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

1. An Introduction to PPI Methodology
that is when b = 0, the Laspeyres index is obtained,
and when quantities are those of the second period,
that is when b = t,—the Paasche index is obtained.
It is necessary to consider the properties of
Laspeyres and Paasche indices, and also the
relationships between them, in more detail.
1.25 The formula for the Laspeyres price index,
P L , is given in equation (1.2).
n
t 0
∑ pq
i i n
i=
1
t
(1.2) = ≡
n ∑( )
0 0 i=
1
∑ pq
i i
P p p s
0 0
L i i i
i=
1
where s 0
i denotes the share of the value of product
i in the total output of goods and services in period
0 0 0 0
0: that is, pi qi / ∑ pi qi
.
1.26 As can be seen from equation (1.2), and as
explained in more detail in Chapter 15, the
Laspeyres index can be expressed in two alternative
ways that are algebraically identical: first, as the ratio
of the values of the basket of producer goods
and services produced in period 0 when valued at
the prices of periods t and 0 respectively; second, as
a weighted arithmetic average of the ratios of the
individual prices in periods t and 0 using the value
shares in period 0 as weights. The individual price
ratios, (p i t /p i 0 ), are described as price relatives. Statistical
offices often calculate PPIs using the second
formula by recording the percentage changes in the
prices of producer goods and services sold and
weighting them by the total value of output in the
base period 0.
1.27 The formula for the Paasche index, P P , is
given in equation (1.3).
n
t t
∑ pq
i i n
i=
1 ⎧ t 0
(1.3)
n ∑( )
0 t i=
1
pq ⎩
∑ i i
−1
t⎫
P = ≡⎨ pi pi si
⎬
⎭
i=
1
t
where s i denotes the actual share of the expenditure
on commodity i in period t: that is, p t i q t i /
∑ p t i q t i . The Paasche index can also be expressed in
two alternative ways, either as the ratio of two
value aggregates or as a weighted average of the
price relatives, the average being a harmonic average
that uses the revenue shares of the later period t
,
−1
,
as weights. However, it follows from equation (1.1)
that the Paasche index can also be expressed as a
weighted arithmetic average of the price relatives
using hybrid expenditure weights in which the
quantities of t are valued at the prices of 0.
1.28 If the objective is simply to measure the
price change between the two periods considered in
isolation, there is no reason to prefer the basket of
the earlier period to that of the later period, or vice
versa. Both baskets are equally relevant. Both indices
are equally justifiable, or acceptable, from a
conceptual point of view. In practice, however,
PPIs are calculated for a succession of time periods.
A time series of monthly Laspeyres PPIs based on
period 0 benefits from requiring only a single set of
quantities (or revenues), those of period 0, so that
only the prices have to be collected on a regular
monthly basis. A time series of Paasche PPIs, on
the other hand, requires data on both prices and
quantities (or revenues) in each successive period.
Thus, it is much less costly, and time consuming, to
calculate a time series of Laspeyres indices than a
time series of Paasche indices. This is a decisive
practical advantage of Laspeyres (as well as Lowe)
indices over Paasche indices and explains why
Laspeyres and Lowe indices are used much more
extensively than Paasche indices. A monthly
Laspeyres or Lowe PPI can be published as soon as
the price information has been collected and processed,
since the base period weights are already
available.
B.1.3 Decomposing current value
changes using Laspeyres and
Paasche
1.29 Laspeyres and Paasche quantity indices are
defined in a similar way to the price indices, simply
by interchanging the ps and qs in formulas (1.2) and
(1.3). They summarize changes over time in the
flow of quantities of goods and services produced.
A Laspeyres quantity index values the quantities at
the fixed prices of the earlier period, while the
Paasche quantity index uses the prices of the later
period. The ratio of the values of the revenues in
two periods (V) reflects the combined effects of
both price and quantity changes. When Laspeyres
and Paasche indices are used, the value change can
be exactly decomposed into a price index times a
quantity index only if the Laspeyres price (quantity)
index is matched with the Paasche quantity (price)
index. Let P L and Q L denote the Laspeyres price
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