poverty measures celia m. reyes introduction to poverty analysis nai, beijing, china

© 2003 By Default!Slide 1

Poverty Measures Poverty Measures

Celia M. ReyesCelia M. Reyes

Introduction to Poverty AnalysisIntroduction to Poverty AnalysisNAI, Beijing, ChinaNAI, Beijing, China

Nov. 1-8, 2005Nov. 1-8, 2005


Headcount IndexHeadcount Index

measures the proportion of the population that measures the proportion of the population that is in povertyis in poverty

the headcount index estimates the percentage the headcount index estimates the percentage of the population living in households with per of the population living in households with per capita consumption below the poverty linecapita consumption below the poverty line

it thus measures the incidence of povertyit thus measures the incidence of poverty


Poverty Indices: Headcount IndexPoverty Indices: Headcount Index

Formally,Formally,

where N = total populationwhere N = total population

(.) = an indicator function that takes on a(.) = an indicator function that takes on a value of 1 (poor); if the bracketed value of 1 (poor); if the bracketed expression is true, and 0 (nonpoor) expression is true, and 0 (nonpoor) otherwiseotherwise yyi i = expenditure = expenditure z z = poverty line = poverty line

N

i

pi N

NzyI

NP

10 ,)(

1



the great virtue of the headcount index is that it the great virtue of the headcount index is that it is simple to construct and easy to understandis simple to construct and easy to understand

adequate measure of assessing overall adequate measure of assessing overall progress in reducing poverty (though preferably progress in reducing poverty (though preferably always calculated for at least two poverty lines)always calculated for at least two poverty lines)



Weaknesses:Weaknesses:

The headcount index does not take the intensity The headcount index does not take the intensity of poverty into account. To see why, suppose of poverty into account. To see why, suppose that a poor person suddenly becomes very that a poor person suddenly becomes very much poorer. What will happen to measured much poorer. What will happen to measured poverty? Nothing. The headcount index is poverty? Nothing. The headcount index is totally insensitive to differences in the depth of totally insensitive to differences in the depth of poverty.poverty.


Example:Example:

There is greater poverty in country A but the There is greater poverty in country A but the

headcount index does not capture this.headcount index does not capture this.

Expenditure for each individual in countryExpenditure for each individual in country Headcount Poverty Rate (PHeadcount Poverty Rate (P00))

Expenditure in Country AExpenditure in Country A 100100 100100 150150 150150 50%50%

Expenditure in Country BExpenditure in Country B 124124 124124 150150 150150 50%50%




As a welfare function, the headcount index violates the As a welfare function, the headcount index violates the transfer principle – an idea first formulated by Dalton transfer principle – an idea first formulated by Dalton (1920) that states that transfers from a richer to a (1920) that states that transfers from a richer to a poorer person should improve the measure of welfare. poorer person should improve the measure of welfare. Here, if a somewhat poor household were to give to a Here, if a somewhat poor household were to give to a very poor household, the headcount index would be very poor household, the headcount index would be unchanged, even though it is reasonable to suppose unchanged, even though it is reasonable to suppose that poverty overall has lessenedthat poverty overall has lessened..

The headcount index implies that there is a “jump” in The headcount index implies that there is a “jump” in welfare, at about the poverty line, In practice, such a welfare, at about the poverty line, In practice, such a jump is not found.jump is not found.



The headcount index is very simple The headcount index is very simple to construct to construct and easy to understand.and easy to understand.

However, However, it does not indicate how poor the poor it does not indicate how poor the poor are, and hence does not change if people below are, and hence does not change if people below the poverty line become poorer.the poverty line become poorer.



The easiest way to reduce the headcount index The easiest way to reduce the headcount index is to target benefits to people just below the is to target benefits to people just below the poverty line, because these are the ones who poverty line, because these are the ones who are cheapest to move across the line - but by are cheapest to move across the line - but by most normative standards, people just below most normative standards, people just below the poverty line are the least deserving of the the poverty line are the least deserving of the poor. Thus, despite its popularity, many poor. Thus, despite its popularity, many problems result from an undue concentration on problems result from an undue concentration on the head-count statistic.the head-count statistic.


Headcount IndexHeadcount Index What we calculate using the headcount index is What we calculate using the headcount index is

the percentage of individuals who are poor and the percentage of individuals who are poor and not the percentage of households.not the percentage of households.

To be able to do so, we make a critical assumption To be able to do so, we make a critical assumption that all household members enjoy the same level that all household members enjoy the same level of well-being. This assumption may not hold in of well-being. This assumption may not hold in many situations.many situations.

For example, some elderly members of a household For example, some elderly members of a household may be much poorer that other members of the same may be much poorer that other members of the same household.household.

In reality, not all consumption is evenly shared across In reality, not all consumption is evenly shared across household members.household members.


Philippines: Poverty incidencePhilippines: Poverty incidence

We calculate poverty incidence among families We calculate poverty incidence among families and population.and population.

Poverty incidence among families – proportion Poverty incidence among families – proportion of families who are poorof families who are poor

Poverty incidence among population – Poverty incidence among population – proportion of individuals who are poorproportion of individuals who are poor


Poverty Gap IndexPoverty Gap Index

A moderately popular measure of poverty which A moderately popular measure of poverty which adds up the extent to which individuals fall below adds up the extent to which individuals fall below the poverty line (if they do) and expresses it as a the poverty line (if they do) and expresses it as a percentage of the poverty line.percentage of the poverty line.

More specifically, define the poverty gap (GMore specifically, define the poverty gap (Gnn) as the ) as the poverty line (z) less actual income (ypoverty line (z) less actual income (y ii) for poor ) for poor individuals; the gap is considered to be zero for individuals; the gap is considered to be zero for everyone else.everyone else.

This measure reflects the average distances of the This measure reflects the average distances of the poor below the poverty line so it gives a better idea poor below the poverty line so it gives a better idea of the depth of poverty.of the depth of poverty.



This measure is also thought of as the cost of This measure is also thought of as the cost of eliminating poverty (relative to the poverty line), eliminating poverty (relative to the poverty line), since it shows how much would have to be since it shows how much would have to be transferred to the poor to bring their incomes (or transferred to the poor to bring their incomes (or expenditure) up to the poverty line).expenditure) up to the poverty line).

The minimum cost of eliminating poverty using The minimum cost of eliminating poverty using targeted transfers is simply the sum of all the targeted transfers is simply the sum of all the poverty gaps in a population: every gap is filled poverty gaps in a population: every gap is filled up to the poverty line.up to the poverty line.



However this interpretation is only reasonable if However this interpretation is only reasonable if the transfers could be made perfectly efficiently, the transfers could be made perfectly efficiently, for instance with lump sum transfers. This for instance with lump sum transfers. This assumes that the policy maker has a lot of assumes that the policy maker has a lot of information and that a very “pro-poor” information and that a very “pro-poor” government would need to spend more than the government would need to spend more than the minimum cost in the name of poverty reduction.minimum cost in the name of poverty reduction.



At the other extreme, one can consider the At the other extreme, one can consider the maximum cost of eliminating poverty, assuming maximum cost of eliminating poverty, assuming that the policy maker knows nothing about who that the policy maker knows nothing about who is poor and who is not. is poor and who is not.

From the form of the index, it can be seen that From the form of the index, it can be seen that the ratio of the minimum cost of eliminating the ratio of the minimum cost of eliminating poverty with perfect targeting to the maximum poverty with perfect targeting to the maximum cost with no targeting is simply the poverty gap cost with no targeting is simply the poverty gap index.index.



Thus, this measure is also an indicator of the Thus, this measure is also an indicator of the potential saving to the poverty alleviation budget potential saving to the poverty alleviation budget from budgeting.from budgeting.

A serious shortcoming of this measure is that it A serious shortcoming of this measure is that it may not convincingly capture differences in the may not convincingly capture differences in the severity of poverty amongst the poor.severity of poverty amongst the poor.



Example:Example:

Consider 2 distributions of consumption for 4 Consider 2 distributions of consumption for 4 persons. Poverty line = 3.01persons. Poverty line = 3.01

– A distribution is (1,2,3,4)A distribution is (1,2,3,4)– B distribution is (2,2,2,4)B distribution is (2,2,2,4)

Thus, headcount index is 0.75 and the poverty Thus, headcount index is 0.75 and the poverty gap index is 0.25 in both cases.gap index is 0.25 in both cases.



However, the poorest person in A has only However, the poorest person in A has only half of the consumption of the poorest in B. half of the consumption of the poorest in B. One can think of B as being generated from A One can think of B as being generated from A by a transfer from the least poor person by a transfer from the least poor person (individual with ‘3’ in A) to the poorest. The (individual with ‘3’ in A) to the poorest. The poverty gap will be unaffected by such a poverty gap will be unaffected by such a transfer.transfer.

Thus, the main drawback of this measure is that Thus, the main drawback of this measure is that it ignores inequality among the poor.it ignores inequality among the poor.



To see this again, consider the following example:To see this again, consider the following example:

Poverty Gap Rates in A and B, assuming poverty Poverty Gap Rates in A and B, assuming poverty line of 125line of 125

For both countries, the poverty gap rate is 0.10, it could be For both countries, the poverty gap rate is 0.10, it could be argued that country B has more serious poverty because it argued that country B has more serious poverty because it has an extremely poor member.has an extremely poor member.

Expenditure for each individual in countryExpenditure for each individual in country Poverty Gap Rate (PPoverty Gap Rate (P11))

Expenditure in Country AExpenditure in Country A 100100 100100 150150 150150 0.100.10

Expenditure in Country BExpenditure in Country B 8080 120120 150150 150150 0.100.10



In summary, In summary,

The Poverty Gap Index is the average over all people, of The Poverty Gap Index is the average over all people, of the gaps between poor people’s standard of living and the the gaps between poor people’s standard of living and the poverty line, expressed as a ratio to the poverty line.poverty line, expressed as a ratio to the poverty line.

The aggregate poverty gap shows the cost of eliminating The aggregate poverty gap shows the cost of eliminating poverty by making perfectly targeted transfers to the poor, poverty by making perfectly targeted transfers to the poor, i.e. closing all poverty gaps, in the absence of i.e. closing all poverty gaps, in the absence of transactions costs and disincentive effects.transactions costs and disincentive effects.



– This is unrealistic but it does convey useful This is unrealistic but it does convey useful information about the minimum scale of the information about the minimum scale of the financial resources needed to tackle the financial resources needed to tackle the poverty problem.poverty problem.

Moreover, the poverty gap index can show the Moreover, the poverty gap index can show the value of using survey information to learn about value of using survey information to learn about the characteristics of the poor. A costly way of the characteristics of the poor. A costly way of eliminating poverty would be to make eliminating poverty would be to make completely untargeted poverty line-sized completely untargeted poverty line-sized transfers to everyone in the population.transfers to everyone in the population.



The poverty gap index gives the ratio of the cost The poverty gap index gives the ratio of the cost of eliminating poverty using perfectly targeted of eliminating poverty using perfectly targeted transfers compared with using completely transfers compared with using completely untargeted transfers.untargeted transfers.

– Thus, the smaller is the poverty gap index, Thus, the smaller is the poverty gap index, the greater the potential economies for a the greater the potential economies for a poverty alleviation budget from identifying the poverty alleviation budget from identifying the characteristics of the poor so as to target characteristics of the poor so as to target benefits and programs.benefits and programs.


Squared Poverty Gap or Severity IndexSquared Poverty Gap or Severity Index

Unlike the head-count and poverty-gap indexes, Unlike the head-count and poverty-gap indexes, the absolute value of the poverty severity index the absolute value of the poverty severity index has no intuitive interpretation and is not easy to has no intuitive interpretation and is not easy to interpret. interpret.

For poverty comparisons, however, the key For poverty comparisons, however, the key point is that a ranking of dates, places or point is that a ranking of dates, places or policies in terms of P2 should reflect their policies in terms of P2 should reflect their ranking in terms of the severity of poverty.ranking in terms of the severity of poverty.



This is simply a weighted sum of poverty gaps (as This is simply a weighted sum of poverty gaps (as a proportion of the poverty line), where the weights a proportion of the poverty line), where the weights are the proportionate poverty gaps themselves; a are the proportionate poverty gaps themselves; a poverty gap of (say) 10% of the poverty line is poverty gap of (say) 10% of the poverty line is given a weight of 10% while one of 50% is given a given a weight of 10% while one of 50% is given a weight of 50%; this is in contrast with the poverty weight of 50%; this is in contrast with the poverty gap index, where they are weighted equally. gap index, where they are weighted equally.

Hence, by squaring the poverty gap index, the Hence, by squaring the poverty gap index, the measure implicitly puts more weight on measure implicitly puts more weight on observations that fall well below the poverty line. observations that fall well below the poverty line.



Formally:Formally:

The measure lacks intuitive appeal, because it The measure lacks intuitive appeal, because it is not easy to interpret and so it is not used very is not easy to interpret and so it is not used very widely. widely.

.)(1

1

22

N

i

n

z

G

NP



It may be thought of as one of a family of measures It may be thought of as one of a family of measures proposed by Foster, Greer and Thorbecke (1984), which may proposed by Foster, Greer and Thorbecke (1984), which may be written asbe written as

, (α ≥ 0), (α ≥ 0)

where α = measure of the sensitivity of the index to where α = measure of the sensitivity of the index to povertypoverty

z = poverty linez = poverty line xxjj = the value of expenditure per capita for the = the value of expenditure per capita for the

j-th person’s householdj-th person’s household GGjj=z-x=z-xjj = the poverty gap for individual j = the poverty gap for individual j

(with G(with Gjj=0 when x=0 when xjj>z)>z)

N

i

n

z

G

NP

1

1



When parameter When parameter =0, P=0, P00 is simply the head-count is simply the head-count index. When α=1, the index is the poverty gap index index. When α=1, the index is the poverty gap index PP11, and when , and when is set equal to 2, P is set equal to 2, P22 is the poverty is the poverty severity index.severity index. For all For all > 0, the measure is strictly decreasing in the > 0, the measure is strictly decreasing in the living standard of the poor (the lower your standard living standard of the poor (the lower your standard of living, the poorer you are deemed to be).of living, the poorer you are deemed to be).

For For > 1 it also has the property that the increase in > 1 it also has the property that the increase in measured poverty due to a fall in one’s standard of measured poverty due to a fall in one’s standard of living will be deemed greater the poorer one is. living will be deemed greater the poorer one is.



The measure is then said to be "strictly convex" The measure is then said to be "strictly convex" in incomes (and "weakly convex" for in incomes (and "weakly convex" for =1). =1).

Another convenient feature of the FGT class of Another convenient feature of the FGT class of poverty measures is that they can be poverty measures is that they can be disaggregated for population sub-groups and disaggregated for population sub-groups and the contribution of each sub-group to national the contribution of each sub-group to national poverty can be calculated. poverty can be calculated.



The measures of depth and severity of poverty The measures of depth and severity of poverty provide complimentary information on the provide complimentary information on the incidence of poverty.incidence of poverty.

Table 4.1 provides an example by Madagascar.Table 4.1 provides an example by Madagascar.


Table 4.1 Poverty Indices by Sub-groups, Table 4.1 Poverty Indices by Sub-groups, Madagascar, 1994 Madagascar, 1994

Small farmers 81.6 1 41.0 1 24.6 1Large farmers 77.0 2 34.6 2 19.0 2Unskilled workers 62.7 3 25.5 4 14.0 5Herders/fishermen 51.4 4 27.9 3 16.1 3Retirees/handicapped 50.6 5 23.6 5 14.1 4

RankPoverty severity:

%Rank

Head count: %

RankPoverty gap: %

Source: Coudouel, Hentschel and Wodon (2001)


Sen IndexSen Index

Sen (1976) proposed an index that sought to Sen (1976) proposed an index that sought to combine the effects of the number of poor, the combine the effects of the number of poor, the depth of their poverty, and the distribution of depth of their poverty, and the distribution of poverty within the group. poverty within the group.

The index is given byThe index is given by

),)1(1(0 zGPP

PP

s


Sen IndexSen Index

where Pwhere P0 0 = headcount index= headcount index μμPP = mean income (or exp) of the poor = mean income (or exp) of the poor

GGPP = Gini coefficient of inequality = Gini coefficient of inequality among the poor. The Gini coefficient ranges among the poor. The Gini coefficient ranges from 0 (perfect equality) to 1 (perfect from 0 (perfect equality) to 1 (perfect inequality), and is discussed further below in inequality), and is discussed further below in the context of measuring inequality. the context of measuring inequality.

The Sen Index can also be written as the The Sen Index can also be written as the average of the headcount and poverty gap average of the headcount and poverty gap measures weighted by the Gini coefficient of measures weighted by the Gini coefficient of the poor, giving:the poor, giving:

)1(10PP

s GPGPP


Sen IndexSen Index

The Sen index has the virtue of taking the The Sen index has the virtue of taking the income distribution among the poor into income distribution among the poor into account. However the index is almost never account. However the index is almost never used outside of the academic literature, perhaps used outside of the academic literature, perhaps because it is lacks the intuitive appeal of some because it is lacks the intuitive appeal of some of the simpler measures of poverty, but also of the simpler measures of poverty, but also because it “cannot be used to decompose because it “cannot be used to decompose poverty into contributions from different poverty into contributions from different subgroups”.subgroups”.


The Sen-Shorrocks-Thon indexThe Sen-Shorrocks-Thon index

The Sen index has been modified by others, The Sen index has been modified by others, and perhaps the most compelling version is the and perhaps the most compelling version is the Sen-Shorrocks-Thon (SST) index, defined as Sen-Shorrocks-Thon (SST) index, defined as

which is the product of the headcount index, the which is the product of the headcount index, the poverty gap index (applied to the poor only), poverty gap index (applied to the poor only), and a term with the Gini coefficient of the and a term with the Gini coefficient of the poverty gap ratios (i.e. of the Gn’s). This Gini poverty gap ratios (i.e. of the Gn’s). This Gini coefficient typically is close to 1, indicating great coefficient typically is close to 1, indicating great inequality in the incidence of poverty gaps.inequality in the incidence of poverty gaps.

)ˆ1(10PP

SST GPPP



ExampleExample

In 1996, 12.4% of the population of Quebec In 1996, 12.4% of the population of Quebec province (Canada) was in poverty. The poverty province (Canada) was in poverty. The poverty gap index, gap index, applied to the poor onlyapplied to the poor only, stood at 0.272. , stood at 0.272. And the Gini coefficient of the poverty gap ratios And the Gini coefficient of the poverty gap ratios was 0.924. Thus the Sen-Shorrocks-Thon index was 0.924. Thus the Sen-Shorrocks-Thon index was 0.065 (=0.124 × 0.272 × (1+0.924)).was 0.065 (=0.124 × 0.272 × (1+0.924)).

ExampleExample

In 1996, 12.4% of the population of Quebec In 1996, 12.4% of the population of Quebec province (Canada) was in poverty. The poverty province (Canada) was in poverty. The poverty gap index, gap index, applied to the poor onlyapplied to the poor only, stood at 0.272. , stood at 0.272. And the Gini coefficient of the poverty gap ratios And the Gini coefficient of the poverty gap ratios was 0.924. Thus the Sen-Shorrocks-Thon index was 0.924. Thus the Sen-Shorrocks-Thon index was 0.065 (=0.124 × 0.272 × (1+0.924)).was 0.065 (=0.124 × 0.272 × (1+0.924)).



One strength of the SST index is that it can help One strength of the SST index is that it can help give a good sense of the sources of change in give a good sense of the sources of change in poverty over time. This is because the index poverty over time. This is because the index may be decomposed intomay be decomposed into

which may be interpreted as, % change in SST which may be interpreted as, % change in SST index = % change in headcount index + % index = % change in headcount index + % change in poverty gap index ( among poor) + % change in poverty gap index ( among poor) + % change in (1+Gini coefficient of poverty gaps).change in (1+Gini coefficient of poverty gaps).

)ˆ1ln(lnlnln 10PP

SST GPPP



In plain English, this allows us to decompose In plain English, this allows us to decompose poverty into three aspects: are there more poor? poverty into three aspects: are there more poor? are the poor poorer? and is there higher are the poor poorer? and is there higher inequality among the poor?inequality among the poor?


Time taken to exitTime taken to exit

It may be useful to show how long it would take It may be useful to show how long it would take for the average poor person to exit poverty at for the average poor person to exit poverty at different potential economic growth rates when different potential economic growth rates when thinking about poverty reduction strategies.thinking about poverty reduction strategies.



For the jth person below the poverty line, the For the jth person below the poverty line, the expected time to exit poverty, i.e. to reach the expected time to exit poverty, i.e. to reach the poverty line, if consumption grows at a positive poverty line, if consumption grows at a positive rate g per year is:rate g per year is:

z = poverty linez = poverty line

xxjj = consumption of the poor = consumption of the poor

.)ln()ln(

g

xzt

jjg



Figure 4 shows the average time it would take Figure 4 shows the average time it would take to raise the consumption level of a poor person to raise the consumption level of a poor person in Cambodia to the poverty line, for various in Cambodia to the poverty line, for various hypothetical growth rates. It is assumed that this hypothetical growth rates. It is assumed that this growth rate is continuous, is in real terms, and is growth rate is continuous, is in real terms, and is distributionally neutral among the poor.distributionally neutral among the poor.


Figure 4.4: Average exit time for povertyFigure 4.4: Average exit time for poverty

0

5

10

15

20

25

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Growth Rate

Ye

ars

0

5

10

15

20

25

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Growth Rate

Ye

ars



Despite the potency of economic growth, it will Despite the potency of economic growth, it will take more than just growth to rapidly improve take more than just growth to rapidly improve the lives of the very poor. the lives of the very poor.

The expected time to exit poverty for those The expected time to exit poverty for those people who are so poor that they are below the people who are so poor that they are below the food poverty line in Cambodia – i.e. they cannot food poverty line in Cambodia – i.e. they cannot even afford enough food, even if they were to even afford enough food, even if they were to devote all their consumption spending to food – devote all their consumption spending to food – is more than 15 years, even at a three percent is more than 15 years, even at a three percent continuous annual growth rate. continuous annual growth rate.



Thus, targeted programs are needed to deliver Thus, targeted programs are needed to deliver benefits to the poor, in the form of benefits to the poor, in the form of improvements in their human and physical improvements in their human and physical assets and through interventions (e.g., assets and through interventions (e.g., infrastructure, markets) that improve the returns infrastructure, markets) that improve the returns they get from those assets.they get from those assets.


Other measuresOther measures

First distribution sensitive poverty measure was First distribution sensitive poverty measure was proposed by Watts (1968), and takes the form:proposed by Watts (1968), and takes the form:

Following Atkinson (1987), one can characterize Following Atkinson (1987), one can characterize a general class of additive measures, a general class of additive measures, encompassing W, the FGT (Foster, Greer and encompassing W, the FGT (Foster, Greer and Thorbecke) class of measures, and some other Thorbecke) class of measures, and some other measures as taking the following form:measures as taking the following form:

N

i iyz

NW

1

)log(1

N

iiyzpN

P1

),(1


Other measuresOther measures

where p(z, ywhere p(z, yii) is the individual poverty ) is the individual poverty measure, taking the value zero for the non-poor measure, taking the value zero for the non-poor (y(yii>z) and some positive number for the poor, >z) and some positive number for the poor, the value of which is a function of both the the value of which is a function of both the poverty line and the individual living standard, poverty line and the individual living standard, non-decreasing in the former and non-non-decreasing in the former and non-increasing in the latter. increasing in the latter.

poverty measures celia m. reyes introduction to poverty analysis nai, beijing, china

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