Index formulae

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

I t =   ∑ i = 1 n W i ( p t / 0 ) i

W i = ( P 0 * Q k ) i ∑ n i = 1 ( P 0 * Q k ) i   ;   ∑ i = 1 n W i = 1.00

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 (Wi), 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, Wi, is given by the following; at the numerator: Total Expenditure (P0 times Qk) 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 (P0 times Qk) 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 (Wi) is equal to 1.

Figure 2: Chain-Laspeyres Index

I t = ∑ n i = 1 I i ( t ) W i ( t − 1 ) ∑ n i = 1 I i ( t − 1 ) W i ( t − 1 ) * ∑ n i = 1 I i ( t − 1 ) W i ( t − 2 ) ∑ n i = 1 I i ( t − 2 ) W i ( t − 2 ) * … = ∑ n i = 1 I i ( t ) W i ( t − 1 ) ∑ n i = 1 I i ( t − 1 ) W i ( t − 1 ) * I ( t − 1 )

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

  1. 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);
  2. 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);
  3. 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;

  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) 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);
  2. Price Index I at time (t minus 1).

Figure 3: The summation over all components

∑ i = 1 n W i = 1.00

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.

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