Price indexes have been calculated using either a fixed weight formula or the Chain-Laspeyres index formula of the following general type.

## Figure 1: Fixed weight

The fixed-weight Laspeyres price index *I* in time *t* and relative to time base period 0 is given by the summation over all components, that is, *i* equal to 1 to *n*, of the relative importance of the *i*-th component (*W _{i}*), times the price relative of the

*i*-th component in time

*t*relative to time base period 0.

The relative importance of the *i*-th component, *W _{i}*, is given by the following; at the numerator: Total Expenditure (P

_{0}times Q

_{k}) in period

*k*on the

*i*-th component expressed in base period 0 prices; and the denominator: the summation over all components,

*i*equal to 1 to

*n*, of the Total Expenditure (P

_{0}times Q

_{k}) in period

*k*on the

*i*-th component expressed in base period 0 prices.

The summation over all components, *i* equal to 1 to *n*, of the relative importance of the *i*-th component (*W _{i}*) is equal to 1.

## Figure 2: Chain-Laspeyres Index

The Chain-Laspeyres price index *I* in time *t* is given by multiplication of the following products;

- at the numerator: summation over all components, that is,
*i*equal to 1 to*n*, of the price index*I*of the*i*-th component in time*t*(which may also be calculated in a similar manner to*It*) times the relative importance*W*of the*i*-th component in time (*t*minus 1); and at the denominator: summation over all components, that is,*i*equal to 1 to*n*, of the price index*I*of the*i*-th component in time (*t*minus 1) times the relative importance*W*of the*i*-th component in time (*t*minus 1); - at the numerator: summation over all components,
*i*equal to 1 to n, of the price index*I*of the*i*-th component in time (*t*minus 1) times the relative importance*W*of the*i*-th component in time (*t*minus 2); and at the denominator: summation over all the components, that is*i*equal to 1 to*n*, of the price index*I*of the*i*-th component in time (*t*minus 2) times the relative importance*W*of the*i*-th component in time (*t*minus 2); - Price index products analogous to (1) and (2) are formed for more distant periods.

The Chain Laspeyres price index *I* at time *t* thus can be simplified to the multiplication of the following two products;

- At the numerator; summation over all components,
*i*equal to 1 to*n*, of the price index*I*of the*i*-th component in time (*t*) times the relative importance*W*of the*i*-th component in time (*t*minus 1); and at the denominator: summation over all components,*i*equal to 1 to*n*, of the price index*I*of the*i*-th component in time (*t*minus 1) times the relative importance*W*of the*i*-th component in time (*t*minus 1); - Price Index
*I*at time (*t*minus 1).

## Figure 3: The summation over all components

Note in the above that the Chain-Laspeyres index formula is used to reflect the changing relative importance of index components. The above example showing a single level of index aggregation can be extended to two or more levels.