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

Producer Price Index Manual
(15.28)
pq
0 b
0b i i
i
≡ ;
n
0 b
∑ p
jq
j
j=
1
b b 0 b
pq
i i
( pi / pi
)
=
n
b b 0 b
⎡pq j j
pj pj
j=
1
s
i =1,...,n
.
∑ ⎣ ( / ) ⎤ ⎦
Equation (15.28) shows how the base-year revenues,
p i b q i b , can be multiplied by the product price
indices, p i 0 /p i b , to calculate the hybrid shares.
15.41 One additional formula for the Lowe index,
P Lo (p 0 ,p t ,q b ), will be exhibited. Note that the
Laspeyres decomposition of the Lowe index defined
by the third line in equation (15.26) involves
the very long-term price relatives, p i t /p i b , that compare
the prices in month t, p i t , with the possibly
distant base-year prices, p i b . Further, the hybrid
share decomposition of the Lowe index defined by
the third line in equation (15.27) involves the longterm
monthly price relatives, p i t /p i 0 , which compare
the prices in month t, p i t , with the base month
prices, p i 0 . Both these formulas are not satisfactory
in practice because of the problem of sample attrition:
each month, a substantial fraction of products
disappears from the marketplace and thus it is useful
to have a formula for updating the previous
month’s price index using just month-over-month
price relatives. In other words, long-term price
relatives disappear at a rate that is too large in
practice to base an index number formula on their
use. The Lowe index for month t + 1,
P Lo (p 0 ,p t+1 ,q b ), can be written in terms of the Lowe
index for month t, P Lo (p 0 ,p t ,q b ), and an updating
factor as follows:
(15.29)
P ( p , p , q ) ≡
Lo
n
t+
1 b
∑ pi
qi
0 t+ 1 b i=
1
n
0 b
∑ pi
qi
i=
1
n
n
t b t+
1 b
∑ piqi ∑pi qi
i= 1 i=
1
n
n
0 b t b
∑pi qi ∑piqi
i= 1 i=
1
⎡ ⎤⎡ ⎤
⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥
⎢
⎣
⎥⎢
⎦⎣
⎥
⎦
n
⎡ t+
1 b⎤
⎢∑
pi
qi
⎥
0 t b i=
1
= PLo
( p , p , q ) ⎢ ⎥
n
⎢ t b
piq
⎥
⎢ ∑ i
⎣
⎥
i=
1 ⎦
= P p p q
0 t b
Lo
( , , )
⎡
⎢
⎢
⎢
⎢
⎢⎣
⎛ p
⎞ ⎤
b
p qi
⎥
⎥
⎥
⎥
⎥⎦
n t+
1
i t
∑⎜
t ⎟ i
i=
1 ⎝ pi
⎠
n
t b
∑ pq
i i
i=
1
n t+
1
⎡
0 t b
⎛ p ⎞ ⎤
i tb
= PLo
( p , p , q ) ⎢∑ ⎜ s
t ⎟ i ⎥,
⎢⎣
i=
1 ⎝ pi
⎠ ⎥⎦
where the hybrid weights s i tb are defined by
(15.30)
pq
t b
tb i i
i
≡
n
t b
∑ p
jq
j
j=
1
s
; i =1,...,n.
Thus, the required updating factor, going from
month t to month t + 1, is the chain link in-
n
tb t+
1 t
dex ∑ si ( pi pi
)
i=
1
, which uses the hybrid share
weights s i tb corresponding to month t and base year
b.
15.42 The Lowe index P Lo (p 0 ,p t ,q b ) can be regarded
as an approximation to the ordinary
Laspeyres index, P L (p 0 ,p t ,q 0 ), that compares the
prices of the base month 0, p 0 , to those of month t,
p t , using the quantity vector of month 0, q 0 , as
weights. There is a relatively simple formula that
relates these two indices. To explain this formula,
it is first necessary to make a few definitions. Define
the ith price relative between month 0 and
month t as
t 0
(15.31) r ≡ p / p ; i =1,...,n.
i i i
The ordinary Laspeyres price index, going from
month 0 to t, can be defined in terms of these price
relatives as follows:
(15.32)
P ( p , p , q ) ≡
L
0 t 0 i=
1
n
⎛ p ⎞
n
∑
∑
i=
1
p q
t 0
i i
p q
0 0
i i
n t
i 0 0
∑⎜
0 ⎟pq
i i n t
i=
1 ⎝ pi ⎠ ⎛ p ⎞
i 0
n ∑ s
0 i
0 0 i=
1 pi
pq ⎝ ⎠
∑ i i
i=
1
= = ⎜ ⎟
382