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

Producer Price Index Manual
B.1 Price indices based on baskets
of goods and services
1.19 The purpose of an index number may be
explained by comparing the values of producer’s
revenues from the production of goods and services
in two time periods. Knowing that revenues have
increased by 5 percent is not very informative if we
do not know how much of this change is due to
changes in the prices of the goods and services and
how much to changes in the quantities produced.
The purpose of an index number is to decompose
proportionate or percentage changes in value aggregates
into their overall price and quantity
change components. A PPI is intended to measure
the price component of the change in producer’s
revenues. One way to do this is to measure the
change in the value of an aggregate by holding the
quantities constant.
B.1.1 Lowe indices
1.20 One very wide, and popular, class of price
indices is obtained by defining the index as the percentage
change between the periods compared in
the total cost of producing a fixed set of quantities,
generally described as a “basket.” The meaning of
such an index is easy to grasp and to explain to users.
This class of index is called a Lowe index in
this Manual after the index number pioneer who
first proposed it in 1823: see Section B.2 of Chapter
15. Most statistical offices make use of some kind
of Lowe index in practice. It is described in some
detail in Sections D.1 and D.2 of Chapter 15.
1.21 In principle, any set of goods and services
could serve as the basket. The basket does not have
to be restricted to the basket actually produced in
one or other of the two periods compared. For practical
reasons, the basket of quantities used for PPI
purposes usually has to be based on a survey of establishment
revenues conducted in an earlier period
than either of the two periods whose prices are
compared. For example, a monthly PPI may run
from January 2000 onward, with January 2000 =
100 as its price reference period, but the quantities
may be derived from an annual revenue survey
made in 1997 or 1998, or even spanning both years.
Because it takes a long time to collect and process
revenue data, there is usually a considerable time
lag before such data can be introduced into the calculation
of PPIs. The basket may also refer to a
year, whereas the index may be compiled monthly
or quarterly
1.22 Let there be n products in the basket with
prices p i and quantities q i . Let period b be the period
to which the quantities refer and periods 0 and
t be the two periods whose prices are being compared.
In practice, it is invariably the case that b ≤ 0
< t when the index is first published, and this is assumed
here. However, b could be any period, including
one between 0 and t, if the index is calculated
some time after t. The Lowe index is defined
in equation (1.1).
n
t b
∑ pq
i i n
i=
1
t
(1.1) ≡ ≡
n ∑( )
0 b i=
1
∑ pq
i i
where
P p p s
0 0b
Lo i i i
i=
1
p q
0 b
0b i i
i
=
n
0 b
∑ pi
qi
i=
11
s
The Lowe index can be written, and calculated, in
two ways: either as the ratio of two value
aggregates, or as an arithmetic weighted average of
the price ratios, or price relatives, p t i / p 0 i , for the
individual products using the hybrid revenue shares
0b
s i as weights. They are described as hybrid
because the prices and quantities belong to two
different time periods, 0 and b, respectively. The
hybrid weights may be obtained by updating the
actual revenue shares in period b, namely p b i q b i / ∑
p b i q b i , for the price changes occurring between
periods b and 0 by multiplying them by the price
relative between b and 0, namely p 0 i / p b i . The
concept of the base period is somewhat ambiguous
with a Lowe index, since either b or 0 might be
interpreted as being the base period. To avoid
ambiguity, b is described as the weight reference
period and 0 as the price reference period.
1.23 Lowe indices are widely used for PPI
purposes.
B.1.2 Laspeyres and Paasche indices
1.24 Any set of quantities could be used in a
Lowe index, but there are two special cases that
figure prominently in the literature and are of
considerable importance from a theoretical point of
view. When the quantities are those of the first of
the two periods whose prices are being compared—
,
6