Page 55 - IJPS-8-2

International Journal of
Population Studies Used versus Offered densities of human population

Moment formulas for log-normal distributions 1
E x kr exp k rm (( ) ( kr s) 22 )

O
The “Truncated Moments” formula 2
Coming to the population of consumed units in a Yielding that
consumption model with power function, we can avail
ourselves of the “truncated moment” formula, namely: E x kr exp rm ( 1 ( r 2 rks ))
O
2
2
k
E x 2
ln b m O
Φ rs
b r mrs / ) 2 s And in turn, a Stairs’ generalized density indicator of
xd F O x e ( kr
2
r
a ln a m (C-2) r E O x r / 1 1 2
Φ
( rs) x ( ) exp ( m s r ( 2 k)
k
s
E O x k 2
An immediate consequence is that M O x ( U x / O ) kr /22
1
2
2
1 since exp( )m is the median M of x and s γ .
E x exp rm ( 2 r s ) Lorenz curve and Gini index O O O
r
22
O

Proof of (C-2). The reason is that Here, the Lorenz function, LF U  F O ( 1 ) , involves
1
.Φ

F exp(ms 1 ) together with
b t b 2 t b O
xd F O x e rm st) e rm rs / ) 2 trsdt F x Φ((ln x m ) s) . It thus is a function of α
1
(
tdt

r

(
a t a t a U s
parameterized by s:
t b
in which we replace trsdt with 1
s
Φ b rs ( t rs) . L ΦΦ( s) 1 (C-4)
Φ t
t a
a

1 The Gini index, G 2 ( L s ) d ,

It follows that F x Φ((ln x m ) rs), i.e., that, s 0 can be
U
s
in the population of consumed units, level x is distributed considered as a function of s. It holds that
LN(mrss+ 2 , ) . s
2
G s 2Φ 1 (C-5)
1 2
Thus E x exp m rs( 2 2 s ) and
2
U
2
s
γ U e 1 γ O . Proof of (C-5). At point s = 0, as ΦΦ 1 , then
1
.

)
The case of r = 1 G 2 ( d 0
0
0
1 Differentiating G with respect to s, we get that:
When r = 1, F x Φ((ln x m ) s) and s
U
s
1


3 G s d G s 2 ϕΦ( 1 sd )
2
E x exp m ( 2 s ) . ds 0
U

It is then easy to obtain Changing variables according to t Φ 1 hence
d ϕ()
t dt , we get that
x U 1 γ 2 exp() (C-3)
s
2

x O O G s 2 ϕ t s ϕ .( t dt)

On Stairs’ generalized population density Rearranging
m s ) then
If the offered density is distributed x ∼ LN(, 2 1 2 1
O
2

2
2
we have that ts t 2 t 2 ts s 2 2 t s s 2
2 2
E x exp km ( 1 k s ) and similarly ( 2 t s 2 ) 2 ( s 2 ) 2
22
k

O
2
Volume 8 Issue 2 (2022) 49 https://doi.org/10.36922/ijps.v8i2.297