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

4  Equivalence Scales and Direct Taxation

Previous sections have examined inequality and poverty measures based on the distribution of total household expenditure. The present section considers the role of equivalence scales and the unit of analysis in the context of the redistributive effect of direct taxation.

Aronson and Lambert (1994) decomposed the redistributive effect of taxation, , in terms of the Gini coefficient, into three components, which they describe as the vertical, horizontal, and reranking effects. The vertical effect measures the progressivity of the effective tax schedule, which incorporates no horizontal or reranking effects and is derived from the actual tax schedule by allocating to each individual the average tax paid by the respective pre-tax equals. The horizontal effect relates to the unequal treatment of equals, and reranking captures the presumably unintended inequitable treatment of unequals by the tax system.[21] The concept of reranking is therefore quite different from that discussed earlier, which was interpreted in terms of a negative correlation between equivalent income and household size.

Using slightly different notation from above, suppose that the tax and transfer system is such that post-tax expenditure, , is given by . Divide the population into groups. Within each group individuals have similar (or ‘near equal’) pre-tax values of , for . Groups are ranked in ascending order. Aronson et al (1994) showed that the reduction in the Gini measure of inequality, , is given by:


where , the redistributive effect, is the difference between Gini measures of pre- and post-tax incomes:


The vertical redistribution, , is the difference between the Gini measure of pre-tax income and the between-group Gini measure of post-tax income, obtained when each individual’s pre-tax income is adjusted by the average tax paid by their respective pre-tax equals (that is, each individual is given the average post-tax income in the group). Hence:


The horizontal inequity, , is given by:


where and is the product of the population and income shares of group and is the within-group Gini inequality measure. Finally, is the measure of reranking, given by:[22]


where is the concentration measure of post-tax income, obtained as a Gini-type measure, but with post-tax incomes ranked by .

In practice few exact pre-tax equals are observed in survey data, so the problem arises of selecting an appropriate group class width. This issue was examined by van de Ven et al (2001), who showed that the measured vertical effect initially increases as the class width is increased, and then falls after reaching a maximum. This suggests a strategy whereby the class width used to combine individuals into groups of near-equals is chosen as the value that maximises the estimated vertical effect. The reranking measure, , can be obtained directly using the ungrouped values and is therefore not affected by the choice of class width. The horizontal effect can then be obtained as a residual using .

In the present context the issue involves the effect of alternative adult equivalence scales on the various components of redistribution. In order to concentrate on direct taxes and transfers, the following results were obtained using information on the pre-tax annual incomes and disposable incomes of households in the 2001 Household Economic Survey.

Figure 28 shows, for four values of , the variation in reranking, expressed as a percentage of the redistributive effect, when individuals are regarded as the basic unit of analysis (income per adult equivalent is weighted by the number of individuals in the household). These results show that the degree of reranking is relatively low. The horizontal inequity measures were found to be negligible, being in virtually all cases less than 0.1 percent of redistribution; for this reason they are not reported here. The profiles of reranking with variations in are U-shaped for the lower value of and become J-shaped for higher values. For the range of values displayed here, reranking is lower for lower , the difference increasing for the higher . However, as is reduced further, below the lowest profile shown, the degree of reranking begins to increase: hence a reranking minimising set of equivalence scales exists for which .

Figure 29 shows a similar pattern of reranking in the case where the number of equivalent adults is used as the weight for each household, that is, the basic income unit is the ‘equivalent adult’. While the variations in reranking are similar to those found in Figure 28, the values, as a percentage of redistribution, are systematically slightly lower.

Reranking of unadjusted incomes is of course a deliberate aim of the tax and transfer system. This is precisely because the size and composition of households are considered as relevant non-income characteristics; value judgements about the desirable redistribution arising from taxes and transfers are closely linked with such differences. In considering the tax ‘treatment of equals’, the equivalence scales determine the meaning attached to ‘equality’. It may be suggested that, since reranking works against redistribution, an implicit set of equivalence scales, reflecting the value judgements of policy makers, is found as the set that minimises reranking.[23] Such reranking cannot be expected to be zero, given that some can also arise as the result of government policy that is tangential to equity objectives. For example, unemployment benefits may be designed to encourage labour market participation, or certain types of income may be treated differently on efficiency grounds. It was mentioned above that a reranking-minimising set of scales exists: where individuals are used as weights, this arises for at around 0.05 and for at around 0.45. These values, as shown below, are quite different from those generally used in empirical studies of inequality and poverty.

Figure 28 – Reranking with Individuals as Income unit
Reranking with Individuals as Income unit.
Figure 29 – Reranking with Equivalent Adults as Income unit


  • [21]The treatment of Aronson and Lambert (1994) was in terms of income taxation, but the method has been applied to indirect taxes by Decoster et al (1997a, b). An application of the approach suggested below to indirect taxes is in Creedy (2002) and Creedy and van de Ven (2001a) which apply the decomposition in a lifetime context.
  • [22]See Atkinson (1979) and Plotnick (1981)
  • [23]van de Ven and Creedy (2003) found that the implicit scales are likely to be in the region of the reranking-minimising scales, though they may not be precisely the same.
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