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

Producer Price Index Manual
theoretical output price index, but that a Laspeyres
index is calculated instead for practical reasons.
One important conclusion to be drawn from this
preliminary analysis is that the PPI may be expected
to have a downward bias. Similarly, a series
of Paasche PPIs used to deflate a series of output
values at current prices generates a series of values
at constant period 0 prices (Laspeyres volume index),
which in turn will also suffer from a downward
bias. The approach informs us that there are
two equally valid theoretical economic price indices,
and that the bound, while useful, only show
how Laspeyres and Paasche indices compare with
their own theoretical counterparts. What we require
are two-sided bounds on the theoretically justified
index.
E.3 Estimating theoretical output
indices by superlative indices
1.99 The next step is to establish whether there
are special conditions under which it may be possible
to exactly measure a theoretical PPI. In Section
B.2 of Chapter 17 theoretical indices based on
weighted “averages” of the period 0 and period 1
technology and similarly weighted averages of the
period 0 and 1 inputs are considered. These theoretical
indices deal adequately with substitution effects;
that is, when an output price increases, the
producer’s supply increases, holding inputs and the
technology constant. Such theoretical indices are
argued to generally fall between the Laspeyres
(lower bound) and Paasche (upper bound) indices.
The Fisher index, as the geometric mean of the
Laspeyres and Paasche indices, is the only symmetric
average of Laspeyres and Paasche that satisfies
the time reversal test. Thus, economic theory was
used to justify Laspeyres and Paasche bounds, and
axiomatic principles led to the Fisher price index as
the best symmetric average of these bounds.
1.100 In Section B.3 of Chapter 17 the case for
the Törnqvist index number formula is presented. It
is assumed that the revenue function takes a specific
mathematical form: a translogarithmic function.
If the price coefficients of this translog form
are equal across the two periods being compared,
then the geometric mean of the economic output
price index that uses period 0 technology and the
period 0 input vector, and the economic output
price index that uses period 1 technology and the
period 1 input vector, are exactly equal to the Törnqvist
output price index. The assumptions required
for this result are weaker than other subsequent assumptions;
in particular, there is no requirement
that the technologies exhibit constant returns to
scale in either period. The ability to relate an actual
index number formula (Törnqvist) to a specific
functional form (translog) for the production technology
is a powerful analytical device. Statisticians
using particular index number formulas are in fact
replicating particular mathematical descriptions of
production technologies. A good formula should
not correspond to a restrictive functional form for
the production technology.
1.101 Diewert (1976) described an index number
formula to be superlative if it is equal to a theoretical
price index whose functional form is flexible—
it can approximate an arbitrary technology to the
second order. That is, the technology by which inputs
are converted into output quantities and revenues
is described in a manner that is likely to be realistic
of a wide range of forms. Relating a class of
index number formulas to technologies represented
by flexible functional forms is another powerful
finding, since it gives credence to this class of index
number formulas. Note also that the translog functional
form is an example of a flexible functional
form, so the Törnqvist output price index number
formula is superlative. In contrast to the theoretical
indices, a superlative index is an actual index number
that can be calculated. The practical significance
of these results is that they give a theoretical
justification for expecting a superlative index to
provide a fairly close approximation to the unknown,
underlying theoretical index in a wide range
of circumstances.
1.102 In Section B.4 the Fisher index is revisited
from a purely economic approach. An additional
assumption is invoked, that outputs are homogeneously
separable from other commodities in the production
function: if the input quantities vary, the
output quantities vary with them, so that the new
output quantities are a uniform expansion of the old
output quantities. It is shown that a homogeneous
quadratic utility function is flexible and corresponds
to the Fisher index. The Fisher output price
index is therefore also superlative. This is one of
the more famous results in index number theory.
Although it is generally agreed that it is not plausible
to assume that a production technology would
have this particular functional form, this result does
at least suggest that, in general, the Fisher index is
likely to provide a close approximation to the un-
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