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

Producer Price Index Manual
Laspeyres and Paasche indices are equally justifiable,
with the Fisher index having support on axiomatic
grounds. If the conditional intermediate input
cost function takes the form of a translog technology,
the theoretical intermediate input price index is
exactly given by a Törnqvist index, which is superlative.
If separability is invoked, Fisher and Walsh
indices are also shown to be superlative, and the
three indices closely approximate each other.
1.119 The third index is the value-added deflator.
The analysis is based on the maximization of a net
revenue function, a function that relates output
revenue less intermediate input costs to sets of output
prices, input prices, and given primary inputs
and technology. The results follow those from using
a revenue function for the output price index.
Laspeyres and Paasche indices are lower and upper
bounds on their respective theoretical value-added
deflators, and a family of theoretical value-added
deflators could be defined that lie between them.
The Fisher index again has some support as a symmetric
average on axiomatic grounds, although the
Törnqvist index is shown, using fairly weak assumptions,
to correspond to a flexible translog
functional form for the net revenue function and is,
therefore, superlative. This finding requires no assumption
of the more restrictive constant returns to
scale that are necessary for Fisher and Walsh indices,
analogous to those for the output price index.
F. Aggregation Issues
1.120 It has been assumed up to this point that
the theoretical PPI is based on the technology of a
single representative establishment. Chapter 18 examines
the extent to which the various conclusions
reached above remain valid for PPIs that are actually
compiled for industries or the overall economy.
The general conclusion is that essentially the same
relationships hold at an aggregate level, although
some additional issues arise that may require additional
assumptions.
1.121 That there are three possible PPIs requires
an examination of how they relate to each other. It
is thus necessary to consider how the value-added
deflator is related to the output price and the intermediate
input price indices, and how the output
price index and the intermediate input price index
can be combined in order to obtain a value-added
deflator. It is shown in Chapter 18 that when the
Laspeyres output price index is used to separately
deflate outputs and the Laspeyres input price index
is used to separately deflate inputs—double deflation—at
each stage of aggregation, the results are
the same as when Laspeyres is used to aggregate in
one single stage. The separate deflation of inputs by
the input price index and outputs by the output price
index make up the components of the doubledeflated
value-added index. The same applies for
the Paasche index. However, if superlative price indices
are used, there are some small inconsistencies.
It was noted previously that unlike superlative indices,
Laspeyres and Paasche indices may suffer from
serious substitution bias. They may add up, but not
to the right number. A value-added deflator equivalent
to the separate Laspeyres (Paasche) deflation of
output and input indices is shown as a weighted
“average” of the Laspeyres (Paasche) output price
index, and the Laspeyres (Paasche) intermediate input
price index, although the weights used to combine
the input and output deflators are rather unusual.
1.122 But how do we derive estimates of doubledeflated
value added There is an equivalence between
a number of methods. Using the product rule,
a value ratio divided (deflated) by a Laspeyres
value-added deflator generates a Paasche valueadded
quantity index; or, correspondingly, a value
ratio divided by a Paasche value-added deflator
generates a Laspeyres value-added quantity index.
An alternative approach yielding equivalent results
is to take value added in, say, period 0 at period 0
prices and escalate (multiply) it by a series of
Laspeyres value-added quantity indices. The resulting
series of value added at constant period 0 prices
will be identical to the results from separately escalating
the value of inputs and outputs by their respective
Laspeyres input and output quantity indices
and subtracting the (escalated) former from the
latter. More usually, estimates of value added at
constant prices are derived by deflation. Deflating a
series of nominal current period value added by a
series of Paasche value-added indices yields a series
of value added at constant prices. This is equivalent
to double deflation: the separate deflation of the inputs
and output current period values by their respective
input and output separate Paasche price indices,
subtracting the former from the latter. Similar
equivalence results can be found using the less
well-known approach for a comparison between periods
0 and 1 of deflating the period 1 values by a
Paasche quantity index to provide a measure of current
period 0 quantities at period 1 prices. These
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