Full text: The Personal Distribution of Income

extent an inversion of the other; in addition, each will be 
influenced by the dispersion f the values r^und the other 
recession line. 
Thus ltfiT. each-.*# 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 ither 
regression line. No 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. Cet 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: 
1) If the regression line income on wealth could 
be regarded as an expression of the allometric law then, 
as it will he remembered, the regression coefficient 
is the rati ■ of the tw P&ret ja efficients if the inc me 
and the wealth distributi n 
2) Foil wing up the idea that wealth can be 
explained fr m saving ver a certain time and saving can 
be explained fr m inc me taking saving r pensity as given, 
we can deirive the distributi n f wealth fr m that of income 
in much the same way as the ther way r undi 
V/e explain the saving distributi n as a c nvnluti n 
ofjthe income distribution and >f thesistribution f the 
propensity to save ( savings ratio ): 
q«(s) - q(y) * g ( Vy - s ), (9) 
and the wealth distribution a s a convalution 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 )s 
q«»( w ) « ^ q'(s) * h ( s - w ) (10) 
From this wealth distribution we should by means f the 
original transformati n (6) c me back to the income 
distribution
	        

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