1
Variable Population Poverty Comparisonsi
I.
Introduction
This paper defends a new desideratum for making variable population poverty
comparisons that lends some support to poverty indexes that satisfy this axiom. It suggests that,
unless the main poverty indexes in the literature are modified, they entail that poverty declines
with the mere addition of a rich person to a population. This paper isolates the problem with
these indexes and proposes a solution. An adequate account of how we should make variable
population poverty comparisons is important for both theory and practice. It is important for
philosophical theory because poverty indexes may inform an account of intergenerational justice.
Poverty indexes may provide the basis for an account of how we should aid the poor in present
and future generations and an account of how we should aid the poor is often central to
philosophical work on intergenerational justice. Work on variable population poverty
comparisons is important for development practice because debates about poverty relief and
foreign aid often hinge on such comparisons. The first Millennium Development Goal’s first
target to “halve, between 1990 and 2015, the proportion of people whose income is less than one
dollar a day,” measures its success partly in terms of the headcount index and the poverty gap
index for the total population (UNDP, 2007).ii Different poverty indexes are also used to measure
national and international development policies’ impact (Cranfield et. al., 2007).
II.
Axiom for a Plausible Account of Poverty in Variable Populations
Historically, poverty indexes were quite simple. Consider, for instance, the indexes used
in the Millennium Development Goals. The headcount index (H) measures the number of poor
people as a percentage of the total population. The poverty gap index for the total population (I)
takes the total aggregate shortfall from the poverty line divided by the number of people and
2
divided by the poverty line itself. Recently, however, economists have suggested several more
complicated alternatives including Sen’s index, the Sen-Shorrocks-Thon (SST) index, and the
Foster-Greer-Thorbecke (FGT) index (which, under some parameterizations is equivalent to H
and I). Although some of these indexes do not suppose full interpersonal comparisons of need,
for simplicity, the rest of this paper will do so.iii
Most work on poverty indexes presupposes an answer to the question: “What is the
proper basis for poverty measurement?” but this paper will simply remain silent on the issue.
That is, this paper will not give an account of what poverty is. Poverty may, for instance, be
deprivation in the space of welfare, capabilities, resources, or opportunities. To stay neutral on
the proper basis for poverty measurement, this paper will refer to this basis as need. It will refer
to whatever alleviates need or brings someone up higher above the poverty line as a good.
Material goods and utility may be goods in this sense. Alternately, opportunities might be
necessary to alleviate need (Miller, 1999).iv
This paper will assume, however, that there are better and worse answers to questions
like: What is poverty? How much poverty is there in a population? Which population is poorer?
There may be room for multiple answers to these questions. What counts as poverty may, for
instance, be different in some contexts (e.g. in the US and Peru). Still, some things do not count
as appropriate answers to the questions above. On any good account, most of the people living in
the slums of New Dehli count as poor, for instance.
The first part of this paper will also be less controversial if one assumes that poverty is
absolute as opposed to relative. Poverty is relative (to some degree) if how much need someone
in a population has is sensitive to how much other people have. People in some societies may
need automobiles to get to work, while those in others do not (Sen, 1997; Townsen, 1985).
3
This paper will just assume, however, that each individual’s poverty level is fixed before
giving some arguments in favor of its preferred methods of aggregating poverty in variable
populations. So, its arguments may be compatible with some relativity in individuals’ needs.
This paper just assumes that any relativity in individuals’ needs is taken into account before
aggregating the poverty in a population. This is not intended to bias the question of whether there
is additional relativity at the population level. To see this, consider an analogy. It is possible to
say we know how good a situation is for everyone without begging the question against those
who think there are non-person affecting reasons for saying situations are better or worse.
(Incidentally, this paper will say nothing about whether there are non-person-affecting reasons
for saying that poverty is better or worse etc.). Assuming that individuals’ poverty levels are
fixed will also obviate the need for a few other simplifying assumptions – namely, that
individuals’ histories and identities are not relevant to their poverty. For if individuals’ poverty
levels are fixed before measuring the poverty in a population, questions like “How do individual
identities and histories matter to their poverty?” have already been answered.v
Keeping these preliminaries in mind, the first part of this paper will consider methods for
aggregating individuals’ poverty in a population. It will argue that, when the main indexes in the
literature are interpreted as providing a measure of how much poverty there is in a population,
these indexes violate an axiom or intuition that (Subramanian and Hassoun, 2010) have called:
(1) Weak Population Focus: Poverty in a population is not reduced by changes in the nonpoor population which leave the income distribution amongst the poor unchanged.
The next section will consider why the main indexes in the literature violate this axiom if they
are interpreted as measures of poverty in a population. The rest of this section will consider this
axiom’s justification.
4
Weak Population Focus says that merely adding a rich person to a population should not
decrease poverty in that population. The rich may, of course, do a lot to alleviate poverty in a
population. They might voluntarily or involuntarily give money (or other things) to the poor or
their money (etc.) might trickle down to the poor. But, their mere existence in a population does
not reduce poverty in a population. To illustrate the import of this axiom consider the following
population:
P1 P2 P3 P4
In this diagram, each rectangle represents a person. The dotted line denotes the level at which
people can meet their needs – the poverty line. Here the poverty line is at three units of good.
The height of each rectangle shows how close a person comes to being able to meet their needs.
The first person, P1, has one unit of good, so needs two units to reach the poverty line. The
second and third person, P2 and P3, have two units of good, so need one more unit to reach the
poverty line. The last person, P4 has more than three units of good, so is not poor.
The Weak Population Focus Axiom suggests that simply adding more people who can
meet their needs to the population does not reduce its poverty. So, the following population does
not have less poverty in it than the initial population as it simply contains three additional people
P5-P7 who can meet their needs:
5
P1 P2 P3 P4 P5 P6 P7
Nor would poverty in this population decline if P5-P7 were much richer.
If some of the rich become poor, the amount of poverty in a population might change.
Even if some poor become rich at the same time, one might say poverty in the population has
increased. This might be so if the newly poor are poorer than the newly rich were when they
were poor or if there are more poor people.
The burden of poverty might also be less if rich people are just added to a population and
nothing else changes. That is, it might be much easier to alleviate poverty by redistributing some
income from the rich to the poor. There may also be some respects in which the new population
is better (or worse) than the initial population. It might be better able to support the arts or
sciences. But the mere addition of P5-P7 does not in itself reduce poverty in this population.vi
The intuition here is along the same lines as that Amartya Sen expresses in objecting to
the idea “that some increase in the income shortfall of the poor may be compensated by a
sufficiently high rise in the income of the non-poor” (Sen, 1981, 190). He says “poverty is a
characteristic of the poor, and a reduction of the incomes of the poor must increase the measure
of poverty, no matter how much the incomes of the non-poor go up at the same time” (Sen, 1981,
190). Similarly, if poverty is a characteristic of the poor, an increase in the number of rich people
should not alone reduce poverty (in a population). For, a change in the number of rich people
does not affect the poor any more than if the incomes of the rich increase.
Poverty indexes are not necessarily measures of how it is best to fulfill needs, but these
things are related. The best explanation for why it is best to meet needs in a particular way may,
for instance, be the fact that doing so meets the most need.
6
Relying on this connection, one might object to Weak Population Focus by adapting an
argument first advanced by David Miller in a slightly different context. Miller was concerned
with distribution according to need and claimed that it is important in deciding who to help to
consider not only P1’s claims vs. P2 and P3’s claims but P1’s claims against P4-P7’s claims (e.g.
to what they deserve). So one could argue that people in the middle of the stack are worse off the
closer to the bottom they fall. Perhaps it is important to consider “the relative position of
everyone” in determining how much poverty there is in a population (Miller, 1999, 219). This
might support the intuition that it is best to “equalize degrees of unmet need, which means
distributing in favor of those in greater need until they are brought up to the same level as others”
(Miller, 1999, 74).
Miller suggests that this intuition underlies some empirical evidence regarding how
people think about distributing according to need in the social psychology literature. In one
experiment, there were two students one of whom needed extra money for books. Subjects had to
decide how to split a set amount of money between them. Most subjects wanted to give the
needy student enough to buy the textbooks before splitting the rest equally (Miller, 1999, 74). In
another experiment, intended to mimic John Rawls’ original position, subjects had to choose the
rules for remuneration for work they were to perform. Most subjects chose to maximize average
income subject to a floor constraint.
The empirical evidence does not support Miller’s claim that people have the intuition that
it is best to “equalize degrees of unmet need, which means distributing in favor of those in
greater need until they are brought up to the same level as others” , however ((Miller, 1999, 74).
So, one cannot use this evidence to argue that the poor fare better when the only change is that
more people who can meet their needs are added to the population. The first experiment only
7
provides evidence that sometimes people will try to help others meet their needs before
distributing the remaining goods equally. The second experiment only provides evidence that
most people want to provide a flat minimum for everyone (Frohlich & Oppenheimer, 1992;
Miller, 2001, 79). The results just show that people are concerned about need, not that “people
will aim to equalize degrees of unmet need” (Miller, 2001, 74).vii The evidence does not support
Miller’s intuition about what distribution according to need requires (Hassoun, 2008). Hence,
one cannot argue that the best explanation for why people share Miller’s intuition about how it is
best to meet need, is that poverty in a population decreases when the only change is that more
people who can meet their needs are added to the population.
A different objection is that the Weak Population Focus Axiom is incompatible with the
idea that poverty in a population should be sensitive to the proportion of people in poverty. One
might argue that large countries with a relatively small proportion of poor people are richer than
small countries with a relatively large proportion of poor people. Or, more generally, one might
endorse what Subramanian has called the “Likelihood Principle,” on which (at least ceteris
perabus) a population in which one is more likely to encounter a poor person is poorer
(Subramanian, 2006). On the Likelihood Principle, large populations with a relatively small
proportion of poor people are richer than small populations with a relatively large proportion of
poor people.
We can consider the relevance of proportions by holding the numbers of the poor people
in a population constant and just manipulating the number of rich people in the population.
Suppose that at time t1 there are two populations that have one-hundred poor people each, so one
hundred percent of each population is poor (the width of the boxes that follow can be used to
represent the size of the populations).
8
Population 1 t1
Population 2 t1
At t2, the first population has stayed the same and the second population has gained one-hundred
rich people. So, the proportion of poor people in the second population has decreased drastically
to fifty percent, but it has the same number of poor people.
Population 1 t2
Population 2 t2
Considering these cases, I hope no one will think poverty has decreased in the second population
when all that has happened is that it has gained rich people.viii
Proponents of the Likelihood Principle might argue, however, that these simple diagrams
decontextualize our judgments about the amount of poverty in a population too much. If one
thinks that it is impermissible to discriminate against smaller countries in deciding which to aid,
for instance, it might make sense to look at the proportion of the population in the country that is
poor. (Though, it is debatable whether it is best to treat countries rather than individuals equally.)
Alternately, if one thinks that it should be easier to alleviate poverty in countries with less
poverty in the population, one might want to say that there is more poverty in a country with a
greater proportion of poor people.
It is possible, however, to account for the force of the motivating intuitions in these
examples without granting that proportions are relevant to how much poverty there is in a
9
population. It is important to distinguish clearly between how much poverty there is in a
population and what we should do about it. A measure of the amount of poverty in a population
is descriptive. It may not be better to give more aid to smaller countries with larger proportions
but smaller numbers of poor people in them because they are poorer. Perhaps it is better to give
aid to such countries because, given their size, they have fewer resources with which to combat
poverty in the population. Second, it is important to distinguish between the potential for a
population to ameliorate poverty and the amount of poverty in that population. Or, as Sen puts it,
between descriptive indexes which are concerned with “‘the state of the poor’” and indexes
intended to measure a population’s “potential ability to meet the challenge of poverty” (Sen,
1981, 190). Indexes sensitive to the proportion of a population in poverty may capture the
potential for the population to ameliorate poverty or the average depth-of-poverty in that
population, rather than the amount of poverty in that population.ix More generally, it is important
to distinguish between the amount of poverty in a population and other features of a population
that constrain how easy it is to ameliorate that poverty.x
Proponents of the Likelihood Principle might argue, however, that poverty in a
population is sensitive to the proportion of a population that is poor in at least some cases. The
proportion of people in a population who are poor can change for many reasons – migration,
birth, and death. Each of these reasons may also be explained by many factors, some endogenous
to a population and some exogenous. Changing internal conditions may, for instance, alter
economic or population growth rates amongst different segments of a population. Alternately,
changing external conditions may increase immigration or encourage emigration. If the
proportion of a population that is poor depends on endogenous factors, one might say poverty in
a population should be sensitive to proportions.
10
It is not clear why the fact that good internal conditions that let a population support a
greater number of rich people is supposed to decrease poverty in a population on this proposal.
Perhaps the motivating intuition is that endogenous changes that allow a population to support
additional rich people suggest that the population is better in some way – so it is fair to say it is
less poor. Populations can, however, be better in many ways without containing less poverty. A
population might be better because it can support a greater number of rich people, even if it
contains more poverty. Furthermore, it may be important to clearly distinguish between a
population’s poverty and how much poverty there is in a population. When internal features
allow a population to support more rich people, the population may be poorer without there
being less poverty in the population. Swaziland might be poorer than China, for instance, if it has
a higher proportion of poor people in it, even if it contains fewer poor people and, so, has less
poverty. Unless there is a better justification for the Likelihood Principle, there is reason to reject
it.xi
III.
Problems with the Existing Poverty Indexes and a Solution
The reason many of the most common poverty indexes fail to satisfy the Weak
Population Focus Axiom is precisely that they consider the proportion of poor people in the
population. These indexes normalize for population size by dividing by the total number of
people in the population. Consider the simplest example -- the headcount index (H) which
measures the proportion of people below the poverty line.
H = np /n
(1)
n is the number of people in the population and n p the number of people who have an amount of
good yi less than the amount necessary to reach the poverty line z . The problem is that adding a
person above the poverty line to the population only increments n and, so, poverty declines.
11
(Hassoun, 2010) demonstrates that Sen’s Index, the Sen-Shorrocks-Thon index and the Foster,
Greer, Thorbeke (FGT) index also fail to satisfy Weak Population Focus.
Consider, here, just the FGT index.
n
FGT  1 / n( xia )
(2)
i 1
When a = 0 this is H, when a = 1 it is I, and when a > 1 it is a weighted poverty gap index for
the whole population. x i is the income of individual i and x i = 0 when the amount of good yi
n
that individual has is less than the amount necessary to reach the poverty line z . So
x
i 1
a
i
is
unchanged by the mere addition of a rich person to a population. Because the addition of a rich
person increases n, however, poverty declines on the FGT when the only change is that a rich
person is added to the population.
One might object that the problems with the main poverty indexes are not genuine.
Perhaps these indexes only have problems when they are interpreted as measures of poverty in a
population. Perhaps it is better to just use them as measures of success in reducing poverty.
The way the main poverty indexes in the literature are often used is indefensible,
however. For, when they are used to measure success in reducing poverty, they suggest it is a
good thing to kill the poor or let them die (Kanbur & Mukherjee, 2007; Kanbur, 2006).xii Many
international organizations measure their programs’ success, as in the Millennium Development
Goals, by looking at indicators like H. Famines, disasters, and the AIDs crisis all decrease the
amount of poverty on these indicators. That should not count as success (Pogge, 2010).
In fact, using the main poverty indexes in the literature as measures of success in
reducing poverty is worse than interpreting them as measures of the amount of poverty in a
population. If one does the later, these indexes only entail (incorrectly) that poverty in a
12
population declines when rich people are added to a population. (There does seem to be less
poverty in the world when poor people die or are killed. Though, again, it is not a good thing that
poverty is reduced in these ways). If the indexes’ creators only intend the indexes to be used as
measures of the amount of poverty in a population, they can argue that using any metric for
policy purposes will require at least a modicum of good judgment.
Interpreting the main poverty indexes in the literature as measures of poverty in a
population, it is easy to see how to modify them so that they avoid the problem sketched above:
Do not divide by the total number of people in the population. So H would just become n and the
n
FGT would become the aggregate gap index: ( xia ) . But what poverty index should one
i 1
accept?
Moreover, there is some reason to accept a version of the aggregate gap index if, as this
paper is assuming in holding individuals’ poverty fixed, researchers can arrive at the correct
account of individuals’ poverty. The aggregate gap index just adds up each individual’s poverty
in a population.xiii
IV.
Conclusion
This paper has argued that there is good reason to reject traditional measures of poverty
in a population and some reason to accept the aggregate gap index instead. If the main poverty
indexes in the literature are interpreted as measures of poverty in a population and are not
modified, they suggest poverty in a population declines with the mere addition of rich people.
Assuming that researchers can arrive at the proper basis for poverty measurement and have fully
characterized individuals’ needs, there is some reason to accept the aggregate gap index. It tells
us the sum-total amount of need in the population.
13
Once poverty indexes are modified, they provide the basis for plausible indexes that tell
n
us how we should aid the poor. If ( x i ) does capture the amount of poverty in a population,
i 1
n
the function Minimize ( x i ) is just some kind of maximizing consequentialism below the
i 1
poverty line; it tells us that we should minimize the amount of poverty in that population. The
n
function Minimize ( xip ) where p > 1, is a version of prioritarianism below the poverty line. It
i 1
tells us to give greater weight to helping the less well off. A more extreme roughly Rawlsian
function might tell us to give ultimate priority to helping the least well off poor person. This
paper’s arguments about how we should measure poverty in variable populations are also
important for debates about aid and trade. These debates often hinge on claims about the extent
of poverty in different populations and success in alleviating it. Further work on variable
population poverty comparisons is pressing and important.
14
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i This paper is a less technical version of (Hassoun, 2008). I owe special thanks to Subbu
Subramanian, John Weymark, Lucio Esposito, Dan Halliday, John Mumma, Teddy Seidenfeld,
Luc Christensen, Alex London, Julian Culp, Imed Drine, Paul Alagdede, Terra Lawsen-Remer,
Madhura Swaminathan, Peter Stone, Allegra McCloud, Keiran Oberman, Augustine Fosu, and
scholars and fellows at Stanford University and the United Nations’ World Institute for
Development Economics Research (WIDER) in Helsinki for their very helpful comments on
related papers. I also owe thanks to those who commented on my Utilitas article as this paper
draws on an argument first published there. Finally, I would like to thank WIDER and Stanford\s
Center for Ethics in Society for their support during the initial stages of this research.
ii It also uses the share of the poorest quintile in national consumption.
iii For one way of doing this see: (Hare, 1981). Recent work on theory of mind may add to
Hare’s account by helping to explicate the process by which such comparisons are made. See:
(Goldman, 1993; Gopnik and Wellman, 1995).
iv Talk about needs and goods does not presuppose that any particular way of distributing goods
(in the relevant broad sense) is best. It might, for instance, alleviate just as much need to give 10
children a year of schooling as to inoculate a single child against measles.
v An individuals’ poverty may be worse if that individual has been poor for a longer period of
time or if the individual was richer (or poorer) in the past. It may also matter who is poor so that
one of two people with identical histories could be poorer than another.
vi The idea here is different from Derek Parfit’s mere addition paradox discussed below (Parfit,
1984).
vii At another point, Miller also suggests subjects are balancing a concern for giving people their
just deserts against a concern for need fulfillment. But, it is not clear that the evidence supports
this contention (Hassoun, 2009).
viii Alternately, it is possible to change the number of poor people in a population, but hold the
proportion of poor people constant. Suppose that the first society looks like this at t3 with twice
as many poor people as it had at t2:
Society 1 t2
Society 1 t3
I hope everyone can agree that there is more poverty in the first society at t3 than at t2, though
the proportions are the same. At least the number of poor people in a population seems relevant
to how much poverty there is. In light of cases like these, those who think proportions are
relevant to the amount of poverty in a population must at least say more. They must explain
exactly how proportions matter relative to numbers (e.g. provide a weighting on proportions and
numbers that generates a composite measure of poverty in a population and can generate their
desired judgments about particular cases).
ix Some standard justifications for tools in economics for distributional, welfare, and poverty
analyses (such as Lorenz Dominance, Generalized Lorenz Dominance, and Stochastic
Dominance) may require one to reject the Weak Population Focus Axiom. The desire to be able
to use these tools may drive intuitions about the relevance of proportions (Hassoun &
Subramanian, 2010).
20
x One reason to think potential matters in some cases (even if it does not map onto the amount of
poverty in a population) is this: If one were behind a Rawlsian veil of ignorance and did not
know who one would be in a society, one might reasonably choose to enter a society with a
smaller proportion of poor people.
xi Furthermore, even if poverty is sensitive to proportions in some cases, that is not enough to
establish the Likelihood Principle, which suggests that poverty is always sensitive to the
proportion of the population that is poor. So, even if proportions are sometimes relevant to how
much poverty there is in a population, it will still be worth examining the Weak Population
Focus Axiom. For, the mere addition of a rich person to a population may not generally decrease
poverty in that population. In fact, even if one is not completely convinced by these debunking
explanations of why one might find the Likelihood Principle compelling, it is worth considering
the Weak Population Focus Axiom further. For, it may be possible to create an index that
captures some of the force of Weak Population Focus Axiom and the Likelihood Principle.
Modifying Subramanian’s suggestion for how to satisfy both the Likelihood principle and the
Strong Focus Axiom in some cases, one might apply the following decision procedure: If two
distributions have the same population size then rank the population using the aggregate gap
index. If they have the same score on the aggregate gap index then rank the populations based on
the proportion of the population that is poor. At best, however, this mitigates the tension between
the Likelihood Principle and Weak Population Focus Axiom (Subramanian, 2006). Nevertheless,
the remainder of this paper will proceed on the assumption that the Weak Population Focus
Axiom is correct.
xii The only objection I am aware of to using the aggregate gap indexes, in particular, is that they
are impractical as they can range from zero to infinity (Paxton, 2003, 3). I am not sure, however,
why this makes the indexes impractical in the real world with finite populations. Julia Paxton’s
suggested alternative scales the aggregate gap indexes by the log of the number of poor people in
a population divided by the total number of poor people. It is not clear, however, that this scaling
is justified. Why care about the percentage of poor people in a country that are being helped at a
decreasing rate? Some justification here is necessary.
xiii If goods have declining marginal needs satisfaction that is already taken into account in
calculating individuals’ needs (recall that individuals’ needs are fixed before any aggregation
takes place). The marginal need-satisfaction of a good for a person is the difference an additional
unit of that good will make to how much need a person has. The marginal need satisfaction a
person gets from a unit of necessary good declines the more units of necessary good the person
already has.