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

16. Axiomatic and Stochastic Approaches to Index Number Theory
425
That is, if current period values are all multiplied
by the number λ, then the price index remains unchanged.
Put another way, the price index function
P(p 0 ,p 1 ,v 0 ,v 1 ) is (positively) homogeneous of degree
zero in the components of the period 1 value
vector v 1 .
T7—Invariance to Proportional Changes in Base
Period Values:
P(p 0 ,p 1 ,λv 0 ,v 1 ) = P(p 0 ,p 1 ,v 0 ,v 1 ) for all λ > 0.
That is, if base period values are all multiplied by
the number λ, then the price index remains unchanged.
Put another way, the price index function
P(p 0 ,p 1 ,v 0 ,v 1 ) is (positively) homogeneous of degree
zero in the components of the period 0 value
vector v 0 .
16.105 T6 and T7 together impose the property
that the price index p does not depend on the absolute
magnitudes of the value vectors v 0 and v 1 . Using
test T6 with λ = 1 ∑ vi
and using test T7
n
1
with
λ = 1
n
∑
i=
1
property:
(16.63)
v
0
i
i=
1
, it can be seen that p has the following
0 1 0 1 0 1 0 1
P( p , p , v , v ) = P( p , p , s , s ),
where s 0 and s 1 are the vectors of revenue shares
for periods 0 and 1; that is, the ith component of s t
is s t t
i ≡ v
i
n
t
∑ vk
for t = 0,1. Thus, the tests T6 and
k = 1
T7 imply that the price index function p is a function
of the two price vectors p 0 and p 1 and the two
vectors of revenue shares, s 0 and s 1 .
16.106 Walsh suggested the spirit of tests T6 and
T7 as the following quotation indicates:
What we are seeking is to average the variations
in the exchange value of one given total sum of
money in relation to the several classes of goods,
to which several variations [i.e., the price relatives]
must be assigned weights proportional to
the relative sizes of the classes. Hence the relative
sizes of the classes at both the periods must
be considered. (Correa Moylan Walsh, 1901, p.
104)
16.107 Walsh also realized that weighting the ith
price relative r i by the arithmetic mean of the value
weights in the two periods under consideration,
(1/2)[v 0 i + v 1 i ], would give too much weight to the
revenues of the period that had the highest level of
prices:
At first sight it might be thought sufficient to add
up the weights of every class at the two periods
and to divide by two. This would give the
(arithmetic) mean size of every class over the
two periods together. But such an operation is
manifestly wrong. In the first place, the sizes of
the classes at each period are reckoned in the
money of the period, and if it happens that the
exchange value of money has fallen, or prices in
general have risen, greater influence upon the result
would be given to the weighting of the second
period; or if prices in general have fallen,
greater influence would be given to the weighting
of the first period. Or in a comparison between
two countries, greater influence would be
given to the weighting of the country with the
higher level of prices. But it is plain that the one
period, or the one country, is as important, in
our comparison between them, as the other, and
the weighting in the averaging of their weights
should really be even. (Correa Moylan Walsh,
1901, pp. 104-105)
16.108 As a solution to the above weighting problem,
Walsh (1901, p. 202; 1921a, p. 97) proposed
the following geometric price index:
(16.64)
P p p v v
0 1 0 1
GW
( , , , )
wi ()
n 1
⎛ p ⎞
i
≡ ∏ ⎜ 0 ⎟
i=
1 ⎝ p
,
i ⎠
where the ith weight in the above formula was defined
as
(16.65)
wi () ≡
( vv)
n
∑
k = 1
k = 1
0 1 1/2
i i
( vv)
0 1 1/2
k k
=
0 1 1/2
( ss
i i)
n
, i = 1,..., n
( ss)
∑
0 1 1/2
k k
The second part of equation (16.65) shows that
Walsh’s geometric price index P GW (p 0 ,p 1 ,v 0 ,v 1 ) can
also be written as a function of the revenue share
vectors, s 0 and s 1 ; that is, P GW (p 0 ,p 1 ,v 0 ,v 1 ) is homogeneous
of degree 0 in the components of the