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

Producer Price Index Manual
is, it can approximate an arbitrary twice continuously
differentiable linearly homogeneous functional
form to the second order.
17.45 Define the quadratic mean of order r quantity
index Q r by
r 0 1 0 1
(17.23) Q ( p , p , q , q )
where
2
1
2
1
1
r
r
1
−r
−r
n
n
0⎛ q ⎞
i
1⎛ q ⎞
i
si
s
0 i 0
i= 1 qi
i=
1 qi
⎡ ⎤ ⎡ ⎤
≡ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥
⎢
⎣ ⎝ ⎠ ⎥
⎦
⎢
⎣ ⎝ ⎠ ⎥
⎦
∑ ∑ ,
N
t t t t t
i i i i i
i=
1
s = pq ∑ pq is the period t revenue
share for output i as usual. It can be verified that
when r = 2, Q r simplifies into Q F , the Fisher ideal
quantity index.
17.46 Using exactly the same techniques as were
used in Section B.3, it can be shown that Q r is exact
for the aggregator function f r defined by equation
(17.22); i.e.,
(17.24) Q r (p 0 ,p 1 ,q 0 ,q 1 r 1 r 0
) = f ( q ) f ( q )
= .
Thus, under the assumption that the producer engages
in revenue maximizing behavior during periods
0 and 1 and has technologies that satisfy a
linearly homogeneous aggregator function for outputs
17 where the output aggregator function f(q) is
defined by eqaution (17.22), then the quadratic
mean of order r quantity index Q F is exactly equal
to the true quantity index, f r (q 1 ) / f r (q 0 ). 18 Since
Q r is exact for f r , and f r is a flexible functional
form, the quadratic mean of order r quantity index
Q r is a superlative index for each r ≠ 0. Thus, there
are an infinite number of superlative quantity indices.
17.47 For each quantity index Q r , the product
test in equation (15.3) can be used to define the
corresponding implicit quadratic mean of order r
price index P r *:
17 This method for justifying aggregation over commodities
is due to Shephard (1953, pp. 61–71). It is assumed that
f(q) is an increasing ,positive, and convex function of q for
positive q. Samuelson and Swamy (1974) and Diewert
(1980, pp. 438–42) also develop this approach to index
number theory.
18 See Diewert (1976; 130).
(17.25) P r *(p 0 ,p 1 ,q 0 ,q 1 )
≡
N
∑
i=
1
( , , , )
p q ⎡
⎣
p q Q p p q q
1 1 0 0 r 0 1 0 1
i i i i
⎤
⎦
r
( ) ( )
r* 1 * 0
= r p r p ,
where r r * is the unit revenue function that corresponds
to the aggregator function f r defined by
equation (17.22). For each r ≠0, the implicit quadratic
mean of order r price index P r * is also a superlative
index.
17.48 When r = 2, Q r defined by equation
(17.23) simplifies to Q F , the Fisher ideal quantity
index, and P r * defined by equation (17.25) simplifies
to P F , the Fisher ideal price index. When r = 1,
Q r defined by equation (17.23) simplifies to:
(17.26) Q 1 (p 0 ,p 1 ,q 0 ,q 1 )
12
1
12
1
1 1
−
−
∑
n
n
0⎛ q ⎞
i
1 qi
s ∑
⎛ ⎞
i
s
0 i 0
i= 1 qi
i=
1 qi
⎡ ⎤ ⎡ ⎤
≡ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥
⎢⎣ ⎝ ⎠ ⎥⎦ ⎢⎣ ⎝ ⎠ ⎥⎦
=
N
1 1
12
1
12
−1
∑ pq
i i 1 1
−
⎡ N N
1
0 0⎛ i
q ⎞ ⎤ ⎡
i 1 1⎛ q ⎞ ⎤
=
i
⎢
N ∑pq
i i ⎜ pq
0 ⎟ ⎥ ⎢∑
i i ⎜ 0 ⎟ ⎥
0 0 i= 1 qi
i=
1 qi
pq
⎢ ⎝ ⎠ ⎥ ⎢ ⎝ ⎠ ⎥
∑ ⎣ ⎦ ⎣ ⎦
i i
i=
1
N
1 1
∑ pq
i i N
1 1
0 0 1
12
N
−
1 0 1
−12
i=
1 ⎡ ⎤ ⎡ ⎤
=
N ⎢∑pi ( qi qi ) ⎥ ⎢∑pi ( qi qi
) ⎥
0 0 i= 1 i=
1
pq ⎣ ⎦ ⎣ ⎦
∑ i i
i=
1
N
1 1
∑ pq
i i N
1 0 1
12
N
0 0 1
12
i=
1
=
N ∑pi ( qi qi ) ∑pi ( qi qi
)
0 0 i= 1 i=
1
∑ pq
i i
i=
1
N
1 1
∑ pq
i i
i=
1
0 1 0 1
= ⎡PW
( p , p , q , q
N
) ⎤
0 0 ⎣ ⎦
,
∑ pq
i i
i=
1
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
where P W is the Walsh price index defined previously
by equation (15.19) in Chapter 15. Thus P 1 *
is equal to P W , the Walsh price index, and hence it
is also a superlative price index.
448