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

9. PPI Calculation in Practice
(9.2)
I
0: t
D
1
=
n
1
n
∑
∑
p
p
t
i
0
i
• The third is the Jevons index, which is defined
as the unweighted geometric mean of the price
relative or relatives, which is identical with the
ratio of the unweighted geometric mean prices.
(9.3)
I
0: t
J
1 n
t
⎛ p ⎞
i
= ∏⎜
=
0 ⎟
⎝ pi
⎠
t
∏( pi
)
0
∏( pi
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1 n
1 n
The properties of the three indices are examined
and explained in some detail in Chapter 20. Here,
the purpose is to illustrate how they perform in
practice, to compare the results obtained by using
the different formulas, and to summarize their
strengths and weaknesses.
9.19 Each month-to-month index shows the
change in the index from one month to the next.
The chained month-to-month index links together
these month-to-month changes by successive multiplication.
The direct index compares the prices in
each successive month directly with those of the
reference month, January. By simple inspection of
the various indices, it is clear that the choice of
formula and method can make a substantial difference
in the results obtained. Some results are striking—in
particular, the large difference between the
chained Carli index for July and each of the direct
indices for July, including the direct Carli.
9.20 The properties and behavior of the different
indices are summarized in the following paragraphs
and explained in more detail in Chapter 20.
First, the differences between the results obtained
by using the different formulas tend to increase as
the variance of the price relatives, or ratios, increases.
The greater the dispersion of the price
movements, the more critical the choice of index
formula and method becomes. If the elementary aggregates
are defined in so that the price movements
within the aggregate are minimized, the results obtained
become less sensitive to the choice of formula
and method.
9.21 Certain features displayed by the data in
Table 9.1 are systematic and predictable and follow
from the mathematical properties of the indices. For
example, it is well known that an arithmetic mean is
always greater than, or equal to, the corresponding
geometric mean—the equality holding only in the
trivial case in which the numbers being averaged
are all the same. The direct Carli indices are therefore
all greater than the Jevons indices, except in
May and July when the four price relatives based on
January are all equal. In general, the Dutot index
may be greater or less than the Jevons index, but
tends to be less than the Carli index.
9.22 One general property of geometric means
should be noted when using the Jevons formula. If
any one observation out of a set of observations is
zero, its geometric mean is zero, whatever the values
of the other observations. The Jevons index is
sensitive to extreme falls in prices, and it may be
necessary to impose upper and lower bounds on the
individual price relatives of, say, 10 and 0.1, respectively,
when using the Jevons. Of course, extreme
observations are often the results of errors of
one kind or another, and so extreme price movements
should be carefully checked in any case.
9.23 Another important property of the indices
illustrated in Table 9.1 is that the Dutot and the
Jevons indices are transitive, whereas the Carli index
is not. Transitivity means that the chained
monthly indices are identical with the corresponding
direct indices. This property is important in
practice, because many elementary price indices are
in fact calculated as chain indices that link together
the month-to-month-indices. The intransitivity of
the Carli index is illustrated dramatically in Table
9.1, in which each of the four individual prices in
May returns to the same level as it was in January,
but the chained Carli index registers an increase of
almost 14 percent over January. Similarly, in July,
although each individual price is exactly 10 percent
higher than in January, the chained Carli index registers
an increase of 29 percent. These results would
be regarded as perverse and unacceptable in the
case of a direct index, but even in the case of the
chained index, the results seems so intuitively unreasonable
as to undermine the credibility of the
chained Carli index. The price changes between
March and April illustrate the effects of “price
bouncing,” in which the same four prices are observed
in both periods, but they are switched between
the different products. The monthly Carli index
from March to April increases, whereas both
the Dutot and the Jevons indices are unchanged.
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