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

15. Basic Index Number Theory
389
these circumstances. Hence with long-run trends in
prices and very inelastic responses of purchasers
to price changes, the Young index is likely to be
less than the corresponding Laspeyres index.
15.59 The previous two paragraphs indicate that
a priori, it is not known what the likely difference
between the Young index and the corresponding
Laspeyres index will be. If elasticities of substitution
are close to 1, then the two sets of revenue
shares, s b i and s 0 i , will be close to each other and
the difference between the two indices will be
close to zero. However, if monthly revenue shares
have strong seasonal components, then the annual
b
shares s i could differ substantially from the
monthly shares s 0 i .
15.60 It is useful to have a formula for updating
the previous month’s Young price index using
only month-over-month price relatives. The Young
index for month t + 1, P Y (p 0 ,p t+1 ,s b ), can be written
in terms of the Lowe index for month t,
P Y (p 0 ,p t ,s b ), and an updating factor as follows:
(15.50)
⎛ p ⎞
PY( p , p , s ) ≡ s ⎜ ⎟
⎠
n t+
1
0 t+
1 b b i
∑ i 0
i=
1 ⎝ pi
n
= P ( p , p , s )
Y
∑
0 t b i=
1
n
= P ( p , p , s )
Y
∑
i=
1
n
∑
0 t b i=
1
n
;
using equation (15.47)
∑
i=
1
s p p
b t+
1 0
i
(
i
/
i
)
b t 0
i
(
i
/
i
)
s p p
p q p p
b b t+
1 0
i i
(
i
/
i
)
b b t 0
i i
(
i
/
i
)
p q p p
⎛ p ⎞⎛ p ⎞
⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
n
t t+
1
b b i i
∑ pq
i i 0 t
0 t b i=
1 pi pi
Y
( , , )
n
b b t 0
∑ pq
i i
( pi / pi
)
i=
1
n
0 t b ⎡ b0t t+
1 t ⎤
Y( , , ) ∑ i
(
i
/
i) ,
i=
1
= P p p s
= P p p s ⎢ s p p ⎥
⎣
⎦
where the hybrid weights s i b0t are defined by
=
b t 0
si ( pi / pi
)
n
b t 0
∑ sk pk pk
k = 1
; i = 1,...,n
( / )
Thus, the hybrid weights s i b0t can be obtained from
the base year weights s i b by updating them; that is,
by multiplying them by the price relatives (or indices
at higher levels of aggregation), p i
t
/ p i 0 . Thus,
the required updating factor, going from month t to
month t + 1, is the chain link index,
n
∑
i=
1
s ( p / p ), which uses the hybrid share
b0t t+
1 t
i i i
weights s i b0t defined by equation (15.51).
15.61 Even if the Young index provides a close
approximation to the corresponding Laspeyres index,
it is difficult to recommend the use of the
Young index as a final estimate of the change in
prices going from period 0 to t, just as it was difficult
to recommend the use of the Laspeyres index
as the final estimate of inflation going from period
0 to t. Recall that the problem with the Laspeyres
index was its lack of symmetry in the treatment of
the two periods under consideration; that is, using
the justification for the Laspeyres index as a good
fixed-basket index, there was an identical justification
for the use of the Paasche index as an equally
good fixed-basket index to compare periods 0 and
t. The Young index suffers from a similar lack of
symmetry with respect to the treatment of the base
period. The problem can be explained as follows.
The Young index, P Y (p 0 ,p t ,s b ), defined by equation
(15.48), calculates the price change between
months 0 and t, treating month 0 as the base. But
there is no particular reason to treat month 0 as the
base month other than convention. Hence, if we
treat month t as the base and use the same formula
to measure the price change from month t back to
month 0, the index P Y (p 0 ,p t ,s b ) =
n
∑
i=
1
s p p
b 0 t
i
(
i
/
i)
would be appropriate. This estimate of price
change can then be made comparable to the original
Young index by taking its reciprocal, leading
to the following rebased Young index, 48
P Y *(p 0 ,p t ,s b ), defined as
(15.51)
b b t 0
b0t pq
i i
( pi / pi
)
i
≡
n
b b t 0
∑ pkqk pk pk
k = 1
s
( / )
48 Using Irving Fisher’s (1922, p. 118) terminology,
P Y *(p 0 ,p t ,s b ) ≡ 1/[P Y (p t ,p 0 ,s b )] is the time antithesis of
the original Young index, P Y (p 0 ,p t ,s b ).