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

Producer Price Index Manual
tity indices is not equal to the revenue ratio, V 1 /V 0 .
Thus, believers in the pure or unequivocal price
and quantity index concepts have to choose one of
these two concepts; they cannot apply both simultaneously.
41
16.71 If the quantity index Q(q 0 ,q 1 ,p 0 ,p 1 ) satisfies
the additivity test in equation (16.30) for some
price weights p i *, then the percentage change in
the quantity aggregate, Q(q 0 ,q 1 ,p 0 ,p 1 ) − 1, can be
rewritten as follows:
(16.35)
n
* 1
∑ pq
i i
0 1 0 1 i=
1
( , , , ) − 1=
−1
n
* 0
∑ pq
m m
m=
1
n
n
* 1 * 0
∑ piqi − ∑ pmqm
i= 1 m=
1
=
n
* 0
∑ pq
m m
m=
1
n
1 0
= ∑ wi( qi −qi
),
i=
1
Q p p q q
where the weight for product i, w i , is defined as
(16.36)
*
pi
wi
≡ ; i = 1,...,n.
n
pq
∑
m=
1
* 0
m m
Note that the change in product i going from situation
0 to situation 1 is q i 1 − q i 0 . Thus, the ith term
on the right-hand side of equation (16.35) is the
contribution of the change in product i to the overall
percentage change in the aggregate going from
period 0 to 1. Business analysts often want statistical
agencies to provide decompositions like equation
(16.35) so they can decompose the overall
change in an aggregate into sector-specific components
of change. 42 Thus, there is a demand on the
part of users for additive quantity indices.
16.72 For the Walsh quantity index defined by
equation (16.34), the ith weight is
41 Knibbs (1924) did not notice this point!
42 Business and government analysts also often demand an
analogous decomposition of the change in price aggregate
into sector-specific components that add up.
(16.37)
p p
w ≡ ; i=
1,..., n.
Wi
0 1
i i
n
0 0 1
∑ qm pmpm
m=
1
Thus, the Walsh quantity index Q W has a percentage
decomposition into component changes of the
form in equation (16.35) where the weights are defined
by equation (16.37).
16.73 It turns out that the Fisher quantity index
Q F defined by equation (15.14) in Chapter 15 also
has an additive percentage change decomposition
of the form given by equation (16.35). 43 The ith
weight w Fi for this Fisher decomposition is rather
complicated and depends on the Fisher quantity
index Q F (p 0 ,p 1 ,q 0 ,q 1 ) as follows 44 :
(16.38)
0 2 1
wi + ( QF)
wi
wF
≡ ; i = 1,..., n,
i
1+
Q
F
where Q F is the value of the Fisher quantity index,
Q F (p 0 ,p 1 ,q 0 ,q 1 ), and the period t normalized price
for product i, w i t , is defined as the period i price p i
t
divided by the period t revenue on the aggregate:
(16.39)
t
t pi
wi
≡
n
; t = 0,1 ; i = 1, …, n.
t t
pq
∑
m=
1
m
m
16.74 Using the weights w Fi defined by equations
(16.38) and (16.39), the following exact decomposition
is obtained for the Fisher ideal quantity
index 45 :
43 The Fisher quantity index also has an additive decomposition
of the type defined by equation (16.30) due to Van
Ijzeren (1987, p. 6). The ith reference price p i * is defined as
p i * ≡ (1/2)p i 0 + (1/2)p i 1 /P F (p 0 ,p 1 ,q 0 ,q 1 ) for i = 1,…,n and
where P F is the Fisher price index. This decomposition was
also independently derived by Dikhanov (1997). The Van
Ijzeren decomposition for the Fisher quantity index is currently
being used by the Bureau of Economic Analysis; see
Moulton and Seskin (1999, p. 16) and Ehemann, Katz, and
Moulton (2002).
44 This decomposition was obtained by Diewert (2002a)
and Reinsdorf, Diewert, and Ehemann (2002). For an economic
interpretation of this decomposition, see Diewert
(2002a).
45 To verify the exactness of the decomposition, substitute
equation (16.38) into equation (16.40) and solve the result-
(continued)
416