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Encyclopedia > Consumer price index

A consumer price index (CPI) is a statistical estimate of the level of prices of goods and services bought for consumption purposes by households. The change in the CPI is a measure of inflation, and can be used for indexation (or evaluation) of wages, salaries, pensions, or regulated or contracted prices. The CPI is one of several major price indices, and along with the population census and the National Income and Product Accounts, it is one of the most important products of national statistical offices. Image File history File links Derived from public domain images featured at: http://commons. ... This article does not cite its references or sources. ... 1870 US Census for New York City A census is the process of obtaining information about every member of a population (not necessarily a human population). ... National Income and Product Accounts (NIPA) use double entry accounting to report the monetary value and sources of output produced in a country and the distribution of incomes that production generates. ...

Two basic types of data are required to construct the CPI: price data and weighting data. The price data are collected for a sample of goods and services from a sample of sales outlets in a sample of locations for a sample of times. The weighting data are estimates of the shares of the different types of expenditure as fractions of the total expenditure covered by the index. These weights are usually based upon expenditure data obtained for sampled periods from a sample of households. Although some of the sampling is done using a sampling frame and probabilistic sampling methods, much is done in a commonsense way (purposive sampling) that does not permit estimation of confidence intervals. Therefore, the sampling variance is normally ignored, since a single estimate is required in most of the purposes for which the index is used. Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ...

The index is usually computed monthly, or quarterly in some countries, as a weighted average of sub-indices for different components of consumer expenditure, such as food, housing, clothing, each of which is in turn a weighted average of sub-sub-indices. At the most detailed level, the elementary aggregate level, (for example, men's trousers sold in department stores in the Northwest), detailed weighting information is unavailable, so elementary aggregate indices are computed using an unweighted arithmetic or geometric mean of the prices of the sampled product offers. (However, the growing use of scanner data is gradually making weighting information available even at the most detailed level.) These indices compare prices each month with prices in the price-reference month. The weights used to combine them into the higher-level aggregates, and then into the overall index, relate to the estimated expenditures during a preceding whole year of the consumers covered by the index on the products within its scope in the area covered. Thus the index is a fixed-weight index, but rarely a Laspeyres index, since the weight-reference period of a year and the price-reference period, usually a more recent single month, do not coincide. It takes time to assemble and process the information used for weighting which, in addition to household expenditure surveys, may include trade and tax data. In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... A typical barcode scanner. ... A price index is any single number calculated from an array of prices and quantities over a period. ...

Ideally, the weights would relate to the composition of expenditure during the time between the price-reference month and the current month. There is a large technical economics literature on index formulae which would approximate this and which can be shown to approximate what economic theorists call a true cost of living index. Such an index would show how consumer expenditure would have to move to compensate for price changes so as to allow consumers to maintain a constant standard of living. Approximations can only be computed retrospectively, whereas the index has to appear monthly and, preferably, quite soon. Nevertheless, in some countries, notably in North America and Sweden,the philosophy of the index is that it is inspired by and approximates the notion of a true cost of living (constant utility) index, whereas in most of Europe it is regarded more pragmatically. A cost-of-living index measures differences in the price of goods and services over time. ...

The coverage of the index may be limited. Consumers' expenditure abroad is usually excluded; visitors' expenditure within the country may be excluded in principle if not in practice; the rural population may or may not be included; certain groups such as the very rich or the very poor may be excluded. Black market expenditure and expenditure on illegal drugs and prostitution are often excluded for practical reasons, although the professional ethics of the statistician require objective description free of moral judgments. Saving and investment are always excluded, though the prices paid for financial services provided by financial intermediaries may be included along with insurance.

The index reference period, usually called the base year, often differs both from the weight-reference period and the price reference period. This is just a matter of rescaling the whole time-series to make the value for the index reference-period equal to 100. Annually revised weights are a desirable but expensive feature of an index, for the older the weights the greater is the divergence between the current expenditure pattern and that of the weight reference-period.

## Weighting

### Weights and sub-indices

Weights can be expressed as fractions summing to unity, as percentages summing to 100 or as per mille numbers summing to 1000.

In the European Union's Harmonised Index of Consumer Prices, for example, each country computes some 80 prescribed sub-indices, their weighted average constituting the national Harmonised Index. The weights for these sub-indices will consist of the sum of the weights of a number of component lower level indexes. The classification is according to use, developed in a national accounting context. This is not necessarily the kind of classification that is most appropriate for a Consumer Price Index. Grouping together of substitutes or of products whose prices tend to move in parallel might be more suitable.

For some of these lower level indexes detailed weights within them may be available, allowing computations where the individual price observations can all be weighted. This may be the case, for example, where all selling is in the hands of a single national organisation which makes its data available to the index compilers. For most lower level indexes, however, the weight will consist of the sum of the weights of a number of elementary aggregate indexes, each weight corresponding to its fraction of the total annual expenditure covered by the index. An 'elementary aggregate' is a lowest-level component of expenditure, one which has a weight but within which, weights of its sub-components are usually lacking Thus, for example: Weighted averages of elementary aggregate indexes (e.g. for men’s shirts, raincoats, women’s dresses etc.) make up low level indexes (e.g. Outer garments),

Weighted averages of these in turn provide sub-indices at a higher, more aggregated level,(e.g. Clothing) and Weighted averages of the latter provide yet more aggregated sub-indices (e.g. Clothing and Footwear).

Some of the elementary aggregate indexes, and some of the sub-indexes can be defined simply in terms of the types of goods and/or services they cover, as in the case of such products as newspapers in some countries and postal services, which have nationally uniform prices. But where price movements do differ or might differ between regions or between outlet types, separate regional and/or outlet-type elementary aggregates are ideally required for each detailed category of goods and services, each with its own weight. An example might be an elementary aggregate for sliced bread sold in supermarkets in the Northern region.

Most elementary aggregate indexes are necessarily 'unweighted' averages for the sample of products within the sampled outlets. However in cases where it is possible to select the sample of outlets from which prices are collected so as to reflect the shares of sales to consumers of the different outlet types covered, self-weighted elementary aggregate indexes may be computed. Similarly, if the market shares of the different types of product represented by product types are known, even only approximately, the number of observed products to be priced for each of them can be made proportional to those shares.

### Estimating weights

The outlet and regional dimensions noted above mean that the estimation of weights involves a lot more than just the breakdown of expenditure by types of goods and services, and the number of separately weighted indexes composing the overall index depends upon two factors:

1. The degree of detail to which available data permit breakdown of total consumption expenditure in the weight reference-period by type of expenditure, region and outlet type.
2. Whether there is reason to believe that price movements vary between these most detailed categories.

How the weights are calculated, and in how much detail, depends upon the availability of information and upon the scope of the index. In the UK the RPI does not relate to the whole of consumption, for the reference population is all private households with the exception of a) pensioner households that derive at least three-quarters of their total income from state pensions and benefits and b) “high income households” whose total household income lies within the top four per cent of all households. The result is that it is difficult to use data sources relating to total consumption by all population groups.

For products whose price movements can differ between regions and between different types of outlet:

• The ideal, rarely realisable in practice, would consist of estimates of expenditure for each detailed consumption category, for each type of outlet, for each region.
• At the opposite extreme, with no regional data on expenditure totals but only on population (e.g. 24% in the Northern region) and only national estimates for the shares of different outlet types for broad categories of consumption (e.g. 70% of food sold in supermarkets) the weight for sliced bread sold in supermarkets in the Northern region has to be estimated as the share of sliced bread in total consumption × 0.24 × 0.7.

The situation in most countries comes somewhere between these two extremes. The point is to make the best use of whatever data are available.

### The nature of the data used for weighting

No firm rules can be suggested on this issue for the simple reason that the available statistical sources differ between countries. However, all countries conduct periodical Household Expenditure surveys and all produce breakdowns of Consumption Expenditure in their National Accounts. The expenditure classifications used there may however be different. In particular:

• Household Expenditure surveys do not cover the expenditures of foreign visitors, though these may be within the scope of a Consumer Price Index.
• National Accounts include imputed rents for owner-occupied dwellings which may not be within the scope of a Consumer Price Index

Even with the necessary adjustments, the National Account estimates and Household Expenditure Surveys usually diverge.

The statistical sources required for regional and outlet-type breakdowns are usually weaker. Only a large-sample Household Expenditure survey can provide a regional breakdown. Regional population data are sometimes used for this purpose, but need adjustment to allow for regional differences in living standards and consumption patterns. Statistics of retail sales and market research reports can provide information for estimating outlet-type breakdowns, but the classifications they use rarely correspond to COICOP categories.

The increasingly widespread use of bar codes and scanners in shops has meant that detailed cash register printed receipts are provided by shops for an increasing share of retail purchases. This development makes possible improved Household Expenditure surveys, as Statistics Iceland has demonstrated. Survey respondents keeping a diary of their purchases need to record only the total of purchases when itemised receipts were given to them and keep these receipts in a special pocket in the diary. These receipts provide not only a detailed breakdown of purchases but also the name of the outlet. Thus response burden is markedly reduced, accuracy is increased, product description is more specific and point of purchase data are obtained, facilitating the estimation of outlet-type weights.

There are only two general principles for the estimation of weights: use all the available information and accept that rough estimates are better than no estimates.

### Reweighting

Ideally, in computing an index, the weights would represent current annual expenditure patterns. In practice they necessarily reflect past expenditure patterns, using the most recent data available or, if they are not of high quality, some average of the data for more than one previous year. Some countries have used a three-year average in recognition of the fact that household survey estimates are of poor quality. In some cases some of the data sources used may not be available annually, in which case some of the weights for lower level aggregates within higher level aggregates are based on older data than the higher level weights.

Infrequent reweighting saves costs for the national statistical office but delays the introduction into the index of new types of expenditure. For example, subscriptions for Internet Service entered index compilation with a considerable time lag in some countries, and account could be taken of digital camera prices between reweightings only by including some digital cameras in the same elementary aggregate as film cameras.

## Aggregative formulas

### Introduction

This section deals with formulas for indexes composed of weighted sub-indexes. At the lowest level of aggregation, most sub-indexes are elementary aggregate indexes computed without weights, using only a sample of prices observed for one or more specified product-types, though the weight for each such sub-index is derived from estimates of consumption values for all the products covered by it. These "elementary aggregate indexes" or "micro indexes", are the subject of another chapter.

This section bypasses the issue of whether a Consumer Price Index should be based on consumption viewed as Transactions, as Expenditure or as Use. The term "consumption value" used here should be interpreted to signify consumption in any one of these meanings.

### Basic formulas

#### Laspeyres indexes

Consumer Price Indexes are often described as Laspeyres indexes. A Laspeyres index for period t with period 0 as reference-period is simply:

$frac{{sum {p_t q_0 } }}{{sum {p_0 q_0 } }} = sum {left( {frac{{p_0 q_0 }}{{sum {p_0 q_0 } }} cdot frac{{p_t }}{{p_0 }}} right)}$

that is to say, a weighted average of the price ratios from 0 to t for every single consumption product where the weight for each is its share in reference-period, 0, total consumption value. However, Consumer Price Indexes are not actually compiled using this formula. There are two reasons for this.

The first is that what is compiled is only a sample estimate:

1. The weights of a Consumer Price Index do not relate to each and every product covered by it but to groupings of products or, with probabilistic product selection, to a sample of products.
2. The prices used to compute a Consumer Price Index do not relate to to each and every product covered by it, but to a sample of products.
3. These prices are generally collected only for selected days in a single month

The second reason is that what is estimated are, in practice, not true Laspeyres indexes. They should be described as "fixed-base" or "Laspeyres-type" indexes. There are a number of reasons for this:

1. The weights in a Consumer Price Index relate to a year, the price reference -period is usually a single month. Thus they relate to periods of different length.
2. Furthermore, the weight reference-year often antedates the price reference-month. For example, the weights may relate to the year 1999 while the price reference-month is December 2001 for the index from January 2002 onward..
3. Whereas the Laspeyres concept is defined in terms of an identical set of products in the price reference-period and the current period, Consumer Price Indexes are computed even though some products disappear from the market and new products appear that were not available in the price reference-period.
4. Some of the products covered in a Consumer Price Index may have no natural quantity units (no q's), for example services for which a percentage fee is charged.

The first two of these last points, the fact that weight and price reference-periods are quite different, mean that theoretical analysis of the relationship between a Laspeyres index and other index concepts, such as a superlative index, cannot be simply applied to the relationship between a monthly Consumer Price Index and such other concepts. The theoretical analysis always assumes that weight and price reference-periods coincide so that the prices of the price reference-period are the prices that help to determine the pattern of weight reference-period consumption. If an index is to be computed with a price reference-period which coincides with its weight reference-period, the price reference-period has to be a whole year since only annual weights are normally available.

#### A weighted sum or mean of sub-indexes

Consumer Price Indexes are compiled in practice as a weighted arithmetic mean of sub-indexes:

$sum {wI_{p:t} }$

where the w are the weights, summing to unity, and the Ip:t are sub-indexes for month t with p as price reference-period. The indexes are almost all what has been called "plutocratic" indexes, as the weights reflect the value shares of its different components in the total value of consumption. A "democratic" index would instead use as weights the average across households of the share of the different components in the consumption value of each household taken separately.

Consumer Price Indexes could also be, but are not, compiled using other formulas, such as a weighted geometric mean of sub-indexes

$prod {left( {I_{p:t}^{} } right)^w }$

The coverage of each sub-index is defined in terms of the type of goods and services covered and also, for many of them, in terms of the location and type of outlet where they are sold.

#### Superlative indexes

Superlative indexes compare prices between two periods, each of whose weight reference-period coincides with its price reference-period, using weights which are a symmetric average of weights from both periods. Under strict and unrealistic assumptions they have been shown to provide a very close approximation to a "True cost of living index" as defined by economic theorists. However, they can be accepted as a standard of reference on the commonsense grounds that, for example, to compare 2000 with 1999 prices, both the 1999 and the 2000 pattern of consumption are relevant. The Fisher, Törnqvist and Walsh indexes, are all superlative. Though defined in price and quantity terms, like all national Consumer Price Indexes they can in practice only be estimated using value weights and sub-indexes.

Since the available weighting data relate to calendar years, estimated superlative indexes have to be whole-year to whole-year comparisons. These, of course, can only be made retrospectively once weights for the second year of the comparison become available. Thus they could be used in three ways:

1. To provide a historical series.
2. For retrospective evaluation of the Consumer Price Index.
3. (As recently introduced in Sweden) to provide the index up to the last of the two successive years for which it can be computed, carrying forward the index from that year to the current month as a fixed-base index, using that year as both price reference-year and weight reference-year.

Consider the Edgeworth index which, though not formally a superlative index, provides practically identical results.[a] This compares annual average prices in two years, say y-1 and y, using a simple average of the annual quantities of those two years, thus relating to a fixed "basket", an obviously meaningful comparison. It is thus defined as: $frac{{sum {p_y left( {q_{y - 1} + q_y } right)} }}{{sum {p_{y - 1} left( {q_{y - 1} + q_y } right)} }}$

Letting V represent the consumption values of the component sub-aggregates and, as above, the I represent the annual sub-indexes, this can be estimated as:

$sum {frac{{V_{y - 1} + frac{{V_y }}{{I_{y - 1:y} }}}}{{sum {V_{y - 1} + frac{{V_y }}{{I_{y - 1:y} }}} }}I_{y - 1:y} }$

Although Fisher, Törnqvist and Walsh indexes yield almost identical results to an Edgeworth index, unlike Fisher and Törnqvist indexes, but like a Laspeyres-type index, the Walsh and Edgeworth indexes have the desirable property that they can be additively decomposed.

As noted above, retrospective comparisons of the Edgeworth or a superlative index with the actual Consumer Price Index can be illuminating. The differences between them are sometimes erroneouslytreated as a measure of the strength of substitution effects between elementary aggregates – shifts in consumption away from those products whose relative prices have risen and towards those whose relative prices have fallen. This would only be legitimate in the extremely unlikely circumstance that all other factors determining the pattern of consumption had remained unchanged over the interval covered by the indexes. These other factors are: changes in disposable income distribution and levels; advertising, fashion, magazine articles, pop stars, TV, weather; the fact that households which bought a durable good or service in the first period usually won't want to repeat the purchase in the second period; and that people and their children are older, some have died and new households have been formed.

#### Indexes and their sub-indexes

Usually, Consumer Price Indexes are compiled in several stages by aggregation, that is to say as weighted averages of sub-indexes, most of which are in turn weighted averages of sub-sub-indexes, some of which in turn may be weighted averages of sub-sub-sub-indexes. These weights are shares in consumption values, and for a given set of overall consumption values, V, it makes no difference how this aggregation is done. Thus, for example, if, for month t with p as price reference-month, there are four sub-sub-indexes A1, A2, B1 and B2, the Consumer Price Index computed from all four of them is identically equal to the index computed from the two sub-indexes for A and B:

$begin{array}{c} {rm{CPI}} = frac{{{rm{V}}^{{rm{A}}_{rm{1}} } }}{{sum V }}I_{p:t}^{A_1 } + frac{{{rm{V}}^{{rm{A}}_{rm{2}} } }}{{sum V }}I_{p:t}^{A_2 } + frac{{{rm{V}}^{{rm{B}}_{rm{1}} } }}{{sum V }}I_{p:t}^{B_1 } + frac{{{rm{V}}^{{rm{B}}_{rm{2}} } }}{{sum V }}I_{p:t}^{B_2 } equiv frac{{{rm{V}}^{rm{A}} }}{{sum V }}I_{p:t}^A + frac{{{rm{V}}^{rm{B}} }}{{sum V }}I_{p:t}^B end{array}$

where $I_{p:t}^A = frac{{{rm{V}}^{{rm{A}}_{rm{1}} } }}{{{rm{V}}^{{rm{A}}_{rm{1}} } + {rm{V}}^{{rm{A}}_{rm{2}} } }}I_{p:t}^{A_1 } + frac{{{rm{V}}^{{rm{A}}_{rm{2}} } }}{{{rm{V}}^{{rm{A}}_{rm{1}} } + {rm{V}}^{{rm{A}}_{rm{2}} } }}I_{p:t}^{A_2 }$ and similarly for B.

The consumption values, V, relate to the weight reference-period, though they may be price-updated as discussed below. As already noted, the weight reference-period is usually a full year antedating the price reference-period which is usually a single month. However the index can be expressed as if the price reference-period were a whole year. Thus if December 1999 is the actual price reference-month, an index for May 2000 can be transformed to express it with 1997 as price reference-year as:

$I_{overline {97} :May2000} = I_{overline {97} :Dec.99} cdot I_{Dec.99:May2000}$

where the bar over 1997 signifies an annual average for that year.

It is rarely possible to compute annual average prices directly. In consequence, price re-referencing has to be done by using an average of monthly sub-indexes. Using indexes based on the price reference-period previously used, say December 1996, the average required for $I_{overline {97} :Dec.99}$ is:

$I_{overline {97} /Dec.99}^{} = frac{{I_{Dec.96:Dec.99}^{} }}{{frac{1}{{12}}left( {I_{Dec.96:Jan.97}^{} + I_{Dec.96:Feb.97}^{} cdots + I_{Dec.96:Dec.97}^{} } right)}}$

or

$I_{overline {97} :Dec.99}^{} = frac{{I_{Dec.96:Dec.99}^{} }}{{sqrt[{12}]{{I_{Dec.96:Jan.97}^{} times I_{Dec.96:Feb.97}^{} cdots times I_{Dec.96:Dec.97}^{} }}}}$

As monthly consumption values are known to differ from month to month, a weighted average would be preferable to these unweighted, i.e. equally weighted, averages. If purchase quantities and prices are positively or negatively correlated within a year, a weighted average of twelve monthly indexes will exceed or fall short of their simple average. Such correlations certainly exist for certain index components:

• The availability of fresh products varies seasonally, their prices moving inversely. Simple averages of monthly fresh product indexes will exceed their weighted averages. Hence weight updating from the weight reference-year to the price reference-month using the simple average will yield lower price-updated fresh product weights than if the weighted averages are used.
• Clothing purchases are particularly large in months when there are Sales.
• December purchases exceed their monthly average for many products, so that in a year with marked inflation there is a positive correlation.

Price-quantity correlations are likely to be small for highly aggregate subindexes, however, except for years with rapid inflation.

Examples for such subindexes are provided by some UK data.

Year Food Household goods Clothing & footwear
Weighted minus simple average % points
1992 0.0 0.1 0.2
1993 0.0 0.1 0.3
1994 0.0 0.1 0.3
1995 0.1 0.3 0.4
1996 0.0 0.1 0.5

These relate to seasonally unadjusted monthly retail sales volumes by three groups of outlets: Predominantly food stores, Household goods stores and Textile and clothing stores. Matching them with three approximately corresponding monthly price sub-indexes, allows calculation of the differences between weighted and simple average annual indexes shown in the table below. They are very small, but they do exist.

In the case of monthly fresh product indexes, it is clear that simple averages will exceed their weighted averages because of the marked negative price-quantity correlations for such items. (I have found no data to illustrate this, though users of the Rothwell method obviously have such data.) Weight updating from the weight reference-year to the price reference-month using the simple average will therefore yield lower price-updated fresh product weights than if the weighted averages are used.

Since monthly quantity data are unavailable for the majority of elementary aggregate indexes and of subindexes, monthly quantity weights have to be monthly deflated expenditures. Denoting the monthly indexes as Im and the monthly expenditures as Vm, the weighted averages will thus have to be computed as:

$sumlimits_{m = 1}^{m = 12} {frac{{frac{{E_m }}{{I_m }}}}{{sumlimits_{m = 1}^{m = 12} {frac{{E_m }}{{I_m }}} }}I_m } = frac{{sumlimits_{m = 1}^{m = 12} {E_m } }}{{sumlimits_{m = 1}^{m = 12} {frac{{E_m }}{{I_m }}} }}$

#### Successive indexes

Whenever weights are revised and/or the sample of prices is revised, a new index is introduced and must be chained onto the old index. The following imaginary example shows this, supposing that it happens in January 2000, with prices collected both for the old sample and for the new sample in December 1999. It also shows how any index can be rescaled to make the Index Reference Period different from the price reference-period by dividing it by the value of the index for the selected Index Reference Period. Such rescaling obviously leaves rates of price change unaltered, but affects the absolute differences between successive index values.

Year Old Index. New Index Chain New to Old Divide by 1.250 to rescale

to

Divide by 1.476 to rescale to
Index Reference Dec. 1995 Index Reference Dec. 1999 Index Reference Dec. 1995 Index Reference average

1997

Index Reference June 2000
1997 average 1.250 1.250 1.000 0.84688
December 1999 1.440 1.000 1.440 1.152 0.97561
June 2000 1.025 1.4760 1.1808 1.000
December 2000   1.066 1.53504 1.22803 1.040

These indexes are expressed with an Index Reference Value of 1. In practice, they are usually expressed with an Index Reference Value of 100. In this case the December 2000 index rescaled to a 1997-average index Reference Value would be $frac{{153.504}}{{125}} times 100 = 122.803$. Similarly, when two indexes with Index Reference Values of 100 are multiplied together, their product has to be divided by 100, for example $frac{{106.6 times 144}}{{100}} = 153.504$.

### Price-updating of weights

#### The method

The weights used in a fixed-base index Consumer Price Index are usually consumption values which have been price-updated from the weight reference-year to the price reference-month. Thus if V97 is the value of consumption in the weight-reference-year of 1997 while the price reference-month is December 1999, the weights used from January 2000 onwards will be:

$frac{{left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} } right)}}{{sumlimits_{} {left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} } right)} }}$

where the bar over 97 signifies that these indexes relate December 1999 prices to annual average prices of 1997. (Preferably each would be a monthly-weighted mean of the twelve indexes Jan 97:Dec.99, Feb.97:Dec.99….. ….Dec.97:Dec.99.)

The index for May 2000 will then be computed by chaining the index computed with these new weights on to its predecessor Consumer Price Index, base b, with December 99 as the link month: $begin{array}{c} CPI_{b:May00} = CPI_{b:Dec.99} times sum {frac{{left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} } right)}}{{sumlimits_{} {left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} } right)} }}I_{Dec.99:May00} } = CPI_{b:Dec.99} times frac{{sum {left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :May00}^{} } right)} }}{{sumlimits_{} {left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} } right)} }} end{array}$

The index can be interpreted in terms of ratios of revalued weight reference-period consumption values. Thus consider the ratio of the May 2000 index to the February 2000 index, the left-hand expression below. The central expression shows this calculated with price-updated weights, but it reduces to the right-hand expression which is the ratio of the sum of consumption values revalued to May to their sum revalued to February.

$frac{{CPI_{Dec.99:May2000} }}{{CPI_{Dec.99:Feb2000} }} = frac{{sum {frac{{V_{overline {97} } I_{overline {97} :Dec,99} }}{{sum {V_{overline {97} } I_{overline {97} :Dec,99} } }}} I_{Dec.99:May2000} }}{{sum {frac{{V_{overline {97} } I_{overline {97} :Dec,99} }}{{sum {V_{overline {97} } I_{overline {97} :Dec,99} } }}} I_{Dec.99:Feb2000} }} = frac{{sum {V_{overline {97} } I_{overline {97} :May2000} } }}{{sum {V_{overline {97} } I_{overline {97} :Feb2000} } }}$

The ratio of the May 2000 index to the May 99 index is $frac{{CPI_{b/Dec.99} times frac{{sum {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :May00}^{} } }}{{sumlimits_{} {left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} } right)} }}}}{{CPI_{b:May.99} }}$

#### International indexes

For consistent aggregation of national indexes into an index for a group of countries, the weights used both within countries and for countries must be expressed at prices for the same point of time. For the European Harmonised Index of Consumer Prices, price-updating from each December to the following December is required, and any real change in the weights may be introduced only in December.

#### Deficient information

The fact that price-updating involves separate price-updating of each component weight by its own sub-index creates difficulties when a new set of weights includes new components. Thus if the new set of weights based on 1997 consumption values included a new component $V_{97}^{B_3 }$, the index $I_{overline {97} :Dec.99}^{B_3 }$ would be needed, and this requires that B3 prices will have had to be collected ever since December 1996. Yet the need to collect these prices may not have been discovered until well after the end of 1997 when the consumption value data were compiled. No estimate of $I_{overline {97} :Dec.99}^{B_3 }$ will then be available and it may be necessary to assume that B3 prices moved in parallel with those of some other sub-aggregate for which a sub-index has been compiled.

#### The significance of price-updating

For Consumer Price Indexes, whose weights necessarily relate to the past, it turns out that the choice between :

1. Price-updating weights to the price reference-period, implying constant quantity ratios;
2. Not updating them, implying constant value shares;

is related to the choice of formula to be used in computing that index.

Indexes for May 2000 calculated using weights price-updated from 1997 to a price reference-period of December 1999, for example, provide estimates of: $frac{{mbox{Value of 1997 annual consumption at May 2000 prices}}}{{mbox{Value of 1997 annual consumption at December 1999 prices}}}$

Price-updating thus preserves the 1997 consumption volume pattern. This result would provide an estimate of $frac{{mbox{Value of December 1999 consumption at May 2000 prices}}}{{mbox{Value of December 1999 consumption at December 1999 prices}}}$

only if December 1999 relative volumes of the different components of consumption were the same as in 1997.

Without any price-updating, the indexes for May 2000 would be estimates of the same thing under a different assumption, namely that December 1999 value shares, wi, were the same as the value shares, wi, in 1997 annual consumption.

This alternative assumption would mean that the ratio of the December 1999 values of each component to its 1997 value was the same for all of them, say a ratio of R. Hence the ratio of the December 1999 total consumption value to its 1997 value would also be R. Expressing the argument in terms of the universe of prices, p, and quantities, q, where the i are the component products; if, for all i: ${rm{ }}w^i = frac{{p_{Dec.99}^i q_{Dec.99}^i }}{{sumlimits_i {p_{Dec.99}^i q_{Dec.99}^i } }} = frac{{p_{overline {97} }^i q_{97}^i }}{{sumlimits_i {p_{overline {97} }^i q_{97}^i } }};{rm{then}};frac{{p_{Dec.99}^i q_{Dec.99}^i }}{{p_{overline {97} }^i q_{97}^i }} = frac{{sumlimits_i {p_{Dec.99}^i q_{Dec.99}^i } }}{{sumlimits_i {p_{overline {97} }^i q_{97}^i } }} = R$

However, since the weights sum to unity, R equals $prodlimits_i {R^{w^i } }$, and so: $prodlimits_i {R^{w_i } } = prodlimits_i {left( {frac{{p_{Dec.99}^i q_{Dec.99}^i }}{{p^i _{overline {97} } q^i _{97} }}} right)^{w^i } = quad } prodlimits_i {left( {frac{{p_{Dec.99}^i }}{{p^i _{overline {97} } }}} right)^{w^i } } times quad prodlimits_i {left( {frac{{q_{Dec.99}^i }}{{q^i _{97} }}} right)^{w^i } }$

(I am indebted to Jörgen Dalén for this formulation).

This decomposition into price and quantity components means that, under this assumption of constant value shares, the estimator of the price index would be $prodlimits_i {left( {I_{overline {97} :Dec.99}^i } right)^{w^i } }$, not $sumlimits_i {w^i I^i _{overline {97} :Dec.99} }$. In this case, the estimator of the price index for May 2000, with December 1999 as price reference-period, should be $prodlimits_i {left( {I_{Dec.99:May.2000}^{} } right)} ^{w^i }$ where the weights wi are the 1997 weights without any price updating.

#### When to price-update

This shows that whether to price-update, and whether to use the weighted arithmetic or geometric mean of sub-indexes, are linked questions which ought to be answered consistently.

• Price-updating and the weighted arithmetic mean of sub-indexes should be chosen when it is expected that the price reference-period pattern of consumption will be closer to the weight reference-period volume pattern of consumption than to its value-share pattern.
• When, alternatively, it is expected to be closer to the weight reference-period value-share pattern, the weights should not be price-updated and the weighted geometric mean of sub-indexes should be chosen.

The first alternative is appropriate when relative price changes and relative quantity changes are uncorrelated. The second is appropriate when they move inversely. This is more likely when the components are close substitutes in consumption. The textbook example of this is chicken and beef, a fall in chicken prices relative to beef prices inducing consumer substitution of chicken for beef.

The choice between these alternatives is not necessarily an either/or choice, because constant relative quantities and the weighted arithmetic mean may be more appropriate for some aggregates than for others. Substitution effects are more likely to be important within lower-level aggregates such as Meat, than between Meat and other food lower-level aggregates, that is to say within the higher-level food aggregate. Substitution effects between, for example, Food and Clothing, are even less likely, so that there is a presumption that constant relative quantities and the weighted arithmetic mean are appropriate when the overall Consumer Price Index is computed as a weighted mean of the highest-level aggregates such as Food, Clothing, Transport etc. Substitution effects are also unlikely in response to differential price changes in different regions of the country. Thus the aggregates for which constant value shares and the weighted geometric mean may be appropriate are regional low-level aggregates.

The decision of when to use arithmetic means and when to use geometric means should be made by investigating, for different aggregates, which would best approximate the most recently estimated weights of their sub-aggregates:

the preceding set of weights of their sub-aggregates with each weight price-updated to the year of the most recently estimated weights and divided by their new sum, implying constant relative volumes

or

the preceding set of weights of their sub-aggregates without any price-updating, implying constant value shares.

#### An alternative to price-updating

There is a good case for using a full year as link period, with chaining year upon year instead of using a single month.

For example, instead of price-updating the 1997 annual weights to the link month of December 1999, the subindexes would be price-backdated to their average for 1997 which would be used as the link year. Thus the index for May 2000 becomes: $begin{array}{c} CPI_{b:May00} = CPI_{b:overline {97} } times sum {frac{{V_{97} }}{{sum {V_{97} } }}I_{overline {97} :May00} } end{array}$

with $I_{bar 9bar 7:May.00} = I_{Dec.99:May00} times frac{{I_{b:Dec.99} }}{{sqrt[{12}]{{I_{b:Jan.97} times I_{b:Feb.97} cdots I_{b:Dec.97} }}}}$ or its arithmetic equivalent.

The ratio of this May 2000 index to the old May 99 index is: $frac{{CPI_{b:overline {97} } times sum {frac{{V_{97} }}{{sum {V_{97} } }}I_{overline {97} :May00} } }}{{CPI_{b:May99} }}$

To understand the difference between using a whole year rather than a single month, compare:

the index for May 00 with monthly linking in December 1999

with

what it would be with yearly linking in 1997 by dividing the former by the latter:

$frac{{CPI_{b:Dec.99} times sum {frac{{{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} }}{{sumlimits_{} {left( {{rm{V}}_{{rm{97}}}^{} times I_{overline {97} :Dec.99}^{} } right)} }}I_{Dec.99:May00} } }}{{CPI_{b:overline {97} } times sum {frac{{V_{97} }}{{sum {V_{97} } }}I_{overline {97} :May00} } }} = frac{{CPI_{b:Dec.99} }}{{CPI_{b:overline {97} } }} div sum {frac{{V_{97} }}{{sum {V_{97} } }}} I_{overline {97} :Dec.99}^{}$

This ratio is seen to equal the year 1997 to December 1999 change calculated with the old b weights, divided by the same change calculated retrospectively using the new 1997 weights. Comparing the May 99 to May 00 twelve-month change in the monthly linked index with that in the yearly linked index yields exactly the same expression. If substitution effects dominate, it may come out slightly below unity. (It can be said that the old index is upward biased on account of substitution effects only if b was both its price reference-period and its weight reference-period, so that b quantities were optimal with respect to b prices.)

The advantages of yearly links in the Consumer Price Index are several:

• The weights, not being price-updated, can be simply described as weight reference-year value proportions.
• The index can be simply described as a Laspeyres index if no geometric mean formulas are used, as it compares the current month's value of weight reference-year consumption with its weight reference-year value.
• The index can accommodate Rothwell-type sub-indexes for seasonal products of constant quality.
• Retrospective comparison can be made with a superlative index.

Since yearly and monthly linking both require year-97 to December-00 indexes for each and every component, V97, the admitted difficulty of estimating them retrospectively for new components introduced into the 1997 weights does not affect the choice between them.

### Chaining and aggregation

#### Decomposition

Index users often want to ask questions of the kind illustrated by the following examples:

• How much of the change in the index was due to food price changes?
• What is the division of the index between its durable and non-durable components?
• What would the index be if shelter costs were omitted?

All of these require some decomposition of the index, but the possibilities are limited.

#### Transitivity and Intransitivity

For indexes that are transitive it makes no difference whether the index for May 2000 is computed by directly relating May 2000 prices to December 1999 prices or whether it is computed by chaining from month to month. Transitivity means that, for example:

${rm{I}}_{{rm{Dec}}{rm{.99:May 2000}}} = {rm{I}}_{{rm{Dec}}{rm{.99:Jan 2000}}} times I_{Jan.2000:Feb.2000} times {rm{I}}_{{rm{Feb}}{rm{.2000:Mar}}{rm{.2000}}} times {rm{I}}_{{rm{Mar}}{rm{.2000:Apr}}{rm{.2000}}} times {rm{I}}_{{rm{Apr}}{rm{.2000:May}}{rm{.2000}}}$

"Elementary aggregate" or "micro" indexes calculated as unweighted ratios of arithmetic or geometric mean prices of the same unchanged set of sampled products are transitive. They are discussed in another chapter. But the indexes under consideration here, which are calculated as weighted sums of lower-level indexes, are not transitive. Chaining January to February with February to March, and so on, using the same value share weights throughout, would amount to altering the implicit quantities from month to month. The indexes therefore have to be calculated as fixed-base indexes by directly relating each month's prices to December 1999 prices.

The result of this is that, for these non-transitive indexes, the price change from, say, February 2000 to May 2000 has to be calculated as:

$frac{{sumlimits_i {w^i I_{Dec.99:May2000}^i } }}{{sumlimits_i {w^i I_{Dec.99:Feb2000}^i } }}$

which could perhaps be described as chain linking May with February indirectly via December 1999.

### Additivity of sub-indexes with given weighting

Such IFeb:May sub-indexes cannot be weighted and summed to obtain the overall ratio of the May CPI to the February CPI. Index ratios are not additive, as can simply be demonstrated for two components, A and B :

${rm{CPI}}_{{rm{Feb:May}}} = frac{{w^A I_{Dec.99:May.2000}^A + w^B I_{Dec.99:May.2000}^B }}{{w^A I_{Dec.99:{rm{Feb}}.2000}^A + w^B I_{Dec.99:{rm{Feb}}.2000}^B }} ne w^A frac{{I_{Dec.99:May.2000}^A }}{{I_{Dec.99:{rm{Feb}}.2000}^A }} + w^B frac{{I_{Dec.99:May.2000}^B }}{{I_{Dec.99:{rm{Feb}}.2000}^B }}$

The proportional increase (or decrease) in the overall Consumer Price Index between February and May 2000 is:

$begin{array}{l} {rm{CPI}}_{{rm{Feb:May}}} - 1 = frac{{w^A I_{Dec.99:May.2000}^A + w^B I_{Dec.99:May.2000}^B - w^A I_{Dec.99:{rm{Feb}}.2000}^A - w^B I_{Dec.99:{rm{Feb}}.2000}^B }}{{w^A I_{Dec.99:{rm{Feb}}.2000}^A + w^B I_{Dec.99:{rm{Feb}}.2000}^B }} quad quad quad quad quad = frac{{w^A left( {I_{Dec.99:May.2000}^A - I_{Dec.99:{rm{Feb}}.2000}^A } right) + w^B left( {I_{Dec.99:May.2000}^B - I_{Dec.99:{rm{Feb}}.2000}^B } right)}}{{{rm{CPI}}_{Dec.99:{rm{Feb}}.2000} }} end{array}$

Separating the absolute increase (or decrease) in the overall index into additive A and B components is simpler :

$begin{array}{l} quad {rm{CPI}}_{{rm{Dec}}{rm{.99:May}}{rm{.2000}}} quad minusquad {rm{CPI}}_{{rm{Dec}}{rm{.99:Feb}}{rm{.2000}}} = left( {w^A I_{Dec.99:May.2000}^A + w^B I_{Dec.99:May.2000}^B } right) - left( {w^A I_{Dec.99:{rm{Feb}}.2000}^A + w^B I_{Dec.99:{rm{Feb}}.2000}^B } right) = w^A left( {I_{Dec.99:May.2000}^A - I_{Dec.99:{rm{Feb}}.2000}^A } right) + w^B left( {I_{Dec.99:May.2000}^B - I_{Dec.99:{rm{Feb}}.2000}^B } right) end{array}$

These expressions, and those that follow, apply equally to decomposition of a sub-index into the sub-sub-indexes for its components.

#### Additivity with reweighting

If a new set of weights based on more up-to-date consumption value data is introduced, the index has to be chained. An aggregation problem then arises for all comparisons that overlap the weight change. Consider, for example, a comparison of May 2000 with May 1999, supposing that:

• Old weights, w0 were used up to December 1999, with December 1997 as price reference-month. The indexes from May 1999 to December 1999, IMay.99:Dec.99, thus equal the ratios $frac{{I_{Dec.97:Dec.99} }}{{I_{Dec.97:May.99} }}$;
• New weights, wn, were used thereafter for the index from January 1999 onwards and December 1999 replaced December 1997 as the price reference-month;

The overall Consumer Price Index for the twelve-month interval from May 1999 to May 2000 must be computed aggregatively by chaining together the overall Consumer Price Index from May 1999 to December 1999 and the overall Consumer Price Index from December 1999 to May 2000: $left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } right)quad times quad left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B } right)$

It cannot be obtained as a weighted sum of the two chained sub-indexes $I_{May.99:Dec.99}^A times I_{Dec.99:May.2000}^A quad {rm{,}}quad I_{May.99:Dec.99}^B times I_{Dec.99:May.2000}^B$

unless the two sets of weights are identical.

This is unfortunate. It means, for example, that for comparisons overlapping a weight change, such as from May 1999 to May 2000, it is impossible to divide the overall Consumer Price Index ratio or its proportional change into an A component and a B component.

It is, however, possible to decompose the absolute change in the overall Consumer Price Index into additive A and B components.

Write: CPIDec 97:May 2000 minus CPIDec 97:May 97 as: $CPI_{Dec97:Dec99} times ;left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B } right); - ,;left( {w_o^A I^A _{Dec.97:May.99} + w_o^B I^B _{Dec.97:May.99} } right)$

and then rearrange as:

$begin{array}{l} quad left( {CPI_{Dec97:Dec99} quad times quad w_n^A I_{Dec.99:May.2000}^A } right) - w_o^A I^A _{Dec.97:May.99} + left( {CPI_{Dec97:Dec99} quad times quad w_n^B I_{Dec.99:May.2000}^B } right) - w_o^B I^B _{Dec.97:May.99} quad end{array}$

A weight-change-overlapping sub-index for the A or B component regarded separately would require a separate computation, using the corresponding two sets of weights for its sub-sub-indexes and chaining through December 1999. There is no single set of higher level weights which will combine such chained lower-level sub-indexes overlapping a reweighting into a higher level index.

An imaginary numeric example, with the figures unrounded, follows. The price reference-period for the New index is December 1999. The indexes excluding Food are calculated using only Clothing, Housing and Everything-else weights. The May to May indexes in the bottom two lines are calculated as the product of an index for May 1999 to December 1999 (derived by dividing the December index by the May index) and an index for December 1999 to May 2000.

Old index New Index
Weight May 1999 index December 1999 index May index * Weight December index * Weight Weight May 2000 index May index * Weight
Food 0.2 1.06 1.07 0.212 0.214 0.15 1.04 0.156
Clothing 0.1 1.05 1.08 0.105 0.108 0.15 1.02 0.153
Housing 0.4 1.01 1.02 0.404 0.408 0.3 1.01 0.303
Everything-else 0.3 1.02 1.04 0.306 0.312 0.4 1.02 0.408
Overall index 1 1.027 1.042 1 1.0200
Excluding Food 0.8 1.01875 1.035 0.85   1.0146706
Overall index May 1999 to May 2000 = 1.042 / 1.027 * 1.0200 1.034898
Index excluding Food May 1999-May 2000 =1.035 /1.01875 * 1.0164706 1.032684

This all assumes an unchanged set of sub-indexes. But sometimes the set is changed. For example, reverting to the case where there are subindexes for A and B, reweighting might entail the introduction of a new and additional sub-index for a component of consumption, C. This may have been too small to have been included in the old weights, wo, or may have been deliberately excluded from coverage when they were estimated, but is now to be included in the index computed with new weights, wn, from December 1999 onward. In this case, the overall Consumer Price Index from May 1999 to May 2000 will be: $left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } right) times left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B + w_n^C I_{Dec.99:May.2000}^C } right)$

#### Prices that were zero

A special case of the introduction of new weights arises when government policy changes so that prices have to be paid for a group of goods or services, C, which were previously provided free. A new sub-index has to be created in order to include them in the index. This, for once, requires information about quantities, which should, however, be available from the branch of government that provided the free goods or services.

Suppose that the new sub-index is introduced from the beginning of 1999 and consider an index for, say, May 2000 calculated using 1997 weights price-updated to December 1999. It provides an estimate of:

$frac{{mbox{Market value of 1997 annual consumption at May 2000 prices}}}{{mbox{Market value of 1997 annual consumption at December 1999 prices}}}$

Writing pc and qc for the prices and quantities of the C goods or services this becomes: $frac{{V_{97}^A I_{overline {97} :Dec.99}^A I_{Dec.99:May2000}^A + V_{97}^B I_{overline {97} :Dec.99}^B I_{Dec.99:May2000}^B + sum {p_{May2000}^c q_{1997}^c } }}{{V_{97}^A I_{overline {97} :Dec.99}^A + V_{97}^B I_{overline {97} :Dec.99}^B }}$

so that $frac{{sum {p_{May2000}^c q_{1997}^c } }}{{V_{97}^A I_{overline {97} :Dec.99}^A + V_{97}^B I_{overline {97} :Dec.99}^B }}$ has to be added to the index computed only for A and B.

The opposite case arises when a product which had its own weight disappears from the market as, for example, when leaded petrol ceases to be sold. This should not be allowed to affect the index. If its weight were simply added to the weight for a similar product or product group, for example unleaded petrol, the index might change even if the prices of everything remaining in the index had not changed. Chaining is therefore necessary, for example, the last petrol price index which included leaded petrol should be moved forward by the index for unleaded petrol.

#### Reweighting without price-updating

Having examined the introduction of new weights which are price-updated to the price reference-month, consider the introduction of new weights without any price updating.

Suppose that:

• Old weights, wo were used up to December 1999, with December 1997 as price reference-month.
• New weights, wn, were used thereafter, but December 1997 continues to be used as the price reference-month;

and consider the overall Consumer Price Index for May 2000. This can be computed by chaining the overall Consumer Price Index from December 1997 to December 1999, computed with the old weights, with the overall Consumer Price Index from December 1999 to May 2000, computed with the new weights: $left( {w_o^A I_{Dec.97:Dec.99}^A + w_o^B I_{Dec.97:Dec.99}^B } right)quad times quad frac{{w_n^A I_{Dec.97:May.2000}^A + w_n^B I_{Dec.97:May.2000}^B }}{{w_n^A I_{Dec.97:Dec.99}^A + w_n^B I_{Dec.97:Dec.99}^B }}$

In this case too, no decomposition into the separate contributions of A prices and B prices to the overall increase is possible for periods overlapping December 1999.

#### Price-updating without real change

Note that these problems attached to reweighting arise only when there are real changes in the weights. Merely price-updating them to a new reference-period, n, does not prevent the combination of sub-indexes to produce a higher-level index.. Price-updating multiplies each old weight, wo, by its sub-index and then divides by the sum of all the weights each multiplied by its sub-index, thus preserving the sum of the price-updated weights, wn, as unity. In this case the index from May 1999 to May 2000, the product of the 1999 May to December index and the December 1999 to May 2000 index $left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } right)quad times quad left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B } right)$

can be expressed as:

$begin{array}{l} quad quad left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } right)quad times quad left( begin{array}{l} quad w_o^A frac{{w_o^A I_{May.99:Dec.99}^A }}{{left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } right)}}I_{Dec.99:May.2000}^A + w_o^B frac{{I_{May.99:Dec.99}^B }}{{left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } right)}}I_{Dec.99:May.2000}^B end{array} right) end{array}$

which simplifies to:

${rm{CPI}}_{{rm{May99:May 2000}}} = left[ {w_o^A left( {I_{May.99:Dec.99}^A I_{Dec.99:May.2000}^A } right) + w_o^B left( {I_{May.99:Dec.99}^B I_{Dec.99:May.2000}^B } right)} right]$

The A and B components are obviously additive.

### Seasonal products

#### The problem

The computation of a Consumer Price Index involves a comparison between:

• The current month' s cost to consumers of (a sample of) the products that constituted the weight reference-year' s consumption.
• The price reference-period cost to consumers of (the sample of) the products that constituted the weight reference-year's consumption.

This creates an obvious problem in months when some products are wholly unavailable, or too largely unavailable to allow their prices to be collected. The most important such "seasonal" products are fresh fruit, vegetables and fish, cut flowers, package holidays and, in many cases, sporting goods and clothing..

#### Practical alternatives

There are three ways out of this dilemma. All three are unsatisfactory, but one of them has unavoidably to be chosen. They are:

1. Completely omit seasonal products from the index, thus limiting its coverage to products that are available throughout the year.
2. Pretend that unavailable seasonal products are available and invent prices for them. There two ways of doing this:
1. Carry forward the last collected prices unchanged,
2. Extrapolate the last collected prices by the month-to-month change in the prices of available products.
In both cases annual weights will be used. Prices can either be compared with prices (including any fictitious prices) for a weight reference-month, or they can be compared with average prices for a weight reference-year
3. Use monthly weights and a price reference-year. The computation for each month will compare the current price for each product available in that month with the corresponding price reference-year weighted average price of that product. For each group of seasonal products a sub-index is computed using monthly weights equalling quantities from that month in the weight reference-year valued at weighted average prices from the price reference-year. Thus the August sub-index for such such a group will compare the current August value of August weight reference-year consumption with its value at price reference-year weighted average prices.
Monthly quantities are required both to compute the weights of the available fruits for each month and to compute the weighted annual average price of each fruit in the price reference-year. It may be better to derive the weights from data for several years, since the seasonal pattern can vary between years.
In the absence of monthly data, weights will simply have to be set at zero for months of non-availability and at one for months of availability.

Which of these is chosen can have a considerable effect upon the index.

Whether the index views consumption as Transactions, as Expenditure or as Use will make a difference for products where the timing of transactions, payments and use can differ. Thus transactions in which consumers book a July package holiday may be made as early as January, payment of part of the price may follow in a later month, and the holiday can only be taken in July.

#### Pros and Cons

Method (2.ii) assumes that, if the products had remained available, their prices would have moved in the same way as the prices of the products within the broader consumption category which did remain available. This is equivalent to giving a greater weight to these other products. But though this way of imputing fictitious prices is algebraically equivalent to the use of monthly weights, its implicit monthly weights are unreasonable. To see why, consider the simple case where annual expenditures on oranges and on apples are each 40, spread evenly over the twelve months, while annual expenditure on cherries is 20, spread evenly over only the two months of July and August. Then, if cherry prices are extrapolated forward according to the movement of apple and orange prices, starting each September and continuing monthly until the following June, apples and oranges each acquire effective weights of 50 per cent during these ten months. This is reasonable. In July and August, apples and oranges have weights of 40 per cent and cherries have a weight of 20 per cent. But if cherries account for 20 per cent of total annual expenditure on the three types of fruit, they must account for vastly more than 20 per cent of expenditure on fruit in July and August! The implicit July and August weights are, therefore, not reasonable for measuring month-to-month changes.

Both methods (2.i) and (2.ii) have the advantage as compared with (3) that they can cope with differences between years in the timing of availability. If cherries become available a month earlier than usual, their actual current price can be used instead of a fictitious price. But method (3) requires that a current cherry price be collected only for each month for which cherries have a weight. If, on the other hand, cherries cease to be available a month earlier than normal, a fictitious price will have to be used under all of these methods.

A drawback with method (3) is that the index may change between two months because of a weight change, even though no price has changed and availabilities are unaltered. Furthermore, it may seem unclear what an index comparison between two successive months in different seasons signifies. Is it meaningful, for example, to say that the June 2001 cost of buying the June 2000 basket of fruit compared with its cost at average 2000 prices exceeds the May 2001 cost of buying the May 2000 basket compared with its cost at average 2000 prices?

$frac{{sum {{rm{P}}_{June01} Q_{June00} } }}{{sum {{rm{P}}_{overline {00} } Q_{June00} } }} > frac{{sum {{rm{P}}_{May01} Q_{May00} } }}{{sum {{rm{P}}_{overline {00} } Q_{May00} } }}$

If method (3) is employed, as many statisticians recommend, the use of a whole year as price reference-period for seasonal products may require the use of a whole year as weight and price reference-period for all other sub-indexes as well. This can be avoided if the sub-indexes for seasonal products are rescaled from their price reference-year to a price reference-period of a single month.

Leaving aside any such rescaling, in method (3), a sub-index for May 2002

with 2000 as both weight reference-year and price reference-year would be an estimate of:

$frac{{sum {p_{2002,May} q_{2000,May} } }}{{sum {bar p_{2000} q_{2000,May} } }} = sum {frac{{p_{2002,May} }}{{bar p_{2000} }}left( {frac{{bar p_{2000} q_{2000,May} }}{{sum {bar p_{2000,May} q_{2000,May} } }}} right)}$

where the summation is over all products and p with a bar signifies the quantity-weighted annual average price. The weight for each product (the bracketed term) is not its share in weight reference-period May value; it is its share calculated using weighted average 2000 prices to value the May quantities of products.

#### No weights available

(The graph can be uploaded as image: Fruit price indexes; Turvey's imaginary data)

### Appendix: An in-between formula

A variant of the $sum {wI_{p:t} }$ formula suggested by Brent Moulton can provide a result which falls somewhere between the two extremes so far examined, providing an approximation to a superlative index – if substitution effects are the only factor causing weights to alter. If adopted, consistency requires that this variant be applied both to the price-updating of weights and the computation of the index using those weights.

This variant is (omitting the i superscripts) $left( {sum {wI_{p:t}^k } } right)^{frac{1}{k}}$ where k is a constant which will be lower the greater is substitutability between the sub-aggregates and hence the greater are demand elasticities.. In the case of no substitutability, implying zero price elasticities, k = 1 and the formula reduces to $sum {wI_{p:t} }$. The corresponding formula for weights price-updated from the weight reference-period, w, to the price reference-period, t, is: $frac{{w_p = left( {w_w I_{w:p} } right)^k }}{{sum {w_p = left( {w_w I_{w:p} } right)^k } }}$

With k = 1 this formula gives the same weights as the price-updated weights normalised to sum to unity, while with k almost equal to 0 the formula gives almost equal weights.

When past weights are available for two years, application of the formula to the earlier weights will allow those values of k to be determined which best predict the actual later weights for each aggregate and its sub-aggregates. There is then a presumption that:

• The formula should apply these values of k to update the latest available weights to the price reference-period currently being used
• Each index and its sub-indexes should then be computed as $left( {sum {wI_{p:t}^k } } right)^{frac{1}{k}}$, using the same values of k.

Where high-level aggregate indexes are combined to update weights, k = 1 will be likely to give the least bad answer. Since such aggregates are not substitutes for one-another, straightforward price-updating is appropriate But k < 1 may well be appropriate for lower-level aggregates where substitutability may exist.

Turning now from the price-updating of weights to the computation of the index, the problem is to find, for any aggregate and its sub-aggregates, that value of k which brings the index for that aggregate computed as, for example, $left( {sum {w_{87} I_{87:94}^k } } right)^{frac{1}{k}}$ closest to a corresponding superlative index that compares the later year with the earlier year. A "superlative" index uses some kind of symmetric average of weights reflecting the consumption patterns of the two terminal years. Such an index can, of course, only be computed retrospectively, so can only be used as a standard against which an actual monthly Consumer Price Index can be checked after the event. If k turns out to have been stable this might justify using its past value in current computations.

Theoretical discussions of ideal indexes and superlativeness assume that the weight and price reference-periods coincide and are of the same length. The pure Laspeyres and Paasche indexes and the (superlative) Fisher index to which this theory relates can be calculated only for whole years and not for months. Moulton proved that the above formula yields an exact true cost of living index for a representative consumer with unchanging homothetic preferences with a constant elasticity of substitution equal to 1-k. A representative consumer is presumably an adult hermaphrodite with a non-integer number of children.

Shapiro, M. and Wilcox, D., in their paper "Alternative strategies for aggregating prices in the CPI"[1] have applied this idea in a one-stage computation of the overall U.S. Consumer Price Index from 9,108 area/product sub-indexes and in a two-stage computation, in which they first applied $left( {sum {w_{} I_{}^k } } right)^{frac{1}{k}}$ to the 207 product indexes, then used $sum {wI_{} }$ to aggregate over the 44 areas. The justification of this two-stage approach was the eminently reasonable assumption that substitutability between areas is extremely low, so that k ≈ 1.

As they computed their indexes from December to December instead of from whole-year to whole-year, they had to use a Fisher- or Törnqvist-type approximation to a pure Fisher or Törnqvist index.. Their Laspeyres-type index weights elementary aggregate indexes based on December in the preceding year by expenditures for the whole of that preceding year; their Paasche-type weights them by expenditures for the whole of the current year. Their Törnqvist-type index uses the simple mean of the weights for the two years. They note that this method could be refined in two ways. One would centre the price reference-period within the period over which the expenditures are calculated, e.g. basing the elementary aggregate indexes on mid-year rather than December prices. The other would price-update or down-date expenditures to the price reference-month (and perhaps the current month).

Percentage points difference between annual change of $left( {sum {w_{} I_{}^k } } right)^{frac{1}{k}}$ index and Törnqvist index annual change
k = 0.4 k = 0.3 k = 0.2
1987-8 .06 .04 .02
1988-9 .02 -.0 -.02
1989-90 -.0 -.05 -.09
1990-1 -.08 -.09 -.09
1991-2 .03 0 -.20
1992-3 .07 .04 .02
1993-4 .06 .04 .02
1994-5 .04 .02 .01
Mean difference .02 0 -.02
Standard Deviation .05 .04 .05
Mean annual index change 3.35 3.32 3.30

Their result, formulated in terms of 1-k, indicated that the value of k which gave the closest approximation to the superlative index was 0.3. This followed from the following comparison of December to December changes in the chained index computed for different values of k, with the changes in an annually chained Törnqvist index:

The difference from the previous calculation is due to the fact that this one relates to a large number of low-level aggregates instead of a small number of high-level aggregates, so that much more substitutability between aggregates would be likely.

The greater complexity of $left( {sum {wI_{p:t}^k } } right)^{frac{1}{k}}$ as compared with $sum {wI_{p:t} }$ and $prod {I_{p:t}^w }$ has two consequences. The first is that it is more difficult to explain to users. The second is that the change in it between a pair of months cannot be decomposed to show the contribution of the various components to the aggregate change. Perhaps it should only be employed to construct sub-indexes for low-level aggregates which would then be combined to compute higher-level indexes and the overall index employing one of the two simpler formulas.

### Notes

• ^[a]  An equally meaningful concept, hitherto applied in Sweden, relates to twelve monthly price changes from the beginning to the end of a year, from pby to pey, using the annual quantities of the whole of that year:
$frac{{sum {p_{ey} q_y } }}{{sum {p_{by} q_y } }}$
Using monthly subindexes from December of year y-1 to December of year y, this can be estimated as:
$sum {frac{{V_y }}{{sum {V_y } }}} I_{y - 1,12:y,12}$

## CPIs around the world

### United States

CPI-U 1913-2004; Source: U.S. Department Of Labor
Annual inflation (and deflation), 1914-2007

In the USA, CPI figures are prepared monthly by the Bureau of Labor Statistics of the United States Department of Labor. CPI-U 1913-2004; Source: U.S. Department Of Labor The U.S. Consumer Price Index is a time series measure of the price level of consumer goods and services. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... The Bureau of Labor Statistics was founded in 1884 by President Chester A. Arthur. ... The United States Department of Labor is a Cabinet department of the United States government responsible for occupational safety, wage and hour standards, unemployment insurance benefits, re-employment services, and some economic statistics. ...

The CPI-U includes expenditures by all urban consumers. The CPI-W includes expenditures by consumer units with clerical workers, sales workers, craft workers, operative, service workers, or laborers. Recently, the Chained Consumer Price Index C-CPI-U, a chained index, was introduced. The C-CPI-U tries to mitigate the substitution bias that is encountered in CPI-W and CPI-U by employing a Tornqvist formula and utilizing expenditure data in adjacent time periods in order to reflect the effect of any substitution that consumers make across item categories in response to changes in relative prices. The new measure, called a "superlative" index, is designed to be a closer approximation to a "cost-of- living" index than the other measures. The use of expenditure data for both a base period and the current period in order to average price change across item categories distinguishes the C-CPI-U from the existing CPI measures, which use only a single expenditure base period to compute the price change over time. In 1999, the BLS introduced a geometric mean estimator for averaging prices within most of the index’s item categories in order to approximate the effect of consumers’ responses to changes in relative prices within these item categories. The geometric mean estimator is used in the C-CPI-U in the same item categories in which it is now used in the CPI-U and CPI-W. Look up substitution in Wiktionary, the free dictionary. ... Look up substitution in Wiktionary, the free dictionary. ... The price of one thing (usually a good) in terms of another; ie, the ratio of two prices. ...

#### Sources of data

Prices for the goods and services used to calculate the CPI are collected in 87 urban areas throughout the country and from approximately 23,000 retail and service establishments. Data on rents are collected from about 50,000 landlords and tenants.

The weight for an item is derived from reported expenditures on that item as estimated by the Consumer Expenditure Survey. Prices are taken throughout the month. The Consumer Expenditure Survey (CE) program consists of two surveysâ€”the quarterly Interview survey and the Diary surveyâ€”that provide information on the buying habits of American consumers, including data on their expenditures, income, and consumer unit (families and single consumers) characteristics. ...

The BLS numbers are available through:

• Monthly news release. Consumer Price Index. Electronic access available.
• Historical data in Handbook of Labor Statistics. Electronic access available.
• Diskettes
• "LABSTAT" database.

#### Major research in progress

• Continuing research on technical improvements in the calculation of the CPI.
• Continuing work on the next major weight revision of the CPI.

In 1996, the Boskin Commission found the CPI to be a biased measure, and gave a quantitative analysis of the bias. The Boskin critique helped to spur some changes in the U.S. CPI, although it was partially disputed by the BLS. Many of the changes were aimed at moving the CPI to a cost of living model which takes consumer substitutions into account and typically reduces the reported level of inflation. The Boskin Commission, formally called the Advisory Commission to Study the Consumer Price Index, was appointed by the United States Senate in 1995 to study possible bias in the computation of the Consumer Price Index (CPI), which is used to measure inflation in the United States. ...

Statistics Canada looks at all the payments and taxes paid by consumers in Canada and then eventually publishes the CPI figures and the data used in the calculation.[2] Statistics Canada is the Canadian federal government bureau commissioned with producing statistics to help better understand Canada, its population, resources, economy, society, and culture. ...

### Eurozone

The European Central Bank publishes the Monetary Union Index of Consumer Prices (MUICP). It is a weighted average of price indices of member states. The method is a HICP or "Harmonized Index of Consumer Prices", where goods are split by final consumption; it is a seasonally adjusted chained index. Headquarters Coordinates , , Established 1 January 1998 President Jean-Claude Trichet Central Bank of Austria, Belgium, France, Finland, Germany, Greece, Ireland, Italy, Luxembourg, Netherlands, Portugal, Slovenia, Spain Currency Euro ISO 4217 Code EUR Reserves â‚¬43bn directly, â‚¬338bn through the Eurosystem (including gold deposits). ... The Harmonised Index of Consumer Prices (HICP) is an idicator of inflation and price stability for the European Central Bank (ECB). ...

### United Kingdom

The traditional measure of inflation in the UK for many years was the Retail Price Index, which was first calculated in the early 20th Century to evaluate the extent to which workers were affected by price changes during the first world war. An explicit inflation target was first set in October 1992 by then-Chancellor of the Exchequer Norman Lamont following the departure of the UK from the Exchange Rate Mechanism. Initially, the target was based on the RPIX, which is the RPI calculated excluding mortgage interest payments. This was felt to be a better measure of the effectiveness of macroeconomic policy. It was argued that if interest rates are used to curb inflation, then including mortgage payments in the inflation measure would be misleading. Until 1997, interest rates were set by the Treasury. The Consumer Price Index is the official inflation measure of the United Kingdom. ... Norman Stewart Hughson Lamont, Baron Lamont of Lerwick, PC (born 8 May 1942) was Conservative Member of Parliament for Kingston-upon-Thames, England from 1972 until 1997. ... The European exchange rate mechanism (or ERM) was a system introduced by the European Community in March 1979, as part of the European Monetary System (EMS), to reduce exchange-rate variability and achieve monetary stability in Europe, in preparation for Economic and Monetary Union and the introduction of a single... It has been suggested that this article or section be merged with Consumer price index. ...

On winning power in May 1997, the New Labour government handed control over interest rates to the Bank of England, whose Monetary Policy Committee now sets rates on the basis of an inflation target set by the Chancellor.[3] If in any month inflation is more than one percentage point off its target, the Governor of the Bank of England is required to write to the Chancellor explaining why. Mervyn King became the first Governor to do so in April 2007, when inflation ran at 3.1% against a target 2%.[4] New Labour is an alternative name of the British political Labour Party. ... Headquarters Coordinates , , Governor Mervyn King Central Bank of United Kingdom Currency Pound Sterling ISO 4217 Code GBP Base borrowing rate 5. ... The Monetary Policy Committee (MPC) is a committee of the Bank of England, which meets every month to decide the official interest rate in the United Kingdom. ... The Governor of the Bank of England is the most senior position in the Bank of England. ...

Since 1996 the United Kingdom has also tracked a Consumer Price Index figure, and in December 2003 its inflation target was changed to one based on the CPI. [5] The CPI target is currently 2%. Both the CPI and the RPI are published monthly by the Office for National Statistics. Office for National Statistics logo The Office for National Statistics (ONS) is the United Kingdom government executive agency charged with the collection and publication of statistics related to the economy, population and society of the United Kingdom at national and local levels. ...

### Belgium

In Belgium, wages, pensions, house rent, insurance premiums, unemployment benefits, health insurance payments, etc. are by law tied to a consumer price index. The Belgian Consumer Price Index (commonly referred to as the Index) is a list of prices of goods and services, kept by the Belgian Federal Government Service Economy. ...

### Mexico

The INPC, which stands for Indice Nacional de Precios al Consumidor (National Consumer Price Index in English), is calculated and published on a monthly basis by Banco de México, the Central Bank. [6]

### Sweden

The index is calculated and published by Statistics Sweden[7] Statistics Sweden, or Statistiska centralbyrån (SCB), is a Government agency responsible of producing the official statistics on Sweden. ...

### Switzerland

Switzerland issues a monthly CPI calculation by the Swiss Federal Statistical Office.[8]

### Australia

The CPI is calculated and posted quarterly by the Australian Bureau of Statistics .[9] Historical figures are available at the Reserve Bank of Australia website.[10] Australian Bureau of Statistics logo The Australian Bureau of Statistics (ABS) is the Australian government agency that collects and publishes statistical information about Australia. ...

### China

China's CPI is published by National Bureau of Statistics of China.[11]

### Israel

The Israeli CPI (Hebrew: מדד המחירים לצרכן Madad HaMechirim Latsarchan) is calculated and published by the Central Bureau of Statistics.[12] â€œHebrewâ€ redirects here. ...

### Ireland

Ireland's CPI is published by the Central Statistics Office Ireland.[13]

### Hong Kong

Hong Kong's CPI is published by the Census and Statistics Department.[14]

### New Zealand

New Zealands CPI is calculated and published quarterly by Statistics New Zealand from prices gathered in a range of surveys at 15 urban areas.[15] Statistics New Zealand (Te Tari Tatau) is a New Zealand government department, and the source of the countrys official statistics. ...

Look up fiat in Wiktionary, the free dictionary. ... A measure of inflation that excludes certain items which face volatile price movements. ... Nominal GDP per person (capita) in 2006. ... The Harmonised Index of Consumer Prices (HICP) is an idicator of inflation and price stability for the European Central Bank (ECB). ... In economics, hedonic regression, or more generally hedonic demand theory, is a method of estimating demand or prices. ... Household final consumption expenditure (HFCE) is a price index which represents consumer spending. ... In economics, the GDP deflator (implicit price deflator for GDP) is a measure of the change in prices of all new, domestically produced, final goods and services in an economy. ... This aims to be a complete list of the articles on economics. ... Market Basket is a grocery chain that serves southeast Texas and Louisiana. ... The Producer Price Index (PPI) measures average changes in prices received by domestic producers for their output. ... Look up substitution in Wiktionary, the free dictionary. ... It has been suggested that this article or section be merged with Consumer price index. ...

## References

1. ^ Shapiro, M. and Wilcox, D. "Alternative strategies for aggregating prices in the CPI" in Federal Reserve Bank of St Louis Review, May/June 1997 Volume 79, no.3
2. ^ "Latest Release from the Consumer Price Index", Statistics Canada
3. ^ "Brown sets Bank of England Free", BBC News
4. ^ FT, 14th May 2007
5. ^ "FAQs: The UK target measure of inflation", Office for National Statistics
6. ^ http://www.banxico.org.mx/sitioIngles/index.html
7. ^ "Consumer Price Index (CPI)", Statistics Sweden
8. ^ Economic and Financial Data for Switzerland, Swiss Federal Statistical Office
9. ^ "Consumer Price Index, Australia", Australian Bureau of Statistics
10. ^ "Consumer Price Index", Reserve Bank of Australia
11. ^ "Consumer Price Index" ,National Bureau of Statistics of China
12. ^ Central Bureau of Statistics
13. ^ "Consumer Price Index", Central Statistics Office Ireland.
14. ^ "Consumer Price Index",Census and Statistics Department of Hong Kong.
15. ^ "Consumer Price Index", Statistics New Zealand

Statistics Canada is the Canadian federal government bureau commissioned with producing statistics to help better understand Canada, its population, resources, economy, society, and culture. ... Office for National Statistics logo The Office for National Statistics (ONS) is the United Kingdom government executive agency charged with the collection and publication of statistics related to the economy, population and society of the United Kingdom at national and local levels. ... Statistics Sweden, or Statistiska centralbyrån (SCB), is a Government agency responsible of producing the official statistics on Sweden. ... Australian Bureau of Statistics logo The Australian Bureau of Statistics (ABS) is the Australian government agency that collects and publishes statistical information about Australia. ...

Results from FactBites:

 Consumer price index - Wikipedia, the free encyclopedia (1504 words) It is a price index that tracks the prices of a specified basket of consumer goods and services, providing a measure of inflation. An example of this adjustment might be in the price of a TV where a new model has replaced the curved screen with a flat one; the price of the new TV would be reduced in the calculation to reflect the additional "value" that the flat screen represents. Prices for the goods and services used to calculate the CPI are collected in 87 urban areas throughout the country and from approximately 23,000 retail and service establishments.
 Consumer Price Index Summary (2847 words) The transportation index increased 1.6 percent in July, reflecting an upturn in the index for motor fuel. In calculating the index, price changes for the various items in each location are averaged together with weights, which represent their importance in the spending of the appropriate population group. For the Nonalcoholic beverages index, the procedure was used to offset the effects of sharp rises in the price of coffee futures.
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