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

1. An Introduction to PPI Methodology
PPI with those in a subsequently calculated superlative
version may be helpful in evaluating and interpreting
movements in the official PPI. It may be
that revenue data can be collected from establishments
alongside price data, and this is to be encouraged
so that Fisher PPI indices may be calculated in
real time for at least some industrial sectors. To the
extent that revenue data are available on an annual
basis, annual chain Laspeyres indices could be produced
initially and Fisher or Törnqvist indices produced
subsequently as the new revenue weights become
available. The advantage of the annual updating
is that chaining helps to reduce the spread between
the Laspeyres and Paasche indices.
1.115 Section B.7 of Chapter 17 notes that, in
practice, PPIs are usually calculated in stages (see
Chapters 9 and 20) and addresses the question of
whether indices calculated this way are consistent
in aggregation—that is, have the same values
whether calculated in a single operation or in two
stages. The Laspeyres index is shown to be exactly
consistent, but superlative indices are not. However,
the widely used Fisher and Törnqvist indices are
shown to be approximately consistent.
E.4 Allowing for substitution
1.116 Section B.8 of Chapter 17 examines one
further index proposed recently, the Lloyd-Moulton
index, P LM , defined as follows:
(1.16)
P
LM
⎧
⎪
≡⎨
⎪⎩
n
∑
i=
1
t
0
⎛ p ⎞
i
si
⎜ 0 ⎟
⎝ pi
⎠
1
1−σ
1−σ
⎫
⎪
⎬
⎪⎭
σ≠1
The parameter σ, which must be nonpositive for the
output PPI, is the elasticity of substitution between
the products covered. It reflects the extent to which,
on average, the various products are believed to be
substitutes for each other. The advantage of this index
is that it may be expected to be free of substitution
bias to a reasonable degree of approximation,
while requiring no more data, except for an estimate
of the parameter σ, than the Laspeyres index. It is
therefore a practical possibility for PPI calculation,
even for the most recent periods. However, it is
likely to be difficult to obtain a satisfactory, acceptable
estimate of the numerical value of the elasticity
of substitution, the parameter used in the formula.
E.5 Intermediate input price indices
and value-added deflators
1.117 Having considered the theory and appropriate
formula for output price indices, Chapter 17
turns to intermediate input price indices (Section C)
and to value added deflators (Section D). The behavioral
assumption behind the theory of the output
price index was one of producers maximizing a
revenue function. An input price index is concerned
with the price changes of intermediate inputs, and
the corresponding behavioral assumption is the
minimization of a conditional cost function. The
producer is held to minimize the cost of intermediate
inputs in order to produce a set of outputs, given
a set of intermediate inputs prices and that primary
inputs and technology are fixed. These are fixed so
that hypothetical input quantities can be generated
from a fixed setup that allows the input quantities in
period 1 to reflect the producer buying more of
those inputs that have become cheaper. Theoretical
intermediate input price indices are defined as ratios
of hypothetical intermediate input costs that the
cost-minimizing producer must pay in order to produce
a fixed set of outputs from technology and
primary inputs fixed to be the same for the comparison
in both periods. As was the case with the
theory of the output price index, theoretical input
indices can be derived on the basis of either fixed
period 0 technology and primary inputs, or fixed
period 1 technology and primary inputs, or some
average of the two. The observable Laspeyres index
of intermediate input prices is shown to be an upper
bound to the theoretical intermediate input price index
based on period 0 technology and inputs. The
observable Paasche index of intermediate input
prices is a lower bound to its theoretical intermediate
input price index based on period 1 fixed technology
and inputs. Note that these inequalities are
the reverse of the findings for the output price index,
but that they are analogous to their counterparts
in the CPI for the theory of the true cost-ofliving
index, which is also based on an expenditure
(cost) minimization problem.
1.118 Following the analysis for the output price
index, a family of intermediate input price indices
can be shown to exist based on an average of period
0 and period 1 technologies and inputs leading
to the result that Laspeyres (upper) and Paasche
(lower) indices are bounds on a reasonable theoretical
input index. A symmetric mean of the two
bounds is argued to be applicable given that
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