var

The Gini index can also be computed as the integral of a simple transformation of the familiar cumulative distribution function. Recall that F(y) and 1 - F(y) are simply the proportions of individuals with incomes below and above y. If we integrate the product of these proportions across all possible values of y, we again obtain the Gini coefficient:

Note also that F(y)(1 – F(y)) is largest at F(y) = 0.5, which also explains why the Gini index is often said to be most sensitive to changes in incomes occurring around the median income.

Now suppose that society can be split into two classes, and that income is equally distributed within each class.

1. Assume that those in the first class hold no income. The Gini index of the total population is then given by the population share of that zero-income class.
   
   
2. Assume that the population share of each group is 0.5. The Gini index of the total population is then given by 0.5 – L(0.5). In other words, the income share of the bottom class is 0.5 minus the Gini coefficient.
   
   
3. Assume that the population share of each group is again 0.5. Denote the incomes of those in the richer class by y_R and of those in the poorer class by y_p We then have:

or alternatively

which gives a simple relationship between incomes and the Gini coefficient. For instance, if y_R = λ_yP, then the Gini index is simply (λ – 1)/(2λ + 2); for λ = 2, we thus have I(ρ = 2) = 1/6.

4.2.3 Gini indices and relative deprivation

A final interesting interpretation of the Gini index is in terms of relative deprivation, which has been linked in the sociological and psychological literature to subjective well-being, social protest and political unrest. Runciman (1966) 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.10)

Sen (1973), Yitzhaki (1979) and Hey and Lambert (1980) follow Runciman's lead to propose for each individual an indicator of relative deprivation that measures the distance between his income and the income of all those relative to whom he feels deprived. For instance, let the relative deprivation of an individual with income Q(p), when comparing himself to another individual with income Q(q), be given by:

The expected relative deprivation of an individual at rank p is then ⁶:

As we did for the "shares deficits" above, we can aggregate the relative deprivation at every percentile p by applying the weights κ(p;ρ). We can show that this gives the S-Gini indices of inequality:

6DAD: Curves|Relative Deprivation.

Hence, the S-Gini indices are also a weighted average of the average relative deprivation felt in a population. By equations (4.8), (4.14) and (4.27), they equal the expected relative deprivation of the poorest individual in a sample of ρ – 1 randomly selected individuals. The greater the value of ρ the more important is the relative deprivation of the poorer in computing I(ρ).

4.3 Social welfare and inequality

We now introduce the concept of a social welfare function. Unlike relative inequality, which considers incomes relative to the mean, social welfare aggregates absolute incomes. We will see that under some popular conditions on the shape of social welfare functions, the measurement of inequality and social welfare can often be nicely linked and integrated, and that the tools used for the two concepts are then similar. This will explain why some inequality indices are sometimes called "normative".

The social welfare functions we consider take the form of:

where for expositional simplicity we restrict ω(p) to be of the special form ω(p;ρ) defined by equation (4.10). U(Q(p)) is a "utility function" of income Q(p). Social welfare is then the expected utility of the poorest individual in a sample of (ρ - 1) individuals.

Another requirement that we wish to impose on the form of W is that it be homothetic. Homotheticity of W is analogous to the requirement for consumer utility functions that the expenditure shares of the different consumption goods be constant as income increases, or the requirement for production functions that the ratios of the marginal products of inputs stay constant as output is increased. For social welfare measurement, homotheticity implies that the ratio of the marginal social utilities (the marginal utility being given by U'(Q(p)) ω(p)) of any two individuals in a population stays the same when all incomes are changed by the same proportion. For (4.28) to be homothetic, we need U(Q(p)) to take the popular form of U(Q(p); ε), which is defined as

Hence, W in equation (4.28) will depend on the parameters ρ and on ε and we will denote this as W(ρ,ε)⁷:

Homotheticity of a social welfare function has an important advantage: the social welfare function can then easily be used to measure relative inequality. To see how this can be done, define ξ(ρ, ε) as the equally distributed income that is equivalent, in terms of social welfare, to the actual distribution of income. We will refer to ξ as the EDE income, the equally distributed equivalent income. ξ(ρ,ε) is implicitly defined as:

Since is also such that

or, alternatively,

where is the inverse utility function:

The index of inequality I corresponding to the social welfare function W is then defined as the distance between the EDE and the mean incomes, expressed as a proportion of mean income:

Using ξ(ρ,ε) in (4.35) gives I(ρ, ε): I(ρ, ε) = 1 - ξ(ρ,ε)/μ ⁸.

Clearly, then, the EDE income is a simple function of average income and inequality, with

⁷ DAD: Welfare|S-Gini Index.

⁸ DAD: Inequality|Atkinson-Gini Index.

Compared to W, ξ has the advantage of being money metric and thus of being easily interpreted. It can, for instance, be compared to other economic indicators that are also expressed in money-metric terms.

To increase social welfare, we can either increase μ or we can increase equality of income 1 - I by decreasing inequality I. Two distributions of income can display the same social welfare even with different average incomes if these differences are offset by differences in inequality. This is shown in Figure 4.4, starting initially with two different levels of mean income μ₀ and μ₁ and common zero inequality. We then have that ξ = μ₀ and ξ = μ₁ To preserve the same level of social welfare in the presence of inequality, mean income must be higher: this is shown by the positive slope of the constant ξ functions. Furthermore, as inequality becomes larger, further increases in I must be matched by higher and higher increases in mean income for social welfare not to fall.

Defined as in (4.35), inequality has an interesting interpretation: it measures the difference between

the mean level of actual income

and the (lower) level that would instead be needed to achieve the same level of social welfare were income distributed equally across the population.

This difference being expressed as a proportion of mean income, I thus shows the per capita proportion of income that is "wasted" in social welfare terms because of its unequal distribution. Society as a whole would be just as well-off with an equal distribution of a proportion of just 1 - I of total actual income. I can thus be interpreted as a unit-free indicator of the social cost of inequality.

Let a distribution B of income be a proportional re-scaling of a distribution A. In other words, for a constant λ > 0, let QB(P) = λQA(P) for all p. If the social welfare function used for the computation of I is homothetic, it must be that I_A = I_B This is illustrated in Figure 4.5 for the case of two incomes and for an initial distribution A, and two incomes and for a "scaled-up" distribution B (since λ > 1). Social welfare in A is given by W_A. The social indifference curve W_A shown in Figure 4.5 also depicts the many other combinations of incomes that would yield the same level of social welfare. The combinations at point F correspond to a situation of equality of income where both individuals enjoy ξ_A.ξ_A is therefore the equally distributed income that is socially equivalent to the distribution (, ).

The average income in A is given by μ_A, which leads to point G = (μ_A,μ_A)in Figure 4.5. Hence two distributions of income, one made of the vector (, ) and the other of the vector (ξ_A, ξ_A), generate the same level W_A of social welfare, the first with an unequally distributed average income μ_A and the other with an equally distributed average income ξ_A. Hence, the vertical (or horizontal) distance between point F and point G in Figure 4.5 can be understood as the "cost of inequality" in A's distribution of income. Taking that distance as a proportion of μ_A (see equation (4.35)) gives the index of inequality I_A

That for the same λ can be seen from the fact that the two vectors of income lie along the same ray from the origin. If the function W is homothetic, then inequality in A must be the same as inequality in B. In other words, the distance between points D and E as a proportion of the distance OE must be the same as the distance between points F and G as a proportion of the distance OG.

4.4 Social welfare

4.4.1 Atkinson indices

Two special cases of W(ρ,ε) are of particular interest in assessing social welfare and relative inequality. The first is when income ranks are not important per se in computing social welfare: this is obtained with ρ = 1, and it yields the well-known Atkinson additive social welfare function, W(ε)⁹

This Atkinson social welfareatk function has had two major interpretations: 1) first, as a utilitarian social welfare function, where U(Q(p);ε) is an individual utility function displaying decreasing marginal utilities of income, and 2) second, as a concave social evaluation of a concave individual utility of income.

It can be argued, however, that "it is fairly restrictive to think of social welfare as a sum of individual welfare components", and that one might feel that "the social value of the welfare of individuals should depend crucially on the levels of welfare (or incomes) of others" (Sen 1973, pp.30 and 41). The unrestricted form W(ρ,ε) allows for such interdependence and may therefore be thought more flexible than the Atkinson additive formulation. In the light of the above, we can indeed interpret W(ρ,ε) as the expected utility of the poorest individual in a group of ρ randomly selected individuals, or the expected social valuation of the utility of such individuals. This interpretation of the social evaluation function W(ρ,ε) confirms why it is not additive or separable in individual welfare: the social welfare weight on U(Q(p);ε) depends on the rank p of the individual in the whole distribution of income. It is only when ε = 1 that W(ρ,ε) gives the average utility U(Q(p);ε) weighted by a function of ranks.

⁹ DAD: Welfare|Atkinson Index.

Figure 4.6 shows the shape of the utility functions U(y,ε) for different values of ε¹⁰ Incomes are shown on the horizontal axis as a proportion of their mean, and utility U(y;ε) can be read on the vertical axis. The normalization U(μ;ε) = 1 has been applied for graphical convenience. Although for all values of ε the slope of U(y;ε) is positive, that slope is not constant. This is made more explicit on Figure 4.7 which shows the marginal social utility of income U⁽¹⁾(y;ε) for different values of ε. Again, a normalization of U⁽¹⁾(μ;ε) = 1 is applied. For ε = 0, the marginal social utility is constant: increasing by a given amount a poor person's income has the same social welfare impact as increasing by the same amount a richer person's income. For ε > 0, however, increasing the poor's income is socially more desirable than increasing the rich's. The larger the value of ε, the faster marginal social utility falls with y.

By (4.33) and (4.35), the Atkinson inequality index is then given by¹¹:

The Atkinson indices are said to exhibit constant relative inequality aversion since the elasticity of U^(l)(Q(p);ε) with respect to Q(p) is constant and equal to ε:

The parameter ε is thus usually called the Atkinson parameter of relative inequality aversion.

Figure 4.8 illustrates graphically the link between the Atkinson social evaluation functions W(ε) and their associated inequality indices. For this, suppose a population of only two individuals, with incomes y₁ and y₂ as shown on the horizontal axis. Mean income is given by μ = (y₁ + y₂)/2 (the middle point between y₁ and y₂). The utility function U(y;ε) has a positive but decreasing slope. W(ε) is then given by (U(y₁) + U(y₂))/2, the average height of U(y₁) and U(y₂).

If equally distributed, a mean income of ξ would be sufficient to generate that same level of social welfare, since on Figure 4.8 we have that W(ε) = U(ξ;ε). The cost of inequality is thus given by the distance between μ and ξ, shown as C on Figure 4.8. Inequality is the ratio C/μ.

Graphically, the more "concave" the function U(y;ε), the greater the cost of inequality and the greater the inequality indices I(ε). This can be seen on Figure 4.9 where two functions U(y;ε) have been drawn, with different relative inequality aversion parameters ε⁰ < ε¹. We have that W(ξ) = U (ξ⁰; ε⁰) and W(ξ) = U(ξ¹;ε¹). The difference in relative inequality aversion parameters nevertheless leads to ξ⁰ > ξ¹, and therefore to I(ε⁰) < I(ε¹). A specification with greater inequality aversion leads to a greater inequality index, and to the judgement that inequality costs socially a greater proportion of average income.

¹⁰ This paragraph draws from Cowell (1995), pp.40-41.

¹¹ DAD: Inequality|Atkinson Index.

4.4.2 S-Gini social welfare indices

The second special case of W(ρ,ε) is obtained when the utility functions U(Q (p);ε) are linear in the levels of income, and thus when ε = 0. This yields the class of S-Gini social welfaresgini functions, W(ρ)¹²:

Social welfare is then the expected income of the poorest individual in a group of ρ randomly selected individuals. By (4.33), this is also the EDE income. Hence, the associated inequality indices are given by:

which is seen by (4.11) to be the same as the S-Gini inequality indices I(ρ). Hence, social welfare and the EDE income equal per capita income corrected by the extent of relative deprivation in those incomes:

4.4.3 Generalized Lorenz curves

A useful curve for the analysis of the distribution of absolute incomes is the Generalized Lorenz curve. It is defined as GL(p)¹³:

and is illustrated on Figure 4.10. The Generalized Lorenz curve has all of the attributes of the Lorenz curve, except for the fact that it does not normalize incomes by their mean. GL(p) gives the absolute contribution to per capita income of the bottom p proportion (the 100p% poorest) of the population. GL(p) is thus also the per capita income that would be available if society could rely only on the income of the bottom p proportion of the population. Assume for instance that μ = $20000 and that GL(0.5) = $5000. Then, per capita income would be only $5000 if we assumed that the richest 50% of the population were suddenly to retire and earn no income... Note also that GL(p)/p gives the average income of the bottom p proportion of the population. In the example just provided, the average income of the 50% poorest would be $10,000, half the level of overall average income.

¹² DAD: Welfare|S-Gini Index.

¹³ DAD: Curves|Generalized Lorenz.

Combining (4.9), (4.35) and (4.40) further shows that the Generalized Lorenz curve has a nice graphical link to the S-Gini indices of social welfare:

4.5 Statistical and descriptive indices of inequality

A popular descriptive index of inequality is the quantile ratio. This is simply the ratio of two quantiles, Q(p2)/Q(p1) using percentiles p1 and p2¹⁴. Popular values of p1 and p2 include p1 = 0.25 and p2 = 0.75 (the quartile ratio), as well as p1 = 0.10 and p2 = 0.90 (the decile ratio). Note that these values of p1 and p2 are often reversed. Median income is also a popular choice for Q(p1). Observe also that these ratios are by definition insensitive to changes that affect quantiles other than Q(p1) and Q(p2). Moreover, none of them is consistent with Lorenz inequality orderings: it can be that the Lorenz curve for a distribution A is always above that of distribution B, but that quantile ratios suggest that B has less inequality than A. For inequality analysis, an arguably better choice for normalizing Q(p2) is mean income — an index such as Q(p2)/μ can indeed be shown to be consistent with first-order (restricted) inequality dominance (we discuss this in Chapter 11).

The coefficient of variation is the ratio of the standard deviation to the mean of income. It is given by¹⁵:

and is therefore a function of the squared distance between incomes and the mean.

¹⁴ DAD: Inequality|Quantiles Ratio.

¹⁵ DAD: Inequality|Coefficient of Variation.

Two other popular measures of inequality use distances in logarithms of income. The first one, which we can call the logarithmic variance, is defined as¹⁶

and the second, the variance of logarithms, as¹⁷

These two last measures do not, however, always obey the Pigou-Dalton principle of transfers — that is, they will sometimes increase following a spread-reducing transfer of income between two individuals.

Finally, the relative mean deviation is the average absolute deviation from mean income, normalized by mean income¹⁸:

Note that this measure is insensitive to transfers made between individuals whose income lies on the same side of the mean.

4.6 Decomposing inequality by population subgroups

A frequent goal is to explain the total amount of inequality in a distribution by the extent of inequality found among socio-economic groups ("intra" or "within" group inequality) and across them ("inter" or "between" group inequality). There are several ways to do this. One method uses the class of inequality indices that are exactly decomposable into terms that account for within- and between-groups inequality. Although that class can be given a justification in terms of social welfare functions, this exercise is less transparent and intuitive than for the classes of relative inequality indices considered hitherto. Another method applies the Shapley decomposition to any type of inequality indices. We discuss these two methods in turn.

4.6.1 Generalized entropy indices of inequality

For most practical purposes, we can express these decomposable inequality indices as Generalized entropy indices. We denote them as I(θ)¹⁹:

16DAD: Inequality|Logarithmic Variance.

17DAD: Inequality|Variance of Logarithms.

18DAD: Inequality|Relative Mean Deviation.

19DAD: Inequality|Entropy Index.

Some special cases of (4.50) are worth noting. First, if we constrain θ to be no greater than 1 and let θ = 1 - ε, I(θ) becomes ordinally equivalent to the family of Atkinson indices. This simply means that if an Atkinson index I(ε) indicates that there is more inequality in a distribution A than in a distribution B, then the index I(θ) with θ = 1 - ε will also necessarily indicate more inequality in A than in B. Second, the special case I(θ = 0) gives the Mean Logarithmic Deviation, since I(θ = 0) can also be expressed as

that is, as the average deviation between the logarithm of the mean and the logarithms of incomes. I(θ = 1) gives the well-known Theil index of inequality. I(θ = 2) is half the square of the coefficient of variation (see (4.46)) since I(θ = 2) can be rewritten as

Now assume that we can split the population into K mutually exclusive population subgroups, k = 1,...,K. The indices in (4.50) can then be decomposed as follows²⁰:

where φ(k) is the proportion of the total population that belongs to subgroup k and μ(k) is the mean income of subgroup k.

I(k;θ) is inequality within subgroup k, defined in exactly the same way as in (4.50) for the total population. The first term in (4.53) can thus be interpreted as a weighted sum of the within-group inequalities in the distribution of income.

²⁰DAD: Decomposition|Entropy: Decomposition by Groups.

is total population inequality when each individual in subgroup k is given the mean income μ,(k) of his subgroup (namely, when within subgroup inequality has been eliminated). I(θ) can thus be interpreted as the contribution of between-group inequality to total inequality.

Note, however, that only when θ = 0 is it the case that the within-group inequality contributions do not depend on mean income in the groups; the terms I(k;θ = 0) are then strictly population-weighted. Otherwise, the within-group inequalities are weighted by weights which depend on the mean income in the subgroups k. Depending on the context, this can make I(θ = 0) a more attractive decomposable index than for other values of θ.

4.6.2 A subgroup Shapley decomposition of inequality indices

This decomposition involves two steps. The first one is to decompose total inequality into global between-group and within-group contributions. The second step is to the express global within-group contribution as a sum of the within-group contribution of each of the groups.

For each of these two steps, we want to assess by how much inequality would be reduced if we removed one of the "factors" that contribute to inequality. Take for instance the first step. It has two factors, within-group and between-group inequality. By how much would inequality fall if between group inequality were eliminated? One estimate would be given by the difference between initial inequality and inequality after the mean income of all groups has been equalized. Another estimate would be given by the inequality that remains once within-group inequality is removed and all that there is left is between-group inequality. These two estimates, however, will generally differ. Which one is better? Since there is no right answer to this question, an alternative is to use the average of the two estimates. Note that the first estimate gives the effect of the first factor when the second factor has not been removed, while the second estimate gives the effect of the first factor after the second factor has been eliminated.

Using the average marginal effect of removing a factor across all factor elimination sequences is what is implied by the choice of the Shapley value as a decomposition procedure. The procedure is detailed in an appendix found below in Section 4.7.

As mentioned above, applying the Shapley decomposition procedure to our sub-group inequality decomposition problem involves two steps. In the first step, we suppose that the two Shapley factors are between-group and within-group inequality. The basic rules followed to compute the marginal contribution of each of these factors are:

1. first, to eliminate within-group inequality and to calculate between-group inequality, we use a vector of incomes in which each observation is assigned the average income μ(k) of the observation's group k;
   
   
2. to eliminate between-group inequality and to calculate within-group inequality, we use a vector of incomes where each observation has its income multiplied by the ratio μ(k)/μ of its group k.

To be more precise, let an inequality index I depend on the incomes of individuals in k = 1,..., K groups, each group with n(k) individuals. Let y(k) be the n(k)-vector of incomes of group k. We want to express total inequality I as a sum of between- and within- group inequality²¹:

To compute the contribution of between-group inequality, we compute the fall of inequality observed when the mean incomes of the groups are equalized. This can be done either before or after within-group inequality has been removed. Hence, the Shapley contribution of between-group inequality is given by:

where l(k) is a unit vector of size n_k. The within-group contribution is then given as

The second step consists in decomposing total within-group inequality as a sum of within-group inequality across groups. To do this, we proceed by replacing the incomes of those in a group k by μ(k) in order to eliminate group k's contribution to total within-group inequality. The fall in inequality induced by this equalization of incomes is the contribution of group k to total within-group inequality. We compute this for each group. Given that this computation depends on the sequence ordering of the groups, we compute the average contribution of a group k over all possible orderings of groups. This gives the Shapley value of group k's contribution to total within-group inequality.

²¹DAD: Decomposition|S-Gini: Decomposition by Groups.

To formalize this, suppose that there are only two groups, k = 1, 2. The first group's contribution to total within-group inequality is given as

and symmetrically for the second group.

4.7 Appendix: the Shapley value

The Shapley value is a solution concept often employed in the theory of cooperative games. Consider a set S of s players who must divide some surplus among themselves. The question to resolve is: how can we divide the surplus between the s players? To see how, suppose that the s players can form coalitions (these coalitions are subsets of S) to extract a part of the surplus and redistribute it between their σ members. Suppose that the function V determines the extracting force of the coalition, viz, that amount of the surplus that it can extract without resorting to an agreement with those players that are outside of the coalition. The value of an additional player I in a coalition is given by

The term MV(,i) equals the marginal value added by player i after his adhesion to the coalition What will then be the expected marginal contribution of player i over the different possible coalitions that can be formed and which he can join? Note that the number of possible permutations of the s players equals s!. Note also that the size of coalitions is limited to σ ε . Out of s! possible permutations of players, the number of times that the same first σ players are located in a same coalition is given by the number of possible permutations of the σ players in coalition that is, by σ!. For every permutation in the coalition we find (s – σ –1)! permutations for the players that complement the coalition (excluding player i). The Shapley value gives the expected marginal value that player i generates after his adhesion to a coalition of any possible size σ. It is thus given by:

This decomposition procedure has two useful properties. The first is symmetry, ensuring that the contribution of each factor is independent of the order in which it appears in the initial list or sequence of factors. The second property is exactness and additivity, from which the total surplus is given by

For decompositions of inequality or poverty indices, say, applying a Shapley procedure consists in computing the marginal effect on such indices of removing each contributing factor (between or within group inequality, inequality in income component, differences in mean income, etc.) in a given sequence of elimination. Repeating the computation for all possible elimination sequences, we estimate the mean of the marginal effects for each factor. This mean provides the contribution of each such factor. The contribution of all factors yield an exact, additive decomposition of distributive indices and variations in them into s contributions.

4.8 References

The literature on the measurement of inequality and social welfare is very large. General references include Atkinson (1983), Atkinson and Bourguignon (2000), Atkinson and Micklewright (1992), Bishop, Formby, and Smith (1993), Chakravarty (1990), Champernowne and Cowell (1998), Cowell (1995), Cowell (2000), Essama Nssah (2000), Foster and Sen (1997), Johnson and Shipp (1997), Lambert (2001), Sen (1973), Sen (1992), Sen (1992), and Saunders (1994).

Applications to real data are very numerous too — among the most influential recent ones feature Bourguignon and Morrisson (2002), Danziger and Gottschalk (1995), Gottschalk and Smeeding (1997), Gottschalk and Smeeding (2000), Jantti (1997) and Milanovic (2002).

Seminal work on inequality measurement and Lorenz curves include Atkinson (1970), Blackorby and Donaldson (1978), Dalton (1920), Dasgupta, Sen, and Starret (1973), Gini (1914) (see Gini 2005 for a recent English translation), Hainsworth (1964), Kakwani (1977a), Kolm (1969), Lorenz (1905) and Rothschild and Stiglitz (1973). Aaberge (2000) rationalizes the use of "moments of Lorenz curves" as measures of inequality, and Aaberge (2001a) presents axiomatic bases for the use of Lorenz curve orderings. Foster and Ok (1999) analyze the concordance of the variance of logarithms with Lorenz dominance.

Discussion and interpretation of linear (or rank-dependent) indices of inequality can be found in Aaberge (1997), Aaberge (2000), Anand (1983), Barrett and Salles (1995), Ben Porath and Gilboa (1994), Blackburn (1989), Blackorby, Bossert, and Donaldson (1994), Bossert (1990), Chakravarty (1988), Chew and Epstein (1989), Donaldson and Weymark (1980) and Donaldson and Weymark (1983) (for S-Ginis), Duclos (1997a), Weymark (1981), Yaari (1988), Yitzhaki (1983) (for extended Ginis, equivalent to S-Ginis — see also Kakwani (1980)), and Wang and Tsui (2000). The most popular member of the class of linear inequality indices is the Gini index: it is discussed in detail in Deutsch and Silber (1997), Milanovic (1994b), Milanovic (1997), Subramanian (2002) and Yitzhaki (1998).

The theory and the economic measurement of relative deprivation is explored inter alia in Berrebi and Silber (1985), Chakravarty and Chakraborty (1984), Clark and Oswald (1996), Davis (1959), Duclos (2000), Ebert and Moyes (2000), Festinger (1954), Hey and Lambert (1980), Merton and Rossi (1957), Paul (1991), Podder (1996), Runciman (1966), Silver (1994), Wang and Tsui (2000), Yitzhaki (1979), Yitzhaki (1982a) and Nolan and Whelan (1996).

Discussion and use of the Theil index appears inter alia in Beblo and Knaus (2001), Duro and Esteban (1998) and Goerlich Gisbert (2001).

Other inequality indices are discussed in Araar and Duclos (2003) and Berrebi and Silber (1981) (a combination of Atkinson and Gini inequality indices), Chakravarty (2001) (a defense of the use of the variance), del Rio and Ruiz Castillo (2001) (for "intermediate inequality measures"), and Foster and Shneyerov (2000) (for "path-independent decomposable measures").

Decomposition of inequality across population subgroups has also been the focus of a large literature. This has mostly involved using additive and Generalized entropy indices — see, for instance, Bourguignon (1979), Cowell (1980), Foster and Shneyerov (1999), Mookherjee and Shorrocks (1982), Shorrocks (1980), Shorrocks (1984), Schwarze (1996) and Zandvakili (1999). Decompositions of the Gini and rank-dependent inequality indices are investigated in Dagum (1997), Deutsch and Silber (1999a), Deutsch and Silber (1999b), Milanovic and Yitzhaki (2002), Sastry and Kelkar (1994), Tsui (1998) and Yitzhaki and Lerman (1991). A money-metric cost-of-inequality approach to decomposing inequality across subpopulations is derived in Blackorby, Donaldson, and Auersperg (1981), Duclos and Lambert (2000) and Ebert (1999). Alternative decomposition approaches are also explored in Cowell and Jenkins (1995), Fields and Yoo (2000), Fournier (2001), Hyslop (2001), Jenkins (1995), Parker (1999), and Schultz (1998).

The Shapley value was introduced by Shapley (1953). See also Owen (1977) for how a two-stage decomposition procedure can be applied to the Shapley value, as well as Shorrocks (1999) and Chantreuil and Trannoy (1999) for its use in distributive analysis.

Figure 4.4: Mean income and inequality for constant social welfare ξ

Figure 4.5: Homothetic social evaluation functions

Figure 4.6: Social utility and incomes

Figure 4.7: Marginal social utility and incomes

Figure 4.8: Atkinson social evaluation functions and the cost of inequality

Figure 4.9: Inequality aversion and the cost of inequality

Figure 4.10: Generalized Lorenz curve

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