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On Economic Inequality$

Amartya Sen

Print publication date: 1973

Print ISBN-13: 9780198281931

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198281935.001.0001

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A.6.2 Classical Poverty Aggregation: Head Count and Income Gap

A.6.2 Classical Poverty Aggregation: Head Count and Income Gap

Source:
On Economic Inequality
Publisher:
Oxford University Press

Perhaps the most widely used measure of poverty is the so‐called ‘head‐count ratio’, which identifies the poverty of the community with the proportion of poor people (i.e., in the case of income poverty, the ratio of total population whose incomes fall below the poverty line). Let x be the vector of personal incomes for the community as a whole, and z the poverty‐line income. If the number of people with income less than (or equal to) z in x is given by q = q(x;z), and the number of people in that community is n = n(x), then the head‐count ratio H is simply q/n.78 Clearly, the head‐count measure H ignores the ‘depth’ as well as the ‘distribution’ of poverty. The marginally poor are counted the same as the truly destitute by this simplest of aggregation methods. While H is surely an important partial index of poverty, which along with other such (p.169) indices can tell us much about poverty, it is not by itself a convincing overall measure of poverty.

The problems with using the head‐count ratio as a unique aggregate index of poverty can be illustrated by the recommendations it offers to policy‐makers seeking to reduce poverty by a maximum amount given a fixed budget allocation.79 For any initial distribution, the solution algorithm is to ‘save first’ the most well‐off poor person, then save the second most well‐off, and so on, until the redistributive budget is exhausted. Indeed, if income could be extracted from the most destitute person and redistributed to the least destitute person just below (or at) the poverty line (to make her ‘cross’ the line), this would appear, in terms of the head‐count measure, as an efficacious method of reducing poverty. Clearly, the head‐count ratio H would need to be supplemented by additional information on the incomes of the poor.80

The ‘depth’ of a poor person's poverty can be measured by the extent of the ‘gap’ (zy i) between the poverty line z and the person's income y i. One can capture the overall ‘distance’ of the incomes of the poor by an aggregate gap measure, based on the total, or per capita, shortfall of poor people's incomes from the poverty line. When μp is the mean income of the poor population, the ‘income‐gap ratio’ I = (z − μp)/z reflects the average shortfall of the incomes of the poor expressed as a share of the poverty‐line income z. Gap measures add a second dimension to the picture of poverty, and they can be extremely useful in poverty evaluation. Indeed, they are the second most commonly used measures of poverty.

However, like the head‐count measure H, the gap measures too are best seen as partial indicators of poverty. The income‐gap ratio does not tell us how many people are poor (a subject on which H exclusively concentrates), and along with H, the income‐gap ratio I also ignores the distribution of income among the poor (in particular, how the total income gap is (p.170) divided among them). For example, if there is a regressive transfer of income from the most destitute person to one who is much richer but still below the poverty line (even after the transfer), then neither the income‐gap ration I, nor the head‐count ratio H, would record any change in the state of the poor, and yet the most deprived would have been made poorer still (benefiting a relatively richer person).

The limitations of the head‐count ratio and the income‐gap ratio, taken separately as well as jointly, led to the proposal of distribution‐sensitive measures of poverty. The particular measure proposed in Sen (1973c, 1976b) included distribution sensitivity through a principle of ‘relative equity’, which gives more and more weight per unit of the income shortfall of poorer and poorer persons.81

Notes:

(78) If we look at ‘distribution functions’, the head‐count ratio is H(F;z) = F(z), the distribution function evaluated at the poverty line (yielding the proportion of the population with incomes at or below the poverty line).

(79) See Bourguignon and Fields (1990) for similar analyses of a variety of poverty measures.

(80) It is, thus, remarkable that most empirical studies of poverty tend, still, to stop at the head‐count ratio.

(81) While the recent literature on distribution‐sensitive poverty measures has tended to respond—by following or extending or disputing—the proposals made in Sen (1973c, 1976b), this issue had also received attention in an earlier, but neglected, paper by Watts (1968).