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

3  Empirical Analysis for New Zealand

This section analyses the sensitivity of New Zealand’s inequality and poverty measures to the parameters of the equivalence scale, and to the chosen unit of analysis. The analysis is conducted using data on the weekly expenditure of households, as opposed to incomes. The use of expenditure data may be thought to eliminate to some extent the effects of short term variations in income; on the use of expenditure rather than income data, see Blundell and Preston (1994, 1997) and Attanasio and Japelli (1997).[14] The main emphasis of the present paper is to consider the sensitivity of measures to alternative scales and units of analysis, rather than attempting to provide an exhaustive study of inequality and poverty.[15] From the Household Economic Survey, household expenditure data for the years 1995, 1996, 1997, 1998 and 2001, were adjusted to 2001 prices using the consumer price index (CPI).[16] The surveys were then pooled to form one large data base containing the weekly total expenditure of each household along with information about household structure.

3.1  Inequality Measures

The results of the sensitivity analysis for the Atkinson inequality measure are displayed in Figures 1 to 6, which show inequality against , the economies of scale parameter, for four values of , the weight attached to children. Figures 1 to 3 use the individual as the unit of analysis, while Figures 4 to 6 use the equivalent adult. For both types of income unit, three levels of aversion to inequality are considered. Inequality obviously increases as the degree of inequality aversion,, is raised.

Coulter et al (1992) found that increasing the value of  has two opposing effects on measures of inequality. The first is the concentration effect whereby  is inversely related to inequality. As the value of  is increased from low values, economies of scale are reduced and as a result equivalent income will fall proportionately more for relatively larger households. It is known that income and total expenditure are positively correlated with household size. This implies that relatively richer households incur proportionately greater falls in equivalent income. The rise in  therefore has an equalising effect.

Over low values of , Figures 1 to 6 all display the inverse relationship produced by the concentration effect. However, over higher values of , inequality is seen to rise with , producing a U-shaped inequality profile. The positive relationship between  and inequality may be attributed to a reranking effect, whereby the rank-order of households (when ranked by equivalent income) changes. The proportionately larger fall in equivalent income for the larger households, as  increases from a relatively high value, eventually leads to the kind of reranking identified by Coulter et al (1992). The figures show that over higher values of , the reranking effect dominates the concentration effect, thereby causing inequality to rise. As observed by Jenkins and Cowell (1994), the reranking effect increases with the parameter  and as a result, the inequality profiles for higher values of  show much greater curvature.

The phenomenon of reranking is in fact closely related to the situation in which the correlation between equivalent income and household size becomes negative. Coulter et al (1992, p.1073) state that the reranking occurs, for the case where , when the scale parameter  exceeds the inverse of the elasticity of household size with respect to income.[17] Although total income, , is known to be positively correlated with household size, , the correlation between equivalent income, , and household size, , is parameter dependent. The appendix derives the condition under which the correlation coefficient, , between  and  is negative, for the case where income  and household size are jointly lognormally distributed and . This turns out to depend on the regression coefficient in the log-linear relationship, and so is precisely the same as the ‘elasticity’ condition mentioned above.

Inequality Sensitivity - Unit of Analysis: Individual

Figure 1 - Figure 1.
Figure 2 -
figure 2.  
Figure 3 -
figure 3.  

Inequality Sensitivity - Unit of Analysis: Equivalent Adult

Figure 4 -
Figure 5 -
Figure 6 -


  • [14]Trigger (2003) reviews alternative approaches to the ‘income’ concept.
  • [15]For example, in calculating the inequality measures, the unweighted sample observations are used. In a study in which precise levels were the main focus, it may be desirable to use the survey weights; however, the vast majority of these lie within a narrow range; see Creedy and Tuckwell (2003).
  • [16]Unfortunately no surveys were carried out in 1999 and 2000.
  • [17]They suggest that this finding follows from a result established by Kakwani (1980) on the relationship between Lorenz and concentration curves.
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