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

Producer Price Index Manual
B.6.2 Some alternative index formulas
9.67 Another widely used type of average is the
harmonic mean. In the present context, there are
two possible versions: either the harmonic mean of
price relatives or the ratio of harmonic mean of
prices.
9.68 The harmonic mean of price relatives, or
relatives, is defined as
(9.6)
I
0: t
HR
=
1
n
1
p
p
∑
0
i
t
i
.
The ratio of harmonic mean prices is defined as
(9.7)
I
n p
∑ 0
0: t
i
RH
=
t
∑n pi
.
Equation (9.7), like the Dutot index, fails the commensurability
test and would be an acceptable possibility
only when the products are all fairly homogeneous.
Neither formula appears to be used much
in practice, perhaps because the harmonic mean is
not a familiar concept and would not be easy to explain
to users. However, at an aggregate level, the
widely used Paasche index is a weighted harmonic
average.
9.69 The ranking of the three common types of
mean is always
arithmetic mean ≥ geometric mean ≥ harmonic
mean.
It is shown in Chapter 20 that, in practice, the Carli
index, the arithmetic mean of the relatives, is likely
to exceed the Jevons index, the geometric mean, by
roughly the same amount that the Jevons exceeds
the harmonic mean, equation (9.6). The harmonic
mean of the price relatives has the same kinds of
axiomatic properties as the Carli but with opposite
tendencies and biases. It fails the transitivity and
time reversal tests discussed earlier. In addition it is
very sensitive to “price bouncing,” as is the Carli
index. As it can be viewed conceptually as the
complement, or rough mirror image, of the Carli
index, it has been argued that a suitable elementary
index would be provided by a geometric mean of
the two, in the same way that, at an aggregate level,
a geometric mean is taken of the Laspeyres and
Paasche indices to obtain the Fisher index. Such an
index has been proposed by Carruthers, Sellwood,
Ward, and Dalén—namely,
(9.8)
I = I ⋅ I
0: t 0: t 0: t
CSWD C HR
I CSWD is shown in Chapter 20 to have very good
axiomatic properties but not quite as good as Jevons
index, which is transitive, whereas the I CSWD is not.
However, it can be shown to be approximately transitive
and, empirically, it has been observed to be
very close to the Jevons index.
9.70 More recently, as attention has focused on
the economic characteristics of elementary aggregate
formulas, consideration has been given to formulas
that allow for substitution between products
within an elementary aggregate. The increasing use
of the geometric mean is an example of this. However,
the Jevons index is limited to a functional
form that reflects an elasticity of demand equal to
one that, while clearly allowing for some substitution,
is unlikely to be applicable to all elementary
aggregates. A logical step is to consider formulas
that allow for different degrees of substitution in
different elementary aggregates. One such formula
is the unweighted Lloyd-Moulton formula
(9.9)
I
0: t
LM
1
1−σ
1−σ
⎡
t
1 ⎛ P ⎞ ⎤
i
= ⎢∑ ⎜ ⎥
0 ⎟ ,
⎢ n⎝P
⎣
i ⎠ ⎥
⎦
where σ is the elasticity of substitution. The Carli
and the Jevons indices can be viewed as special
cases of the I LM in which σ = 0 and σ = 1. The advantage
of the I LM formula is that σ is unrestricted.
Provided a satisfactory estimate can be made of σ,
the resulting elementary price index is likely to approximate
the Fisher and other superlative indices.
It reduces substitution bias when the objective is to
estimate an economic index. The difficulty is in the
need to estimate elasticities of substitution, a task
that will require substantial development and maintenance
work. The formula is described in more detail
in Chapter 20.
B.7 Unit value indices
9.71 The unit value index is simple in form. The
unit value in each period is calculated by dividing
total revenue on some product by the related total
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