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. 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. 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:
1) 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
kC
7 - 70
5
is the ratio of the two Pareto,coefficients of the income
i
and the wealth distribution. *
2) Following up the idea that wealth can be
explained from saving over a certain time and saving cam
be explained from income, taking saving propensity as given,
we can darive 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 convnlution
of^the income distribution amd of the(distribution of the
» (
propensity to save ( savings ratio ):
q'(s) - q(y) * g ( iry -is ), \ (9)
amd the wealth distribution a s a convulution 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''(w) - q'(s)*h(s-w) (lO)
From this wealth distribution we should by means of the
original transformation (o) come back to the income
distribution