9
q (Y) - cdY
OO
f* (¥ - Y) e _<7CW dW *»
= c(f(oC) e -oCY dY
(w> HO)
(3)
where Cf> (oo) is the Laplace transform of i*(x).
The first result is thus: If the rate of return is inde
pendent of wealth, the Pareto law will he reproduced in
the distribution of income, and the coefficient will be
the same as for wealth. The particular form of the return
rate distribution is of no account, except for a scale
»>
factor.
This result can be somewhat generalized. If the rate of
return is not ^dependent of wealth - and indeed it will
hardly be in reality - then there will be a correlation
between income and wealth. If this dependence of income
on wealth is linear in the logs and if the linear re
gression is also homoscedastic we can virtually re-establish
the previous case of equation (3) by stretching or con
tracting the scale of W.(see fig. 1). That is to say, we
/
- k- W ^