The Experimental CPI using Geometric Means (CPIUXG)
I. Overview
Effective April 10, 1997, the Bureau of Labor Statistics (BLS)
is releasing an experimental Consumer Price Index for All Urban
Consumers that uses a geometric mean formula to combine
individual price quotations at the lower level of aggregation
while keeping the current Laspeyres arithmetic mean formula for
higher level aggregation. This experimental index, called the
experimental CPI using geometric means, or CPIUXG, is an
additional index and does not replace the official CPI. The
CPIUXG indexes are available from December 1990 through the
present, and BLS currently is evaluating the full or partial
adoption of the geometric mean formula in the official CPI.
The official CPI uses a modified version of the Laspeyres
formula at both levels of aggregation. In a Laspeyres index,
relative price change R from the base period (time 0) to
comparison time t is measured by comparing the sum of the
weighted prices in the comparison period with the sum of the
weighted prices in the base period. Algebraically, the formula
is:
where P_{t,i} is the price of the ith
item in comparison period t, P_{0,i}
is the price in the base period 0,
Q_{0,0i }is the quantity consumed in the
base period 0, and indicates the summation across items. Since price
times quantity equals expenditure, the denominator is the total
amount of money people spent on items at time 0, and the
numerator is the total amount of money they would have spent, had
they chosen to buy the exact same items in the exact same
quantities they did at time 0.
With geometric means, price change R is measured as
where S_{0,i}is the baseperiod
expenditure share associated with each item in the sample, and
indicates the product of the relative price changes across items.
The relationship to the Laspeyres index can be seen by rewriting
the Laspeyres formula as an arithmetic mean of price relatives,
also weighted by baseperiod expenditure shares:
For a given set of expenditure weights and prices, the
geometric mean measure of average price change R^{G}will
lie below the Laspeyres index R^{L}(unless all prices
change in the same proportion, in which case both formulas yield
the same answer). For example, suppose there are two
equallyweighted items sampled within an individual item
category, a pound of iceberg lettuce and a pound of Romaine
lettuce. Assume that both prices are $1.00 in time period 0, but
that the price of Romaine lettuce increases to $1.50 in time
period t:




price relative 
base period
expenditures 
relative
expenditure share 

Q_{0,i}

P_{0,i}

P _{t,i }

(P _{t,i }/P _{0,i }) 
(Q _{0,i }*P _{0,i }) 
in time 0 (S _{0,i }) 
Iceberg lettuce 
1 
$1.00 
$1.00 
1.0 
$1.00 
0.5 
Romaine lettuce 
1 
$1.00 
$1.50 
1.5 
$1.00 
0.5 
In the official Laspeyres CPI, the expenditure for the fixed
marketbasket would increase from $2.00 in the base period to
$2.50 in month t, as [($1.00*1 + $1.50*1)]/[($1.00*1 +
$1.00*1)] = $2.50/$2.00 = 1.25, representing a price increase of
25 percent. Using geometric means, the price change would be 22.5
percent, as =
(1.00/1.00) ^{0.5 }*(1.50/1.00) ^{0.5 }=
1.0*1.225 = 1.225. If both types of lettuce had increased from
$1.00 to $1.50, then both formulas would have shown a 50 percent
increase.
From a theoretical standpoint, the potential advantage of the
geometric mean index would be as a closer approximation to a
costofliving index. A costofliving index measures the change
in the cost of purchasing goods and services that will yield a
constant level of satisfaction to consumers. As a fixedweight
index, the Laspeyres formula can be viewed as an upper bound to a
true costofliving index because it does not reflect the fact
that consumers can and do change spending patterns as relative
prices change. If consumers respond to changes in relative prices
in such a way that relative expenditure shares remain constant,
an index using geometric means would equal the costofliving
index ^{1}. In the example above, the geometric mean
estimate equals the change in the costofliving index if the
purchase of 1.225 pounds of iceberg lettuce and 0.816 pounds of
Romaine lettuce in period t (so that the same dollar
amount is spent on each) yields the same satisfaction as the one
pound of each type purchased in period 0. Within any CPI item
category, however, the precise relationships among a Laspeyres
index, an index using geometric means, and a true costofliving
index are not known.
Section II of this announcement reviews the concepts and
construction of the official CPI in more detail. Section III
summarizes the concepts and construction of the experimental
CPIUXG. Section IV discusses the limitations of comparing the
official CPI to the CPIUXG and introduces a comparison
Laspeyres index called the CPIUXL. Section V presents the
results of the CPIUXG and compares those results to the
CPIUXL from December 1990 to the present. Finally, section VI
outlines future plans by the BLS for the evaluation of geometric
means.
II. How the official (Laspeyres) CPI is currently
constructed
To compute the CPI, the universe of consumer goods and
services is stratified into 9,108 basic indexes (207 item strata
times 44 geographic strata)^{ 2}. A twotiered weighting
system is used to calculate the CPI. The first tier of weights
are used during the calculation of the basic indexes to weight
the individual price quotations together. These weights are
derived from the PointofPurchase Survey (POPS). Basic indexes
are updated each month by the Laspeyresbased price relative:
where P is the price of an item and Q _{b} is the
quantity consumed during the base period corresponding to the
POPS survey. Only expenditures are collected during the outlet
and item selection process; however, base period quantities (Q _{b}
) are not directly available. To handle this, the base period
quantities are rewritten in terms of base period expenditures X,
as in
where
is the estimated "base period price" of the ith
sampled item. The base period price is estimated, not collected,
because the base period price is typically not available when the
new item samples are introduced into the CPI.
Each month in the official CPI, these expenditure estimates
are updated by the price relatives calculated for each of the
9,108 basic cells using the Laspeyres framework. It is at this
initial or basic level where the CPIUXG uses geometric means
instead of the Laspeyres arithmetic means in the estimation of
price relatives.
At the second or aggregate tier, base period expenditure
weights for each of the 9,108 basic indexes of the CPI are
derived from the Consumer Expenditure Survey. The current market
basket represents expenditures drawn from the Consumer
Expenditure Survey during 198284; these weights were introduced
into the CPI in December 1986. In both the official CPI as
well as the CPIUXG, basic indexes are aggregated across
basic itemarea categories using the Laspeyres method (of
arithmetic means) and not geometric means.
III. Geometric Means and the CPIUXG
As noted above, the Laspeyres approach currently used in the
CPI can be interpreted as an upper bound to a true costofliving
index in that it does not account for changes in consumption
induced by changes in relative prices. Item substitution occurs
at both the basic level (e.g., priceinduced substitution may
occur within an individual item category such as apples) and at
the aggregate level (e.g., between chicken and beef).
Historically, analysis of the effect of consumer substitution
in the CPI has focused on substitution at the second or aggregate
level. Most recently, Aizcorbe, Cage and Jackman (1996) and
Shapiro and Wilcox (1996), for example, have estimated the
substitution effect at the aggregate level by constructing
superlative Fisher or Tornqvist indexes, which use current as
well as base period Consumer Expenditure Survey weights^{ 3}.
At the basic index level, the calculation of a CPI using
geometric means is a potential mechanism for reflecting consumer
substitution behavior and thereby eliminating what has been
termed "lowerlevel substitution bias" in the Laspeyres
CPI^{ 4}. Superlative indexes cannot be constructed at
the basic level because only baseperiod expenditure information
is available corresponding to the individual prices being
aggregated. For the same reason, the geometric mean's implicit
assumption of constant expenditure shares is not testable, but in
many markets it is likely to be more plausible than the
assumption of fixed consumption quantities. The possibility of
using the geometric mean formula to calculate basic indexes in
the CPI was first raised by BLS researchers in 1993^{ 5}.
To assess the potential impact of using geometric means, Moulton
(1993) recalculated most nonshelter basic indexes from June 1992
through June 1993 utilizing geometric means instead of arithmetic
means. Later estimates by Moulton and Smedley (1995), which
include the shelter component of the CPI, cover the period from
June 1992 through December 1994. Differences between the two
indexes were most significant for food, apparel, and
entertainment items.
The earlier estimates by BLS researchers were not of
productiongrade quality in that some item categories were not
included in the estimates, and in that the methods of estimation
differed slightly from those used for the official CPI. The BLS
now has created the near productiongrade quality CPIUXG using
geometric means to calculate basic indexes from December 1990 to
the present.^{ 6} As noted earlier, to distinguish these
indexes from earlier estimates by BLS researchers, we call these
near productionquality indexes the experimental CPI using
geometric means, or CPIUXG.
IV. Comparability of series
It is important to note that there are several factors that
seriously reduce the meaningfulness of directly comparing the
official CPI with the CPIUXG, especially before 1996. In
particular, the CPIUXG incorporates changes other than that
associated with the geometric mean formula. These additional
factors can be characterized as "bounding" and
"methodological" effects. The geometric mean estimator
is quite sensitive to extremely large price decreases (the
geometric mean is undefined when any price equals zero, as can
occur in the CPI in rare cases when formerly priced items are
offered without charge). Unlike the official CPI, therefore, the
CPIUXG has special "bounding" rules for handling
extremely large percentage price changes. Specifically, within
most commodity and service categories, monthly prices are bounded
from above at 20 times the item's base period price, and from
below at 5 percent of the base price, to avoid extremely large
percentage price changes in the index. Similarly, any rents less
than $20 are reset to $20 in the CPIUXG.
In addition, there are significant "methodological"
effects present in the CPIUXG. BLS began the process of
constructing the CPIUXG in early 1996. Hence, CPIUXG indexes
from December 1990 through 1995 use the estimation methods in
place as of early 1996. In other words, any methodological
changes made between 1990 and 1995 potentially affect the
CPIUXG over the entire 19901995 period. These same
changes affect the official CPI only as they were implemented.
For example, in January 1995, improvements were made in the
imputation of owners' equivalent rents to eliminate an upward
bias in that estimator; at the same time, a sixmonth chain
estimator was incorporated to estimate rents (Bureau of Labor
Statistics, 1994b). While these owners' equivalent rent and
residential rent methodological changes affect the CPIUXG over
the entire period from 1990 forward, these same changes affect
the official CPI from January 1995 forward.
In order to better isolate the impact of using a Laspeyres
arithmetic means estimator versus a geometric means estimator in
the calculation of basic indexes, BLS created a Test Laspeyres
index, the CPIUXL, which incorporates the same bounding rules
described above for the CPIUXG. In addition, both the CPIUXL
and CPIUXG use the calculation methods in effect at the
beginning of 1996. That is, methodological changes are introduced
into use for the CPIUXL and the CPIUXG at the same time^{
7}. For these reasons, we will focus on the differences
between the CPIUXL and the CPIUXG.
V. Results
From December 1990 through February 1997, the CPIUXG rose
16.2 percent, which is equivalent to an annual growth rate of
2.46 percent. During that same time, the CPIUXL rose 18.6
percent, which is equivalent to an annual growth rate of 2.80
percent, for an annualized difference of 0.34 percent. Tables 1
and 2 give index levels from December 1990 through February 1997
for the CPIUXG and CPIUXL, respectively, for All Items, each
of the 7 major groups, and the categories for food, energy, and
all items less food and energy. Table 3 displays the differences
between the geometric mean and Test Laspeyres indexes by year for
the major CPI item groups. As in previous BLS research, the
differences between the two indexes prior to 1995 were greatest
for food, apparel, and entertainment items^{ 8}. The
relatively small differences in the housing component reflect, in
part, the incorporation in the CPIUXL throughout the study
period of the January 1995 changes to the shelter component of
the official CPIU^{ 9}.
It is important to note that the difference in the rate of
increase between the CPIUXG and the CPIUXL (as well as the
official CPI) has fallen since January 1995, when several
methodological improvements to the official CPI were made. For
example, in January 1995, an improvement was made in the
estimation of quantity weights for newly initiated items within
the CPI foodathome component (Bureau of Labor Statistics,
1994a). That reduced the rate of growth in the CPIUXL (as well
as the official CPI) indexes from January 1995 forward, but had
no effect on the CPIUXG, which is weighted not by quantities
but by expenditure shares. From December 1990 through December
1994, the annualized difference between the CPIUXL and CPIUXG
was approximately 0.37 percent, but from December 1994 through
February 1997, this difference fell to 0.28 percent. For the
foodathome category, the annualized difference between the
CPIUXL and CPIUXG fell from approximately 0.88 percent (Dec.
1990 to Dec. 1994) to 0.26 percent (Dec. 1994 to Feb. 1997).
In June 1996, BLS extended the foodathome improvement in the
estimation of quantity weights to other commodities and services;
in July 1996, a functional form bias occurring at item
substitution for all commodities and services was eliminated
(Bureau of Labor Statistics, 1996a and 1996b). The differences
between the CPIUXG and the CPIUXL (and official CPI) are
expected to decline further as these methodological changes
reduce the rate of growth in the CPIUXL (and official) indexes
without affecting the CPIUXG.
Graph 1 shows the difference
in the 12month changes for the CPI All Items index for the
CPIUXL and the CPIUXG. For comparison purposes, this graph
also shows the difference between the official CPI and the
CPIUXG over the same time frame. This graph shows that changes
in the official CPI over time have reduced the difference between
the official CPI and the CPIUXG.
Graph 2 shows the same
differences for the foodathome component of the CPI. In this
instance, the running 12month changes between the official CPI
and the CPIUXL are virtually indistinguishable. The effect of
the January 1995 functional form correction is very clear, as
differences between the Laspeyres indexes (both the official and
the CPIUXL) and the CPIUXG decline significantly.
As demonstrated in Graph 3,
the case of housing contrasts with that of food at home. The Test
Laspeyres index for housing exceeds the Geometric Mean index by a
relatively small and consistent amount over the study period. The
various changes since 1990 in the CPI treatment of residential
rent, owners' equivalent rent, and lodging while out of town are
visible in Graph 3 as differences between the official CPI series
and the CPIUXL for housing^{ 10}.
Finally, graphs 4 through 11 show,
for the CPIUXL and CPIUXG, the running 12month percent
changes for All Items and each of the seven major groups.
VI. Summary and future BLS plans for geometric means
While the official CPI reflects no consumer substitution
behavior, the CPIUXG reflects an assumed, but in many cases
reasonable, degree of substitution within individual item
categories. Methodological improvements made to the official CPI
since January 1995 have reduced the differences between it and
the CPIUXG. Nevertheless, the BLS estimates that an index using
geometric means for all basic indexes would increase
approximately onequarter of one percent a year less rapidly than
the official CPI, given the current environment of relatively
modest inflation. Partial adoption of the geometric mean formula
would be expected to have a downward impact of between zero and
onequarter of one percent per year, depending on how many, and
which, indexes continue to be based on the Laspeyres approach.
The BLS is now evaluating the adoption of a geometric mean
formula as the official CPI, and the CPIUXG will be used to
generate further research. Scanner data, studies of substitutions
between brands, and other information will be used to assess
elasticity of demand as the relative prices of items within
individual item categories change. The experiences of other
countries using geometric means also will be considered.
By the end of 1997, BLS will announce the findings of its
research, including its determination of which CPI basic indexes
are best calculated with the geometric mean formula and when the
implementation of any change will take place. The likely date for
that implementation is with the release of January 1999 CPI data.
Footnotes
^{1 }In mathematical terms, the geometric mean index
equals the costofliving index if all elasticities of
substitution in consumption between items equal minus one. The
Laspeyres index exceeds the costofliving index unless all the
elasticities of substitution equal zero.
^{2 }We will use the term "basic indexes"
for the sake of consistency. Such indexes have also been called
subindexes, elementary aggregates, basic itemarea component
indexes, lowlevel indexes, firstlevel indexes, itemarea
indexes, stratum indexes, etc.
^{3 }Under certain conditions, superlative indexes can
be shown to be closer approximations than Laspeyres indexes to
true costofliving indexes (see Diewert, 1987). However, they
can be produced only with a significant time lag (i.e., current
period expenditures are not typically available in real time for
most items).
^{4 }See, for example, Reinsdorf (1994), Advisory
Commission to Study the Consumer Price Index (1996) and Moulton
(1996).
^{5} Many countries employ the geometric mean formula
within some or all index categories, although differences in
sample item selection procedures limit the formula’s general
interpretation as a costofliving approximation.
^{6 }The CPIUXG is not considered a productiongrade
index largely because the quote weights within the shelter
component are based not on the theoretically appropriate
expenditures but on quantities. Baseperiod expenditure data for
geometric mean weighting will not be available until January
1999, when the new CPI housing sample is introduced. In addition,
due to a flaw in computing the weights within the commodity and
service components of the CPIUXG, the basic geometric mean
indexes were inefficiently estimated prior to the index for
January 1997.
^{7 }Even though the Test Laspeyres index (CPIUXL)
uses early 1996 calculation methods retroactively to December
1990, some improvements made between 1990 and 1996 are not shown
in the CPIUXL from its inception, because certain data are not
available. For example, an improvement made in January 1995 to
the estimation of quantity weights included the use of
"overlap samples," or the collection of two samples
during the same (overlap) time period. Since these overlap
samples did not exist before January 1995, the effect of this
improvement is not incorporated into the CPIUXL until January
1995, when it was incorporated into the official CPI. Since this
January 1995 improvement affects the estimation of quantity
weights, the CPIUXG, which is weighted by expenditures, is not
affected by this change. The table below identifies important
methodological changes that have occurred since 1990 in which the
impact of the change is different for the official CPI, the
CPIUXL, and the CPIUXG.
Methodological changes between 1990 and 1996 affecting the
official Laspeyres, test Laspeyres, and experimental CPI using
geometric means at different times.
Methodological
change 
Official CPIU
affected starting 
CPIUXL affected
starting 
CPIUXG affected
starting 
Improvements to residential rent and
owners’ equivalent rent 
January 1995 
December 1990 
December 1990 
Improved estimates of quantity weights
for foodathome categories 
January 1995 
January 1995 
has no effect 
Improved estimates of quantity weights
for other commodities and services 
June 1996 
June 1996 
has no effect 
Improved estimates of quantity weights
for item substitutions 
July 1996 
July 1996 
has no effect 
bounding 
N/A 
December 1990 
December 1990 
^{8} Baskin and Leaver (1996) have estimated variances
for the CPIUXG, CPIUXL, and official CPI for the major
shelter components, and concluded that differences in measures of
cumulative index change for the two test series are statistically
significant.
^{9} The unavoidable reliance on quantity weights
rather than the theoretically appropriate expenditure weights in
constructing the CPIUXG housing indexes also affect their
movements relative to the corresponding CPIUXL series. Rent
changes for less expensive housing units have a greater influence
in the CPIUXG. In addition, 1991 and 1995 changes in the
treatment of landlordprovided furniture had a greater effect on
the shelter components of the CPIUXG than the corresponding
components of the CPIUXL.
^{10} The differences between the CPIUXL and
official CPIU housing series reflect not only the earlier
incorporation of methodological changes in the shelter component
of the CPIUXL, but also the special computational procedures
used during 1991 and 1992 in the CPIU’s lodging while out
of town index to compensate for small sample sizes in that
component. These procedures were not replicated in the CPIUXL.
References
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Last Modified Date: October 16, 2001