12
the condition is even more restrictive: The rate of return
must not be larger than w k ' 1 '.loo p.c. whithin the range of
wealth sizes in which the empirical data lie.
The restriction is unavoidable because the Laplace transform
in defined only for positive values of the argument of f^ w ^.
For negative values the derq^iity f is by definition zero.
If the argument is shifted to the right by Y the transform will
be defined only for d&wities of a rate of return below loo p.c.
Similarly, for an argument of kW-Y the transform will be defined
k -"I
only for rates of return below w .loo p.c.
In reality, rates of return in excess of the limit given may
exist. In this case we can, however, always ensure that the
above condition is fulfilled and the transformation (4) remains
valid provided we make the unit of welfath, W Q (Wo = 0) sufficient
ly large.
Indeed the condition
y <w
w
k-i
k
or
y \w
(5)
will be more easily fulfilled if w and y are both measured in
a large unit, because then their values will be both lower in
the same proportion, and that will automatically make it easier
to fulfill the condition (5), if k <1.
The choice of a large unit, however, will mean that the conclusions
with regard to the distribution of income which are implied in (4)