Poverty Measures Celia M. Reyes Introduction to Poverty Analysis NAI, Beijing, China

1. Poverty Measures Celia M. Reyes Introduction to Poverty Analysis NAI, Beijing, China Nov. 1-8, 2005

2. Headcount Index • measures the proportion of the population that is in poverty • the headcount index estimates the percentage of the population living in households with per capita consumption below the poverty line • it thus measures the incidence of poverty

3. Poverty Indices: Headcount Index Formally, where N = total population (.) = an indicator function that takes on a value of 1 (poor); if the bracketed expression is true, and 0 (nonpoor) otherwise yi = expenditure z = poverty line

4. Headcount Index • the great virtue of the headcount index is that it is simple to construct and easy to understand • adequate measure of assessing overall progress in reducing poverty (though preferably always calculated for at least two poverty lines)

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

6. Headcount Index Example: There is greater poverty in country A but the headcount index does not capture this.

7. Headcount Index • As a welfare function, the headcount index violates the transfer principle – 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 “jump” in welfare, at about the poverty line, In practice, such a jump is not found.

8. Headcount Index • The headcount index is very simple to construct and easy to understand. • However, it does not indicate how poor the poor are, and hence does not change if people below the poverty line become poorer.

9. Headcount Index • The easiest way to reduce the headcount index is to target benefits to people just below the poverty line, because these 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.

10. Headcount Index • What we calculate using the headcount 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 that other members of the same household. • In reality, not all consumption is evenly shared across household members.

11. Philippines: Poverty incidence • We calculate poverty incidence among families and population. • Poverty incidence among families – proportion of families who are poor • Poverty incidence among population – proportion of individuals who are poor

12. Poverty Gap Index • A moderately popular measure of poverty 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. • This measure reflects the average distances of the poor below the poverty line so it gives a better idea of the depth of poverty.

13. Poverty Gap Index • This measure is also thought of as the cost of eliminating poverty (relative to the poverty line), since 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.

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

15. Poverty Gap Index • At the other extreme, one can consider the maximum cost of eliminating poverty, assuming that the policy maker 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 to the maximum cost with no targeting is simply the poverty gap index.

16. Poverty Gap Index • Thus, this measure is also an indicator of the potential saving to the poverty alleviation budget from budgeting. • A serious shortcoming of this measure is that it may not convincingly capture differences in the severity of poverty amongst the poor.

17. Poverty Gap Index Example: Consider 2 distributions of consumption for 4 persons. Poverty line = 3.01 • A distribution is (1,2,3,4) • B distribution is (2,2,2,4) Thus, headcount index is 0.75 and the poverty gap index is 0.25 in both cases.

18. Poverty Gap Index However, the poorest person in A has only half of the consumption of the poorest in B. One can think of B as being generated from A by a transfer from the least poor person (individual with ‘3’ in A) to the poorest. The poverty gap will be unaffected by such a transfer. • Thus, the main drawback of this measure is that it ignores inequality among the poor.

19. Poverty Gap Index To see this again, consider the following example: Poverty Gap Rates in A and B, assuming poverty line of 125 For both countries, the poverty gap rate is 0.10, it could be argued that country B has more serious poverty because it has an extremely poor member.

20. Poverty Gap Index In summary, The Poverty Gap Index is the average over all people, of the gaps between poor people’s 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.

21. Poverty Gap Index • This is 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.

22. Poverty Gap Index • 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.

23. Squared Poverty Gap or Severity Index • Unlike the head-count and poverty-gap indexes, the absolute value of the poverty severity index has no intuitive interpretation and is not easy to interpret. • For poverty comparisons, however, the key point is that a ranking of dates, places or policies in terms of P2 should reflect their ranking in terms of the severity of poverty.

24. Squared Poverty Gap or Severity 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.

25. Squared Poverty Gap or Severity Index Formally: The measure lacks intuitive appeal, because it is not easy to interpret and so it is not used very widely.

26. Squared Poverty Gap or Severity Index • It may be thought of as one of a family of measures proposed by Foster, Greer and Thorbecke (1984), which may be written as , (α ≥ 0) where α = measure of the sensitivity of the index to poverty z = poverty line xj = the value of expenditure per capita for the j-th person’s household Gj=z-xj = the poverty gap for individual j (with Gj=0 when xj>z)

27. Squared Poverty Gap or Severity Index When parameter =0, P0 is simply the head-count index. When α=1, the index is the poverty gap index P1, and when  is set equal to 2, P2 is the poverty severity index. For all  > 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). For  > 1 it also has the property that the increase in measured poverty due to a fall in one’s standard of living will be deemed greater the poorer one is.

28. Squared Poverty Gap or Severity Index • The measure is then said to be "strictly convex" in incomes (and "weakly convex" for =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.

29. Squared Poverty Gap or Severity Index • The measures of depth and severity of poverty provide complimentary information on the incidence of poverty. • Table 4.1 provides an example by Madagascar.

30. Table 4.1 Poverty Indices by Sub-groups, Madagascar, 1994 Source: Coudouel, Hentschel and Wodon (2001)

31. 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

32. Sen Index where P0 = headcount index μP = mean income (or exp) of the poor GP = 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:

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

34. 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 • 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 Gn’s). This Gini coefficient typically is close to 1, indicating great inequality in the incidence of poverty gaps.

35. 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)). The Sen-Shorrocks-Thon index

36. The Sen-Shorrocks-Thon index • 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 • 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).

37. The Sen-Shorrocks-Thon index • 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?

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

39. Time taken to exit For the jth person below the poverty line, the expected time to exit poverty, i.e. to reach the poverty line, if consumption grows at a positive rate g per year is: z = poverty line xj = consumption of the poor

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

41. Figure 4.4: Average exit time for poverty

42. Time taken to exit • Despite the potency of economic growth, it will take more than just growth to rapidly improve the lives of the very poor. • The expected time to exit poverty for those people who are so poor that they are below the food poverty line in Cambodia – i.e. they cannot even afford enough food, even if they were to devote all their consumption spending to food – is more than 15 years, even at a three percent continuous annual growth rate.

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

44. Other measures • First distribution sensitive poverty measure was proposed by Watts (1968), and takes the form: • Following Atkinson (1987), one can characterize a general class of additive measures, encompassing W, the FGT (Foster, Greer and Thorbecke) class of measures, and some other measures as taking the following form:

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