# List of price index formulas

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List of price index formulas

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 ($p_t/p_0$).

Fixed base indexes

Laspeyres

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

Paasche

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

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. [PPI manual, 598.] At these lower levels, weights do not matter since only one type of good is being aggregated.

Carli

Developed in 1764 by Carli, a Italian economist, this formula is the arithmetic average of the price relative between a period "t" and a base period "0". :$P_C = frac \left\{1\right\}\left\{n\right\}cdotsumleft\left(frac \left\{p_\left\{t\left\{p_0\right\} ight\right)$

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".:$P_D = frac \left\{frac\left\{1\right\}\left\{n\right\}cdotsum \left(p_\left\{t\right\}\right)\right\}\left\{frac\left\{1\right\}\left\{n\right\}cdotsum \left(p_\left\{0\right\}\right)\right\}= frac \left\{sum \left(p_\left\{t\right\}\right)\right\}\left\{sum \left(p_\left\{0\right\}\right)\right\}$

Jevons

In 1863, English economist Jevons proposed taking the geometric average of the price relative of period "t" and base period "0". [PPI manual, 602.] 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. [PPI manual, 596.] :$P_J = prodleft\left(frac\left\{p_\left\{t\left\{p_\left\{0 ight\right)^\left\{1/n\right\}$

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. [Export and Import manual, Chapter 20 p. 8] :$P_\left\{HR\right\} = frac \left\{1\right\}\left\{frac\left\{1\right\}\left\{n\right\}cdotsumleft\left(frac \left\{p_0\right\}\left\{p_t\right\} ight\right)\right\}$

Carruthers, Sellwood, Ward, Dalén index

Is the geometric mean of the Carli and the harmonic price indexes. [PPI manual, 597.] 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. [Export and Import manual, Chapter 20, p. 8] :$P_\left\{CSWD\right\} = sqrt \left\{P_C cdot P_\left\{HR$

Ratio of harmonic means

The ratio of harmonic means or "Harmonic means" price index is the harmonic average counterpart to the Dutot index. [PPI manual, 600.] :$P_\left\{RH\right\} = frac \left\{sumleft\left(frac \left\{n\right\}\left\{p_0\right\} ight\right)\right\}\left\{sumleft\left(frac \left\{n\right\}\left\{p_t\right\} ight\right)\right\}$

Non-Fixed base indexes

Marshall-Edgeworth

The Marshall-Edgeworth index, credited to Marshall (1887) and Edgeworth(1925), [ PPI manual, Chapter 15, p. 378.] 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. [PPI manual, 620.] 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. [PPI manual, Chapter 15, p. 378] :$P_\left\{ME\right\}=frac\left\{sum \left[p_\left\{c,t_n\right\}cdot frac\left\{1\right\}\left\{2\right\}cdot\left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right] \right\}\left\{sum \left[p_\left\{c,t_0\right\}cdot frac\left\{1\right\}\left\{2\right\}cdot\left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right] \right\}=frac\left\{sum \left[p_\left\{c,t_n\right\}cdot \left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right] \right\}\left\{sum \left[p_\left\{c,t_0\right\}cdot \left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right] \right\}$

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. [ILO CPI manual, Chapter 1, p. 2.] 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 othertwice-differentiable functions around the same point." [Export and Import manual, Chapter 18, p. 23.]

Fisher

:$P_F = sqrt\left\{P_Pcdot P_L\right\}$

Törnqvist

The Törnqvist or Törnqvist-Theil index is the geometric average of the price relative of the current to base period prices weighted by the arithmetic average of the value shares for the two periods. [PPI manual, p. 610]

:$P_T = prodleft\left(frac\left\{p_\left\{t\left\{p_\left\{0 ight\right)^\left\{frac\left\{1\right\}\left\{2\right\} cdot left \left[frac\left\{p_\left\{0\right\}cdot q_\left\{0\left\{sumleft\left(p_\left\{0\right\}cdot q_\left\{0\right\} ight\right)\right\}+ frac\left\{p_\left\{t\right\}cdot q_\left\{t\left\{sumleft\left(p_\left\{t\right\}cdot q_\left\{t\right\} ight\right)\right\} ight\right] \right\}$ [ [http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/tornqvist+index+and+other+log-change+index+numbers “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:

:$P_W = frac \left\{sumleft \left[p_\left\{t\right\} cdot \left(q_\left\{0\right\}cdot q_\left\{t\right\}\right)^\left\{frac \left\{1\right\}\left\{2 ight\right] \right\}\left\{sumleft \left[p_\left\{0\right\} cdot \left(q_\left\{0\right\}cdot q_\left\{t\right\}\right)^\left\{frac \left\{1\right\}\left\{2 ight\right] \right\}$

Notes

References

* [http://www.internationalmonetaryfund.com/external/np/sta/tegeipi/index.htm "Export and Import Price Index Manual"]
* [http://www.imf.org/external/np/sta/tegppi/index.htm "PPI Manual"]

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