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

Producer Price Index Manual
• The harmonic average of the price relatives, or
P H .
As just noted, P C ≥ P J ≥. P H .
1.144 Using the second of the options, three possible
indices are:
• The ratio of the arithmetic average prices,
known as the Dutot index, or P D ;
• The ratio of the geometric averages, again the
Jevons index, or P J ;
• The ratio of the harmonic averages, or R H .
The ranking of ratios of different kinds of average
are not predictable. For example, the Dutot, P D ,
could be greater or less than the Jevons, P J .
1.145 The Dutot index can also be expressed as a
weighted average of the price relatives, in which the
prices of period 0 serve as the weights:
(1.17)
n
n
t
0
∑ pi
n ∑ pi
⎜
i=
1
i=
1
D
≡ =
⎝
n n
0 0
∑pi
n ∑pi
i= 1 i=
1
P
t
⎛ p ⎞
i
0 ⎟
pi
⎠
.
As compared with the Carli, which is a simple average
of the price relatives, the Dutot index gives
more weight to the price relatives for the products
with high prices in period 0. However, it is difficult
to provide an economic rationale for this kind of
weighting. Prices are not revenues. If the products
are homogeneous, very few quantities are likely to
be purchased at high prices if the same products can
be purchased at low prices. If the products are heterogeneous,
the Dutot should not be used anyway,
since the quantities are not commensurate and not
additive.
1.146 Noting that P C ≥ P J ≥. P H , it is shown in
Section D of Chapter 20 that the gaps between these
indices widen as the variance of the price relatives
increases. The choice of formula becomes more
important the greater the diversity of the price
movements. Moreover, both P D and P J can be expected
to lie approximately halfway between P C
and P H . While it is useful to establish the interrelationships
between the various indices, they do not
actually help decide which index to choose. However,
because the differences between the various
formulas tend to increase with the dispersion of the
price relatives, it is clearly desirable to define the
elementary aggregates in such a way as to try to
minimize the variation in the price movements
within each aggregate. The less variation there is,
the less difference the choice of index formula
makes. Since the elementary aggregates also serve
as strata for sampling purposes, minimizing the
variance in the price relatives within the strata will
also reduce the sampling error.
I.4 The axiomatic approach to elementary
indices
1.147 One way to decide between the various elementary
indices is to exploit the axiomatic approach
outlined earlier. A number of tests are applied to the
elementary indices in Section E of Chapter 20.
1.148 The Jevons index, P J , satisfies all the selected
tests. It dominates the other indices in the
way that the Fisher tends to dominate other indices
at an aggregative level. The Dutot index, P D , fails
only one, the commensurability test. This failure
can be critical, however. It reflects the point made
earlier that when the quantities are not economically
commensurate, their prices should not be averaged.
However, P D performs well when the sampled
products are homogeneous. The key issue for
the Dutot is therefore how heterogeneous are the
products within the elementary aggregate. If the
products are not sufficiently homogeneous for their
quantities to be additive, the Dutot index should not
be used.
1.149 The Carli index, P C , is widely used, but the
axiomatic approach shows that it has some undesirable
properties. In particular, as the unweighted
version of the Young index, it fails the commodity
reversal, the time reversal, and the transitivity tests.
These are serious disadvantages, especially when
month-to-month indices are chained. A consensus
has emerged that the Carli may be unsuitable because
it is liable to have a significant upward bias.
This is illustrated by numerical example in Chapter
9. Its use is not sanctioned for the Harmonized Indices
of Consumer Prices used within the European
Union. Conversely, the harmonic average of the
price relatives, P H , is liable to have an equally significant
downward bias, although it does not seem
to be used in practice anyway.
1.150 On the axiomatic approach, the Jevons index,
P J , emerges as the preferred index. However,
its use may not be appropriate in all circumstances.
28