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Adult Equivalence Scales, Inequality and Poverty in New Zealand - WP 04/21

2  Alternative Concepts and Measures

This section describes the equivalence scales, along with the inequality and poverty measures used in later sections. Subsection 2.1 describes the functional form of the equivalence scale. This function, while essentially pragmatic, is highly flexible and depends on only two easily-interpreted parameters. Subsection 2.2 compares the use of alternative units of analysis, and reviews the difficulty arising from the fact that for heterogeneous populations the basic equity principle involved in the ‘principle of transfers’ is not consistent with an alternative principle of ‘Pareto indifference’ or anonymity. Subsection 2.3 presents the Atkinson inequality measure and its associated social welfare function, while subsection 2.4 describes the poverty measures used.[2]

2.1  Adult Equivalence Scales

Let denote the income of the th household, for . The number of individuals in the household is , while the household’s demographic structure is described by the vector . This vector provides the number of individuals in various demographic groups based on age and gender classifications. Using these definitions, the adult equivalent size of household may be expressed as:[3]


This size is normalised so that . Household income is adjusted to obtain the equivalent income or ‘living standard’, given by:


A household consisting of one adult with an income of therefore has the same ‘living standard’ as an n-person household with an income of . Further progress requires the form of to be specified

If there are individuals of demographic type in the th household, the adult equivalent size may be written as:


The term is regarded as a measure of economies of scale within the household. This formulation is an extension of the simple form, , used by Buhmann et al (1988) and Coulter et al (1992) and modified by, for example, Cutler and Katz (1992), Banks and Johnson (1994) and Jenkins and Cowell (1994) who distinguished the number of adults, , and children, , such that:


The parameter, , measures the size of children relative to adults, while again reflects economies of scale in consumption. This two-parameter form is used below.[4]

2.2  Units of Analysis

A number of empirical studies have taken the household itself as the basic unit of analysis, usually with little justification. This approach simply assigns to each household, the equivalent income,, of that household and therefore makes no further allowance for the household’s demographic structure. In calculating measures of inequality and poverty, the equivalent incomes of households are all each given the same weight of . While this approach appears to have little rationale, it is included here for comparative purposes.

An alternative approach is to use the ‘adult equivalent person’ as the unit of analysis.[5] This approach assigns to each of the adult equivalent persons in household , an equivalent income of . The income concept and the unit of analysis are treated consistently, ensuring that each individual’s contribution to inequality and poverty depends on the demographic structure of the household to which they belong. An adult in a one-person household for example will ‘count for one’, whereas the same adult in a multi-adult household, will count for ‘less than one’.

This approach also satisfies the basic equity principle, associated with the principle of transfers, such that a transfer of income from a poorer to a relatively richer household, which leaves the position of the richer household unchanged, causes inequality to rise. The fulfilment of this principle enables Lorenz and Generalised Lorenz curve analyses to be conducted from the resulting distribution.[6]

A third approach is to treat the individual as the basic unit of analysis.[7] This approach assigns to each of the individuals in household an equivalent income of . Every individual effectively ‘counts for one’, irrespective of the demographic nature of the household to which they belong. This approach consequently has the property of anonymity, in that inequality and poverty measures remain unchanged when one individual in the population is replaced by another individual who has the same living standard but belongs to a different demographically structured household. This property was called the ‘compensation principle’ by Shorrocks (1997) and the ‘Pareto indifference principle’ by Decoster and Ooghe (2002).

The individual unit of analysis does not in general satisfy the equity principle (of transfers). As shown by Glewwe (1991), a transfer of income from a poor to a relatively richer (and larger) household may actually reduce inequality and raise social welfare.[8] Despite being based on individuals, the anonymity of the individual income unit can lead to a preference for inequality, with the presence of economies of scale causing a large household to be regarded as being ‘more efficient’ at generating welfare.

An important implication is that in the context of heterogeneous populations, the basic equity principle inherent in the principle of transfers and the concept of Lorenz dominance (whereby one Lorenz curve lies unambiguously closer to the diagonal of equality) are no longer equivalent. This equivalence is a fundamental component of welfare analysis for homogeneous populations. Consequently, the choice between individuals and adult equivalents as the basic unit of analysis involves a choice between two incompatible value judgements. They can in principle lead to opposite conclusions about the effects on inequality of a tax policy change.[9]


  • [2]In fact a wider range of inequality measures were obtained, including generalised Gini measures. The results are not reported here as the behaviour is similar to that of the Atkinson measures.
  • [3]The scales are considered to be independent of prices.
  • [4]It has been used recently by Trigger (2003) to examine poverty, and by Creedy and Scutella (2003) where the emphasis was on the income unit.
  • [5]This approach was proposed by Ebert (1997).
  • [6]Despite explicitly not treating individuals as the unit, but instead using adult equivalents, this actually leads to a recommendation for equal standards of living; see Ebert (1997, p.242).
  • [7]Jorgenson and Slesnick (1984) and Slesnick (1994) use this method, as does Glewwe (1991), who dismisses the use of adult equivalents in a footnote (1991, p.213). It is also preferred by Shorrocks (1997), Danziger and Taussig (1979) and Ringen (1991).
  • [8]Transfers of money do not correspond to transfers of ‘living standard' units between individuals. Glewwe (1991, p.213) used a numerical example with three households. Decoster and Ooghe (2002, pp.3-4) also construct some illustrative examples using three persons.
  • [9]Shorrocks (1997) suggested that if concern is with equity, the use of adult equivalents is recommended, whereas if concern is primarily with social welfare, individuals should be the basic income unit. This places the disinterested economist in the position of being required to report results using both approaches.
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