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

15. Basic Index Number Theory
373
1 was to specify an approximate representative
product basket, 6 which is a quantity vector q ≡
[q 1 ,…,q n ] that is representative of purchases made
during the two periods under consideration, and
then to calculate the level of prices in period 1
relative to period 0 as the ratio of the period 1 cost
of the basket,
basket,
n
∑
i=
1
p q
0
i i
n
∑
i=
1
p q
1
i i
, to the period 0 cost of the
. This fixed–basket approach to the
determination of the price index leaves open the
following question: How exactly is the fixed–
basket vector q to be chosen
15.13 As time passed, economists and price statisticians
demanded a bit more precision with respect
to the specification of the basket vector q.
There are two natural choices for the reference
basket: the base period 0 product vector q 0 or the
current period 1 product vector q 1 . These two
choices led to the Laspeyres (1871) price index 7 P L
defined by equation (15.5) and the Paasche (1874)
price index 8 P P defined by equation (15.6): 9
6 Joseph Lowe (1823, Appendix, 95) suggested that the
product basket vector q should be updated every five years.
Lowe indices will be studied in more detail in Section D.
7 This index was actually introduced and justified by
Drobisch (1871a, p. 147) slightly earlier than Laspeyres.
Laspeyres (1871, p. 305) in fact explicitly acknowledged
that Drobisch showed him the way forward. However, the
contributions of Drobisch have been forgotten for the most
part by later writers because Drobisch aggressively pushed
for the ratio of two unit values as being the best index
number formula. While this formula has some excellent
properties, if all the n products being compared have the
same unit of measurement, the formula is useless when,
say, both goods and services are in the index basket.
8 Again, Drobisch (1871b, p. 424) appears to have been
the first to explicitly define and justify this formula. However,
he rejected this formula in favor of his preferred formula,
the ratio of unit values, and so again he did not get
any credit for his early suggestion of the Paasche formula.
9 Note that P L (p 0 ,p 1 ,q 0 ,q 1 ) does not actually depend on q 1 ,
and P P (p 0 ,p 1 ,q 0 ,q 1 ) does not actually depend on q 0 . However,
it does no harm to include these vectors, and the notation
indicates that the reader is in the realm of bilateral index
number theory; that is, the prices and quantities for a
value aggregate pertaining to two periods are being compared.
(15.5)
(15.6)
1 0
piqi
0 1 0 1 i=
1
P(p,p,q,q
L
) ≡ ;
n
p q
n
∑
∑
i=
1
n
∑
i=
1
0 0
i i
1 1
piqi
0 1 0 1 i=
1
P(p,p,q,q
P
) ≡ .
n
p q
∑
0 1
i i
15.14 The above formulas can be rewritten in a
manner that is more useful for statistical agencies.
Define the period t revenue share on product i as
follows:
(15.7)
= 0,1.
n
t t t t t
i i i
/
j j
j=
1
s ≡ pq ∑ pq for i = 1,...,n and t
Then, the Laspeyres index equation (15.5), can be
rewritten as follows: 10
(15.8)
n
n
0 1 0 1 1 0 0 0
L( , , , ) = ∑ i i
/ ∑ j j
i= 1 j=
1
P p p q q p q p q
using definitions in equation (15.7).
=
=
n
n
1 0 0 0 0 0
∑( pi / pi ) pi qi / ∑pjqj
i= 1 j=
1
n
1 0 0
∑( pi / pi ) si
i=
1
Thus, the Laspeyres price index P L can be written
as a base–period revenue share weighted arithmetic
average of the n price ratios, p 1 i /p 0
i . The
Laspeyres formula (until the very recent past) has
been widely used as the intellectual base for PPIs
around the world. To implement it, a statistical
agency needs only to collect information on revenue
shares s 0 n for the index domain of definition
for the base period 0 and then collect information
on item prices alone on an ongoing basis. Thus, the
Laspeyres PPI can be produced on a timely basis
without current period quantity information.
10 This method of rewriting the Laspeyres index (or any
fixed–basket index) as a share weighted arithmetic average
of price ratios is due to Irving Fisher (1897, p. 517; 1911, p.
397; 1922, p. 51) and Walsh (1901, p. 506; 1921a, p. 92).