List of price index formulas

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A number of different formulas, at least hundreds, have been proposed as means of calculating price indexes. While price index formulas all use price and quantity data, they amalgamate this data in different ways. A price index generally aggregates using 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 formulas can be framed as comparing expenditures (An expenditure is a price times a quantity) or taking a weighted average of price relatives ( $p t / p 0$ ).

1 Fixed base indexes
- 1.1 Laspeyres
- 1.2 Paasche
2 Unweighted indexes
3 Non-Fixed base indexes
- 3.1 Fisher price index
4 Notes
5 References

[edit] Fixed base indexes

[edit] Laspeyres

$P_L = \frac{\sum (p_{t}\cdot q_{0})}{\sum (p_{0}\cdot q_{0})}$

[edit] Paasche

$P_P = \frac{\sum (p_{t}\cdot q_{t})}{\sum (p_{0}\cdot q_{t})}$

[edit] Unweighted indexes

Unweighted price indexes or elementary price indexes only compare prices between two periods. They do not make any use of quantities or expenditure weights. These indexes are called "elementary" because they are often used at the lower levels of aggregation for more comprehensive price indexes.^[1] At these lower levels, weights do not matter since only one type of good is being aggregated.

[edit] Carli

Developed in 1764 by Carli, and Italian economist, this formula is the arithmetic average of the price relative between a period t and a base period 0.

$P_C = \frac {1}{n} \sum (\frac {p_{t}}{p_0})$

[edit] Dutot

In 1738 French economist Dutot proposed using an index calculated by dividing the average price in period t by the average price in period 0.

$P_D = \frac {\frac{1}{n}\sum (p_{t})}{\frac{1}{n}\sum (p_{0})} = \frac {\sum (p_{t})}{\sum (p_{0})}$

[edit] Jevons

In 1863, English economist Jevons proposed taking the geometric average of the price relative of period t and base period 0.^[2] 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.^[3]

$P_J = \prod(\frac{p_{t}}{p_{0}})^{1/n}$

[edit] Harmonic mean of price relatives

The harmonic average counterpart to the Carli index.^[4] The index was proposed by Jevons in 1865 and by Coggeshall in 1887.^[5]

$P_{HR} = \frac {1}{\frac{1}{n} \sum (\frac {p_0}{p_t})}$

[edit] Carruthers, Sellwood, Ward, Dalén index

Is the geometric mean of the Carli and the harmonic price indexes.^[6] In the 1922 Fisher wrote that this and the Jevons were the two best unweighted indexes based on Fisher's test approach to index number theory.^[7]

$P_{CSWD} = \sqrt {P_C \cdot P_{HR}}$