Producer Price Index Manual: Theory and Practice ... - METAC

16. Axiomatic and Stochastic Approaches to Index Number Theory
433
where 1 n is a vector of ones of dimension n. Using
a result due to Eichhorn (1978, p. 66), it can be
seen that these properties of P* are sufficient to
imply that there exist positive functions α i (s 0 ,s 1 )
for i = 1,…,n such that P* has the following representation:
∗ 0 1 0 1
(A16.4) ln P ( rs , , s) = ∑ αi( s, s)ln
ri.
n
i=
1
16.141 The continuity test T2 implies that the
positive functions α i (s 0 ,s 1 ) are continuous. For λ >
0, the linear homogeneity test T4 implies that
0 1 0 1
(A16.5) ln P ∗
∗
( λ rs , , s) = ln λ+ ln P( rs , , s)
n
0 1
∑ i( s , s )ln ri, using equation (A16.4)
i=
1
n
n
0 1 0 1
∑ i( s , s )ln ∑ i( s , s )lnri
i= 1 i=
1
n
0 1 ∗ 0 1
∑ i
( s , s )ln ln P ( r, s , s ),
i=
1
= α λ
= α λ+ α
= α λ+
using equation (A16.4)
Equating the right hand sides of the first and last
lines in (A15.5) shows that the functions α i (s 0 ,s 1 )
must satisfy the following restriction:
n
0 1
(A16.6) ∑ α
i
( s , s ) = 1,
i=
1
for all strictly positive vectors s 0 and s 1 .
16.142 Using the weighting test T16 and the commodity
reversal test T8, equation (16.69) hold.
Equation (16.69) combined with the commensurability
test T9 implies that P* satisfies the following
equation:
(A16.7)
∗
0 1 0 1
P (1,...,1, ri,1,...,1 ; s , s ) = f(1, ri, s , s ) ; i = 1,..., n
,
for all r i > 0 where f is the function defined in test
T16.
16.143 Substitute equation (A16.7) into equation
(A16.4) in order to obtain the following system of
equations:
(A16.8)
P r s s
∗
0 1
(1,...,1,
i
,1,...,1 ; , )
0 1
= f (1, ri
, s , s )
0 1
i(s ,s )ln
i
; 1,...,
=α r i = n.
But the first part of equation (A16.8) implies that
the positive continuous function of 2n variables
α i (s 0 ,s 1 ) is constant with respect to all of its arguments
except s 0 i and s 1 i , and this property holds for
each i. Thus, each α i (s 0 ,s 1 ) can be replaced by the
positive continuous function of two variables
β i (s 0 i ,s 1 i ) for i = 1,…,n. 77 Now replace the α i (s 0 ,s 1 )
in equation (A16.4) by the β i (s 0 i ,s 1 i ) for i = 1,…,n
and the following representation for P* is obtained:
∗ 0 1 0 1
(A16.9) ln P ( rs , , s) = β ( s, s)ln r.
n
∑
i=
1
i i i i
16.144 Equation (A16.6) implies that the functions
β i (s 0 i ,s 1 i ) also satisfy the following restrictions:
(A16.10)
n
n
0 1
∑si
∑ si
i= 1 i=
1
= 1 ; = 1
n
0 1
implies β
i( si , si) = 1
i=
1
∑ .
16.145 Assume that the weighting test T17 holds
and substitute equation (16.71) into (A16.9) in order
to obtain the following equation:
1
⎛ p ⎞
i
(A16.11) β
i
(0,0)ln ⎜ 0 ; i 1,..., n
0 ⎟= = .
⎝ pi
⎠
Since the p 1 i and p 0 i can be arbitrary positive numbers,
it can be seen that equation (A16.11) implies
(A16.12) β
i
(0,0) = 0 ; i = 1,..., n.
16.146 Assume that the number of commodities
n is equal to or greater than 3. Using equations
(A16.10) and (A16.12), Theorem 2 in Aczél (1987,
77 More explicitly, β 1 (s 0 1 ,s 1 1 ) ≡ α 1 (s 0 1 ,1,…,1;s 1 1 ,1,…,1)
and so on. That is, in defining β 1 (s 0 1 ,s 1 1 ), the function
α 1 (s 0 1 ,1,…,1;s 1 1 ,1,…,1) is used where all components of
the vectors s 0 and s 1 except the first are set equal to an arbitrary
positive number like 1.