A number of different formulae, more than a hundred, have been proposed as means of calculating
price indexes. While price index formulae all use price and possibly quantity data, they aggregate these in different ways. A price index aggregates various combinations of
base period prices (
), later period prices (
), base period quantities (
), and later period quantities (
). Price index numbers are usually defined either in terms of (actual or hypothetical) expenditures (expenditure = price * quantity) or as different
weighted average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
s of price relatives (
). These tell the relative change of the price in question. Two of the most commonly used price index formulae were defined by German economists and statisticians
Étienne Laspeyres
Ernst Louis Étienne Laspeyres (; 28 November 1834 – 4 August 1913) was a German economist. He was ''Professor ordinarius'' of economics and statistics or '' State Sciences'' and cameralistics (public finance and administration) in Basel, Riga ...
and
Hermann Paasche
Hermann Paasche (; February 24, 1851 in Burg bei Magdeburg – April 11, 1925 in Detroit) was a German statistician and economist. He is known for his Paasche Index, which provides a calculation of the Price Index. Paasche studied economics, agr ...
, both around 1875 when investigating price changes in Germany.
Laspeyres
Developed in 1871 by
Étienne Laspeyres
Ernst Louis Étienne Laspeyres (; 28 November 1834 – 4 August 1913) was a German economist. He was ''Professor ordinarius'' of economics and statistics or '' State Sciences'' and cameralistics (public finance and administration) in Basel, Riga ...
, the formula:
:
compares the total cost of the same basket of
final good
A final good or consumer good is a final product ready for sale that is used by the consumer to satisfy current wants or needs, unlike a intermediate good, which is used to produce other goods. A microwave oven or a bicycle is a final good, but ...
s
at the old and new prices.
Paasche
Developed in 1874 by
Hermann Paasche
Hermann Paasche (; February 24, 1851 in Burg bei Magdeburg – April 11, 1925 in Detroit) was a German statistician and economist. He is known for his Paasche Index, which provides a calculation of the Price Index. Paasche studied economics, agr ...
, the formula:
:
compares the total cost of a new basket of goods
at the old and new prices.
Geometric means
The geometric means index:
:
incorporates quantity information through the share of expenditure in the base period.
Unweighted indices
Unweighted, or "elementary", price indices only compare prices of a single type of good between two periods. They do not make any use of quantities or expenditure weights. They are called "elementary" because they are often used at the lower levels of aggregation for more comprehensive price indices. In such a case, they are not indices but merely an intermediate stage in the calculation of an index. At these lower levels, it is argued that weighting is not necessary since only one type of good is being aggregated. However this implicitly assumes that only one type of the good is available (e.g. only one brand and one package size of frozen peas) and that it has not changed in quality etc between time periods.
Carli
Developed in 1764 by
Gian Rinaldo Carli, an Italian economist, this formula is the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
of the price relative between a period ''t'' and a base period ''0''.
:
On 17 August 2012 the
BBC Radio 4
BBC Radio 4 is a British national radio station owned and operated by the BBC that replaced the BBC Home Service in 1967. It broadcasts a wide variety of spoken-word programmes, including news, drama, comedy, science and history from the BBC' ...
program
''More or Less'' noted that the Carli index, used in part in the British
retail price index
In the United Kingdom, the Retail Prices Index or Retail Price Index (RPI) is a measure of inflation published monthly by the Office for National Statistics. It measures the change in the cost of a representative sample of retail goods and servic ...
, has a built-in bias towards recording inflation even when over successive periods there is no increase in prices overall.
Dutot
In 1738 French economist
Nicolas Dutot
Nicolas or Nicolás may refer to:
People Given name
* Nicolas (given name)
Mononym
* Nicolas (footballer, born 1999), Brazilian footballer
* Nicolas (footballer, born 2000), Brazilian footballer
Surname Nicolas
* Dafydd Nicolas (c.1705–1774), ...
proposed using an index calculated by dividing the average price in period ''t'' by the average price in period ''0''.
:
Jevons
In 1863, English economist
William Stanley Jevons
William Stanley Jevons (; 1 September 183513 August 1882) was an English economist and logician.
Irving Fisher described Jevons's book ''A General Mathematical Theory of Political Economy'' (1862) as the start of the mathematical method in ec ...
proposed taking the
geometric average
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of the price relative of period ''t'' and base period ''0''. When used as an elementary aggregate, the Jevons index is considered a constant elasticity of substitution index since it allows for product substitution between time periods.
:
This is the formula that was used for the old ''
Financial Times
The ''Financial Times'' (''FT'') is a British daily newspaper printed in broadsheet and published digitally that focuses on business and economic current affairs. Based in London, England, the paper is owned by a Japanese holding company, Nik ...
'' stock market index (the predecessor of the
FTSE 100 Index
The Financial Times Stock Exchange 100 Index, also called the FTSE 100 Index, FTSE 100, FTSE, or, informally, the "Footsie" , is a share index of the 100 companies listed on the London Stock Exchange with (in principle) the highest market ...
). It was inadequate for that purpose. In particular, if the price of any of the constituents were to fall to zero, the whole index would fall to zero. That is an extreme case; in general the formula will understate the total cost of a basket of goods (or of any subset of that basket) unless their prices all change at the same rate. Also, as the index is unweighted, large price changes in selected constituents can transmit to the index to an extent not representing their importance in the average portfolio.
Harmonic mean of price relatives
The harmonic average counterpart to the Carli index.
[PPI manual, 600.] The index was proposed by Jevons in 1865 and by Coggeshall in 1887.
:
Carruthers, Sellwood, Ward, Dalén index
Is the geometric mean of the Carli and the harmonic price indexes. In 1922 Fisher wrote that this and the Jevons were the two best unweighted indexes based on Fisher's test approach to index number theory.
:
Ratio of harmonic means
The ratio of harmonic means or "Harmonic means" price index is the harmonic average counterpart to the Dutot index.
:
Bilateral formulae
Marshall-Edgeworth
The Marshall-Edgeworth index, credited to
Marshall
Marshall may refer to:
Places
Australia
* Marshall, Victoria, a suburb of Geelong, Victoria
Canada
* Marshall, Saskatchewan
* The Marshall, a mountain in British Columbia
Liberia
* Marshall, Liberia
Marshall Islands
* Marshall Islands, an i ...
(1887) and
Edgeworth
Edgeworth may refer to:
People
* Edgeworth (surname)
Places
* Edgeworth, Gloucestershire, England
* Edgeworth, New South Wales, Australia
* Edgeworth, Pennsylvania, USA
* Edgeworth Island, Nunavut, Canada
* Edgeworthstown, County Longford, Repu ...
(1925), is a weighted relative of current period to base period sets of prices. This index uses the arithmetic average of the current and based period quantities for weighting. It is considered a pseudo-superlative formula and is symmetric. The use of the Marshall-Edgeworth index can be problematic in cases such as a comparison of the price level of a large country to a small one. In such instances, the set of quantities of the large country will overwhelm those of the small one.
:
Superlative indices
Superlative indices treat prices and quantities equally across periods. They are symmetrical and provide close approximations of
cost of living indices and other theoretical indices used to provide guidelines for constructing price indices. All superlative indices produce similar results and are generally the favored formulas for calculating price indices. A superlative index is defined technically as "an index that is exact for a flexible functional form that can provide a
second-order approximation to other twice-differentiable functions around the same point."
Fisher
The change in a Fisher index from one period to the next is the
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of the changes in Laspeyres's and Paasche's indexes between those periods, and these are chained together to make comparisons over many periods:
:
This is also called Fisher's "ideal" price index.
Törnqvist
The Törnqvist or Törnqvist-Theil index is the geometric average of the n price relatives of the current to base period prices (for n goods) weighted by the arithmetic average of the value shares for the two periods.
"Tornqvist Index and other Log-change Index Numbers"
, Statistics New Zealand Glossary of Common Terms.
:
Walsh
The Walsh price index is the weighted sum of the current period prices divided by the weighted sum of the base period prices with the geometric average of both period quantities serving as the weighting mechanism:
:
Notes
{{reflist
References
''Export and Import Price Index Manual''
Price indices
Price indices
Price indices