A graph of the US CPI from 1913 to 2006 (in blue), and its percent annual change (in red)

A consumer price index (CPI) measures changes in the price level of consumer goods and services purchased by households. The CPI in the United States is defined by the Bureau of Labor Statistics as "a measure of the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services."^[1]

The CPI is a statistical estimate constructed using the prices of a sample of representative items whose prices are collected periodically. Sub-indexes and sub-sub-indexes are computed for different categories and sub-categories of goods and services, being combined to produce the overall index with weights reflecting their shares in the total of the consumer expenditures covered by the index. It is one of several price indices calculated by most national statistical agencies. The annual percentage change in a CPI is used as a measure of inflation. A CPI can be used to index (i.e., adjust for the effect of inflation) the real value of wages, salaries, pensions, for regulating prices and for deflating monetary magnitudes to show changes in real values. In most countries, the CPI is, along with the population census and the USA National Income and Product Accounts, one of the most closely watched national economic statistics.

Introduction[link]

Two basic types of data are needed 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 in the total expenditure covered by the index. These weights are usually based upon expenditure data obtained from expenditure surveys for a sample of households or upon estimates of the composition of consumption expenditure in the National Income and Product Accounts. Although some of the sampling of items for price collection is done using a sampling frame and probabilistic sampling methods, many items and outlets are chosen in a commonsense way (purposive sampling) that does not permit estimation of confidence intervals. Therefore, the sampling variance cannot be calculated. In any case, a single estimate is required in most of the purposes for which the index is used.

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 shirts sold in department stores in San Francisco), detailed weighting information is unavailable, so 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 true 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.

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 the United States 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.

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. 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.

Calculating the CPI for a single item[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{CPI_2}{CPI_1}= \frac{Price_2}{Price_1}

Where 1 is usually the comparison year and CPI₁ is usually an index of 100.

Alternatively, the CPI can be performed as Failed to parse (Missing texvc executable; please see math/README to configure.): CPI= \frac{\text{updated cost}}{\text{base period cost}} \times 100 . The "updated cost" (i.e. the price of an item at a given year, e.g.: the price of bread in 1982) is divided by the initial year (the price of bread in 1970), then multiplied by one hundred.^[2]

Calculating the CPI for multiple items[link]

Example: The prices of 95,000 items from 22,000 stores, and 35,000 rental units are added together and averaged. They are weighted this way: Housing: 41.4%, Food and Beverage: 17.4%, Transport: 17.0%, Medical Care: 6.9%, Other: 6.9%, Apparel: 6.0%, Entertainment: 4.4%. Taxes (43%) are not included in CPI computation.^[3]

Failed to parse (Missing texvc executable; please see math/README to configure.): CPI = \sum_{i=1}^{n} CPI_i*weight_i

Weighting[link]

Weights and sub-indices[link]

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

on the European Union's Harmonised Index of Consumer Prices (HICP), for example, each country computes some 80 prescribed sub-indices, their weighted average constituting the national HICP. The weights for these sub-indices will consist of the sum of the weights of a number of component lower level indices. 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 reweighing to make them 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[link]

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[link]

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, 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.

Reweighing[link]

Ideally, in computing an index, the weights would represent current annual expenditure patterns. In practice they necessarily reflect past sing 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 reweighing 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 re-weightings only by including some digital cameras in the same elementary aggregate as film cameras.

Owner-occupiers and the price index[link]

The way in which owner-occupied dwellings should be dealt with in a Consumer Price Index has been, and remains, a subject of heated controversy in many countries. Various approaches have been considered, each with their advantages and disadvantages.

The economists' approach[link]

Leaving aside the quality of public services, the environment, crime and so forth, and regarding the standard of living as a function of the level and composition of individuals’ consumption, this standard depends upon the amount and range of goods and services they consume. These include the service provided by rented accommodation, which can readily be priced, and the similar services yielded by a flat or house owned by the consumer who occupies it. Its cost to a consumer is, according to the economic way of thinking, an “opportunity cost”, namely what he or she sacrifices by living in it. This cost, according to many economists, is what should form a component of a Consumer Price Index.

Opportunity cost can be looked at in two ways, since there are two alternatives to continuing to live in an owner-occupied dwelling. One — supposing that it is one year’s cost that is to be considered — is to sell it, earn interest on the owner’s capital thus released, and buy it back a year later, making an allowance for its physical depreciation. This can be called the “alternative cost” approach. The other, the “rental equivalent” approach, is to let it to someone else for the year, in which case the cost is the rent that could be obtained for it. Most people do not think about their dwelling in either of these ways, but this does not bother the theoretical economist for whom consistent logic is what matters.

There are, of course, practical problems in implementing either of these economists’ approaches. Thus, with the alternative cost approach, if house prices are rising fast the cost can be negative and then become sharply positive once house prices start to fall, so such an index would be very volatile. On the other hand, with the rental equivalent approach, there may be difficulty in estimating the movement of rental values of types of property which are not actually rented. If one or other of these measures of the consumption of the services of owner-occupied dwellings is included in consumption, then it must be included in income too, for income equals consumption plus saving. This means that if the movement of incomes is to be compared with the movement of the Consumer Price Index, incomes must be expressed as money income plus this imaginary consumption value. That is logical, but it may not be what users of the index want.

Although the argument has been expressed in connection with owner-occupied dwellings, the logic applies equally to all durable consumer goods and services. Furniture, carpets and domestic appliances are not used up soon after purchase in the way that food is. Like dwellings, they yield a consumption service that can continue for years. Furthermore, since strict logic is to be adhered to, there are durable services as well that ought to be treated in the same way; the service consumers derive from appendectomies or crowned teeth continue for a long time. Since estimating values for these components of consumption has not been tackled, the economic theorists are torn between their desire for intellectual consistency and their recognition that inclusion of the opportunity cost of the use of durables is impracticable.

Spending[link]

Another approach is to concentrate on spending.[3] Everyone agrees that repairs and maintenance expenditure of owner-occupied dwellings should be covered in a Consumer Price Index, but the spending approach would include mortgage interest too. This turns out to be quite complicated, conceptually as well as in practice.

To explain what is involved, consider a Consumer Price Index computed with reference to 2009 for just one sole consumer who bought her house in 2006, financing half of this sum by raising a mortgage. The problem is to compare how much interest such a consumer would now be paying with the interest that was paid in 2009. Since the aim is to compare like with like, that requires an estimate of how much interest would be paid now in the year 2010 on a similar house bought and 50% mortgage-financed three years ago, in 2007. It does not require an estimate of how much that identical person is paying now on the actual house she bought in 2006, even though that is what personally concerns her now.

A Consumer Price Index compares how much it would cost now to do exactly what consumers did in the reference-period with what it cost then. Application of the principle thus requires that the index for our one house owner should reflect the movement of the prices of houses like hers from 2006 to 2007 and the change in interest rates. If she took out a fixed-interest rate mortgage it is the change in interest rates from 2006 to 2007 that counts; if she took out a variable interest mortgage it is the change from 2009 to 2010 that counts. Thus her current index with 1999 as reference-period will stand at more than 100 if house prices or, in the case of a fixed-interest mortgage, interest rates rose between 2006 and 2007.

The application of this principle in the owner-occupied dwellings component of a Consumer Price Index is known as the “debt profile” method. It means that the current movement of the index will reflect past changes in dwelling prices and interest rates. Some people regard this as odd. Quite a few countries use the debt profile method, but in doing so most of them behave inconsistently. Consistency would require that the index should also cover the interest on consumer credit instead of the whole price paid for the products bought on credit if it covers mortgage interest payments. Products bought on credit would then be treated in the same way as owner-occupied dwellings.

Variants of the debt profile method are employed or have been proposed. One example is to include down payments as well as interest. Another is to correct nominal mortgage rates for changes in dwelling prices or for changes in the rest of the Consumer Price Index to obtain a “real” rate of interest. Also, other methods may be used alongside the debt profile method. Thus several countries include a purely notional cost of depreciation as an additional index component, applying an arbitrarily estimated, or rather guessed, depreciation rate to the value of the stock of owner-occupied dwellings. Finally, one country includes both mortgage interest and purchase prices in its index.

Transaction prices[link]

The third approach simply treats the acquisition of owner-occupied dwellings in the same way as acquisitions of other durable products are treated. This means: Taking account of the transaction prices agreed;

• Ignoring whether payments are delayed or are partly financed by borrowing;
• Leaving out second-hand transactions. Second-hand purchases correspond to sales by other consumers. Thus only new dwellings would be included.

Furthermore, expenditure on enlarging or reconstructing an owner-occupied dwelling would be covered, in addition to regular maintenance and repair. Two arguments of an almost theological character are advanced in connection with this transactions approach.

One argument is that purchases of new dwellings are treated as Investment in the System of National Accounts, so should not enter a consumption price index. It is said that this is more than just a matter of terminological uniformity. For example it may be thought to help understanding and facilitate economic analysis if what is included under the heading of Consumption is the same in the Consumer Price Index and in the national income and expenditure accounts. Since these accounts include the equivalent rental value of owner-occupied dwellings, the equivalent rental approach would have to be applied in the Consumer Price Index too. But the national accounts do not apply it to other durables, so the argument demands consistency in one respect but accepts its rejection in another.

The other argument is that the prices of new dwellings should exclude that part reflecting the value of the land, since this is an irreproducible and permanent asset that cannot be said to be consumed. This would presumably mean deducting site value from the price of a dwelling, site value presumably being defined as the price the site would fetch at auction if the dwelling were not on it. How this is to be understood in the case of multiple dwellings remains unclear.

Confusion[link]

It is apparent that much of the muddle in discussing the merits of the different approaches arises from the promiscuous mixing up of arguments about feasibility, about dislike or approval of the way the index would move under a particular approach and about principles of various, often incompatible, sorts. Feasibility is naturally important. The difficulty of dealing with site values is obvious.

Statisticians in a country lacking a good dwelling price index (which is required for all except the rental equivalent method) will go along with a proposal to use such an index only if they can obtain the necessary additional resources that will enable them to compile one. Even obtaining mortgage interest rate data can be a major task in a country with a multitude of mortgage lenders and many types of mortgage. Dislike of the effect upon the behaviour of the Consumer Price Index arising from the adoption of some methods can be a powerful, if sometimes unprincipled, argument.

Dwelling prices are volatile and so, therefore, would be an index incorporating the current value of a dwelling price sub-index which, in some countries, would have a large weight under the third approach. Furthermore, the weight for owner-occupied dwellings could be altered considerably when reweighting was undertaken. (It could even become negative under the alternative cost approach if weights were estimated for a year during which house prices had been rising steeply).

Then, there is the point that a rise in interest rates designed to halt inflation could paradoxically make inflation appear higher if current interest rates showed up in the index. Economists’ principles are not acceptable to all; nor is insistence upon consistency between the treatment of owner-occupied dwellings and other durables.

Clarity[link]

Much would be gained if two sets of problems were distinguished.*

• What is the Consumer Price Index to measure?
• How can that be achieved?

Another way of putting this is to distinguish:

• What is the question that should be answered? This is a matter for policy makers and other users of the Consumer Price Index.
• How can it best be answered? This is a matter for the statisticians.#

The three approaches should not be regarded as rivals, they are different answers to different questions. One, or possibly more, should be chosen. The three questions can be formulated as follows:

1. Opportunity cost. What is the change through time in what would be the opportunity cost of the reference-period consumption of the services of owner-occupied dwellings?
2. Spending. What is the change through time in the cash outlays that would correspond to the reference-period cash outlays in respect of owner-occupied dwellings?
3. Transactions. What is the change through time in what would be the purchase value of the reference-period net acquisition of owner-occupied dwellings by consumers?

Which question is to be answered is, as just stated, a policy matter, depending upon the purposes the index is to serve. It is not an issue for statisticians to decide. Their job is the technical, professional one of compiling one or more indexes that answer the selected question or questions as well as possible, given the resources at their disposal. In a perfect world this is how the owner-occupied dwellings issue would be resolved. But the world is not perfect.

History[link]

Between 1971 and 1977, the United States CPI increased 47%.^[4]
In 2009, the Consumer Price Index fell for the first time since 1955.^[5]

Criticisms[link]

Bias due to product selection[link]

Some critics^[who?] claim that the US consumer price index's market basket does not account for certain common household items which did not exist or were less common when the tracking began. If the prices on these items are dropping, then the CPI could be artificially high by failing to account for them. Some examples are computers or DVD players, which generally decrease in cost based on time since they were first developed. The market basket does include primarily items which do not see price spikes due to innovation, such as food, apparel, utilities, and education.

Bias due to quality improvement[link]

The CPI measures the cost of purchasing a product without accounting for the quality of the product. As a result, when consumers choose to buy higher-quality cars or computers, the CPI can increase despite the fact that lower-cost products are still available. While this occurred because the consumer was able to afford higher quality products, the resulting CPI increase can be a sign of economic difficulty. Additionally, the value to the consumer may increase by a greater factor than the cost, again resulting in an apparent decrease in purchasing power when measured in cost per car, as in the CPI, but an actual increase in purchasing power when measured in cost over value.

See also[link]

References[link]

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (October 2008)

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (January 2009)

Further reading[link]

• W.E. Diewert, 1993. The early history of price index research. Chapter 2 of Essays in Index Number Theory, Volume I, W.E. Diewert and A.O. Nakamura, editors. Elsevier Science Publishers, B.V.

External links[link]

• ILO CPI manual This large manual produced co-operatively by a number of international organizations is the standard work on the methods of compiling consumer price indexes and on the underlying economic and statistical theory.
• myCPI.info Project for establishing alternative CPI data based on users inputting and tracking their personal CPI.
• "The Consumer Price Index and index number purpose": A technical article by economist W.E. Diewert
• [1] Bureau of Labor Statistics - Consumer Price Index (CPI)
• [2] SGS Alternate Inflation Charts
• [3] SGS CPI calculation criticisms
• [4] BLS rebuttal to SGS CPI calculation criticisms
• [5] SGS rebuttal to BLS rebuttal regarding CPI

Specific countries

• International:
  • Worldwide Consumer Price Index Per City User-Updated
• Australia:Australian CPI
• France:
  • Consumer Price Index, INSEE
  • A description of the French consumer price index (in French)
• Canada; [6]
• Germany:[7], Federal Statistical Office of Germany
• Hong Kong: Hong Kong Consumer Price Index (CPI) - Census and Statistics Department
• Italy: Indici dei prezzi al consumo, ISTAT
• Italy: Rivalutazione monetaria con gli Indici dei prezzi al consumo per le famiglie di operai ed impiegati
• Spain: CPI information, Instituto Nacional de Estadística
• New Zealand: CPI information from Statistics New Zealand
• Norway;
  • CPI
• Sweden:
  • A Consumer Price Index for Sweden 1290-2006
  • A detailed account in English of the Swedish consumer price index
• Denmark:
  • Oldmoney.dk - A Consumer Price Index-calculator for Denmark 1900-2010
• United Kingdom:
  • National Statistics Online: Consumer Prices, UK Office for National Statistics
  • A detailed account of the UK index
• United States:
  • Consumer Price Index Home Page, U.S. Department of Labor, Bureau of Labor Statistics
  • A detailed account of the US index
  • Inflation Calculator
  • Historical US CPI chart, monthly data since 1947
  • CPI categories 1993-2010
  • Alternate non official CPI measures
  • An inflation research site and personal inflation rate calculator

A price index (plural: “price indices” or “price indexes”) is a normalized average (typically a weighted average) of prices for a given class of goods or services in a given region, during a given interval of time. It is a statistic designed to help to compare how these prices, taken as a whole, differ between time periods or geographical locations.

Price indices have several potential uses. For particularly broad indices, the index can be said to measure the economy's price level or a cost of living. More narrow price indices can help producers with business plans and pricing. Sometimes, they can be useful in helping to guide investment.

Some notable price indices include:

• Consumer price index
• Producer price index
• GDP deflator

History of early price indices[link]

No clear consensus has emerged on who created the first price index. The earliest reported research in this area came from Welshman Rice Vaughan who examined price level change in his 1675 book A Discourse of Coin and Coinage. Vaughan wanted to separate the inflationary impact of the influx of precious metals brought by Spain from the New World from the effect due to currency debasement. Vaughan compared labor statutes from his own time to similar statutes dating back to Edward III. These statutes set wages for certain tasks and provided a good record of the change in wage levels. Vaughan reasoned that the market for basic labor did not fluctuate much with time and that a basic laborers salary would probably buy the same amount of goods in different time periods, so that a laborer's salary acted as a basket of goods. Vaughan's analysis indicated that price levels in England had risen six to eightfold over the preceding century.^[1]

While Vaughan can be considered a forerunner of price index research, his analysis did not actually involve calculating an index.^[1] In 1707 Englishman William Fleetwood created perhaps the first true price index. An Oxford student asked Fleetwood to help show how prices had changed. The student stood to lose his fellowship since a fifteenth century stipulation barred students with annual incomes over five pounds from receiving a fellowship. Fleetwood, who already had an interest in price change, had collected a large amount of price data going back hundreds of years. Fleetwood proposed an index consisting of averaged price relatives and used his methods to show that the value of five pounds had changed greatly over the course of 260 years. He argued on behalf of the Oxford students and published his findings anonymously in a volume entitled Chronicon Preciosum.^[2]

Formal calculation[link]

Given a set Failed to parse (Missing texvc executable; please see math/README to configure.): C

of goods and services, the total market value of transactions in Failed to parse (Missing texvc executable; please see math/README to configure.): C
in some period Failed to parse (Missing texvc executable; please see math/README to configure.): t
would be

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{c\,\in\, C} (p_{c,t}\cdot q_{c,t})

where

Failed to parse (Missing texvc executable; please see math/README to configure.): p_{c,t}\,

represents the prevailing price of Failed to parse (Missing texvc executable; please see math/README to configure.): c
in period Failed to parse (Missing texvc executable; please see math/README to configure.): t

Failed to parse (Missing texvc executable; please see math/README to configure.): q_{c,t}\,

represents the quantity of Failed to parse (Missing texvc executable; please see math/README to configure.): c
sold in period Failed to parse (Missing texvc executable; please see math/README to configure.): t

If, across two periods Failed to parse (Missing texvc executable; please see math/README to configure.): t_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): t_n

, the same quantities of each good or service were sold, but under different prices, then

Failed to parse (Missing texvc executable; please see math/README to configure.): q_{c,t_n}=q_c=q_{c,t_0}\, \forall c

and

Failed to parse (Missing texvc executable; please see math/README to configure.): P=\frac{\sum (p_{c,t_n}\cdot q_c)}{\sum (p_{c,t_0}\cdot q_c)}

would be a reasonable measure of the price of the set in one period relative to that in the other, and would provide an index^{[disambiguation needed ]} measuring relative prices overall, weighted by quantities sold.

Of course, for any practical purpose, quantities purchased are rarely if ever identical across any two periods. As such, this is not a very practical index formula.

One might be tempted to modify the formula slightly to

Failed to parse (Missing texvc executable; please see math/README to configure.): P=\frac{\sum (p_{c,t_n}\cdot q_{c,t_n})}{\sum (p_{c,t_0}\cdot q_{c,t_0})}

This new index, however, doesn't do anything to distinguish growth or reduction in quantities sold from price changes. To see that this is so, consider what happens if all the prices double between Failed to parse (Missing texvc executable; please see math/README to configure.): t_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): t_n
while quantities stay the same: Failed to parse (Missing texvc executable; please see math/README to configure.): P
will double. Now consider what happens if all the quantities double between Failed to parse (Missing texvc executable; please see math/README to configure.): t_0
and Failed to parse (Missing texvc executable; please see math/README to configure.): t_n
while all the prices stay the same: Failed to parse (Missing texvc executable; please see math/README to configure.): P
will double. In either case the change in Failed to parse (Missing texvc executable; please see math/README to configure.): P
is identical. As such, Failed to parse (Missing texvc executable; please see math/README to configure.): P
is as much a quantity index as it is a price index.

Various indices have been constructed in an attempt to compensate for this difficulty.

Paasche and Laspeyres price indices[link]

The two most basic formulae used to calculate price indices are the Paasche index (after the German economist Hermann Paasche [ˈpaːʃɛ]) and the Laspeyres index (after the German economist Etienne Laspeyres [lasˈpejres]).

The Paasche index is computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): P_P=\frac{\sum (p_{c,t_n}\cdot q_{c,t_n})}{\sum (p_{c,t_0}\cdot q_{c,t_n})}

while the Laspeyres index is computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): P_L=\frac{\sum (p_{c,t_n}\cdot q_{c,t_0})}{\sum (p_{c,t_0}\cdot q_{c,t_0})}

where Failed to parse (Missing texvc executable; please see math/README to configure.): P

is the relative index of the price levels in two periods, Failed to parse (Missing texvc executable; please see math/README to configure.): t_0
is the base period (usually the first year), and Failed to parse (Missing texvc executable; please see math/README to configure.): t_n
the period for which the index is computed.

Note that the only difference in the formulas is that the former uses period n quantities, whereas the latter uses base period (period 0) quantities.

When applied to bundles of individual consumers, a Laspeyres index of 1 would state that an agent in the current period can afford to buy the same bundle as he consumed in the previous period, given that income has not changed; a Paasche index of 1 would state that an agent could have consumed the same bundle in the base period as he is consuming in the current period, given that income has not changed.

Hence, one may think of the Laspeyres index as one where the numeraire is the bundle of goods using current year prices but base year quantities. Similarly, the Paasche index can be thought of as a price index taking the bundle of goods using current prices and current quantities as the numeraire.

The Laspeyres index tends to overstate inflation (in a cost of living framework), while the Paasche index tends to understate it, because the indices do not account for the fact that consumers typically react to price changes by changing the quantities that they buy. For example, if prices go up for good Failed to parse (Missing texvc executable; please see math/README to configure.): c

then, ceteris paribus, quantities of that good should go down.

Fisher index and Marshall–Edgeworth index[link]

A third index, the Marshall–Edgeworth index (named for economists Alfred Marshall and Francis Ysidro Edgeworth), tries to overcome these problems of under- and overstatement by using the arithmetic means of the quantities:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_{ME}=\frac{\sum [p_{c,t_n}\cdot \frac{1}{2}\cdot(q_{c,t_0}+q_{c,t_n})]}{\sum [p_{c,t_0}\cdot \frac{1}{2}\cdot(q_{c,t_0}+q_{c,t_n})]}=\frac{\sum [p_{c,t_n}\cdot (q_{c,t_0}+q_{c,t_n})]}{\sum [p_{c,t_0}\cdot (q_{c,t_0}+q_{c,t_n})]}

A fourth, the Fisher index (after the American economist Irving Fisher), is calculated as the geometric mean of Failed to parse (Missing texvc executable; please see math/README to configure.): P_P

and Failed to parse (Missing texvc executable; please see math/README to configure.): P_L

Failed to parse (Missing texvc executable; please see math/README to configure.): P_F = \sqrt{P_P\cdot P_L}

Fisher's index is also known as the “ideal” price index.

However, there is no guarantee with either the Marshall–Edgeworth index or the Fisher index that the overstatement and understatement will exactly cancel the other.

While these indices were introduced to provide overall measurement of relative prices, there is ultimately no way of measuring the imperfections of any of these indices (Paasche, Laspeyres, Fisher, or Marshall–Edgeworth) against reality.

Practical measurement considerations[link]

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Normalizing index numbers[link]

Price indices are represented as index numbers, number values that indicate relative change but not absolute values (i.e. one price index value can be compared to another or a base, but the number alone has no meaning). Price indices generally select a base year and make that index value equal to 100. You then express every other year as a percentage of that base year. In our example above, let's take 2000 as our base year. The value of our index will be 100. The price

• 2000: original index value was $2.50; $2.50/$2.50 = 100%, so our new index value is 100
• 2001: original index value was $2.60; $2.60/$2.50 = 104%, so our new index value is 104
• 2002: original index value was $2.70; $2.70/$2.50 = 108%, so our new index value is 108
• 2003: original index value was $2.80; $2.80/$2.50 = 112%, so our new index value is 112

When an index has been normalized in this manner, the meaning of the number 108, for instance, is that the total cost for the basket of goods is 4% more in 2001, 8% more in 2002 and 12% more in 2003 than in the base year (in this case, year 2000).

Relative ease of calculating the Laspeyres index[link]

As can be seen from the definitions above, if one already has price and quantity data (or, alternatively, price and expenditure data) for the base period, then calculating the Laspeyres index for a new period requires only new price data. In contrast, calculating many other indices (e.g., the Paasche index) for a new period requires both new price data and new quantity data (or, alternatively, both new price data and new expenditure data) for each new period. Collecting only new price data is often easier than collecting both new price data and new quantity data, so calculating the Laspeyres index for a new period tends to require less time and effort than calculating these other indices for a new period.^[3]

Calculating indices from expenditure data[link]

Sometimes, especially for aggregate data, expenditure data is more readily available than quantity data.^[4] For these cases, we can formulate the indices in terms of relative prices and base year expenditures, rather than quantities.

Here is a reformulation for the Laspeyres index:

Let Failed to parse (Missing texvc executable; please see math/README to configure.): E_{c,t_0}

be the total expenditure on good c in the base period, then (by definition) we have

Failed to parse (Missing texvc executable; please see math/README to configure.): E_{c,t_0} = p_{c,t_0}\cdot q_{c,t_0}

and therefore also Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{E_{c,t_0}}{p_{c,t_0}} = q_{c,t_0} . We can substitute these values into our Laspeyres formula as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_L =\frac{\sum (p_{c,t_n}\cdot q_{c,t_0})}{\sum (p_{c,t_0}\cdot q_{c,t_0})} =\frac{\sum (p_{c,t_n}\cdot \frac{E_{c,t_0}}{p_{c,t_0}})}{\sum E_{c,t_0}} =\frac{\sum (\frac{p_{c,t_n}}{p_{c,t_0}} \cdot E_{c,t_0})}{\sum E_{c,t_0}}

A similar transformation can be made for any index.

Chained vs non-chained calculations[link]

So far, in our discussion, we have always had our price indices relative to some fixed base period. An alternative is to take the base period for each time period to be the immediately preceding time period. This can be done with any of the above indices. Here's an example with the Laspeyres index, where Failed to parse (Missing texvc executable; please see math/README to configure.): t_n

is the period for which we wish to calculate the index and Failed to parse (Missing texvc executable; please see math/README to configure.): t_0
is a reference period that anchors the value of the series:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_{t_n}= \frac{\sum (p_{c,t_1}\cdot q_{c,t_0})}{\sum (p_{c,t_0}\cdot q_{c,t_0})} \times \frac{\sum (p_{c,t_2}\cdot q_{c,t_1})}{\sum (p_{c,t_1}\cdot q_{c,t_1})} \times \cdots \times \frac{\sum (p_{c,t_n}\cdot q_{c,t_{n-1}})}{\sum (p_{c,t_{n-1}}\cdot q_{c,t_{n-1}})}

Each term

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sum (p_{c,t_n}\cdot q_{c,t_{n-1}})}{\sum (p_{c,t_{n-1}}\cdot q_{c,t_{n-1}})}

answers the question "by what factor have prices increased between period Failed to parse (Missing texvc executable; please see math/README to configure.): t_{n-1}

and period Failed to parse (Missing texvc executable; please see math/README to configure.): t_n

". When you multiply these all together, you get the answer to the question "by what factor have prices increased since period Failed to parse (Missing texvc executable; please see math/README to configure.): t_0 ".

Nonetheless, note that, when chain indices are in use, the numbers cannot be said to be "in period Failed to parse (Missing texvc executable; please see math/README to configure.): t_0 " prices.

Index number theory[link]

Price index formulas can be evaluated based on their relation to economic concepts (like cost of living) or on their mathematical properties. Several different tests of such properties have been proposed in index number theory literature. W.E. Diewert summarized past research in a list of nine such tests for a price index Failed to parse (Missing texvc executable; please see math/README to configure.): I(P_{t_0}, P_{t_m}, Q_{t_0}, Q_{t_m}) , where Failed to parse (Missing texvc executable; please see math/README to configure.): P_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): P_n
are vectors giving prices for a base period and a reference period while Failed to parse (Missing texvc executable; please see math/README to configure.): Q_{t_0}
and Failed to parse (Missing texvc executable; please see math/README to configure.): Q_{t_m}
give quantities for these periods.[5]

1. Identity test:

   Failed to parse (Missing texvc executable; please see math/README to configure.): I(p_{t_m},p_{t_m},\alpha \cdot q_{t_m},\beta\cdot q_{t_n})=1~~\forall (\alpha ,\beta )\in (0,\infty )^2

1. The identity test basically means that if prices remain the same and quantities remain in the same proportion to each other (each quantity of an item is multiplied by the same factor of either Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

, for the first period, or Failed to parse (Missing texvc executable; please see math/README to configure.): \beta , for the later period) then the index value will be one.

1. Proportionality test:

   Failed to parse (Missing texvc executable; please see math/README to configure.): I(p_{t_m},\alpha \cdot p_{t_m},q_{t_m},q_{t_n})=\alpha \cdot I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})

1. If each price in the original period increases by a factor α then the index should increase by the factor α.

2. Invariance to changes in scale test:

   Failed to parse (Missing texvc executable; please see math/README to configure.): I(\alpha \cdot p_{t_m},\alpha \cdot p_{t_n},\beta \cdot q_{t_m}, \gamma \cdot q_{t_n})=I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})~~\forall (\alpha,\beta,\gamma)\in(0,\infty )^3

1. The price index should not change if the prices in both periods are increased by a factor and the quantities in both periods are increased by another factor. In other words, the magnitude of the values of quantities and prices should not affect the price index.

2. Commensurability test:

   The index should not be affected by the choice of units used to measure prices and quantities.

3. Symmetric treatment of time (or, in parity measures, symmetric treatment of place):

   Failed to parse (Missing texvc executable; please see math/README to configure.): I(p_{t_n},p_{t_m},q_{t_n},q_{t_m})=\frac{1}{I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})}

1. Reversing the order of the time periods should produce a reciprocal index value. If the index is calculated from the most recent time period to the earlier time period, it should be the reciprocal of the index found going from the earlier period to the more recent.

2. Symmetric treatment of commodities:

   All commodities should have a symmetric effect on the index. Different permutations of the same set of vectors should not change the index.

3. Monotonicity test:

   Failed to parse (Missing texvc executable; please see math/README to configure.): I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \le I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\Leftarrow~~p_{t_n} \le p_{t_r}

1. A price index for lower later prices should be lower than a price index with higher later period prices.

2. Mean value test:

   The overall price relative implied by the price index should be between the smallest and largest price relatives for all commodities.

3. Circularity test:

   Failed to parse (Missing texvc executable; please see math/README to configure.): I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \cdot I(p_{t_n},p_{t_r},q_{t_n},q_{t_r})=I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\Leftarrow~~t_m \le t_n \le t_r

1. Given three ordered periods Failed to parse (Missing texvc executable; please see math/README to configure.): t_m

, Failed to parse (Missing texvc executable; please see math/README to configure.): t_n , Failed to parse (Missing texvc executable; please see math/README to configure.): t_r , the price index for periods Failed to parse (Missing texvc executable; please see math/README to configure.): t_m

and Failed to parse (Missing texvc executable; please see math/README to configure.): t_n
times the price index for periods Failed to parse (Missing texvc executable; please see math/README to configure.): t_n
and Failed to parse (Missing texvc executable; please see math/README to configure.): t_r
should be equivalent to the price index for periods Failed to parse (Missing texvc executable; please see math/README to configure.): t_m
and Failed to parse (Missing texvc executable; please see math/README to configure.): t_r

.

Quality change[link]

Price indices often capture changes in price and quantities for goods and services, but they often fail to account for improvements (or often deteriorations) in the quality of goods and services. Statistical agencies generally use matched-model price indices, where one model of a particular good is priced at the same store at regular time intervals. The matched-model method becomes problematic when statistical agencies try to use this method on goods and services with rapid turnover in quality features. For instance, computers rapidly improve and a specific model may quickly become obsolete. Statisticians constructing matched-model price indices must decide how to compare the price of the obsolete item originally used in the index with the new and improved item that replaces it. Statistical agencies use several different methods to make such price comparisons.^[6]

Theproblem discussed above can be represented as attempting to bridge the gap between the price for the old item in time t, Failed to parse (Missing texvc executable; please see math/README to configure.): P(M)_{t} , with the price of the new item in the later time period, Failed to parse (Missing texvc executable; please see math/README to configure.): P(N)_{t+1} .^[7]

• The overlap method uses prices collected for both items in both time periods, t and t+1. The price relative Failed to parse (Missing texvc executable; please see math/README to configure.): {P(N)_{t+1}}

/Failed to parse (Missing texvc executable; please see math/README to configure.): {P(N)_{t}}

is used.

• The direct comparison method assumes that the difference in the price of the two items is not due to quality change, so the entire price difference is used in the index. Failed to parse (Missing texvc executable; please see math/README to configure.): P(N)_{t+1}

/Failed to parse (Missing texvc executable; please see math/README to configure.): P(M)_t

is used as the price relative.

• The link-to-show-no-change assumes the opposite of the direct comparison method; it assumes that the entire difference between the two items is due to the change in quality. The price relative based on link-to-show-no-change is 1.^[8]
• The deletion method simply leaves the price relative for the changing item out of the price index. This is equivalent to using the average of other price relatives in the index as the price relative for the changing item. Similarly, class mean imputation uses the average price relative for items with similar characteristics (physical, geographic, economic, etc.) to M and N.^[9]

See also[link]

• Aggregation problem
• Inflation
• Chemical plant cost indexes
• GDP deflator
• Etienne Laspeyres
• Hermann Paasche
• Hedonic index
• Indexation
• Irving Fisher
• Real versus nominal value (economics)
• U.S. Import Price Index
• Volume index

Notes[link]

References[link]

• Chance, W.A. “A Note on the Origins of Index Numbers“, The Review of Economics and Statistics, Vol. 48, No. 1. (Feb., 1966), pp. 108–10. Subscription URL
• Diewert, W.E. Chapter 5: “Index Numbers” in Essays in Index Number Theory. eds W.E. Diewert and A.O. Nakamura. Vol 1. Elsevier Science Publishers: 1993. (Also online.)
• McCulloch, James Huston. Money and Inflation: A Monetarist Approach 2e, Harcourt Brace Jovanovich / Academic Press, 1982.
• Triplett, Jack E. “Economic Theory and BEA's Alternative Quantity and Price indices”, Survey of Current Business April 1992.
• Triplett, Jack E. [http://www.oecd.org/dataoecd/37/31/33789552.pdf Handbook on Hedonic Indexes and Quality Adjustments in Price Indexes: Special Application to Information Technology Products. OECD Directorate for Science, Technology and Industry working paper. October 2004.
• U.S. Department of Labor BLS “Producer Price Index Frequently Asked Questions”.
• Vaughan, Rice. A Discourse of Coin and Coinage (1675). (Also online by chapter.)

External links[link]

Manuals[link]

• IMF Export and Import price index
• IMF PPI manual
• ILO CPI manual

Data[link]

• PPI data from BLS