Document(s) 11 of 20

9.1 Ordering distributions

We have, up to now, focussed mostly on measuring and comparing cardinal indices of poverty and equity. As discussed in Chapter 4, this has several expositional advantages. The greatest of these advantages is probably that of focussing on only one (or a few) numerical assessments of poverty and equity. It is then relatively straightforward to compare poverty and equity across distributions just by comparing the values of these cardinal indices. The conclusions are then (seemingly) "clear-cut".

There are, however, important reasons to consider instead ordinal comparisons of poverty and equity. The most important one is that comparisons of cardinal poverty and equity indices (comparisons across time, regions, sociodemographic groups, or comparisons of policy regimes, for instance) may be disturbingly sensitive to the choice of indices and poverty lines. For instance, we might find for some poverty lines and indices that poverty is greater in a region A than in a region B, but we then find the opposite for other lines and indices. We could support the introduction of a particular fiscal policy or macroeconomic adjustment program for some social welfare indices, but could be in doubt as to whether the same support would be warranted with other indices. Since there is rarely unanimity as to the right choice of poverty lines and distributive indices, it is clear that such sensitivity can seriously undermine one's confidence in comparing distributions or in making policy recommendations.

9.2 Sensitivity of poverty comparisons

To see this better in the context of poverty comparisons, consider the hypothetical example of Table 9.1. The second, third and fourth lines in the table show the incomes of three individuals in two hypothetical distributions, A and B. Thus, distribution A contains three incomes of 4, 11 and 20 respectively. The bottom 3 lines of the table show the value of the two most popular indices of poverty, the headcount F(z) and the average poverty μ,^g(z) indices, at two alternative poverty lines, z = 5 and z = 10. Recall from Section 5.1.2 that the poverty headcount gives the proportion of individuals in a population whose income falls underneath a poverty line. At a poverty line of 5, there is only one such person in poverty in distribution A, and the headcount is thus equal to 0.33. The average poverty gap index is the sum of the distances of the poor's incomes from the poverty line, divided by the total number of people in the population. For instance, at a poverty line of 10, there are 2 people in poverty in B, and the sum of their distances from the poverty line is (10-6)+(10-9)=5. Divided by 3, this gives 1.66 as the average poverty gap in B for a poverty line of 10.

At a poverty line of 5, the headcount in A is clearly greater than in B, but this ranking is spectacularly reversed if we consider instead the same headcount index but at a poverty line of 10. The ranking changes again if we use the same poverty line of 10 but now focus on the average poverty gap μ^g(z): Clearly, the poverty ranking A and B can be quite sensitive to the precise choice of measurement assumptions.

Table 9.1: Sensitivity of poverty comparisons to choice of poverty indices and poverty lines

9.3 Ordinal comparisons

The alternative to comparing the value of one or a few cardinal indices is to check whether rankings of poverty and equity are valid for a class of ethical judgments. These classes are defined over classes of indices as well as over ranges of poverty lines (for poverty comparisons). In other words, we do not wish to quantify poverty or equity. We only want to determine whether poverty and equity is higher or lower in one distribution than in another, for a class of ethical judgments. When inferred, an ordinal ranking of poverty and equity across distributions or policies establishes the sign of the differences across these distributions or policies of everyone of the cardinal poverty and equity indices of that class. Note that it can say only whether poverty and equity is higher in one distribution or for one policy than for another, but not by how much. In the article in which he introduces his famous inequality index (or "concentration ratio"), Gini (1914) criticizes the curve introduced earlier by Lorenz (1905) exactly along those lines:

This graphical approach presented two drawbacks (...):

1. it does not provide a precise measurement of concentration
   
   
2. it does not allow to assess, not even in some circumstances, when or where concentration is stronger. In fact, if two curves cross each other (...), it is not always possible to say if one denotes a stronger concentration than the other, (translated in Gini 2005, p. 24.)

Ordinal comparisons of poverty do not, therefore, provide precise numerical values to compare with numerical indicators of other aspects or effects of government policy, such as the policy's administrative or efficiency cost. This is seemingly their main defect. It is arguably also their greatest advantage. As seen above in the context of Table 9.1, differences in simple poverty indices can be deceptive when it comes to ranking distributions. They can also quantify deceptively differences across distributions. To illustrate this, consider Table 9.2 with distributions A and B and a poverty line z = 1. The three FGT poverty indices agree that poverty has not increased in moving from A to B. But the quantitative change in poverty varies significantly with the value of α. With the poverty headcount, poverty remains the same, but the average poverty gap falls by 33% and the index falls by 56%.

Table 9.2: Sensitivity of differences in poverty to choice of indices

^aFirst individual's income.
^bSecond individual's income.
^cChanges in poverty from A to B.

A focus on ordinal comparisons can save most of the considerable energy and time often spent on selecting poverty lines and poverty indices. It can avoid inter alia the difficult debate on the choice of appropriate theoretical and econometric models for estimating poverty lines. It can also escape arguments on the relative merits and properties of the many distributive indices that have been proposed in the social welfare literature, and of which the previous chapters introduced only a few. Again, this is because of ordinal distributive comparisons simply order distributions, and for this, differences in numerical indices do not need to be estimated. For instance, we will see later in Section 10.1 that we can order robustly distributions A and B in Table 9.1 for all "distribution-sensitive" poverty indices and for any choice of poverty line. If such an ordering is considered sufficiently strong and informative, then, in comparing A and B, we can effectively stop quibbling on whether we should use the Watts index or the average poverty gap as a poverty index, and on whether the poverty line should be 5 or 10.

In short, ordinal poverty comparisons can sometimes be robust to the choice of measurement assumptions, since they will sometimes be valid for wide classes and ranges of such assumptions. When the problem is simply of resolving which of two policies will better alleviate poverty, or determining which of two distributions displays the greatest level of social welfare, or assessing which of two distributions is the most equal, ordinal comparisons can sometimes be sufficiently informative, and cardinal estimates will then not be needed.

9.4 Ethical judgements

9.4.1 Dominance tests

As we will see in detail below, ordinal comparisons of poverty and equity involve using classes of distributive indices. It is useful to define these classes by referring to "orders of normative (or ethical) judgements", an order being denoted as s = 0, 1, 2,.... An ethical judgement of order s thus serves to define a class of indices also of order s. Whether an ordering of poverty and equity is valid for all of the indices that are members of a class of order s is empirically tested through dominance tests, which happen to be convenient variants of well-known stochastic dominance tests also of order s. When two dominance curves of a given order do not intersect, all indices that obey the ethical principles associated to this order of dominance then rank identically the two distributions. Hence, a dominance test of order s serves to test whether some distributive ranking is valid for all of the indices of a class of order s, and that class of order s can be interpreted through the use of ethical judgements of the same order s.

9.4.2 Paretian judgments

A first natural property of normative judgements is that a society should be judged improved whenever the income of one of its members increases and no one else's income decreases. For poverty, this would mean that indices of poverty should (weakly) fall whenever someone's income increases, everything else being the same. ("Weakly fall" means that the index should at the very least not increase following the change, and conversely for "weakly increase". This caveat applies to all of the ethical statements considered in this book.) For social welfare comparisons, this would imply that social welfare indices should increase following this improvement in someone's income. Such indices thus obey the Pareto principle: they must respond favorably to Pareto-improving changes in the distribution of income.

To see this formally, consider the case of a social welfare function, W(y), that depends on a vector y = (y₁,..., y_n) of n income levels.

Pareto principle

Let y = (y₁, ..., y_n),η > 0 be any positive constant, and = (y₁,..., y^j + η,..y_n). Then the social welfare function W obeys the Pareto principle if and only if W (y) ≤ W () for all possible pairings of y and .

Because the ethical condition imposed by the Pareto principle is very weak, we can consider all of the indices that obey that principle to be members of a class of ethical order 0. The poverty indices belonging to a class of order 0 would for instance all fall whenever someone's income increases, everything else being the same. Note that the case of relative poverty might seem to provide an exception to this principle, since an increase in someone's income could increase the relative poverty line and possibly also increase the poverty index. To deal with this possible exception, it is best to think of the poverty line as constant in the current discussion of ethical principles.

All of the indices which obey the Pareto ethical condition then belong to (poverty or social welfare) classes of order s = 0. It has, however, long been recognized that searches for strict Pareto improvements in distributions of incomes are generally doomed to failure, because of fundamental randomness in economic status and because of strong heterogeneity in preferences, endowments and markets. For a distributive change to be strictly Pareto improving, it must indeed not decrease anyone's income, whatever one's peculiar circumstances. This is unlikely ever to be empirically observable, even if we were to focus only on those with incomes below some poverty line. Besides, checking for Pareto -improving temporal changes would require the use of panel data in order to observe individual-specific changes in incomes. Such panel data are rare, and even if we had access to them, they would still not enable us to infer Pareto -improvements over an entire population (as opposed to only over an available sample). To be valid, searches for strict Pareto improvements also plausibly require no change in population size and composition, a difficulty with which we deal below through the use of the anonymity and population invariance principles.

9.4.3 First-order judgments

It is thus natural and logical to consider ethical principles of order higher than that of the Pareto principle. In the light of the above, a plausible higher-order ethical judgement would require that the distributive indices be anony­mous in the incomes of the individuals. That is, ceteris paribus, whether it is an individual named α rather than b that enjoys some level of income should not affect the value of a distributive index. It also follows from this property that interchanging two income levels should not affect distributive indices: these indices thus obey the symmetry or anonymity principle. Formally, we have (for a social welfare function W):

Anonymity principle

Let M be an n × n permutation matrix (a permutation matrix is composed of 0's and 1's, with each row and each column summing to 1) and let = My'. Then the social welfare function W obeys the anonymity principle if and only if W (y) = W () for all possible pairings of y and

Clearly, this principle would not be acceptable for an index of horizontal equity, but it would seem relatively uncontroversial for comparing inequality, social welfare or poverty across anonymous distributions.

There is another principle that we have implicitly imposed since the beginning of this book and that also goes beyond the Pareto principle. It is usually called the population invariance principle, and it simply states that adding an exact replicate of a population to that same population should not affect distributive comparisons. For a social welfare function W, we thus have:

Population invariance principle

Let be a vector of size 2n, with and with y_j = y_j, j = 1,..., n. Then the social welfare function W obeys the population invariance principle if and only if W (y) = W () for all possible pairings of y and .

As indicated on page 40, imposing this principle simplifies exposition significantly by enabling the use of quantiles and the normalization of population size to 1. The population invariance principle is thus implicitly imposed everywhere throughout the book.

First-order classes of distributive indices then regroup all of the indices that show a social improvement when the income at some percentile in the population increases and when no other income changes. These indices have properties that are analogous to those of Paretian indices: ceteris paribus, the larger the individual incomes, the better off is society. They are in addition symmetric in income since they obey the anonymity principle.

9.4.4 Higher-order judgments

Even with the above anonymity constraint, it is likely that some of the first-order distributive indices will clash in their distributive ranking. Some of the first-order poverty indices could declare a policy reform to worsen poverty, while others might indicate that the reform improves poverty. To resolve this ambiguity, we may move to a second-order class of distributive indices. As above, this is done by constraining distributive indices to obey additional ethical principles.

To do this, assume that distributive indices must show a social improvement whenever a mean-preserving redistributive transfer from a richer to a poorer individual occurs. This corresponds to imposing the well-known Pigou-Dalton principle on social judgements. To see this formally, consider again the case of a social welfare function W(y).

Pigou-Dalton principle

Let η > 0 be any positive constant, and let = (y₁,... , y_j + η,..., y_k – η,..., y_n), with y_j + η ≤ yk – η. Then the social welfare function W obeys the Pigou-Dalton principle if and only if W (y) ≤ W () for all possible pairings of y and .

The second-order classes of distributive indices thus contain those indices that have a greater ethical preference for the poorer than for the richer. They display a preference for equality of income and are therefore said to be distribution-sensitive. For instance, all other things being the same, the more equal the distribution of income among the poor, the lower the level of poverty. Ceteris paribus, if a transfer from a richer to a poorer person takes place, all second-order social welfare indices will increase and all second-order inequality and poverty indices will fall. Note again that all indices that belong to a second-order class of poverty and welfare indices also belong to the first-order class of relevant indices.

There are often sound ethical reasons to be socially more sensitive to what happens towards the bottom of the distribution of income than higher up in it. We may thus be less concerned about a "bad" disequalizing transfer higher up in the distribution of income than lower down. To make this more precise, imagine four levels of income, for individuals 1, 2, 3, and 4, such that y₂ – y₁ = y₄ - y₃ > 0 and y₁ < y₃. Let a marginal transfer of $1 of income be made from individual 2 to individual 1 (an equalizing transfer) at the same time as an identical marginal $1 is transferred from individual 3 to individual 4 (a disequalizing transfer). This is called in the literature a "favorable composite transfer".

Note that the equalizing transfer is made lower down in the distribution of income than the disequalizing transfer. This can be seen by the fact the recipient of the first transfer, individual 1, has a lower income than the donor of the second transfer, individual 3, since y₃ > y₁. For a given distance between recipients and donors, the social improvement effect of equalizing transfers is decreasing in the income of the recipient. Said differently, Pigou-Dalton transfers lose their social improvement effects when recipients are more affluent.

Second-order indices which respond favorably to such a "favorable composite transfer" obey the transfer-sensitivity principle and therefore belong to the third-order class of indices. Again, such a favorable composite transfer is made of a beneficial Pigou-Dalton transfer within a lower part of the distribution, coupled with a reverse Pigou-Dalton transfer within an upper part of the distribution. Third-order welfare indices will increase following this change, and third-order poverty and inequality indices will fall. Formally, we have (for a social welfare function W):

Transfer-sensitivity principle

Let η > 0 and y_j - y_i = y_i - y_k > 2η with yi < y_k.

Also let = (y₁,..., y_i + η,..., y_j - η,..., y_k - η,..., y_i + η,..., y_n). Then the social welfare function W obeys the transfer-sensitivity principle if and only if W (y) ≤ W () for all possible pairings of y and .

Note that the favorable composite transfer considered above involves no change in the variance of the distribution since y_j – y_i = y_l – y_k

We can, if we wish, define subsequent classes of indices in an analogous manner. To define fourth-order indices, for instance, we consider a combination of two exactly opposite and symmetric composite transfers, the first one being favorable and occurring within a lower part of the distribution, and the second one being unfavorable and occurring within a higher part of the distribution. The indices that respond favorably to this combination of composite transfers can then belong to the class of fourth-order indices.

As can be seen, higher-order transfer principles essentially postulate that, as the order increases, the relative ethical weight assigned to the effect of income changes occurring at the bottom of the distribution also increases. Thus, as the order s of the class of distributive indices increases, the indices become more and more sensitive to the distribution of income among the poorest. At the limit, as s becomes very large, only the income of the poorest individual matters in comparing poverty and social welfare across two distributions. In that sense, the poverty and social welfare indices become more and more Rawlsian as s increases.

9.5 References

Much of normative welfare economics has been influenced by the philosophical work of Nozick (1974), Rawls (1971) (see Rawls 1974 for a very short synthesis addressed to economists) and Sen (1982). The combined work of Kolm (1976a) and Kolm (1976b) was the first to introduce the transfer-sensitivity condition into the inequality literature, and Kakwani (1980) subsequently adapted it to poverty measurement. See also Davies and Hoy (1994) (who describe that condition as a Rawlsian extension of the Lorenz criterion), Shorrocks (1987) for a complete characterization of the transfer-sensitivity principle, and Zheng (1997) for an informative discussion of it. Higher-order principles can be interpreted using the generalized transfer principles of Fishburn and Willig (1984) — see also Blackorby and Donaldson (1978) for a description of these principles as becoming "more Rawlsian". surveys of the normative and axiomatic foundations of modern inequality measurement can be found in Blackorby, Bossert, and Donaldson (1999) and Chakravarty (1999).

Other papers which explore variations to the normative principles typically used in distributive analysis are Mosler and Muliere (1996) (for an alternative principle of transfers), Ok (1995) (for a "fuzzy" measurement of inequality), Ok (1997) (for ranking over opportunity sets), Salas (1998) (for marginal population invariance), Zoli (1999) (for a positional transfer principle when Lorenz curves intersect), and Tam and Zhang (1996) (for an alternative Pareto principle defined in terms of growth over the poor).

Experimental evidence on the normative attitudes of individuals and societies towards the measurement of poverty and equity has also grown fast in the last decades. Methods and results can be found in Amiel and Cowell (1992) (on attitudes to inequality — which question the acceptability of transfer and decomposability principles), Amiel and Cowell (1999) (on attitudes to poverty, social welfare and inequality), Amiel and Cowell (1997) (on attitudes towards poverty measurement), and in Amiel, Creedy, and Hurn (1999) (on quantifying inequality aversion using Okun (1975)'s "leaky bucket experiment"). A survey of such attitudes can be found in Corneo and Gruner (2002).

Fong (2001) tests whether normative attitudes can be explained by self-interest or by values about distributive justice. Dolan and Robinson (2001) further explore whether there is a "reference point" problem in such studies, and Ravallion and Lokshin (2002) reports that expectations about future levels of well-being can influence individuals' desire for redistributive policies.

See also Stodder (1991) for empirical evidence as to why inequality aversion can matter for ranking distributions, and Christiansen and Jansen (1978) for an example of the estimation of social preferences using the revealed structure of an existing tax system (the Norwegian one).

A number of studies have recently also attempted to distinguish between attitudes towards inequality and towards risk aversion: see inter alia Amiel, Cowell, and Polovin (2001), Beck (1994), Cowell and Schokkaert (2001), and Kroll and Davidovitz (2003).

Document(s) 11 of 20