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

15. Basic Index Number Theory
383
n
0
∑ si
ri
r ∗ ,
i=
1
= ≡
where the month 0 revenue shares s i 0 are defined
as follows:
(15.33)
s
0
i
≡
pq
0 0
i i
n
0 0
∑ p
jq
j
j=
1
; i =1,...,n.
15.43 Define the ith quantity relative t i as the ratio
of the quantity of product i used in the base
year b, q i b , to the quantity used in month 0, q i 0 , as
follows:
b 0
(15.34) t ≡ q / q ; i =1,...,n.
i i i
The Laspeyres quantity index, Q L (q 0 ,q b ,p 0 ), that
compares quantities in year b, q b , with the corresponding
quantities in month 0, q 0 , using the prices
of month 0, p 0 , as weights can be defined as a
weighted average of the quantity ratios t i as follows:
n
0 b
∑ pi
qi
0 0 i=
1
L
n
0 0
∑ pi
qi
i=
1
n b
⎛q
⎞
i 0 0
∑⎜
0 ⎟pi
qi
i=
1 ⎝qi
⎠
n
0 0
∑ pq
i i
i=
1
∑
n b
⎛q
⎞
i 0
s
0 i
i=
1 ⎝q
i ⎠
n 0
s
1 it
i=
i
t*
b
(15.35) Q ( q , q , p ) ≡
=
= ⎜ ⎟
∑
= ≡
15.44 Using equation (A15.2.4) in Appendix 2,
the relationship between the Lowe index
P Lo (p 0 ,p t ,q b ) that uses the quantities of year b as
weights to compare the prices of month t to month
0 and the corresponding ordinary Laspeyres index
P L (p 0 ,p t ,q 0 ) that uses the quantities of month 0 as
weights is defined as:
(15.36)
P ( p , p , q ) ≡
Lo
0 t b i=
1
n
n
∑
∑
i=
1
= P ( p , p , q ) +
L
n
∑
0 t 0 i=
1
p q
t
i
p q
b
i
0 b
i i
( r −r )( t −t ) s
∗ ∗ 0
i i i
Q q q p
0 b 0
L
( , , )
Thus, the Lowe price index using the quantities of
year b as weights, P Lo (p 0 ,p t ,q b ), is equal to the
usual Laspeyres index using the quantities of
month 0 as weights, P L (p 0 ,p t ,q 0 ), plus a covariance
term
n
∑
i=
1
( r −r )( t −t ) s
∗ ∗ 0
i i i
between the price relatives
r i ≡ p i t / p i 0 and the quantity relatives t i ≡ q i
b
/
q i 0 , divided by the Laspeyres quantity index
Q L (q 0 ,q b ,p 0 ) between month 0 and base year b.
15.45 Equation (15.36) shows that the Lowe
price index will coincide with the Laspeyres price
index if the covariance or correlation between the
month 0 to t price relatives r i ≡ p t i /p 0
i and the
month 0 to year b quantity relatives t i ≡ q b i /q 0 i is
zero. Note that this covariance will be zero under
three different sets of conditions:
• If the month t prices are proportional to the
month 0 prices so that all r i = r*,
• If the base year b quantities are proportional to
the month 0 quantities so that all t i = t*, and
• If the distribution of the relative prices r i is independent
of the distribution of the relative
quantities t i .
The first two conditions are unlikely to hold empirically,
but the third is possible, at least approximately,
if purchasers do not systematically change
their purchasing habits in response to changes in
relative prices.
15.46 If this covariance in equation (15.36) is
negative, then the Lowe index will be less than the
Laspeyres, and, finally, if the covariance is positive,
then the Lowe index will be greater than the
Laspeyres index. Although the sign and magnitude
of the covariance term is ultimately an empirical
matter, it is possible to make some reasonable conjectures
about its likely sign. If the base year b
precedes the price reference month 0 and there are
long-term trends in prices, then it is likely that this
covariance is positive, and hence that the Lowe in-
.