There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper proposes a new methodology for multidimensional poverty measurement consisting of an identification method ρk that extends the traditional intersection and union approaches, and a class of poverty measures Mα. Our identification step employs two forms of cutoff: one within each dimension to determine whether a person is deprived in that dimension, and a second across dimensions that identifies the poor by ‘counting’ the dimensions in which a person is deprived. The aggregation step employs the FGT measures, appropriately adjusted to account for multidimensionality. The axioms are presented as joint restrictions on identification and the measures, and the methodology satisfies a range of desirable properties including decomposability. The identification method is particularly well suited for use with ordinal data, as is the first of our measures, the adjusted headcount ratio. We present some dominance results and an interpretation of the adjusted headcount ratio as a measure of unfreedom. Examples from the US and Indonesia illustrate our methodology.

Counting and Multidimensional Poverty Measurement

https://www.ophi.org.uk/wp-content/uploads/OPHI-wp32.pdf

Counting and multidimensional poverty measurement

Poverty and Famines: An Essay on Entitlement and Deprivation

Many authors have insisted on the necessity of defining poverty as a multidimensional concept rather than relying on income or consumption expenditures per capita. Yet, not much has actually been done to include the various dimensions of deprivation into the practical definition and measurement of poverty. Existing attempts along that direction consist of aggregating various attributes into a single index through some arbitrary function and defining a poverty line and associated poverty measures on the basis of that index. This is merely redefining more generally the concept of poverty, which then essentially remains a one dimensional concept. The present paper suggests that an alternative way to take into account the multi-dimensionality of poverty is to specify a poverty line for each dimension of poverty and to consider that a person is poor if he/she falls below at least one of these various lines. The paper then explores how to combine these various poverty lines and associated one-dimensional gaps into multidimensional poverty measures. An application of these measures to the rural population in Brazil is also given with poverty defined on income and education.

The Measurement of Multidimensional Poverty

This paper introduces a new approach to rank and measure socioeconomic inequality in health. A novel feature of the approach is the use of an income-health matrix that relates socioeconomic class with health status; each row of the matrix corresponds to a socioeconomic class and contains the respective probability distribution of health. By invoking a monotone assumption on the income-health matrix, which is a direct translation of the well-known hypothesis of social gradients in health outcomes, we derive a set of welfare-dominance and inequality-dominance conditions for ranking health distributions. Unlike other existing health inequality measures, our conditions require no cardinal specification of ordinal heath data and, hence, are robust. We then apply the dominance conditions to compare socioeconomic health inequality in the US and Canada using the newly released JCUSH data.

A new approach to measure socioeconomic inequality in health

This paper investigates the possibility of increasing the ordering power of additively separable poverty measures beyond the condition of second degree stochastic dominance by considering third degree stochastic dominance. For a fixed poverty line, the ordering power can be significantly enhanced by using the third degree criterion. For a range of poverty lines, the marginal power of the third degree criterion over the second degree depends critically upon the lower bound of the range; if the lower bound poverty line is arbitrarily close to zero, the two criteria coincide. The implications of a strong version of the transfer sensitivity axiom are also considered.

On the power of poverty orderings

http://darp.lse.ac.uk/papersdb/Zheng_(JET2000).pdf

Minimum Distribution-Sensitivity, Poverty Aversion, and Poverty Orderings☆

BEGINNING WITH THE WORK OF A. K. SEN (1976) poverty research has made extensive use of axioms to construct and evaluate distribution-sensitive poverty measures. An index is distribution-sensitive if it satisfies Sen's (1976) weak transfer axiom; poverty decreases when income is transferred from a high income poor person to a low income poor person. A distribution-sensitive measure is appealing because it reflects the extent of deprivation among the poor and incorporates aspects of poverty that are ignored by standard measures such as the headcount ratio and poverty gap ratio. In addition to the weak transfer axiom, a number of other axioms that have considerable intuitive appeal when considered in isolation have been identified. However, several researchers (e.g., Thon (1979), Kundu and Smith (1983), and Donaldson and Weymark (1986)) have shown that in some cases it is impossible to devise a poverty index that is consistent with the joint implications of otherwise "reasonable" axioms. Foster (1984, p. 233) interprets the difficulties caused by the combined effects of two or more axioms ". . . as a warning not to measure too many aspects of poverty at the same time." As a consequence, it is now generally recognized that poverty axioms can have joint effects and it is important to explore the implications of such influences. In a comprehensive investigation of subgroup consistent poverty measures, Foster and Shorrocks (1991) examine the interaction between the scale invariance and translation invariance axioms.2 The first axiom requires a poverty index to be unaffected by the doubling of all incomes and the poverty line. The second axiom requires a poverty index to be unaffected by equal increments to each income and to the poverty line.3 Poverty indices that satisfy the first axiom are considered to be "relative poverty indices," while those satisfying the second are "absolute poverty indices." If a distribution-sensitive index satisfying both axioms could be found it would provide a single measure of relative and absolute poverty and have considerable intuitive appeal in poverty measurement. But, one implication of Foster and Shorrocks' (1991) work (Proposition 6) is that for the class of poverty measures they consider, a distribution-sensitive poverty index that is both relative and absolute cannot exist. This note follows Foster and Shorrocks (1991) but concentrates on the implications of the joint effects of scale and translation invariance. We drop the assumption of subgroup consistency and derive a proposition that has several important implications. First, for all poverty indices, only those relying upon simple headcounts of the poor satisfy both the scale and translation invariance axioms. This result clearly illustrates the strong restrictions imposed by combining scale and translation invariance with other standard axioms; there can be no distribution-sensitive poverty index that is relative and absolute. Second, the result provides a simple characterization of the entire class of the headcount-related poverty indices (defined below). Third, the result leads to a direct proof of the result

Can a poverty index be both relative and absolute

The paper develops simple statistical test procedures for generalized decomposable poverty measures. It is shown that the estimates of the decomposable poverty indices, including the overall and various subgroup poverty indices, have (asymptotic) jointly normal distributions. The full variance-covariance structure is derived and can be estimated consistently without prior specification of the population density underlying the sample data. The (overall/decomposed) poverty indices can thus be used as tools for statistical inference instead of simply as descriptive statistics. We illustrate the test procedure by examining the contributions of whites and non-whites to total US poverty in 1975 and 1985.

Buhong Zheng

Papers

A new approach to measure socioeconomic inequality in health

On the power of poverty orderings

Minimum Distribution-Sensitivity, Poverty Aversion, and Poverty Orderings☆

Can a poverty index be both relative and absolute

Statistical inference and decomposable poverty measures