# 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 **p**_{i }_{0}, at the observation period to be **p**_{i t}, and quantity at the base period to be **q**_{i }_{0}, the following equation is called "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;

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 **p**_{i}_{ 0}, at the observation period to be **p**_{i t}, and quantity at the base period to be **q**_{i t}, the following equation is called "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;

## 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 **p**_{i }_{0} and **q**_{i }_{0}, at the observation period to be **p**_{i t} and **q**_{i t}, the following equation is called "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.