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

17. Economic Approach
17.49 Suppose the producer’s unit revenue function
19 is the following quadratic mean of order r
unit revenue function: 20
(17.27) r r ⎡
(p 1 ,…,p n ) ≡ ⎢
⎣
N N
1 r
r 2 r 2⎤
∑∑ bik pi pk
⎥
i= 1 k=
1
where the parameters b ik satisfy the symmetry conditions
b ik = b ki for all i and k and the parameter r
satisfies the restriction r ≠ 0. Diewert (1976, p.
130) showed that the unit revenue function r r defined
by equation (17.27) is a flexible functional
form; i.e., it can approximate an arbitrary twice
continuously differentiable linearly homogeneous
functional form to the second order. Note that
when r = 2, r r equals the homogeneous quadratic
function defined by equation (17.16) above.
17.50 Define the quadratic mean of order r price
index P r by:
(17.28) P r (p 0 ,p 1 ,q 0 ,q 1 )
where
2
1
2
1
1
r
r
1
−r
− r
∑
n
n
0⎛ p ⎞
i
1 pi
s ∑
⎛ ⎞
i
s
0 i 0
i= 1 pi
i=
1 pi
⎡ ⎤ ⎡ ⎤
≡ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥
⎢
⎣ ⎝ ⎠ ⎥
⎦
⎢
⎣ ⎝ ⎠ ⎥
⎦
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, P r simplifies into P F , the Fisher ideal
price index.
17.51 Using exactly the same techniques as were
used in section C.3 above, it can be shown that P r
is exact for the unit revenue function r r defined by
(17.27); i.e.,
(17.29) P r (p 0 ,p 1 ,q 0 ,q 1 r t r 0
) r ( p ) r ( p )
⎦
= .
riods 0 and 1 and has technologies that are homogeneously
weakly separable where the output aggregator
function f(q) corresponds to the unit revenue
function r r (p) defined by (17.27), then the
quadratic mean of order r price index P r is exactly
equal to the true output price index, r r (p 1 )/r r (p 0 ). 21
Since P r is exact for r r and r r is a flexible functional
form, that the quadratic mean of order r
price index P r is a superlative index for each r ≠ 0.
Thus there are an infinite number of superlative
price indices.
17.52 For each price index P r , the product test
(15.3) can be used in order to define the corresponding
implicit quadratic mean of order r quantity
index Q r *:
(17.30) Q r *(p 0 ,p 1 ,q 0 ,q 1 )
N
r
≡ ∑ piqi pi qi
P p , p , q , q
1 1 { 0 0 ( 0 1 0 1
)}
r
( ) ( )
i=
1
r* 1 * 0
= f p f p
where f r * is the aggregator function that corresponds
to the unit cost function r r defined by
(17.27) above. 22 For each r ≠ 0, the implicit quadratic
mean of order r quantity index Q r * is also a
superlative index.
17.53 When r = 2, P r defined by (17.28) simplifies
to P F , the Fisher ideal price index and Q r * defined
by (17.30) simplifies to Q F , the Fisher ideal
quantity index. When r = 1, P r defined by (17.28)
simplifies to:
(17.31) P 1 (p 0 ,p 1 ,q 0 ,q 1 )
12
1
12
1
1 1
−
−
∑
n
n
0⎛ p ⎞
i
1⎛ p ⎞
i
si
⎜ s
0 ⎟ ∑ i ⎜ 0 ⎟
i= 1 pi
i=
1 pi
⎡ ⎤ ⎡ ⎤
≡ ⎢ ⎥ ⎢ ⎥
⎢⎣ ⎝ ⎠ ⎥⎦ ⎢⎣ ⎝ ⎠ ⎥⎦
Thus under the assumption that the producer engages
in revenue maximizing behavior during pe-
19 Again the approach here is by way of a unit revenue
function and, though an alternative is via a quadratic mean
of order r superlative quantity index which, using the product
rule, in turn defines an implicit quadratic mean of order
r price index is also a superlative index.
20 This terminology is credited to Diewert (1976, p. 130).
This functional form was first defined by Denny (1974) as
a unit cost function.
21 See Diewert (1976, pp. 133–34).
22 The function f r * can be defined by using r r as follows: f
n
⎧
r ⎫
⎨∑ pq
i i: r p = 1⎬
.
⎩ i=
1
⎭
r *(q) ≡ max p ( )
449