List of price index formulas

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.

${\displaystyle P_{C}={\frac {1}{n}}\cdot \sum {\frac {p_{t}}{p_{0}}}}$

On 17 August 2012 the BBC Radio 4 program More or Less[3] 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.

In 1738 French economist Nicolas Dutot[4] proposed using an index calculated by dividing the average price in period t by the average price in period 0.

${\displaystyle P_{D}={\frac {{\frac {1}{n}}\cdot \sum p_{t}}{{\frac {1}{n}}\cdot \sum p_{0}}}={\frac {\sum p_{t}}{\sum p_{0}}}}$

In 1863, English economist William Stanley Jevons proposed taking the geometric average of the price relative of period t and base period 0.[5] 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.[6]

${\displaystyle P_{J}=\left(\prod {\frac {p_{t}}{p_{0}}}\right)^{1/n}}$

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.

The harmonic average counterpart to the Carli index.[7] The index was proposed by Jevons in 1865 and by Coggeshall in 1887.[8]

${\displaystyle P_{HR}={\frac {1}{{\frac {1}{n}}\cdot \sum {\frac {p_{0}}{p_{t}}}}}}$

Is the geometric mean of the Carli and the harmonic price indexes.[9] 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.[10]

${\displaystyle P_{CSWD}={\sqrt {P_{C}\cdot P_{HR}}}}$

The ratio of harmonic means or "Harmonic means" price index is the harmonic average counterpart to the Dutot index.[7]

${\displaystyle P_{RH}={\frac {\sum {\frac {n}{p_{0}}}}{\sum {\frac {n}{p_{t}}}}}}$

The Marshall-Edgeworth index, credited to Marshall (1887) and Edgeworth (1925),[11] 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.[12] 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.[13]

${\displaystyle P_{ME}={\frac {\sum \left[p_{t}\cdot {\frac {1}{2}}\left(q_{0}+q_{t}\right)\right]}{\sum \left[p_{0}\cdot {\frac {1}{2}}(q_{0}+q_{t})\right]}}={\frac {\sum \left[p_{t}\cdot \left(q_{0}+q_{t}\right)\right]}{\sum \left[p_{0}\cdot \left(q_{0}+q_{t}\right)\right]}}}$

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.[14] 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."[15]

The change in a Fisher index from one period to the next is the geometric mean of the changes in Laspeyres's and Paasche's indexes between those periods, and these are chained together to make comparisons over many periods:

${\displaystyle P_{F}={\sqrt {P_{L}\cdot P_{P}}}}$

This is also called Fisher's "ideal" price index.

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.[16][17]

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

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:

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