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

1. An Introduction to PPI Methodology
output of products from the technology—it would
be a mathematical representation of the technology
that converts inputs to outputs. The prevailing relative
prices would dictate exactly how much of each
product is produced. The economic approach to the
PPI relies on the assumption of optimizing behavior
on the part of producers in competitive, price-taking
markets so that they respond to relative price
changes. In this approach, while actual prices are
considered for both periods, the quantities in each
period may not be the observed ones. They are generated
from a given period’s production function
(with fixed technology) and level of inputs, using
assumptions of maximizing behavior and dictated
by relative prices, which may be the ones in another
period. This is a powerful analytical framework because
it allows us to consider, at least in theory,
how quantities would respond to different price regimes
(say, period 1 prices) under constant (say, period
0) reference technologies and inputs. They are
hypothetical quantities that cannot be observed, but
are generated in a mathematical model so that their
formulation can be compared with real index number
formulas based on observable prices and quantities.
1.95 Pure price index number formulas (based
on observed data) and theoretical indices have in
common that they may both be defined as the ratios
of revenues in two periods. However, by definition,
while the quantities are fixed in pure price indices,
they vary in response to changes in relative prices
in theoretical indices. In contrast to the axiomatic
approach to index theory, the economic approach
recognizes that the quantities produced are actually
dependent on the prices. In practice, rational producers
may be expected to adjust the relative quantities
they produce in response to changes in relative
prices. A theoretical PPI assumes that a producer
seeking to maximize revenues will make the
necessary adjustments. The baskets of goods and
services in the numerator and denominator of a
theoretical PPI are not, therefore, exactly the same.
E.2 Upper and lower bounds on a
theoretical output price index
1.96 The theoretical price index between periods
0 and 1 is the ratio of revenues in those periods
using fixed technology and inputs. Consider an index
that held the technology and inputs constant in
period 0. The revenue generated in period 0 from
period 0 prices using period 0 technology and inputs
is what actually happened: the denominator of
the theoretical ratio is the observed revenue, assuming
the producer was optimizing revenue. The numerator
is period 1 prices multiplied by the hypothetical
quantities that would have been produced
using the same period 0 technology and inputs, had
period 1 prices prevailed. It is not, as in the
Laspeyres index, period 1 prices multiplied by the
actual quantities produced at period 0 prices using
period 0 technology and inputs. Both the theoretical
and the Laspeyres indices use the same period 0
technology and inputs, but the theoretical index
generates quantities from it as if period 1 prices
prevailed, whereas the Laspeyres index uses the actual
period 0 quantities. In practice, relative prices
may change between the two periods, so the quantities
generated will be different. Higher revenue
could be achieved by substituting, at least marginally,
some products that have relatively high price
changes for some that have relatively low ones. The
theoretical index based on period 0 technology and
inputs will take account of this and will increase by
more than the Laspeyres index. The theoretical index
will be at least equal to or greater than the
Laspeyres, since the producer has the possibility of,
at worst, producing the same set of products as in
period 0. Being a revenue maximizer, it is assumed
the producer will substitute products with relatively
high price changes—the Laspeyres index thus incurs
a ”substitution bias.”
1.97 By a similar line of reasoning, it can be
shown that when relative prices change, the theoretical
output price index based on period 1 technology
and inputs will increase by less than the
Paasche index. In other words, as shown in Chapter
17, Section B.1, the Laspeyres index provides a
lower bound to its (period 0) theoretical index, and
the Paasche an upper bound to its (period 1) theoretical
index. Note that these inequalities are in the
opposite direction to their CPI cost-of-living index
counterparts. This is because the optimization problem
in the cost-of-living theory is a cost minimization
problem as opposed to the present revenue
maximization problem.
1.98 The practical significance of these results
stems from the fact that the Laspeyres and Paasche
indices can be calculated directly from the observed
prices and quantities, whereas the theoretical indices
cannot, thus giving some insight into the bias
involved in the use of these two formula. Suppose
the official objective is to estimate a base period
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