چهارشنبه ۲۸ شهریور ۰۳ | ۲۰:۵۸ ۱۰ بازديد
The cost of living is certainly higher in urban areas, but by a factor of 1.2-1.5 rather than by a factor
of 2. Thus the urban self-rated poverty line is, in real terms, higher than its rural counterpart. Why?
• One possibility is that there is more inequality in the urban areas, and that this raises expectations.
• Another plausible explanation is that households in urban areas may have more exposure to the
media, and may have been affected more thoroughly by consumerism.
• A third explanation is that urban households may be more attuned to political processes, and their
estimates of the poverty line may include an element of strategic behavior ñ trying to influence
policy makers.
Self-rated measures of poverty are rarely collected. If the Filipino experience is at all
representative, it is clear that they cannot usefully supplant the more traditional ìobjectiveî measures of
poverty.
CHAPTER 4
Measures of Poverty
What does Aggregate Poverty Measure? Given information on (say) per capita consumption, and
a poverty line, then the only remaining problem is deciding on an appropriate summary measure of
aggregate poverty. There are a number of aggregate measures of poverty that can be computed.
4.1 Headcount index
By far the most widely-used measure is the headcount index, which simply measures the
proportion of the population that is counted as poor, often denoted by P0. Formally,
(4.1) ¦ =≤= =
N
i
p
i N
N
zyI
N
P 1
0 ,)(
1
where N is the total population and I(.) is an indicator function that takes on a value of 1 if the bracketed
expression is true, and 0 otherwise. So if expenditure (yi) is less than the poverty line (z), then I(.) equals
to 1 and the household would be counted as poor. Np is the total number of the poor.
The great virtue of the headcount index is that it is simple to construct and easy to understand.
These are important qualities. However the measure has at least three weaknesses:
The headcount index does not take the intensity of poverty into account. Consider the following two
income distributions:
Headcount Poverty Rates in A and B, assuming poverty line of 125
Expenditure for each individual in
country
Headcount poverty
rate (P 0)
Expenditure in country A 100 100 150 150 50%
Expenditure in country B 124 124 150 150 50%
Clearly there is greater poverty in country A, but the headcount index does not capture this.
As a welfare function, the headcount index violates the transfer principle ñ an idea first formulated by
Dalton (1920) that states that transfers from a richer to a poorer person should improve the measure of
welfare. Here if a somewhat poor household were to give to a very poor household, the headcount
index would be unchanged, even though it is reasonable to suppose that poverty overall has lessened.
The headcount index implies that there is a ìjumpî in welfare, at about the poverty line, so it is
meaningful to speak of the poor and the non-poor. In practice, such a jump is not found (Ravallion
1996, p.1330).
• The head-count index does not indicate how poor the poor are, and, hence, does not change if people
below the poverty line become poorer. Moreover, the easiest way to reduce the headcount index is to
target benefits to people just below the poverty line, because they are the ones who are cheapest to
move across the line. But by most normative standards, people just below the poverty line are the
least deserving of the poor. Thus, despite its popularity, many problems result from an undue
concentration on the head-count statistic.
• It is also important to note that the poverty estimates should be calculated for individuals and not
households. What we calculate using the head-count index is the percentage of individuals who are
poor and not the percentage of households. To be able to do so, we make a critical assumption that
all household members enjoy the same level of well-being. This assumption may not hold in many
situations. For example, some elderly members of a household may be much poorer than other
members of the same household. In reality, not all consumption is evenly shared across household
members.
4.2 Poverty gap index
A moderately popular measure of poverty is the poverty gap index, which adds up the extent to
which individuals fall below the poverty line (if they do), and expresses it as a percentage of the poverty
line. More specifically, define the poverty gap (Gn) as the poverty line (z) less actual income (yi) for poor
individuals; the gap is considered to be zero for everyone else. Using the index function, we have
(4.2) ).().( zyIyzG iin ≤−=
Then the poverty gap index (P1) may be written as
(4.3) ¦= =
N
i
n
z
G
N
P 1
1 .
1
This measure is the mean proportionate poverty gap in the population (where the non-poor have
zero poverty gap). Some people think of this measure as the cost of eliminating poverty (relative to the
poverty line), because it shows how much would have to be transferred to the poor to bring their incomes
(or expenditure) up to the poverty line. The minimum cost of eliminating poverty using targeted transfers
is simply the sum of all the poverty gaps in a population; every gap is filled up to the poverty line.
However this interpretation is only reasonable if the transfers could be made perfectly efficiently, for
instance with lump sum transfers, which is implausible. Clearly this assumes that the policymaker has a
lot of information; one should not be surprised to find that a very ìpro-poorî government would need to
spend far more than this in the name of poverty reduction.
At the other extreme, one can consider the maximum cost of eliminating poverty, assuming that
the policymaker knows nothing about who is poor and who is not. From the form of the index, it can be
seen that the ratio of the minimum cost of eliminating poverty with perfect targeting (i.e. Gn) to the
maximum cost with no targeting (i.e. z, which would involve providing everyone with enough to ensure
they are not below the poverty line) is simply the poverty gap index. Thus this measure is an indicator of
the potential saving to the poverty alleviation budget from targeting.
The poverty gap measure has the virtue that it does not imply that there is a discontinuity
(ìjumpî) at the poverty line. Yet a serious shortcoming of this measure is that it may not convincingly
capture differences in the severity of poverty amongst the poor. For example, consider two distributions
of consumption for four person; the A distribution is (1,2,3,4) and the B is (2,2,2,4). For a poverty line
z=3 (so that the headcount index is 0.75 in both cases), A and B have the same value for the poverty gap
index (i.e. 0.25). However, the poorest person in A has only half the consumption of the poorest in B.
One can think of B as being generated from A by a transfer from the least poor of the poor persons (the
individual with ë3í in A) to the poorest. The poverty gap will be unaffected by such a transfer. Its main
drawback is that it ignores inequality among the poor. To see this, consider the following example:
Poverty Gap Poverty Rates in A and B, assuming poverty line of 125
Expenditure for each individual in country Poverty gap rate (P 1)
Expenditure in country A 100 100 150 150 0.10
Expenditure in country B 80 120 150 150 0.10
For both of these countries, the poverty gap rate is 0.10, but most people would argue that country B has
more serious poverty because it has an extremely poor member.
To summarize, the Poverty Gap Index is the average over all people, of the gaps between poor
peopleís standard of living and the poverty line, expressed as a ratio to the poverty line. The aggregate
poverty gap shows the cost of eliminating poverty by making perfectly targeted transfers to the poor (i.e.,
closing all poverty gaps), in the absence of transactions costs and disincentive effects. This is clearly
unrealistic but it does convey useful information about the minimum scale of the financial resources
needed to tackle the poverty problem. Moreover, the poverty gap index can show the value of using
survey information to learn about the characteristics of the poor. A costly way of eliminating poverty
would be to make completely untargeted poverty line-sized transfers to everyone in the population. The
poverty gap index gives the ratio of the cost of eliminating poverty using perfectly targeted transfers
compared with using completely untargeted transfers. Thus, the smaller is the poverty gap index, the
greater the potential economies for a poverty alleviation budget from identifying the characteristics of the
poor so as to target benefits and programs.
4.3 Squared poverty gap index
To solve the problem of inequality among the poor, some researchers use the squared poverty gap
index. This is simply a weighted sum of poverty gaps (as a proportion of the poverty line), where the
weights are the proportionate poverty gaps themselves; a poverty gap of (say) 10% of the poverty line is
given a weight of 10% while one of 50% is given a weight of 50%; this is in contrast with the poverty gap
index, where they are weighted equally. Hence, by squaring the poverty gap index, the measure
implicitly puts more weight on observations that fall well below the poverty line. Formally:
(4.4) .)(
1
1
2
2 ¦=
=
N
i
n
z
G
N
P
The measure lacks intuitive appeal, because it is not easy to interpret and so it is not used very
widely. It may be thought of as one of a family of measures proposed by Foster, Greer and Thorbecke
(1984), which may be written as
(4.5) ¦ ¸¸
¹
·
¨¨
©
§
= =
N
i
n
z
G
N
P 1
1
α
α , (α ≥ 0)
where α is a measure of the sensitivity of the index to poverty and the poverty line is z, the value of
expenditure per capita for the j-th personís household is xj, and the poverty gap for individual j is Gj=z-
xj(with Gj=0 when xj>z) When parameter
α=0, P0 is simply the head-count index. When α=1, the index is
the poverty gap index P1, and when
α is set equal to 2, P2 is the poverty severity index. For all α > 0, the
measure is strictly decreasing in the living standard of the poor (the lower your standard of living, the
poorer you are deemed to be). Furthermore, for
α > 1 it also has the property that the increase in
measured poverty due to a fall in oneís standard of living will be deemed greater the poorer one is. The
measure is then said to be "strictly convex" in incomes (and "weakly convex" for
α=1). Another
convenient feature of the FGT class of poverty measures is that they can be disaggregated for population
sub-groups and the contribution of each sub-group to national poverty can be calculated.
The work by Foster, Greer and Thorbecke provides an elegant unifying framework for measures
of poverty. However it begs the question of what the best value of α. Some of these measures also lack
emotional appeal.
The measures of depth and severity of poverty provide complementary information on the
incidence of poverty. It might be the case that some groups have a high poverty incidence but low
poverty gap (when numerous members are just below the poverty line), while other groups have a low
poverty incidence but a high poverty gap for those who are poor (when relatively few members are below
the poverty line but with extremely low levels of consumption). Table 4.1 provides an example from
Madagascar. According to the headcount, unskilled workers show the third highest poverty rate, while
the group is in the fifth rank according to the poverty severity. Comparing them with the herders shows
that they have a higher risk of being in poverty, but that the poverty tends to be less sever or deep. The
types of interventions needed to help the two groups are therefore likely to be different.
Table 4.1: Poverty Indices By sub-groups, Madagascar, 1994
Head count: % Rank Poverty gap: % Rank Poverty severity: % Rank
Small farmers 81.6 1 41.0 1 24.6 1
Large farmers 77.0 2 34.6 2 19.0 2
Unskilled workers 62.7 3 25.5 4 14.0 5
Herders/fishermen 51.4 4 27.9 3 16.1 3
Retirees/handicapped 50.6 5 23.6 5 14.1 4
Source: Coudouel, Hentschel and Wodon (2001)
4.4 Sen Index.
Sen (1976) proposed an index that sought to combine the effects of the number of poor,
the depth of their poverty, and the distribution of poverty within the group. The index is given
by
(4.6) ),)1(1(0 z
GPP
P
P
s
µ
−−=
where P0 is the headcount index, µP is the mean income (or expenditure) of the poor, and GP is the Gini
coefficient of inequality among the poor. The Gini coefficient ranges from 0 (perfect equality) to 1
(perfect inequality), and is discussed further below in the context of measuring inequality. The Sen Index
can also be written as the average of the headcount and poverty gap measures weighted by the Gini
coefficient of the poor, giving:
(4.7) )1(10
PP
s GPGPP −+=
The Sen index has been widely discussed, and has the virtue of taking the income distribution
among the poor into account. However the index lacks intuitive appeal, and cannot be decomposed
satisfactorily into its constituent components, which explains why it is rarely used in practice.
4.5 The Sen-Shorrocks-Thon index.
The Sen index has been modified by others, and perhaps the most compelling version is the Sen-
Shorrocks-Thon (SST) index, defined as
(4.8) )à1(10
PP
SST GPPP += ,
which is the product of the headcount index, the poverty gap index (applied to the poor only), and a term
with the Gini coefficient of the poverty gap ratios (i.e. of the Gnís). This Gini coefficient typically is
close to 1, indicating great inequality in the incidence of poverty gaps.
Example
In 1996, 12.4% of the population of Quebec province (Canada) was in poverty. The poverty gap
index, applied to the poor only, stood at 0.272. And the Gini coefficient of the poverty gap ratios
was 0.924. Thus the Sen-Shorrocks-Thon index was 0.065 (=0.124 ◊ 0.272 ◊ (1+0.924)).
Osberg and Xu (1999) use the SST index to compare poverty across the 10 Canadian provinces
for 1984, 1989, 1994, 1995 and 1995, as well as to put the degree of Canadian provincial poverty into an
international context. A number of graphs from their study are reproduced below. Figure 1 provides an
international comparison, using the SST index, and shows that the US is an outlier with its relatively high
poverty rate (as measured by the SST). A comparison of the US and Canada over time (figure 2) shows
that while poverty was similar in the two countries a generation ago, it is now higher in the US. Figure 3
provides information on some Canadian provinces: Newfoundland was the poorest in 1984, but by 1996
had become much less of an outlier.
One strength of the SST index is that it can help give a good sense of the sources of change in
poverty over time. This is because the index may be decomposed into
(4.9) )à1ln(lnlnln 10
PP
SST GPPP +∆+∆+∆=∆ ,
which may be interpreted as, % change in SST index = % change in headcount index + % change in
poverty gap index ( among poor) + % change in (1+Gini coefficient of poverty gaps).
In plain English, this allows us to decompose poverty into three aspects: are there more poor? are
the poor poorer? and is there higher inequality among the poor?
Example:
The information in table 4.2 comes from Osberg and Xu, and traces the evolution of poverty in
the Canadian province of Newfoundland between 1984 and 1996. It is clear that most of the
change in the poverty rate over time was due to variations in the number of people in poverty
(P1), rather than in the size of the poverty gap per poor person (P1P
) or the distribution of poverty
among the poor (GP
).
Table 4.2: Decomposition of poverty, and changes in poverty, in Newfoundland
SST
index
P0 PP1 1+GP ∆lnSST
index
∆LnP0 ∆lnPP1 ∆ln(1+GP
)
1984 .137 .245 .304 1.844
1989 .095 .169 .296 1.897 -.370* -.372* -.027 .028
1994 .105 .184 .304 1.884 .104 .086 .026 -.007
1995 .125 .212 .316 1.864 .168 .141 .038 -.010
1996 .092 .164 .294 1.897 -.307 -.254 -.071 .018
Notes: * denotes statistically significant at the 95% level. Poverty line is half of median equivalent income, using the ìOECD
scaleî (i.e. equivalent income = 1 + 0.7(Nadults-1)+0.5(Nchildren).
Source: Osberg and Xu, 1999.
Note that the values of the Sen-Shorrocks-Thon index provided by Osberg and Xu do not give
just a single point estimate for each province; they also provide a confidence interval. Because the SST
index is complex, it is not possible to compute these confidence intervals analytically. Instead, they are
computed artificially using bootstrapping. The basic idea behind the bootstrap is straightforward and
clever. Suppose we have a survey sample of 2,000 households. Now pick a sample of 2,000 from this
sample with replacement ñ i.e. pick a household, then put it back into the sample, pick another household,
put it back into the sample, and so on, until you have picked 2,000 households. Compute the SST index
using this artificial sample. Then repeat the process many times; Osberg and Xu use 300 repetitions. The
result is a distribution of values of the SST, from which it is easy to find (say) the 95% confidence
interval. A sample code to generate confidence intervals for the SST index is given in the exercise for
Chapter Five.
4.6 Time taken to exit
Previous poverty profiles for
dfs3434