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3.1 Inequality Measures (continued)

However, the present empirical analysis allows also for variations in the weight attached to children. Figure 7 shows the correlation coefficient, , between and as varies, for four values of and for the case where the unit of analysis is the individual. The correlation is initially positive, but clearly falls as economies of scale are reduced, and it eventually becomes negative. Furthermore, the correlation falls faster for higher values of the parameter . This is turn causes the correlation to turn negative (introducing reranking) earlier, so that the profile of inequality turns up earlier for higher values of . This is clearly reflected in Figures 1 to 6.

Figure 7 – Correlation Between Equivalent Income, and Household Size,

The correlation also affects comparisons between the inequality profiles for different income units. For any given household, which leads the equivalent adult unit to give proportionately more weight to smaller households when compared with the individual unit. As rises and the correlation between and falls, smaller households enjoy increasingly larger equivalent incomes relative to larger households. Consequently, as rises, inequality measures based on the equivalent adult unit fall relative to those based on the individual unit. This is clearly seen in Figure 8, which shows the inequality measures for the equivalent adult and individual units in the case of an inequality aversion coefficient of 1.2 when the weight attached to children is .

Figure 8 – Inequality Measures, and

Another finding from the sensitivity analysis is that, for all values of and for a given unit of analysis, inequality is positively related to the weight attached to children, . This result is independent of the inequality aversion coefficient and the chosen unit of analysis. This feature was suggested by Banks and Johnson (1994), but Cowell and Jenkins (1994, pp. 892-893) argued that the relationship need not necessarily be monotonic, although it ‘may be difficult to characterise precisely from theoretical analysis alone’. However, some insight may be obtained as follows. The adult equivalent size of a household may be written, where for convenience subscripts for the household have been omitted, as:

(12)    

where:

(13)    

Assuming that the number of adults and the number of children are independent of each other, the variance of is described by:[18]

(14)    

Hence rises with .

Taking logs of equation gives and the variance of is thus:

(15)     

Movements in and are monotonic, so the rise in and hence in as a result of an increase in leads to increase. It is then necessary to consider the effect of such an increase on the dispersion of equivalent income, given by . Taking logs gives . The variance of logarithms of equivalent income is therefore:

(16)    

Hence, the dispersion, measured by the variance of logarithms of equivalent income, , rises with , which has been seen to rise with . But the covariance term is also affected positively by . Hence, although a positive effect has been found using the present data, it is possible in principle, over some range of parameter values, for the dispersion of equivalent income to fall as the weight attached to children increases.

Notes

  • [18]Allowing for a positive correlation strengthens the effect of .
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