Full text: The Personal Distribution of Income

19 
extent an inversion of the other; in addition, each will be 
influenced by the dispersion of the values round the other 
regression line. 
Thus . . each of the regression lines will 
represent a compromise between the two underlying relations 
the weight of them being different in the one and the other 
regression line. Ho regression line therefore will be 
a true reflection of an underlying causal (or rather stochastic ) 
relation. We shall have a better chance of understanding the 
meaning of joint size distributions of this type if we regard 
them as-residues of a growth process. Set us therefore return 
to the allometric law. As far as its relation to the joint 
distribution wealth-income is concerned we have to make 
two observations: 
l) If the regression line income on wealth could 
be regarded as an expression of the allometric law then, 
as it will be remembered, the regression coefficient 
is the ratio of the two Pareto,coefficients of the income 
i 
and the wealth distribution. 1 
2) Following up the idea that wealth can be 
explained from saving over a certain time and saving can 
be explained from income, taking saving propensity as given, 
we can dBxive the distribution of wealth from that of income 
in much the same way as the other way round: 
We explain the saving distribution as a convolution 
of/the income distribution and of thejdistribution of the 
propensity to save ( savings ratio ): 
l'(s) = q(y) * g ( Y7 s ), v ‘> 1 (9) 
and the wealth distribution a s a convolution of this and 
the time the saving has accumulated ( which will be finite 
in the case of earned income but not necessarily for unearned 
income ): 
q ,f (w) = q'(s) * h ( s - w ) (10) 
From this wealth distribution we should by means of the 
original transformation (6) come back to the income 
distribution
	        

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