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Q&A about the Consumer Price Index

Laspeyres formula 

  Laspeyres suggested this index formula in 1871. In case of calculating the price index, assuming that for individual item i, price at the base period to be pi 0, at the observation period to be pi t, and quantity at the base period to be qi 0, the following equation is called "Laspeyres formula".

Laspeyres formula

  where, denominator and numerator are total expenditures for all items, at the base and the observation period, respectively, assuming that consumers purchase the same amount of commodities both at the base period and the observation period. In this formula, quantities are fixed at the base period. 

  For practical use, eq.(1) is transformed as follows;

Laspeyres formula

  This is weighted average of price ratios of each item, weighted by expenditures at the base period.

Paasche formula

  Paasche suggested this index formula in 1874. In case of calculating the price index, assuming that for individual item i, price at the base period to be pi 0, at the observation period to be pi t, and quantity at the base period to be qi t, the following equation is called "Paasche formula".

Paasche formula

  where, denominator and numerator are total expenditures for all items, at the base and the observation period, respectively, assuming that consumers purchase the same amount of commodities both at the base period and the observation period. In this formula, quantities are fixed at the observation period.

  For practical use, eq.(3) is transformed as follows;

Paasche formula

Fisher formula

  This index formula is suggested by Fisher and called "ideal formula". Assuming that for individual item i, prices and quantities at the base period to be pi 0 and qi 0, at the observation period to be pi t and qi t, the following equation is called "Fisher formula".

Fisher formula

  This is a geometric mean of Laspeyres and Paasche formula. Normally, the following inequality holds; Laspeyres >= Fisher >= Paasche. Fisher formula is called ideal formula in a sense that the time reversal test and the factor reversal test are satisfied. This formula is used in the case when prices and quantities at the base and the observation period are quite different.

  In Japan, base period = price reference period = weight reference period.

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