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This subsection considers a method of dealing with the heterogeneity of households. A common approach, given only the market and disposable incomes, is to make comparisons on the basis, not of observed actual income either of households or individuals, but of an artificial income construct which reflects the differences in the demographic structure of the households. The simplest way to convert household income into a measure of income per person is clearly to divide total income by the number of individuals in the household. But the view is widely taken that not all members of the household have the same consumption needs. Furthermore, there may be economies of scale within a household. The latter can arise because some goods (including some durables and goods like heating and lighting) may be 'public goods' within the household; thus those goods can be consumed simultaneously by several people. In addition, there may be economies from purchasing larger quantities of some goods.

Instead of dividing total household income by the number of people in the household (irrespective of their ages or gender), a measure of household size can be obtained by giving each person a separate weight, using a set of 'adult equivalence scales'. Such scales are typically imposed by the analyst and are based, loosely speaking, on perceived relative needs of different types of individual in the household and economies of scale within the household. However, value judgements cannot be avoided in the choice of scales. In practice they are often taken from other studies, often relating to other countries, without consideration of their rationale. Equivalence scales are also implicit in the tax and transfer system, but of course it could not be implied that they are consciously used (and the system is typically made up of a range of overlapping benefits).[6]

A simple but very flexible adult equivalence scale is the following, where, na and nc denote respectively the number of adults and children in the household, and m is the adult equivalent size of the household:[7]

Here θ and a ≤1 are parameters reflecting the relative 'cost' of a child and economies of scale respectively. Using (1), with θ = 0.5 and a = 0.8, the equivalent sizes of the four hypothetical households are respectively: 1; 1.74; 2.08; and 2.41.

Having obtained the adult equivalent size of each household, it is then a simple matter to obtain the total income per adult equivalent person. The resulting gross and disposable income per adult equivalent for each household is shown in Table 2. It can be seen that the movement from market income per adult equivalent to disposable income per adult equivalent involves a 'reranking' of households 2 and 3, if the households are ranked in ascending order.

Table 2 - Household Income per Adult Equivalent Person
Household m Gross income/m Disposable income/m
1 1 100 80
2 1.74 57.5 46.0
3 2.08 48.1 48.1
4 2.41 41.5 37.34

The further assumption is made that each member of the household receives the income per adult equivalent person; this is the new welfare metric. It is clearly an artificial contruct. Comparisons then depend on the choice of unit of analysis in combination with this welfare metric. It turns out that this choice is not as straightforward as has often been assumed. In fact, three further pairs of distributions may be considered. First, comparisons can be made using the household as the basic unit of analysis (as with the first two distributions considered in the previous subsection): this approach compares

[100, 57.5, 48.1, 41.5]

with

[80, 46.0, 48.10, 37.34].

However, the rationale for this choice of unit is not entirely clear.

Second, perhaps the simplest and most natural choice is to make comparisons using the individual as the basic unit of analysis. This compares the distribution

[100, 57.5, 57.5, 48.1, 48.1, 48.1, 41.5, 41.5, 41.5, 41.5]

with

[80, 46.0, 46.0, 48.1, 48.1, 48.1, 37.34, 37.34, 37.34, 37.34].

Here the elements are ordered by taking each household and individual in turn from Table 1. Again it can be seen that some reranking of individuals is involved when comparing the two distributions: this is discussed further in Section 3 below.

When using the individual as the unit of analysis, each person 'counts for one' irrespective of the household to which they belong. Inequality remains unchanged when one person is replaced by another person with the same metric (income per adult equivalent) but belonging to a different type of household. It thereby satisfies an 'anonymity principle'. However, it does not necessarily satisfy the 'principle of transfers'.[8] This principle is the inequality-disliking value judgement which takes the view (in the context of homogeneous individuals) that an income transfer from a richer to a poorer individual (which leaves the relative rank of the two people unchanged) is judged to reduce inequality and represent an improvement.[9] But if rich large households are highly efficient at generating welfare (in terms of the choice of this metric), given large economies of scale, it is possible, when using the individual as unit, for evaluations to be inequality-preferring.

A third possibility uses the equivalent adult as the income unit. This artificial income unit is thus combined with its corresponding artificial income measure, income per adult equivalent. In this case there are not necessarily integer numbers of equivalent adults (except for the single-adult, who is household number 1 in Table 1) and the distributions cannot be written simply as vectors. Thus the equivalent adult size must be treated as a household weight in obtaining inequality or other measures. To illustrate this case, the arithmetic mean gross adult equivalent income per adult equivalent person, denoted ȳ, is:

The use of the artificial equivalent adult as the unit of analysis means that the income unit and the income concept are treated consistently. Each individual's contribution to inequality depends on the demographic structure of the household to which that individual belongs.[10] Thus an adult in a one-person household 'counts for one'. But an adult counts for 'less than one' (has a weight less than 1) when placed in a multi-person household.

Importantly, the use of this income unit is consistent with the principle of transfers, described above. This can be useful because there are general results linking this value judgement to Lorenz curves, which are widely used to depict distributions. For an individual income distribution, first arrange individuals in ascending order, that is from lowest to highest income. The Lorenz curve plots the cumulative proportion of people (on the horizontal axis) against the cumulative proportion of total income (on the vertical axis of the diagram).

Consider the Lorenz curves of two distributions which have the same arithmetic mean income. Suppose the Lorenz curve of one distribution, say A, lies everywhere inside the other distribution, say B: that is, A's curve is closer to the upward sloping diagonal line of equality which arises if all incomes are equal. One way that distributions can be evaluated is as follows. For a distribution (x1,x2,…,xn), suppose the evaluation function – representing the value judgements of an independent judge - takes the form, , where U(x1) is a concave function representing the contribution of individual i's income to W. The concavity of U reflects adherence to the principle of transfers and the degree of concavity reflects the extent of aversion to inequality.[11] It has been established that all functions of this general kind would judge the first distribution to be better than the second in that it gives a higher value of W.[12] This result is true irrespective of the precise extent of aversion to inequality.

If the arithmetic means of the two distributions differ, the same result applies instead to the concept of the Generalised Lorenz curve: this plots the product of the proportion of total income and the arithmetic mean income against the corresponding proportion of people. Thus the vertical axis of the Lorenz curve is 'stretched' by an amount depending on the arithmetic mean.

Importantly, it cannot be assumed that comparisons are insensitive to the choice of income unit. Indeed, it is quite possible for a tax reform to be judged differently, changing inequality and welfare comparisons in opposite directions, when using the individual and the equivalent adult as income units.[13]

The discussion has so far been in terms of distributions of market and disposable incomes. Some household surveys contain detailed information about household expenditures, and this can be used to compute an additional metric, that of disposable income after the deduction of indirect taxes. If the indirect tax system has considerable selectivity, this task is complicated by the need for detailed expenditure data for each category. But if there is a broad-based goods and services tax (such as a value-added tax), perhaps combined with limited excises (for example, on tobacco, alcohol and petrol), the allocation is less complex.[14] Hence additional distributions can be produced in terms of a welfare metric described as 'income after direct taxes and transfers and after indirect taxes'. However, no new basic issues arise in terms of choice of income unit or equivalence scale. For this reason, this metric is not considered separately here.

### Notes

• [6]For an example of the calculation of implicit scales, see van de Ven and Creedy (2005).
• [7]This form was suggested by Cutler and Katz (1992) and Jenkins and Cowell (1994). For an analysis of a wide range of scales using this formula, see Creedy and Sleeman (2005).
• [8]This was first pointed out by Glewwe (1991).
• [9]Although the implications of adopting this principle have been widely investigated, it is important always to keep in mind that it is a value judgement.
• [10]Its use was first suggested by Ebert (1997). For detailed analysis of the choice of income unit, see Shorrocks (2004).
• [11]An evaluation function of this type is commonly referred to as an additive, individualistic and Paretian 'social welfare function’. Despite the name, it does not represent society’s views, but those of a single person who is not part of the distribution but who is making the evaluation.
• [12]This result was established by Atkinson (1970).
• [13]Examples are given by Decoster and Ooge (2002) and Creedy and Scutella (2004).
• [14]For example, if x denotes expenditure by an individual and v is the tax-exclusive indirect tax rate, then the corresponding tax-inclusive rate is v/(1+v) and expenditure after the deduction of tax is x/(1+v).
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