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

20. Elementary Indices
approach helps identify what the target index
should be.
20.46 Suppose that each establishment producing
products in the elementary aggregate has a set
of inputs, and the linearly homogeneous aggregator
function f(q) describes what output vector q ≡
[q 1 ,...,q M ] can be produced from the inputs. Further
assume that each establishment engages in revenue-maximizing
behavior in each period. Then, as
was seen in Chapter 17, it can be shown that that
certain specific functional forms for the aggregator
f(q) or its dual unit revenue function R(p) 13 lead to
specific functional forms for the price index,
P(p 0 ,p 1 ,q 0 ,q 1 ), with
(20.31) P(p 0 ,p 1 ,q 0 ,q 1 ) =
1
( )
R p
0
( )
R p
20.47 Suppose that the establishments have aggregator
functions f defined as follows 14 :
(20.32) f(q 1 ,...,q M ) ≡ max m {q m /α m : m = 1,...,M},
where the α m are positive constants. Then under
these assumptions, it can be shown that Equation
(20.31) becomes 15
(20.33)
1
( )
R p
0
( )
R p
=
pq
1 0
0 0
p q = 1 1
.
pq
0 1
p q ,
and the quantity vector of products produced during
the two periods must be proportional; that is,
(20.34) q 1 = λq 0 for some λ > 0.
20.48 From the first equation in formula (20.33),
it can be seen that the true output price index,
R(p 1 ) /R(p 0 ), under assumptions of formula (20.32)
about the aggregator function f, is equal to the
Laspeyres price index, P L (p 0 ,p 1 ,q 0 ,q 1 ) ≡ p 1 ⋅q 0 /
p 0 ⋅q 0 . The Paasche formula P P (p 0 ,p 1 ,q 0 ,q 1 ) ≡
p 1 q 1 /p 0 q 1 is equally justified under formula (20.34).
20.49 Formula (20.32) on f thus justifies the
Laspeyres and Paasche indices as being the “true”
13 The unit revenue function is defined as R(p) ≡ max q
{p⋅q : f(q) = 1}.
14 The preferences that correspond to this f are known as
Leontief (1936) or no substitution preferences.
15 See Pollak (1983).
elementary aggregate from the economic approach
to elementary indices. Yet this is a restrictive assumption,
at least from an economic viewpoint,
that relative quantities produced do not vary with
relative prices. Other less restrictive assumptions
on technology can be made. For example, as
shown in Section B.3, Chapter 17, certain assumptions
on technology justify the Törnqvist price index,
P T , whose logarithm is defined as
( )
0 1
(20.35) ln P T (p 0 ,p 1 ,q 0 ,q 1 1
) ≡ ∑ M si + si ⎛ p ⎞
i
ln ⎜ 0 ⎟
i=
1 2 ⎝ p
.
i ⎠
20.50 Suppose now that product revenues are
proportional for each product over the two periods
so that
(20.36) p m 1 q m 1 = λ p m 0 q m 0 for m = 1,...,M and for
some λ > 0.
Under these conditions, the base period revenue
shares s m 0 will equal the corresponding period 1
revenue shares s m 1 , as well as the corresponding
β(m); that is, formula (20.36) implies
(20.37) s m 0 = s m 1 ≡ β(m) ; m = 1,...,M.
Under these conditions, the Törnqvist index reduces
to the following weighted Jevons index:
β( m)
(20.38) P J (p 0 ,p 1 1
,β(1),…,β(M)) = ∏ M
⎛ p ⎞
m
⎜ 0
p ⎟
m=
1 ⎝ m ⎠
.
20.51 Thus, if the relative prices of products in a
Jevons index are weighted using weights proportional
to base (which equals current) period revenue
shares in the product class, then the Jevons index
defined by formula (38) is equal to the following
approximation to the Törnqvist index:
0
s m
(20.39) P J (p 0 ,p 1 ,s 0 1
) ≡ ∏ M
⎛ p ⎞
m
⎜ 0
p ⎟
m=
1 ⎝ m ⎠
.
20.52 In Section G, the sampling approach
showS how, under various sample designs, elementary
index number formulas have implicit
weighting systems. Of particular interest are sample
designs where products are sampled with probabilities
proportionate to quantity or revenue
shares in either period. Under such circumstances,
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