Note that the two empirical relations are of an entirely
different kind. One determines the top salary in
various companies. The other determines the salary
corresponding to the frequency with which it is paid in
a certain company ( the more natural way of putting it
would be N = m' C -5 tL*
Simonas point ( howefer ( is that with his explanation the
parameter is the same in thejtwo very different equations
which squares with the empirical isx result.
Simon's theory follows the pattern of the usual Pareto
explanation - the two exponential distributions -
but it is unsatisfactory because, like Roys esplanation of t
lognormal distribution , it does not contain the time element.
We must get history iito it. v
A way by which it can enter: the size of organisations grows
with time.
Incidentally the siye distribution of companies is already
pareto distributed, therefore the-top salurj would bo so ‘
rH-a^ri bo ted, s-inde—irt^-drs—a—linear —function of the s i z W7~7 ^
Elaboration of the algebra:
Substituting from 3 in 7•
('LOln b . _ T
In ^ --n’T"" 1 " r - In N
In n
lnb
In n
( In ^ _ constanti ) + constant
P
In "b
In C' =“ t ( In N ^ In S -l-constant.) + constant,, j
-*-^1 n I i0l0 ^r
The results of Simon imply that th e same parameter In b/ In n
occurs in both relations and the fact that the parame&er is
empirically the same serves him as confirmation.
He has implicitly assumed that the parameter is the same for
all companies, that is implied in his argument.
We stuck to this assumption in deriving the gneral distribution 9
which can be more conventionally put as follows:
In N § In C 1 In S ^constanti j xX chnstant2
In b
N = C '
(ET
exp(constanti -constant2 )
If we now mix the above distribution of N with the frequency function
of S, the size distribution^we ought to get the frequency of
salaries for the whole industry. The distribution of S is Pareto.