# Price index

A price index (plural: "price indices" or "price indexes") is a normalized average (typically a weighted average) of price relatives for a given class of goods or services in a specific region over a defined time period. It is a statistic designed to measure how these price relatives, as a whole, differ between time periods or geographical locations, often expressed relative to a base period set at 100.
Price indices serve multiple purposes. Broad indices, like the Consumer price index, reflect the economy’s general price level or cost of living, while narrower ones, such as the Producer price index, assist producers with pricing and business planning. They can also guide investment decisions by tracking price trends.

## Types of price indices

Some widely recognized price indices include:
* Consumer price index – Measures retail price changes for consumer goods and services.
* Producer price index – Tracks wholesale price changes for producers.
* Wholesale price index – Monitors price changes at the wholesale level (historical in some regions).
* Employment cost index – Gauges changes in labor costs.
* Export price index – Tracks export price trends.
* Import price index – Monitors import price changes.
* GDP deflator – Reflects price changes across all goods and services in GDP.

## History of early price indices

The origins of price indices are debated, with no clear consensus on their inventor. The earliest reported research in this area came from Rice Vaughan, who in his 1675 book A Discourse of Coin and Coinage analyzed price level changes in England. Vaughan sought to distinguish inflation from precious metals imported by Spain from the New World from effects of currency debasement. By comparing labor statutes from his era to those under Edward III (e.g., Statute of Labourers of 1351), he used wage levels as a proxy for a basket of goods, concluding prices had risen six- to eight-fold over a century. Though a pioneer, Vaughan did not actually compute an index.
In 1707, Englishman William Fleetwood developed perhaps the first true price index. Responding to an Oxford student facing loss of a fellowship due to a 15th-century income cap of five pounds, Fleetwood used historical price data to create an index of averaged price relatives. His work, published anonymously in Chronicon Preciosum, showed the value of five pounds had shifted significantly over 260 years.

## Basic formula

Price indices measure relative price changes using price ($p$) and quantity ($q$) data for a set of goods or services ($C$). The total market value in period $t$ is: :$\sum _{c\in C}(p_{c,t}\cdot q_{c,t})$ where $p_{c,t}$ is the price and $q_{c,t}$ the quantity of item $c$ in period $t$. If quantities remain constant across two periods ($q_{c,t_{n}}=q_{c,t_{0}}=q_{c}$), the price index simplifies to: :$P={\frac {\sum (p_{c,t_{n}}\cdot q_{c})}{\sum (p_{c,t_{0}}\cdot q_{c})}}$ .
This ratio, weighted by quantities, compares prices between periods $t_{0}$ (base) and $t_{n}$. In practice, quantities vary, requiring more complex formulas.

## Price index formulas

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 ($p_{0}$), later period prices ($p_{t}$), base period quantities ($q_{0}$), and later period quantities ($q_{t}$). Price index numbers are usually defined either in terms of (actual or hypothetical) expenditures (expenditure = price * quantity) or as different weighted averages of price relatives ($p_{t}/p_{0}$). 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 and Hermann Paasche, both around 1875 when investigating price changes in Germany.
The following categories cover widely used indices today, such as the Laspeyres index for its simplicity in official statistics, and superlative indices like the Fisher index, favored in modern economic measurement (e.g., U.S. GDP calculations), alongside less common historical or specialized formulas like unweighted indices used at lower aggregation levels.

### Laspeyres and Paasche indices

The two most basic formulae used to calculate price indices are the Paasche index (after the economist Hermann Paasche) and the Laspeyres index (after the economist Etienne Laspeyres).
Developed in 1871 by Étienne Laspeyres, the Laspeyres index:
    $P_{L}={\frac {\sum (p_{c,t_{n}}\cdot q_{c,t_{0}})}{\sum (p_{c,t_{0}}\cdot q_{c,t_{0}})}}$
compares the total cost of the same basket of final goods $q_{0}$ at the old and new prices. $P$ is the relative index of the price levels in two periods, $t_{0}$ is the base period (usually the first year), and $t_{n}$ the period for which the index is computed.
Developed in 1874 by Hermann Paasche, the Paasche index:
    $P_{P}={\frac {\sum (p_{c,t_{n}}\cdot q_{c,t_{n}})}{\sum (p_{c,t_{0}}\cdot q_{c,t_{n}})}}$
compares the total cost of a new basket of goods $q_{t}$ at the old and new prices.
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. A helpful mnemonic device to remember which index uses which period is that L comes before P in the alphabet so the Laspeyres index uses the earlier base quantities and the Paasche index the final 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 she 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 she is consuming in the current period, given that income has not changed.
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 $c$ then, ceteris paribus, quantities demanded of that good should go down.

### Lowe indices

Many price indices are calculated with the Lowe index procedure. In a Lowe price index, the expenditure or quantity weights associated with each item are not drawn from each indexed period. Usually they are inherited from an earlier period, which is sometimes called the expenditure base period. Generally, the expenditure weights are updated occasionally, but the prices are updated in every period. Prices are drawn from the time period the index is supposed to summarize." Lowe indexes are named for economist Joseph Lowe. Most CPIs and employment cost indices from Statistics Canada, the U.S. Bureau of Labor Statistics, and many other national statistics offices are Lowe indices. Lowe indexes are sometimes called a "modified Laspeyres index", where the principal modification is to draw quantity weights less frequently than every period. For a consumer price index, the weights on various kinds of expenditure are generally computed from surveys of households asking about their budgets, and such surveys are less frequent than price data collection is. Another phrasings is that Laspeyres and Paasche indexes are special cases of Lowe indexes in which all price and quantity data are updated every period.
Comparisons of output between countries often use Lowe quantity indexes. The Geary-Khamis method used in the World Bank's International Comparison Program is of this type. Here the quantity data are updated each period from each of multiple countries, whereas the prices incorporated are kept the same for some period of time, e.g. the "average prices for the group of countries".

### 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 of the price relative between a period t and a base period 0.
    $P_{C}={\frac {1}{n}}\cdot \sum {\frac {p_{t}}{p_{0}}}$
On 17 August 2012 the BBC Radio 4 program More or Less noted that the Carli index, used in part in the British retail price index, 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 proposed using an index calculated by dividing the average price in period t by the average price in period 0.
    $P_{D}={\frac {{\frac {1}{n}}\cdot \sum p_{t}}{{\frac {1}{n}}\cdot \sum p_{0}}}={\frac {\sum p_{t}}{\sum p_{0}}}$

#### Jevons

In 1863, English economist William Stanley Jevons proposed taking the geometric average 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.
    $P_{J}=\left(\prod {\frac {p_{t}}{p_{0}}}\right)^{1/n}$
This is the formula that was used for the old Financial Times stock market index (the predecessor of the FTSE 100 Index). 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. The index was proposed by Jevons in 1865 and by Coggeshall in 1887.
    $P_{HR}={\frac {1}{{\frac {1}{n}}\cdot \sum {\frac {p_{0}}{p_{t}}}}}$

#### 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.
    $P_{CSWD}={\sqrt {P_{C}\cdot P_{HR}}}$

#### Ratio of harmonic means

The ratio of harmonic means or "Harmonic means" price index is the harmonic average counterpart to the Dutot index.
    $P_{RH}={\frac {\sum {\frac {n}{p_{0}}}}{\sum {\frac {n}{p_{t}}}}}$

### Geometric means

The geometric means index:
    $P_{GM}=\prod _{i=1}^{n}\left({\frac {p_{i,t}}{p_{i,0}}}\right)^{\frac {p_{i,0}\cdot q_{i,0}}{\sum \left(p_{0}\cdot q_{0}\right)}}$
incorporates quantity information through the share of expenditure in the base period.

### Bilateral formulae

Bilateral formulae calculate price indices by comparing data between two specific periods or locations, typically using both base and current period prices and quantities in a symmetric or averaged manner to reduce bias. These differ from unilateral indices like Laspeyres or Paasche, which rely solely on one period’s quantities. This category includes the Marshall-Edgeworth index and superlative indices, the latter distinguished by their alignment with flexible economic models.

#### Marshall-Edgeworth

The Marshall–Edgeworth index (named for economists Alfred Marshall and Francis Ysidro Edgeworth), credited to Marshall (1887) and Edgeworth (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.
    $P_{ME}={\frac {\sum [p_{c,t_{n}}\cdot {\frac {1}{2}}\cdot (q_{c,t_{0}}+q_{c,t_{n}})]}{\sum [p_{c,t_{0}}\cdot {\frac {1}{2}}\cdot (q_{c,t_{0}}+q_{c,t_{n}})]}}={\frac {\sum [p_{c,t_{n}}\cdot (q_{c,t_{0}}+q_{c,t_{n}})]}{\sum [p_{c,t_{0}}\cdot (q_{c,t_{0}}+q_{c,t_{n}})]}}$

#### Superlative indices

Superlative indices, introduced by W. Erwin Diewert in 1976, are price index formulas that treat prices and quantities from both periods symmetrically, providing close approximations to theoretical economic measures like the cost-of-living index. Unlike simpler indices such as Laspeyres or Paasche, which use quantities from only one period and may over- or understate price changes, superlative indices adjust for substitution behavior, making them preferred in modern applications like inflation and GDP measurement.

#### Superlative indices

Superlative indices, introduced by W. Erwin Diewert in 1976, are price index formulas that 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. Unlike simpler indices such as Laspeyres or Paasche, which use quantities from only one period and may over- or understate price changes, superlative indices adjust for substitution behavior, making them preferred in modern applications like inflation and GDP measurement. 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 Fisher index, named for economist Irving Fisher, also known as the Fisher ideal index, is calculated as the geometric mean of $P_{P}$ and $P_{L}$:
    $P_{F}={\sqrt {P_{P}\cdot P_{L}}}$
The change in a Fisher index from one period to the next is the geometric mean of the changes in Laspeyres' and Paasche's indices between those periods, and these are chained together to make comparisons over many periods.

##### 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.
    $P_{T}=\prod _{i=1}^{n}\left({\frac {p_{i,t}}{p_{i,0}}}\right)^{{\frac {1}{2}}\left[{\frac {p_{i,0}\cdot q_{i,0}}{\sum \left(p_{0}\cdot q_{0}\right)}}+{\frac {p_{i,t}\cdot q_{i,t}}{\sum \left(p_{t}\cdot q_{t}\right)}}\right]}$

##### 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:
    $P_{W}={\frac {\sum \left(p_{t}\cdot {\sqrt {q_{0}\cdot q_{t}}}\right)}{\sum \left(p_{0}\cdot {\sqrt {q_{0}\cdot q_{t}}}\right)}}$

## Calculation methods
### Practical measurement considerations
#### Normalizing index numbers

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. Every other year is expressed as a percentage of that base year. In this example, let 2000 be the base year:
* 2000: original index value was $2.50; $2.50/$2.50 = 100%, so new index value is 100
* 2001: original index value was $2.60; $2.60/$2.50 = 104%, so new index value is 104
* 2002: original index value was $2.70; $2.70/$2.50 = 108%, so new index value is 108
* 2003: original index value was $2.80; $2.80/$2.50 = 112%, so new index value is 112
When an index has been normalized in this manner, the meaning of the number 112, for instance, is that the total cost for the basket of goods is 4% more in 2001 than in the base year (in this case, year 2000), 8% more in 2002, and 12% more in 2003.

#### Relative ease of calculating the Laspeyres index

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.
In practice, price indices regularly compiled and released by national statistical agencies are of the Laspeyres type, due to the above-mentioned difficulties in obtaining current-period quantity or expenditure data.

#### Calculating indices from expenditure data

Sometimes, especially for aggregate data, expenditure data are more readily available than quantity data. For these cases, the indices can be formulated in terms of relative prices and base year expenditures, rather than quantities.
Here is a reformulation for the Laspeyres index:
Let $E_{c,t_{0}}$ be the total expenditure on good c in the base period, then (by definition) we have $E_{c,t_{0}}=p_{c,t_{0}}\cdot q_{c,t_{0}}$ and therefore also ${\frac {E_{c,t_{0}}}{p_{c,t_{0}}}}=q_{c,t_{0}}$. We can substitute these values into our Laspeyres formula as follows:
    $P_{L}={\frac {\sum (p_{c,t_{n}}\cdot q_{c,t_{0}})}{\sum (p_{c,t_{0}}\cdot q_{c,t_{0}})}}={\frac {\sum (p_{c,t_{n}}\cdot {\frac {E_{c,t_{0}}}{p_{c,t_{0}}}})}{\sum E_{c,t_{0}}}}={\frac {\sum ({\frac {p_{c,t_{n}}}{p_{c,t_{0}}}}\cdot E_{c,t_{0}})}{\sum E_{c,t_{0}}}}$
A similar transformation can be made for any index.

### Chained vs unchained calculations

The above price indices were calculated relative to a 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 is an example with the Laspeyres index, where $t_{n}$ is the period for which we wish to calculate the index and $t_{0}$ is a reference period that anchors the value of the series:
    $P_{t_{n}}={\frac {\sum (p_{c,t_{1}}\cdot q_{c,t_{0}})}{\sum (p_{c,t_{0}}\cdot q_{c,t_{0}})}}\times {\frac {\sum (p_{c,t_{2}}\cdot q_{c,t_{1}})}{\sum (p_{c,t_{1}}\cdot q_{c,t_{1}})}}\times \cdots \times {\frac {\sum (p_{c,t_{n}}\cdot q_{c,t_{n-1}})}{\sum (p_{c,t_{n-1}}\cdot q_{c,t_{n-1}})}}$
Each term
    ${\frac {\sum (p_{c,t_{n}}\cdot q_{c,t_{n-1}})}{\sum (p_{c,t_{n-1}}\cdot q_{c,t_{n-1}})}}$
answers the question "by what factor have prices increased between period $t_{n-1}$ and period $t_{n}$". These are multiplied together to answer the question "by what factor have prices increased since period $t_{0}$". The index is then the result of these multiplications, and gives the price relative to period $t_{0}$ prices.
Chaining is defined for a quantity index just as it is for a price index.

## Index number theory

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 $I(P_{t_{0}},P_{t_{m}},Q_{t_{0}},Q_{t_{m}})$, where $P_{t_{0}}$ and $P_{t_{m}}$ are vectors giving prices for a base period and a reference period while $Q_{t_{0}}$ and $Q_{t_{m}}$ give quantities for these periods.
* Identity test:
    $I(p_{t_{m}},p_{t_{n}},\alpha \cdot q_{t_{m}},\beta \cdot q_{t_{n}})=1~~\forall (\alpha ,\beta )\in (0,\infty )^{2}$
    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 $\alpha$, for the first period, or $\beta$, for the later period) then the index value will be one.
* Proportionality test:
    $I(p_{t_{m}},\alpha \cdot p_{t_{n}},q_{t_{m}},q_{t_{n}})=\alpha \cdot I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})$
    If each price in the original period increases by a factor α then the index should increase by the factor α.
* Invariance to changes in scale test:
    $I(\alpha \cdot p_{t_{m}},\alpha \cdot p_{t_{n}},\beta \cdot q_{t_{m}},\gamma \cdot q_{t_{n}})=I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})~~\forall (\alpha ,\beta ,\gamma )\in (0,\infty )^{3}$
    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.
* Commensurability test:
    The index should not be affected by the choice of units used to measure prices and quantities.
* Symmetric treatment of time (or, in parity measures, symmetric treatment of place):
    $I(p_{t_{n}},p_{t_{m}},q_{t_{n}},q_{t_{m}})={\frac {1}{I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})}}$
    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.
* 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.
* Monotonicity test:
    $I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})\leq I(p_{t_{m}},p_{t_{r}},q_{t_{m}},q_{t_{r}})~~\Leftarrow ~~p_{t_{n}}\leq p_{t_{r}}$
    A price index for lower later prices should be lower than a price index with higher later period prices.
* Mean value test:
    The overall price relative implied by the price index should be between the smallest and largest price relatives for all commodities.
* Circularity test:
    $I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})\cdot I(p_{t_{n}},p_{t_{r}},q_{t_{n}},q_{t_{r}})=I(p_{t_{m}},p_{t_{r}},q_{t_{m}},q_{t_{r}})~~\Leftarrow ~~t_{m}\leq t_{n}\leq t_{r}$
    Given three ordered periods $t_{m}$, $t_{n}$, $t_{r}$, the price index for periods $t_{m}$ and $t_{n}$ times the price index for periods $t_{n}$ and $t_{r}$ should be equivalent to the price index for periods $t_{m}$ and $t_{r}$.

## Quality change

Price indices often capture changes in price and quantities for goods and services, but they often fail to account for variation in the quality of goods and services. This could be overcome if the principal method for relating price and quality, namely hedonic regression, could be reversed. Then quality change could be calculated from the price. Instead, 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.
The problem discussed above can be represented as attempting to bridge the gap between the price for the old item at time t, $P(M)_{t}$, with the price of the new item at the later time period, $P(N)_{t+1}$.
* The overlap method uses prices collected for both items in both time periods, t and t+1. The price relative ${P(N)_{t+1}}$/${P(N)_{t}}$ 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. $P(N)_{t+1}$/$P(M)_{t}$ 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.
* 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.