author's proof - DARP

AUTHOR'S PROOF
Reference distributions and inequality measurement
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Q2
Lemma 2 Given Axioms 1 to 6 ≽ is representable by φ
( ∑n
i=1 x ih i
(
xi
y i
))
where h i is a 329
real-valued function. 330
Proof Using the function introduced in the proof of Lemma 1 Axiom 6 implies 331
(z) = ( z ′) and (tz) = ( tz ′) ; since this has to be true for arbitrary z, z ′ we have 332
(tz)
(z) = ( tz ′)
(z ′ ) = ψ (t)
where ψ is a continuous function R → R. Hence,usingtheφ i given in Lemma 1 we 333
have for all i : φ i (tz i ) = ψ (t) φ i (z i ) or, equivalently φ i (tx i , ty i ) = ψ (t) φ i (x i , y i ) . In 334
view of Aczél and Dhombres [2], page 346 there must exist c ∈ R and a function 335
h i : R + → R such that 336
( )
φ i (x i , y i ) = xi c h xi
i . (14)
y i
From Lemma 1 and Eq. 14: φ i (x i , x i ) = xi ch i (1) = a i + b i x i , which, ifφ i (x, x) is non- 337
constant in x, implies c = 1. Noting that Lemma 1 implies that ≽ is also representable 338
by φ (∑ n
i=1 φ i (z i ) ) (whereφ : R → R is continuous and monotonic increasing, and 339
taking Eq. 14 with c = 1 gives the result. ⊓⊔ 340
Theorem 1 Given Axioms 1 to 7 ≽ is representable by φ (∑ n
)
i=1 xα i y1−α i where α ̸= 1 341
is a constant. 342
Proof (Cf [21]) Take the special case where, in distribution z ′ 0
the income discrep- 343
ancy takes the same value r for all n income pairs. If (x i , y i ) represents a typical 344
component in z 0 then z 0 ∼ z ′ 0
implies 345
( n∑ ( ) )
xi
r = ψ x i h i (15)
y
i=1 i
where ψ is the solution in r to 346
n∑
( )
xi
n∑
x i h i = x i h i (r) (16)
y i
UNCORRECTED PROOF
i=1
i=1
In Eq. 16 we take the x i as fixed weights. Using Axiom 7 in Eq. 15 requires 347
( n∑ (
tr = ψ x i h i t x ) )
i
,forallt > 0. (17)
y
i=1
i
Using Eq. 16 we have 348
(
n∑
n∑ ( )
x i h i
(tψ
)) xi
n∑
(
x i h i = x i h i t x )
i
(18)
y
i=1
i=1 i y
i=1
i
Introduce the following change of variables 349
( )
xi
u i := x i h i , i = 1, ..., n (19)
y i