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

1. An Introduction to PPI Methodology
duced on a timely basis. Thus, these geometric indices
must be treated as serious practical possibilities
for purposes of PPI calculations. As explained
later, the geometric indices are likely to be less subject
than their arithmetic counterparts to the kinds
of index number biases discussed in later sections.
Their main disadvantage may be that, because they
are not fixed-basket indices, they are not so easy to
explain or justify to users.
B.3 Symmetric indices
1.50 When the base and current periods are far
apart, the index number spread between the numerical
values of a Laspeyres and a Paasche price
index is liable to be quite large, especially if relative
prices have changed a lot (as shown Appendix
15.1 and illustrated numerically in Chapter 19). Index
number spread is a matter of concern to users
because, conceptually, there is no good reason to
prefer the weights of one period to those of the
other. In these circumstances, it seems reasonable
to take some kind of symmetric average of the two
indices. More generally, it seems intuitive to prefer
indices that treat both of the periods symmetrically
instead of relying exclusively on the weights of
only one of the periods. It will be shown later that
this intuition can be backed up by theoretical arguments.
There are many possible symmetric indices,
but there are three in particular that command much
support and are widely used.
1.51 The first is the Fisher price index, P F , defined
as the geometric average of the Laspeyres and
Paasche indices; that is,
(1.10) PF ≡ PL × P
.
1.52 The second is the Walsh price index, P W , a
pure price index in which the quantity weights are
geometric averages of the quantities in the two periods;
that is
(1.11)
P
W
≡
n
∑
i=
1
n
∑
i=
1
p
p
qq
t t 0
i i i
qq
0 t 0
i i i
.
The averages of the quantities need to be geometric
rather than arithmetic for the relative quantities in
both periods to be given equal weight.
1.53 The third index is the Törnqvist price index,
P T , defined as a geometric average of the price
relatives weighted by the average revenue shares in
the two periods:
n
t 0
(1.12)
T ( i i )
P p p σ
= ∏
i
,
i=
1
where σ i is the arithmetic average of the share of
revenue on product i in the two periods, and
t 0
si
+ si
(1.13) σ
i
= ,
2
where the s i s are defined as in equation (1.2) and
above.
1.54 The theoretical attractions of these indices
become apparent in the following sections on the
axiomatic and economic approaches to index numbers.
B.4 Fixed-base versus chain indices
B.4.1 Fixed-basket indices
1.55 This topic is examined in Section F of
Chapter 15. When a time series of Lowe or
Laspeyres indices is calculated using a fixed set of
quantities, the quantities become progressively out
of date and increasingly irrelevant to the later periods
whose prices are being compared. The base period
whose quantities are used has to be updated
sooner or later, and the new index series linked to
the old. Linking is inevitable in the long run.
1.56 In a chain index, each link consists of an
index in which each period is compared with the
preceding one, the weight and price reference periods
being moved forward each period. Any index
number formula can be used for the individual links
in a chain index. For example, it is possible to have
a chain index in which the index for t + 1 on t is a
Lowe index defined as ∑ p t+1 q t–j / ∑ p t q t–j . The
quantities refer to some period that is j periods earlier
than the price reference period t. The quantities
move forward one period as the price reference period
moves forward one period. If j = 0, the chain
Lowe becomes a chain Laspeyres, while if j = –1,
[that is, t – (–1) = t + 1], it becomes a chain
Paasche.
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