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

Producer Price Index Manual
and quantity indices and let P P and Q P denote the
Paasche price and quantity indices. As shown in
Chapter 15, P L • Q P ≡ V and P P • Q L ≡ V.
1.30 Suppose, for example, a time series of industry
output in the national accounts is to be deflated
to measure changes in output at constant
prices over time. If it is desired to generate a series
of output values at constant base period prices
(whose movements are identical with those of the
Laspeyres volume index), the output at current
prices must be deflated by a series of Paasche price
indices. Laspeyres-type PPIs would not be appropriate
for the purpose.
B.1.4 Ratios of Lowe and Laspeyres
indices
1.31 The Lowe index is transitive. The ratio of
two Lowe indices using the same set of q b s is also a
Lowe index. For example, the ratio of the Lowe index
for period t + 1 with price reference period 0
divided by that for period t also with price reference
period 0 is:
(1.4)
n n n
t+ 1 b 0 b t+
1 b
∑pi qi ∑pi qi ∑pi qi
i= 1 i= 1 i=
1
n n n
t b 0 b t b
∑pq i i ∑pq i i ∑ pq
i i
i= 1 i= 1 i=
1
= = P
tt , + 1
Lo
1.32 This is a Lowe index for period t + 1, with
period t as the price reference period. This kind of
index is, in fact, widely used to measure short-term
price movements, such as between t and t + 1, even
though the quantities may date back to some much
earlier period b.
1.33 A Lowe index can also be expressed as the
ratio of two Laspeyres indices. For example, the
Lowe index for period t with price reference period
0 is equal to the Laspeyres index for period t with
price reference period b divided by the Laspeyres
index for period 0 also with price reference period
b. Thus,
n n n
∑ ∑ ∑
t b t b b b
pq
i i
pq
i i
pq
i i t
i= 1 i= 1 i=
1
PL
(1.5) PLo = = = .
n n n
0
0 b 0 b b b PL
pq pq pq
∑ ∑ ∑
i i i i i i
i= 1 i= 1 i=
1
.
B.1.5 Updated Lowe indices
1.34 It is useful to have a formula that enables a
Lowe index to be calculated directly as a chain index
in which the index for period t + 1 is obtained
by updating the index for period t. Because Lowe
indices are transitive, the Lowe index for period t +
1 with price reference period 0 can be written as
the product of the Lowe index for period t with
price reference period 0 multiplied the Lowe index
for period t + 1 with price reference period t. Thus,
(1.6)
⎡ ⎤⎡ ⎤
⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥
⎢
⎣
⎥⎢
⎦⎣
⎥
⎦
n
⎡ t b ⎤
⎢∑
pq
i i n t + 1 ⎥⎡ ⎛
i 1
p ⎞ ⎤
=
i tb
= ⎢ ⎥ s
n ⎢∑⎜
t ⎟ i
,
⎢ 0 b i=
1 pi
pq
⎥⎢⎣
⎝ ⎠ ⎥⎦
⎢∑
i i
⎣
⎥
i=
1 ⎦
n n n
t+ 1 b t b t+
1 b
∑pi qi ∑piqi ∑pi qi
i= 1 i= 1 i=
1
n n n
0 b 0 b t b
∑pi qi ∑pi qi ∑piqi
i= 1 i= 1 i=
1
where the revenue weights s i
tb
are hybrid weights
defined as:
tb t b t b
(1.7) s ≡ pq ∑ pq .
i i i i i
i=
1
n
1.35 Hybrid weights of the kind defined in
equation (1.7) are often described as price updated
weights. They can be obtained by adjusting the
original revenue weights p i b q i b / ∑p i b q i b by the price
relatives p i t / p i b . By price updating the revenue
weights from b to t in this way, the index between t
and t + 1 can be calculated directly as a weighted
average of the prices relatives p i t+1 / p i
t
without referring
back to the price reference period 0. The index
can then be linked on to the value of the index
in the preceding period t.
B.1.6 Interrelationships between fixed
basket indices
1.36 Consider first the interrelationship between
the Laspeyres and the Paasche indices. A wellknown
result in index number theory is that if the
price and quantity changes (weighted by values) are
negatively correlated, then the Laspeyres index exceeds
the Paasche. Conversely, if the weighted
price and quantity changes are positively correlated,
then the Paasche index exceeds the Laspeyres. The
proof is given in Appendix 15.1 of Chapter 15.
8