The Canadian Consumer Price Index Reference Paper
Chapter 6 – Calculation of the Consumer Price Index
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6.1 The Consumer Price Index (CPI) is calculated in two stages, termed
the lower level and the upper level.
6.2 At the lower level of
calculation , price change is estimated for elementary aggregates . These are found at the lowest level in the product and geographical classifications of the CPI and are most often
calculated using a Jevons (geometric mean) index
number formula. Elementary aggregates consist of similar groups of products in a geographical stratum .Note
6.3 At the upper level, an asymmetrically-weighted fixed-basket Lowe price index
formula (Laspeyres-type) is
used to
combine elementary aggregates in order to obtain upper level aggregate indices.
6.4 This chapter will discuss the two-stage calculation of the CPI ,
first explaining the computation of elementary indices at the lower level. While the chapter
will focus on the standard method for computing indices, some non-standard methods used in the CPI will
also be discussed. Then the chapter will explain the method used to aggregate elementary price indices to the upper level.
Calculation of Elementary
Indices (lower level)
6.5 At the lower level, elementary price indices are calculated for about 645 elementary product classes in each of the 19 geographical strata of the CPI .Note Elementary indices can be
understood as the building blocks of the CPI and represent the lowest level of the fixed-basket index hierarchy.
Estimation of price change at this level is usually done via the standard approach for elementary
price index calculation. Exceptions are made for special cases addressed later in this
chapter.Note
6.6 Not all elementary indices are derived directly from observed
prices. At the Canada level, about 75% of elementary indices, by basket weight, are derived
directly from observed prices within their product class and geography. The proportion of elementary indices
estimated with direct price observation varies across geography. The remaining portion of
elementary indices is imputed, either from another closely related product class, or from the same product class
in another geographic stratum.Note
6.7 Most of the elementary aggregates that are not calculated using
observed prices are catch-all product classes; as such, they represent more
marginal and diverse varieties of products which do not fit neatly into any of the other elementary product classes.
Typically these catch-all product classes would also be significantly more expensive to estimate via
direct price observation. Their price change is usually estimated by imputing the price movement from another
elementary price index for which prices are observed.
6.8 While it would appear ideal that all elementary price indices be
calculated using observed prices within their product class, this is not always necessary.
Since the goal of the CPI is to measure price change, and not absolute price levels, sampling strategies
are developed to reflect which Product
Offers (POs) are the most important to capture directly, and which
others may be suitably estimated via imputation .Note
6.9 The CPI follows the matched-model approach for calculating
elementary price indices whereby identical (unchanging quantity and quality) POs are
followed through time. However, it is not always possible to follow the same products across time, as new goods
and services are constantly emerging and old ones disappearing. When an identical PO cannot be
collected in a subsequent period, a replacement PO must be observed. This chapter will not discuss
situations where POs are replaced.Note
6.10 Examples where the calculation of elementary price indices is a
relatively simple matter are the few elementary aggregates for which there is one
product having a single price. These product classes typically have goods or services for which prices are
determined by a level of government, such as drivers’ licences or passport fees. In such cases, the ratio of
one month’s price over the previous month is the best estimate of price change. However, for the majority of
elementary product classes reality is more complex, mainly because of the availability of many competing
and continuously changing product types.
6.11 In the majority of cases, elementary price indices are based on a sample of prices for one or more goods
or services belonging to the
elementary product class. The sampled POs receive equal weighting in this elementary calculation, because consumer expenditure weighting information is
usually not available at this
level.
6.12 The following section describes the standard approach for
calculating elementary price indices. The chapter will then go on to discuss several of the
elementary price indices for which estimation methods differ from the standard approach either because of the
complex nature of estimating price change for the goods and services within the elementary product class or because
additional information is available that can be used to produce an improved elementary price
index.
The Standard Approach for
Calculating Elementary Price Indices
6.13 The standard approach refers to the most commonly used method of
combining prices, in order to estimate price change for elementary aggregates in the
CPI . Typically consumer expenditure patterns below the elementary aggregate level are not known and
therefore the implicitly weighted geometric mean, known as the Jevons formula (6.1), is used to
calculate an average price relative from the sample of the collected POs . This means the price relative of each
collected PO is assigned equal importance in the calculation. The Jevons formula has been
used by Statistics Canada since 1995 as its primary formula for the calculation
of elementary price indices in
the CPI .
I
J , a
t − 1 : t
=
∏
i = 1
n
(
p
i
t
p
i
t − 1
)
1
/
n
(6.1)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
Gaamysa8aadaqhaaWcbaWdbiaadQeacaGGSaGaamyyaaWdaeaapeGa
aiiDaiabgkHiTiaaigdacaGG6aGaamiDaaaakiabg2da9maarahaba
WaaeWaaeaadaWcaaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadsha
aaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaG
ymaaaaaaaakiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaa
baGaamOBaaqdcqGHpis1aOWdamaaCaaaleqabaWaaWbaaWqabeaape
GaaGymaaaaliaac+capaWaaSbaaWqaa8qacaWGUbaapaqabaaaaaaa
@5195@
where:
I
J , a
t − 1 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGkbGaaiilaiaadggaaeaacaGG0bGaeyOeI0IaaGymaiaa
cQdacaWG0baaaaaa@3DAD@
is the implicitly weighted Jevons price index for elementary
aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
between period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
and period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@
is the number of POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
in elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
; and
p
i
t
p
i
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGWbWaa0baaSqaaiaadMgaaeaacaWG0baaaaGcbaGaamiCamaaDaaa
leaacaWGPbaabaGaamiDaiabgkHiTiaaigdaaaaaaaaa@3DCA@
is the price relative for PO
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
between period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
and period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
.
6.14 The Jevons formula (6.1) can also be calculated by taking the
ratio of the implicitly weighted geometric mean prices of the observed POs in the two periods
being compared (6.2).
I
J , a
t − 1 : t
=
∏
i = 1
n
(
p
i
t
)
1
n
∏
i = 1
n
(
p
i
t − 1
)
1
n
(6.2)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGkbGaaiilaiaadggaaeaacaGG0bGaeyOeI0IaaGymaiaa
cQdacaWG0baaaOGaeyypa0ZaaSaaaeaadaqeWbqaaiaacIcacaWGWb
Waa0baaSqaaiaadMgaaeaacaWG0baaaOGaaiykamaaCaaaleqabaWa
aSGaaeaacaaIXaaabaGaamOBaaaaaaaabaGaamyAaiabg2da9iaaig
daaeaacaWGUbaaniabg+GivdaakeaadaqeWbqaaiaacIcacaWGWbWa
a0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaakiaacMcada
ahaaWcbeqaamaaliaabaGaaGymaaqaaiaad6gaaaaaaaqaaiaadMga
cqGH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aaaaaaa@58C1@
where:
∏
i = 1
n
(
p
i
t
)
1
n
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaaca
GGOaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaaaakiaacMcadaah
aaWcbeqaamaaliaabaGaaGymaaqaaiaad6gaaaaaaaqaaiaadMgacq
GH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aaaa@4217@
is the geometric mean price for
all POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
for elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
in period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
; and
∏
i = 1
n
(
p
i
t − 1
)
1
n
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaaca
GGOaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTiaaigda
aaGccaGGPaWaaWbaaSqabeaadaWccaqaaiaaigdaaeaacaWGUbaaaa
aaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey4dIunaaaa@43BF@
is the geometric mean price for
all POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
for elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
in period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
.
6.15 The Jevons
formula was adopted because it has advantages over the previously used Dutot
formula.Note Firstly, the geometric mean of price relatives (Jevons) is less influenced
by extreme prices than is the ratio of
arithmetic
mean prices (Dutot). The resulting elementary price indices are less volatile.Note Secondly, elementary price indices that are calculated as geometric mean of price relatives (Jevons) can be interpreted in two ways; first, as an average of price changes (6.1) and
second as a change in average prices (6.2). The first interpretation, which is only applicable to the Jevons
formula, is convenient for explaining the composition of aggregate price changes.
Other Methods for
Calculating Elementary Price Indices
6.16 Among the elementary product indices there are several departures
from the standard approach. Exceptions to the standard approach
are usually made because more complete information is available on the universe of transactions within the
elementary aggregate.
6.17 Post-1995,
arithmetic formulae were retained for the calculation of a few elementary price
indices (Rent, Passenger vehicle insurance premiums and Tuition fees). What
sets these elementary aggregates
apart is
that the sampled POs are drawn from a population frame and there is confidence
that the sample sufficiently represents the universe of consumer
expenditures for these product classes. Furthermore, the contractual nature of the expenditures in these product classes
means that it is likely that product substitution will not take place over the period of price comparison. The unweighted arithmetic formula used in
the Canadian CPI is the Dutot
(6.3).Note
I
D , a
t − 1 : t
=
∑
i = 1
n
1
n
p
i
t
∑
i = 1
n
1
n
p
i
t − 1
(6.3)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGebGaaiilaiaadggaaeaacaWG0bGaeyOeI0IaaGymaiaa
cQdacaWG0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaamaalaaabaGaaG
ymaaqaaiaad6gaaaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaaaa
aeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOqaam
aaqahabaWaaSaaaeaacaaIXaaabaGaamOBaaaacaWGWbWaa0baaSqa
aiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaeaacaWGPbGaeyypa0
JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@55BB@
where:
I
D , a
t − 1 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGebGaaiilaiaadggaaeaacaGG0bGaeyOeI0IaaGymaiaa
cQdacaWG0baaaaaa@3DA7@
is the Dutot price index for elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
between period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
and period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@
is the number of POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
in elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
;
∑
i = 1
n
1
n
p
i
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada
WcaaqaaiaaigdaaeaacaWGUbaaaiaadchadaqhaaWcbaGaamyAaaqa
aiaadshaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaniabgg
HiLdaaaa@4097@
is the arithmetic mean price for
all POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
for elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
in period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
; and
∑
i = 1
n
1
n
p
i
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada
WcaaqaaiaaigdaaeaacaWGUbaaaiaadchadaqhaaWcbaGaamyAaaqa
aiaadshacqGHsislcaaIXaaaaaqaaiaadMgacqGH9aqpcaaIXaaaba
GaamOBaaqdcqGHris5aaaa@423F@
is the arithmetic mean price for
all POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
for elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
in period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
.
6.18 An explicitly weighted Jevons formula (6.4) is used in a few
special cases where more detailed expenditure information is available below the elementary
aggregate level. Examples where an explicitly weighted Jevons formula is used are the indices for postal
fees, newspapers and magazines, urban transit and parking rates.
I
W J , a
t − 1 : t
=
∏
i = 1
n
(
p
i
t
)
w
i
/
∑
i = 1
n
w
i
∏
i = 1
n
(
p
i
t − 1
)
w
i
/
∑
i = 1
n
w
i
(6.4)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGxbGaamOsaiaacYcacaWGHbaabaGaaiiDaiabgkHiTiaa
igdacaGG6aGaamiDaaaakiabg2da9maalaaabaWaaebCaeaadaqada
qaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakiaawIcacaGL
PaaadaahaaWcbeqaamaalyaabaGaam4DamaaBaaameaacaWGPbaabe
aaaSqaamaaqahabaGaam4DamaaBaaameaacaWGPbaabeaaaeaacaWG
PbGaeyypa0JaaGymaaqaaiaad6gaa4GaeyyeIuoaaaaaaaWcbaGaam
yAaiabg2da9iaaigdaaeaacaWGUbaaniabg+GivdaakeaadaqeWbqa
amaabmaabaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTi
aaigdaaaaakiaawIcacaGLPaaadaahaaWcbeqaamaalyaabaGaam4D
amaaBaaameaacaWGPbaabeaaaSqaamaaqahabaGaam4DamaaBaaame
aacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa4Ga
eyyeIuoaaaaaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGUbaani
abg+Givdaaaaaa@6AE5@
where:
I
W J , a
t − 1 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGxbGaamOsaiaacYcacaWGHbaabaGaaiiDaiabgkHiTiaa
igdacaGG6aGaamiDaaaaaaa@3E89@
is the explicitly weighted Jevons price index for elementary
aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
between period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
and period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@
is the number of collected POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
in elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
;
∏
i = 1
n
(
p
i
t
)
w
i
/
∑
i = 1
n
w
i
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaada
qadaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakiaawIca
caGLPaaadaahaaWcbeqaamaalyaabaGaam4DamaaBaaameaacaWGPb
aabeaaaSqaamaaqahabaGaam4DamaaBaaameaacaWGPbaabeaaaeaa
caWGPbGaeyypa0JaaGymaaqaaiaad6gaa4GaeyyeIuoaaaaaaaWcba
GaamyAaiabg2da9iaaigdaaeaacaWGUbaaniabg+Givdaaaa@4ABB@
is the explicitly weighted
geometric mean price for all POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
in elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
in period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
∏
i = 1
n
(
p
i
t − 1
)
w
i
/
∑
i = 1
n
w
i
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaada
qadaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshacqGHsislcaaI
XaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaadaWcgaqaaiaadEhada
WgaaadbaGaamyAaaqabaaaleaadaaeWbqaaiaadEhadaWgaaadbaGa
amyAaaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaoiabgg
HiLdaaaaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGH
pis1aaaa@4C63@
is the explicitly weighted
geometric mean price for all POs
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
for elementary aggregate
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
in period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
; and
w
i
/
∑
i = 1
n
w
i
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabe
aadaahaaadbeqaamaalyaabaGaam4DamaaBaaabaGaamyAaaqabaaa
baWaaabCaeaacaWG3bWaaSbaaeaacaWGPbaabeaaaeaacaWGPbGaey
ypa0JaaGymaaqaaiaad6gaaiabggHiLdaaaaaaaaaaaa@4049@
is the weight of PO
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
as a proportion of the
aggregate weight for all POs .
6.19 The weights
used in the calculation do not have to relate to the period of price
comparison, however in each comparison period they are fixed. The weights
are obtained from administrative records or other data sources. These cases can be seen as improvements on the standard
approach because rather than giving
implicit
equal importance to each price relative (6.1) they make use of additional
information about the relative importance, or size, of each
group of transactions.
6.20 In cases
where there are different product types available within one elementary
aggregate, but each product type is homogeneous, a unit value index is a
preferred method for calculating elementary price indices. A unit value index is simply the quantity-weighted average
transaction price for all products within an elementary aggregate in one period, divided by the
quantity-weighted average transaction price in the previous period. The rationale for using a unit value calculation
must be based on a reasonable assumption that the changes in these average prices do not reflect a change
in quality over time. Otherwise the index could be prone to bias .Note
6.21 The CPI uses a unit value calculation for the spectator
entertainment index, which includes prices for stadium sports seating and live staged performances.
The assumption behind this index is that if the stadium or theatre is full in each of the two periods
being compared, there is likely to be no change in the overall quality, even though seats may be valued differently.
In effect, the price of all seats in the stadium or theatre is used rather than a few individual seats. A similar
approach is used to calculate the Air transportation index.
6.22 A unit value calculation is also used in the property taxes
elementary price index. A sample of properties is drawn so that the average annual property tax
paid in a given municipality can be calculated. These calculated average annual taxes are then
multiplied by the total stock of dwellings in each municipality in order to obtain the average annual property
tax paid in each CPI geographical stratum. No attempt is made to control for differences in the quality of services that homeowners
receive in exchange for their tax payments from one municipality to another.
Additionally, there is no treatment to control for changes in the quality of
municipal services from one period to another. Accounting for these differences
is impractical as there are no data available which associate specific
municipal services to proportions of property taxes paid.Note
Calculation of the Consumer
Price Index Above Elementary Indices (upper level)
6.23 The calculation of the CPI at the upper level
is relatively straightforward compared to the lower level. It involves
aggregating calculated elementary price indices by applying an asymmetrically
weighted arithmetic fixed-basket formula in order to obtain aggregate indices which culminate in the All-items CPI .Note
6.24 The Laspeyres formula (6.5) is a basic method for calculating
price indices and is consistent with the CPI ’s fixed basket concept. It expresses the change
in the cost between period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
of buying a fixed basket of products, by aggregating the prices of the products in the basket using
quantities consumed from the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
as
weights.
I
L , A
0 : t
=
∑
i = 1
n
p
i
t
q
i
0
∑
i = 1
n
p
i
0
q
i
0
(6.5)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaGc
cqGH9aqpdaWcaaqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaaba
GaamiDaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaaicdaaaaabaGa
amyAaiabg2da9iaaigdaaeaacaWGUbaaniabggHiLdaakeaadaaeWb
qaaiaadchadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0ba
aSqaaiaadMgaaeaacaaIWaaaaaqaaiaadMgacqGH9aqpcaaIXaaaba
GaamOBaaqdcqGHris5aaaaaaa@5402@
where:
I
L , A
0 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaaa
aa@3BA9@
is the Laspeyres price index of
aggregate class
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
between
period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@
is the number of elementary aggregates
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
in the aggregate class
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
;
p
i
t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa
aaleaacaWGPbaabaGaamiDaaaaaaa@38FF@
is the price of elementary
aggregate
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
, in time
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
p
i
0
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa
aaleaacaWGPbaabaGaaGimaaaaaaa@38C0@
is the price of elementary
aggregate
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
, in time
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
; and
q
i
0
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaDa
aaleaacaWGPbaabaGaaGimaaaaaaa@38C1@
is the quantity weight of
elementary aggregate
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
, in the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
.
6.25 In practice, the Laspeyres index is not commonly used to calculate
the CPI because it requires information on the quantities consumedNote in the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and these data are not available in a timely manner. This has to do with the fact that
household expenditure surveys are typically produced with a lag. Therefore, since
Statistics Canada aims to produce a CPI that is timely, in that it measures
changes in prices for recent periods,
the Laspeyres formula must be altered to use quantities from a period preceding
the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
. This
transformation is the Lowe formula (6.6), a more general form of a Laspeyres index because the quantities come from a chosen weight reference period
b
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@
which
precedes the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
.
I
L o , A
0 : t
=
∑
i = 1
n
p
i
t
q
i
b
∑
i = 1
n
p
i
0
q
i
b
(6.6)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaaiaadchadaqhaaWcbaGaam
yAaaqaaiaadshaaaGccaWGXbWaa0baaSqaaiaadMgaaeaacaWGIbaa
aaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaGcba
WaaabCaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaaIWaaaaOGaamyC
amaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWGPbGaeyypa0JaaG
ymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@5550@
where:
I
L o , A
0 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0baaaaaa@3C9D@
is the Lowe price index of aggregate class
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
between
period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@
is the number of elementary
aggregates
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
in the aggregate class
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
;
p
i
t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa
aaleaacaWGPbaabaGaamiDaaaaaaa@38FF@
is the price of elementary aggregate
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
, in time
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
p
i
0
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa
aaleaacaWGPbaabaGaaGimaaaaaaa@38C0@
is the price of elementary aggregate
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
, in time
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
; and
q
i
b
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaDa
aaleaacaWGPbaabaGaamOyaaaaaaa@38EE@
is the quantity weight of elementary aggregate
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
, in the weight reference period
b
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@
, with
b ≤ 0 < t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgs
MiJkaaicdacqGH8aapcaWG0baaaa@3B4A@
.
6.26 The Lowe index can also be expressed as the
weighted sum of elementary price indices (6.7) with the weights expressed as expenditure shares.
I
L o , A
0 : t
=
∑
i = 1
n
(
p
i
t
/
p
i
0
)
s
i
0 b
(6.7)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0baaaOGaeyypa0ZaaabCaeaacaGGOaGaamiCamaaDaaaleaacaWGPb
aabaGaamiDaaaakiaac+cacaWGWbWaa0baaSqaaiaadMgaaeaacaaI
WaaaaOGaaiykaiaadohadaqhaaWcbaGaamyAaaqaaiaaicdacaWGIb
aaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa
@4F2D@
where:
p
i
t
/
p
i
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca
WGWbWaa0baaSqaaiaadMgaaeaacaWG0baaaaGcbaGaamiCamaaDaaa
leaacaWGPbaabaGaaGimaaaaaaaaaa@3BEA@
is the price index of elementary aggregate
(
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
) between periods
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
, and;
s
i
0 b
≡
p
i
0
q
i
b
∑
i = 1
n
p
i
0
q
i
b
(6.8)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa
aaleaacaWGPbaabaGaaGimaiaadkgaaaGccqGHHjIUdaWcaaqaaiaa
dchadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0baaSqaai
aadMgaaeaacaWGIbaaaaGcbaWaaabCaeaacaWGWbWaa0baaSqaaiaa
dMgaaeaacaaIWaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaa
aaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaa
aa@4D08@
6.27 The expenditure shares
s
i
0 b
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa
aaleaacaWGPbaabaGaaGimaiaadkgaaaaaaa@39AA@
in the Lowe formula (6.7) are hybrid expenditures because the
prices and quantities (that equal the
expenditures when multiplied) are from different periods,
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
b
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@
.
6.28 Hybrid expenditure shares (6.8) are obtained by updating
the original expenditure weights
p
i
b
q
i
b
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa
aaleaacaWGPbaabaGaamOyaaaakiaadghadaqhaaWcbaGaamyAaaqa
aiaadkgaaaaaaa@3BEF@
(observed in the weight reference period
b
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@
) to reflect the prices of the
price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
using the price relatives
p
i
0
/
p
i
b
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDa
aaleaacaWGPbaabaGaaGimaaaakiaac+cacaGGWbWaa0baaSqaaiaa
dMgaaeaacaWGIbaaaaaa@3C73@
. This process is often referred to as price-updating and thus
hybrid expenditure weights are frequently termed price-updated weights.Note
The use of price-updated or hybrid expenditure weights is essential to the
fixed-quantity basket concept of the CPI .
6.29 Because the weights used in the calculation of the CPI are
obtained from consumer expenditure data with a weight reference period that precedes the
price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
, the Lowe index formula is the
practical option for computing a timely
CPI .
6.30 Notwithstanding this practical advantage, the Lowe formula also
has many desirable properties. One
is its consistency in aggregation. This means that no matter in which order the
elementary price indices are aggregated (for example first by geographical stratum
and then by product class, or the reverse) the aggregate index results are the
same.
6.31 Another desirable property of the Lowe formula is its transitivityNote , whereby the ratio of two Lowe
indices using the same set of basket reference
quantities
q
b
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaCa
aaleqabaGaamOyaaaaaaa@3801@
is also a Lowe index (6.9).Note This property is useful because it enables index compilers to calculate
short-term price movements. For example, price change between period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
and period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
can be estimated by taking the ratio of two
long-term Lowe price indices, one comparing periods
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
and the
other comparing periods
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
.
I
L o , A
t − 1 : t
=
∑
i = 1
n
p
i
t
q
i
b
∑
i = 1
n
p
i
t − 1
q
i
b
=
∑
i = 1
n
p
i
t
q
i
b
∑
i = 1
n
p
i
0
q
i
b
∑
i = 1
n
p
i
t − 1
q
i
b
∑
i = 1
n
p
i
0
q
i
b
= (
I
L o , A
0 : t
I
L o , A
0 : t − 1
) (6.9)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaamiDaiabgkHiTiaa
igdacaGG6aGaamiDaaaakiabg2da9maalaaabaWaaabCaeaacaWGWb
Waa0baaSqaaiaadMgaaeaacaWG0baaaOGaamyCamaaDaaaleaacaWG
PbaabaGaamOyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0
GaeyyeIuoaaOqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaabaGa
amiDaiabgkHiTiaaigdaaaGccaWGXbWaa0baaSqaaiaadMgaaeaaca
WGIbaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5
aaaakiabg2da9maalaaabaWaaSaaaeaadaaeWbqaaiaadchadaqhaa
WcbaGaamyAaaqaaiaadshaaaGccaWGXbWaa0baaSqaaiaadMgaaeaa
caWGIbaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHri
s5aaGcbaWaaabCaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaaIWaaa
aOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWGPbGaey
ypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaakeaadaWcaaqaamaa
qahabaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTiaaig
daaaGccaWGXbWaa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMga
cqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaGcbaWaaabCaeaaca
WGWbWaa0baaSqaaiaadMgaaeaacaaIWaaaaOGaamyCamaaDaaaleaa
caWGPbaabaGaamOyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6
gaa0GaeyyeIuoaaaaaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiaadMea
daqhaaWcbaGaamitaiaad+gacaGGSaGaamyqaaqaaiaaicdacaGG6a
GaamiDaaaaaOqaaiaadMeadaqhaaWcbaGaamitaiaad+gacaGGSaGa
amyqaaqaaiaaicdacaGG6aGaamiDaiabgkHiTiaaigdaaaaaaaGcca
GLOaGaayzkaaaaaa@9CE9@
where:
I
L o , A
t − 1 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaamiDaiabgkHiTiaa
igdacaGG6aGaamiDaaaaaaa@3E84@
is the short-term Lowe index for aggregate
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
between
period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
and period;
I
L o , A
0 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0baaaaaa@3C9D@
is the long-term Lowe index for aggregate
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
between
period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
, and;
I
L o , A
0 : t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0bGaeyOeI0IaaGymaaaaaaa@3E45@
is the long-term Lowe index for aggregate
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
between period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
.
6.32 The transitivity property of the Lowe formula
also enables index compilers to calculate long-term price change by chaining
together short-term price indices. For example, a Lowe index comparing prices
in period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
to prices in the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
is obtained by multiplying the Lowe index
comparing period t to period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
by the Lowe index comparing period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
with the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
(6.10). The product of monthly chained indices
provides identical results to an index that directly compares prices in period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
to prices in the price reference period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
.
I
L o , A
0 : t
=
[
∑
i = 1
n
p
i
1
q
i
b
∑
i = 1
n
p
i
0
q
i
b
]
︸
I
L o , A
0 : 1
×
[
∑
i = 1
n
p
i
2
q
i
b
∑
i = 1
n
p
i
1
q
i
b
]
︸
I
L o , A
1 : 2
× .... ×
[
∑
i = 1
n
p
i
t − 2
q
i
b
∑
i = 1
n
p
i
t − 3
q
i
b
]
︸
I
L o , A
t − 3 : t − 2
×
[
∑
i = 1
n
p
i
t − 1
q
i
b
∑
i = 1
n
p
i
t − 2
q
i
b
]
︸
I
L o , A
t − 2 : t − 1
︸
I
L o , A
0 : t − 1
=
∑
i = 1
n
p
i
t − 1
q
i
b
∑
i = 1
n
p
i
0
q
i
b
×
[
∑
i = 1
n
p
i
t
q
i
b
∑
i = 1
n
p
i
t − 1
q
i
b
]
︸
I
L o , A
t − 1 : t
=
∑
i = 1
n
(
p
i
t
p
i
t − 1
)
s
i
t − 1 b
(6.10)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0baaaOGaeyypa0ZaaGbaaeaadaagaaqaamaadmaabaWaaSaaaeaada
aeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaaigdaaaGccaWGXbWa
a0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqpcaaIXa
aabaGaamOBaaqdcqGHris5aaGcbaWaaabCaeaacaWGWbWaa0baaSqa
aiaadMgaaeaacaaIWaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaam
OyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoa
aaaakiaawUfacaGLDbaaaSqaaiaadMeadaqhaaadbaGaamitaiaad+
gacaGGSaGaamyqaaqaaiaaicdacaGG6aGaaGymaaaaaOGaayjo+dGa
ey41aq7aaGbaaeaadaWadaqaamaalaaabaWaaabCaeaacaWGWbWaa0
baaSqaaiaadMgaaeaacaaIYaaaaOGaamyCamaaDaaaleaacaWGPbaa
baGaamOyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey
yeIuoaaOqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaabaGaaGym
aaaakiaadghadaqhaaWcbaGaamyAaaqaaiaadkgaaaaabaGaamyAai
abg2da9iaaigdaaeaacaWGUbaaniabggHiLdaaaaGccaGLBbGaayzx
aaaaleaacaWGjbWaa0baaWqaaiaadYeacaWGVbGaaiilaiaadgeaae
aacaaIXaGaaiOoaiaaikdaaaaakiaawIJ=aiabgEna0kaac6cacaGG
UaGaaiOlaiaac6cacqGHxdaTdaagaaqaamaadmaabaWaaSaaaeaada
aeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshacqGHsislcaaI
YaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWGPb
Gaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOqaamaaqahabaGa
amiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTiaaiodaaaGcca
WGXbWaa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqp
caaIXaaabaGaamOBaaqdcqGHris5aaaaaOGaay5waiaaw2faaaWcba
GaamysamaaDaaameaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaiiD
aiabgkHiTiaaiodacaGG6aGaaiiDaiabgkHiTiaaikdaaaaakiaawI
J=aiabgEna0oaayaaabaWaamWaaeaadaWcaaqaamaaqahabaGaamiC
amaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTiaaigdaaaGccaWGXb
Waa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqpcaaI
XaaabaGaamOBaaqdcqGHris5aaGcbaWaaabCaeaacaWGWbWaa0baaS
qaaiaadMgaaeaacaWG0bGaeyOeI0IaaGOmaaaakiaadghadaqhaaWc
baGaamyAaaqaaiaadkgaaaaabaGaamyAaiabg2da9iaaigdaaeaaca
WGUbaaniabggHiLdaaaaGccaGLBbGaayzxaaaaleaacaWGjbWaa0ba
aWqaaiaadYeacaWGVbGaaiilaiaadgeaaeaacaGG0bGaeyOeI0IaaG
OmaiaacQdacaGG0bGaeyOeI0IaaGymaaaaaOGaayjo+daaleaacaWG
jbWaa0baaWqaaiaadYeacaWGVbGaaiilaiaadgeaaeaacaaIWaGaai
OoaiaadshacqGHsislcaaIXaaaaSGaeyypa0ZaaSaaaeaadaaeWbqa
aiaadchadaqhaaadbaGaamyAaaqaaiaadshacqGHsislcaaIXaaaaS
GaamyCamaaDaaameaacaWGPbaabaGaamOyaaaaaeaacaWGPbGaeyyp
a0JaaGymaaqaaiaad6gaa4GaeyyeIuoaaSqaamaaqahabaGaamiCam
aaDaaameaacaWGPbaabaGaaGimaaaaliaadghadaqhaaadbaGaamyA
aaqaaiaadkgaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaoi
abggHiLdaaaaGccaGL44pacqGHxdaTdaagaaqaamaadmaabaWaaSaa
aeaadaaeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaGcca
WGXbWaa0baaSqaaiaadMgaaeaacaWGIbaaaaqaaiaadMgacqGH9aqp
caaIXaaabaGaamOBaaqdcqGHris5aaGcbaWaaabCaeaacaWGWbWaa0
baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaakiaadghadaqh
aaWcbaGaamyAaaqaaiaadkgaaaaabaGaamyAaiabg2da9iaaigdaae
aacaWGUbaaniabggHiLdaaaaGccaGLBbGaayzxaaaaleaacaWGjbWa
a0baaWqaaiaadYeacaWGVbGaaiilaiaadgeaaeaacaWG0bGaeyOeI0
IaaGymaiaacQdacaWG0baaaSGaeyypa0ZaaabCaeaadaqadaqaamaa
laaabaGaamiCamaaDaaameaacaWGPbaabaGaamiDaaaaaSqaaiaadc
hadaqhaaadbaGaamyAaaqaaiaadshacqGHsislcaaIXaaaaaaaaSGa
ayjkaiaawMcaaiaadohadaqhaaadbaGaamyAaaqaaiaadshacqGHsi
slcaaIXaGaamOyaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6ga
a4GaeyyeIuoaaOGaayjo+daaaa@420C@
where:
I
L o , A
0 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0baaaaaa@3C9D@
is the long-term Lowe index for
aggregate class
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
between period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
;
I
L o , A
t − 1 : t
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaamiDaiabgkHiTiaa
igdacaGG6aGaamiDaaaaaaa@3E84@
is the monthly short-term Lowe
index for aggregate
A
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@
; and
s
i
t − 1 b
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa
aaleaacaWGPbaabaGaamiDaiabgkHiTiaaigdacaWGIbaaaaaa@3B91@
is the hybrid expenditure share
of elementary aggregate
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@
,
with quantities from the basket reference period
b
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@
expressed at period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
prices, derived
as (6.11).
s
i
t − 1 b
≡
p
i
t − 1
q
i
b
∑
i = 1
n
p
i
t − 1
q
i
b
(6.11)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa
aaleaacaWGPbaabaGaamiDaiabgkHiTiaaigdacaWGIbaaaOGaeyyy
IO7aaSaaaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0
IaaGymaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaadkgaaaaakeaa
daaeWbqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshacqGHsislca
aIXaaaaOGaamyCamaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWG
PbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@52BD@
6.33 In any given period
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
the hybrid expenditure shares price-updated to
period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
are used to aggregate elementary price indices. Since hybrid
expenditure weights are an estimate of the value of purchasing the quantities from the weight
reference period b expressed in period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
prices, they do not reflect changes in consumer purchasing patterns. These are
necessary in order to maintain the fixed quantity concept of the Lowe formula.
6.34 In the ongoing practice of compiling the CPI , hybrid expenditure
shares (6.11) are not explicitly calculated. Instead, the equivalent Lowe
formula is used (6.12), where monthly price relatives
(
p
i
t
p
i
t − 1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada
WcaaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakeaacaWG
WbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaaaaki
aawIcacaGLPaaaaaa@3F5D@
multiplied by hybrid expenditure weights expressed at
period
t − 1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk
HiTiaaigdaaaa@3898@
prices are compared to the hybrid expenditures
expressed at period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
prices
in order to obtain price change between period
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@36B1@
and
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36F0@
.
I
L o , A
0 : t
=
∑
i = 1
n
(
p
i
t
p
i
t − 1
) (
p
i
t − 1
q
i
b
)
∑
i = 1
n
(
p
i
0
q
i
b
)
=
∑
i = 1
n
(
p
i
t
p
i
t − 1
) (
p
i
t − 1
p
i
0
)
(
p
i
0
q
i
b
)
∑
i = 1
n
(
p
i
0
q
i
b
)
(6.12)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa
aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG
0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaamaabmaabaWaaSaaaeaaca
WGWbWaa0baaSqaaiaadMgaaeaacaWG0baaaaGcbaGaamiCamaaDaaa
leaacaWGPbaabaGaamiDaiabgkHiTiaaigdaaaaaaaGccaGLOaGaay
zkaaWaaeWaaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOe
I0IaaGymaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaadkgaaaaaki
aawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqd
cqGHris5aaGcbaWaaabCaeaadaqadaqaaiaadchadaqhaaWcbaGaam
yAaaqaaiaaicdaaaGccaWGXbWaa0baaSqaaiaadMgaaeaacaWGIbaa
aaGccaGLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6
gaa0GaeyyeIuoaaaGccqGH9aqpdaWcaaqaamaaqahabaWaaeWaaeaa
daWcaaqaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakeaaca
WGWbWaa0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaaaa
kiaawIcacaGLPaaadaqadaqaamaalaaabaGaamiCamaaDaaaleaaca
WGPbaabaGaamiDaiabgkHiTiaaigdaaaaakeaacaWGWbWaa0baaSqa
aiaadMgaaeaacaaIWaaaaaaaaOGaayjkaiaawMcaaaWcbaGaamyAai
abg2da9iaaigdaaeaacaWGUbaaniabggHiLdGcdaqadaqaaiaadcha
daqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0baaSqaaiaadM
gaaeaacaWGIbaaaaGccaGLOaGaayzkaaaabaWaaabCaeaadaqadaqa
aiaadchadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0baaS
qaaiaadMgaaeaacaWGIbaaaaGccaGLOaGaayzkaaaaleaacaWGPbGa
eyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@91DA@
6.35 Despite all the practical advantages of using the Lowe formula for
calculating
the upper level of the CPI , it is
an
asymmetrically weighted price index, meaning that the weights used to aggregate
elementary price indices refer to a period preceding the
price reference month. For this reason the Lowe formula does not
represent the current spending patterns
of consumers and therefore is subject to substitution bias .Note
Notes
Note
Classifications of the
Consumer Price Index are discussed in Chapter 4.
Return to note referrer
Note
There are slightly fewer elementary aggregates (that is, “building
blocks”) to the Consumer Price Index than the maximum of the product of the
total number of elementary product classes and the number of geographical
strata because not all of the 19 geographic strata have the full set of
elementary product classes available. The absence of product classes occurs
mainly in the small geographic strata. Each elementary aggregate has a
corresponding expenditure weight used in the upper-level calculation.
Return to note referrer
Note
Some common index formulae used to
calculate elementary price indices can be found in Appendix A.
Return to note referrer
Note
Of these elementary aggregates estimated
by proxy, roughly half, by basket weight, are product imputations (e.g. price
movements for college tuition fees are imputed from the price movement of
university tuition fees within each geographic stratum) and the other half are
geographic imputations (e.g. price movements for baseball game admission fees
in Prince Edward Island are imputed from those in Toronto).
Return to note referrer
Note
The
sampling strategy for the Consumer Price Index is discussed in Chapter 5.
Return to note referrer
Note
The
ways in which adjustments are made for the quality changes that may occur when product
offers (PO) are replaced
are discussed in Chapter 7.
Return to note referrer
Note
The Dutot formula was
used as the standard method for calculating elementary
price indices in the Consumer Price Index prior to 1995.
Return to note referrer
Note
The
geometric mean of price relatives (Jevons) can be more volatile than the ratio
of arithmetic mean prices (Dutot). This occurs in the case of very steep price
drops as with liquidation sales. Liquidation sale prices, although they are
part of the universe of consumer expenditures which the Consumer Price Index
(CPI ) aims to measure, are excluded from the CPI sample. This is because liquidation sales
are deemed less representative of the average consumer transaction.
Return to note referrer
Note
The
use of the Dutot formula is appropriate when product offers are expressed in a
homogenous unit of measure. ILO et al.
(2004), paragraphs 20.64-20.68. When quantity or expenditure information is
available, an explicitly weighted Laspeyres-type formula (6.5) can be used,
with the same weights appearing in the numerator and the denominator.
Return to note referrer
Note
Balk
(2002) showed that unit value ratios require special consideration, as they
are not only driven by price change but can also be driven by changing
quantities.
Return to note referrer
Note
The treatment of owned accommodation in the
Consumer Price Index is discussed in Chapter 10.
Return to note referrer
Note
Some common formulae for calculating
aggregate price indices (above the elementary level) can be found in Appendix
A.
Return to note referrer
Note
In practice, what are observed are the
expenditures, which contain the implicit
p
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36EC@
and
q
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaaaa@36ED@
terms.
Return to note referrer
Note
ILO et
al. (2004), paragraph 1.29.
Return to note referrer
Note
Transitivity is an axiomatic property of index
number formulae. Satisfying this property enables price indices to be
calculated via chained or direct price comparison. For more information on this
property, ILO et al. (2004),
paragraphs 9.25 and 15.88.
Return to note referrer
Note
ILO et al. (2004), paragraph 1.26.
Return to note referrer
Note
The
topic of substitution bias in a Consumer Price Index, as well as the
efforts by Statistics Canada to reduce it, are discussed in Chapter 9.
Return to note referrer