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

Appendix: Covariance Between Adult Equivalent Income and Number of Persons

This appendix examines the covariance between household size (the number of persons, ) and equivalent income , in the simplified case where . First, suppose that income and the number of persons in the income unit are jointly log-normally distributed as:

(1)    

Further results require the following general properties of the lognormal distribution. If is :

(2)     

The power is distributed as;

(3)    

and for two variables jointly distributed as , the ratio is distributed as:

(4)    

The covariance between household size and equivalent income is, by definition:

(5)    

Using the three properties give above, average adult equivalent income is:

(6)    

A similar result holds for . It can be shown, after some manipulation, that:

(7)    

where:

(8)    

and:

(9)    

Thus the covariance is positive if , that is if . Hence there is a positive correlation between adult equivalent income and the number of adults if:

(10)    

This condition is therefore simply interpreted in terms of the size of the regression coefficient in the linear regression relationship between logarithms of household size and income. It is clearly equivalent to the elasticity condition mentioned by Coulter et al (1992).

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