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In what follows in this book, we will denote living standards by the variable y. The indices we will use will sometimes require these living standards to be strictly positive, and, for expositional simplicity, we may assume that this is always the case. Strictly positive values of y are required, for instance, for the Watts poverty index and for many of the decomposable inequality indices. It is of course reasonable to expect indicators of living standards such as consumption or expenditures to be strictly positive. This assumption is less natural for other indicators, such as income, for which capital losses or retrospective tax payments can generate negative values. Also recall that, for expositional simplicity, we will also usually refer to living standards as incomes.

Let p = F(y) be the proportion of individuals in the population who enjoy a level of income that is less than or equal to y. F(y) is called the cumulative distribution function (cdf) of the distribution of income; it is non-decreasing in y, and varies between 0 and 1, with F(0) = 0 and F(∞)=1¹. For expositional simplicity, we will sometimes implicitly assume that F(y) is continuously differentiable and strictly increasing in y. These are reasonable approximations for large-population distributions of income. They are also reasonable assumptions from the point of view of describing the data generating processes that generate the distributions of income observed in practice. The density function, which is the first-order derivative of the cdf, is denoted as f(y) = F'(y) and is strictly positive when F(y) is assumed to be strictly increasing in y.

¹ DAD: Distribution|Distribution Function.

3.1 Continuous distributions

A useful tool throughout the book will be "quantile functions". The use of quantiles will help simplify greatly the exposition and the computation of several distributive measures. Quantiles will also sometimes serve as direct tools to analyze and compare distributions of living standards (to check first-order dominance for instance). The quantile function Q(p) is defined implicitly as F(Q(p)) = p, or using the inverse distribution function, as Q(p)= F(⁻¹(p)². Q(p) is thus the living standard level below which we find a proportion p of the population. Alternatively, it is the income of that individual whose rank — or percentile — in the distribution is p. A proportion p of the population is poorer than he is; a proportion 1 – p is richer than him.

These tools are illustrated in Figure 3.1. The horizontal axis shows percentiles p of the population. The quantiles Q(p) that correspond to different p values are shown on the vertical axis. The larger the rank p, the higher the corresponding income Q(p). Alternatively, incomes y appear on the vertical axis of Figure 3.1, and the proportions of individuals whose income is below or equal to those y are shown on the horizontal axis. At the maximum income level, y_max, that proportion F(y_max) equals 1. The median is given by Q(0.5), which is the income value which splits the distribution exactly in two halves.

Note that an important expositional advantage of working with quantiles is to normalize the population size to 1. This also means that everyone's income and contribution to this book's poverty and equity analysis can then appear on an interval of percentiles ranging from 0 to 1. In a sense, the population size is thus scaled to that of a socially representative individual. Normalizing all population sizes to 1 also makes comparisons of poverty and equity accord with the population invariance principle. This principle says that adding an exact replicate of a population to that same population should not change the value of its distributive indices. Putting everyone's income within a common total population scale of 1 is a handy descriptive way of comparing populations of different sizes. It also ensures that adding exact replications to these populations will not change the distributive picture.

We will define most of the distributive measures (indices and curves) in terms of integrals over a range of percentiles. This is a familiar procedure in the context of continuous distributions. We will see below why this procedure is also generally valid in the context of discrete distributions, even though the use of summation signs is often more familiar in that context. Using integrals will make the definitions and the exposition simpler, and will help focus on what matters more, namely, the interpretation and the use of the various measures.

² DAD: Curves|Quantile.

The most common summary index of a distribution is its mean. Using integrals and quantiles, it is defined simply as:

μ is therefore the area underneath the quantile curve. This corresponds to the grey area shown on Figure 3.1. Since the horizontal axis varies uniformly from 0 to 1, μ is also the average height of the quantile curve Q(p), and this is given by μ on the vertical axis, μ is thus the income of the population's "average individual".

The computation of the average income μ gives equal weight to all incomes in the population. We will see later in the book alternative weighting schemes for computing socially representative incomes. As for most distributions of income, the one shown on Figure 3.1 is skewed to the right, which gives rise to a mean μ that exceeds the median Q(p). Said differently, the proportion of individuals whose income falls underneath the mean, F(μ), exceeds one half.

3.2 Discrete distributions

To see how to rewrite the above definitions using familiar summation signs for discrete distributions, we need a little more notation. Say that we are interested in a distribution of n incomes. We first order the n observations of y_i in increasing values of y, such that y₁≤ y₂≤ y₃≤ … ≤ y_n−1≤ y_n. We then associate to these n discrete quantiles over the interval of p between 0 and 1. For p such that (i − 1)/n < p ≤ i/n, we then have Q(p) = y_i. Technically, this is equivalent to defining quantiles as Q(p) = min. This is illustrated in Table 3.1 for n = 3 and where the three income values are 10, 20 and 30. Figure 3.2 graphs those quantiles as a function of p. p values between 0 and 1/3 give a quantile of 10, the second income, 20, covers percentile 1/3 to 2/3, and the highest incomes, 30, covers percentile 2/3 to 1.

The formulae for discrete distributions are then computed in practice by replacing the integral sign in the continuous case by a summation sign, by summing across all quantiles, and by dividing that sum by the number of observations n. Thus, the mean μ of a discrete distribution can be expressed as:

Thus, whenever an expression like (3.1) arises, we can think of the integral sign as standing for a summation sign and of dp as standing for 1/n.

Using (3.2), the mean of the discrete distribution of Table 3.1, which is 20, is then simply the integral of the quantile curve shown on Figure 3.2. In other words, it is the sum of the area of the three boxes each of length 1/3 that can be found underneath the filled curve. For completeness, we will mention from time to time how indices and curves can be estimated using the more familiar summation signs. For more information, the reader can also consult DAD's User Guide, where the estimation formulae shown use summation signs and thus apply to discrete distributions.

Table 3.1: Incomes and quantiles in a discrete distribution

3.3 Poverty gaps

For poverty comparisons, we will also need the concept of quantiles censored at a poverty line z. These are denoted by Q*(p; z) and defined as:

Censored quantiles are therefore just the incomes Q(p) for those in poverty (below z) and z for those whose income exceeds the poverty line. This is illustrated on Figure 3.3, which is similar to Figure 3.1. Quantiles Q(p) and censored quantiles Q*(p; z) are identical up to p = F(z), or up to Q(p) = z. After this point, censored quantiles equal a constant z and therefore diverge from the quantiles Q(p).

The mean of the censored quantiles is denoted as μ* (z):

This is the area underneath the curve of censored incomes Q* (p; z). Censoring income at z helps focus attention on poverty, since the precise value of those living standards that exceed z is irrelevant for poverty analysis and poverty comparisons (at least so long as we consider absolute poverty).

The poverty gap at percentile p, g(p; z), is the difference between the poverty line and the censored quantile at p, or, equivalently, the shortfall (when applicable) of living standard Q(p) from the poverty line. Let f₊ = max (f, 0).

E:18.7.6

Poverty gaps can then be defined as³:

When income at p exceeds the poverty line, the poverty gap equals zero. A shortfall g(q; z) at rank q is shown on Figure 3.3 by the distance between z and Q(q). The larger one's rank p in the distribution — the higher up in the distribution of income — the lower the poverty gap g(p; z). The proportion of individuals with a positive poverty gap is given by F(z). The average poverty gap then equals μ^g(z):

μ^g(z) is then the size of the area in grey shown on Figure 3.3.

3.4 Cardinal versus ordinal comparisons

There are two types of poverty and equity comparisons: cardinal and ordinal ones. Cardinal comparisons involve comparing numerical estimates of poverty and equity indices. Ordinal comparisons rank broadly poverty and equity across distributions, without attempting to quantify the precise differences in poverty and equity that exist between these distributions. They can often say where poverty and equity is larger or smaller, but not by how much.

Consider for instance the case of cardinal poverty comparisons. Numerical poverty estimates attach a single number to the extent of aggregate poverty in a population, e.g., 40% or $200 per capita. But calculating cardinal poverty estimates requires making a number of very specific assumptions. These include, inter alia, assumptions on the form of the poverty index, the definition of the indicator of well-being, the choice of equivalence scales, the value of the poverty line, and how that poverty line varies precisely across space and time.

Once these assumptions are made, cardinal poverty estimates can tell, for instance, that the consumption expenditures of 30% of the individuals in a population lie underneath a poverty line, but that a proposed government program could decrease that proportion to 25%. Cardinal poverty estimates can also be used to carry out a money-metric cost-benefit analysis of the effects of social programs. Thus, if the above government program involved yearly expenditures of $500 million, then we would know immediately that a 1% fall in the proportion of the poor would cost on average $100 million to the government. That amount could then be compared to the poverty alleviation cost of other forms of government policy.

The main advantage of cardinal estimates of poverty and equity is their ease of communication, their ease of manipulation, and their (apparent) lack of ambiguity. Government officials and the media often want the results of distributive comparisons to be produced in straightforward and seemingly precise terms, and will often feel annoyed when this is not possible. As hinted above, cardinal estimates of poverty and equity are, however, necessarily (and often highly) sensitive to the choice of a number of arbitrary measurement assumptions.

³ DAD: Curves|Poverty gap.

It is clear, for example, that choosing a different poverty line will almost always change the estimated numerical value of any index of poverty. The elasticity of the poverty headcount index with respect to the poverty line is, for example, often significantly larger than 1 (see Section 12.2). This implies that a variation of 10% in the poverty line will then change by more than 10% the estimated proportion of the poor in the population; this sensitivity is substantial, especially since poverty lines are rarely convincingly bounded within a narrow interval.

Another source of cardinal variability comes from the choice of the form of a distributive index. Many procedures have been proposed for instance to aggregate individual poverty. Depending on the chosen procedure, numerical estimates of aggregate poverty will end up larger or lower. As we will see later, for instance, identifying a "socially representative poverty gap" will hinge particularly on the relative weight given to the more deprived among the poor. There is little objective guidance in choosing that weight; the greater its value, however, the greater the socially representative poverty gap, and the greater the estimate of aggregate poverty.

Ordinal comparisons, on the other hand, do not attach a precise numerical value to the extent of poverty or equity, but only try to rank poverty and equity across all indices that obey some generally-defined normative (or ethical) principles. This can be useful when it suffices to know which of two policies will better alleviate poverty, or which of two distributions has more inequality, but not precisely by how much. Because of this lower information requirement, ordinal rankings can prove robust to the choice of a number of measurement assumptions. For instance, ordinal poverty orderings can often rank poverty over general classes of possible poverty indices and wide ranges of possible poverty lines.

It is thus useful to consider in turn cardinal and ordinal comparisons of poverty and equity. We first see how to construct aggregate cardinal distributive indices. Ordinal comparisons are considered in Part III.

Figure 3.1: Quantile curve for a continuous distribution

Figure 3.2: Quantile curve for a discrete distribution

Figure 3.3: Incomes and poverty at different percentiles

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