Given this long-term price relative, a fixed-base index is obtained by aggregating over cells using the formulas in exhibit 1.

Given that the substitution effect problem arises because the expenditure weights are fixed, one would think that the chained Laspeyres-- where weights change every year--would provide an index closer to the Tornqvist than would the fixed-base index. However, a comparison of the Laspeyres indexes in table 1 reveals that the chained Laspeyres increases at a faster rate over the 1982-91 period than does the fixed-base Laspeyres: by December 1991, the chained Laspeyres was 144.3, compared with 143.8 for the fixed-base Laspeyres(19) Second, the differences are numerically small.

Year-end values for fixed-base indexes of long-term price change relative to December 1982 are shown in chart 1.

The degree of the substitution effect in chained indexes is somewhat larger than that found using the fixed-base indexes. (See table 1.) Using the Tornqvist index as the basis for comparison,(18) the estimated degree of the substitution effect is 3.4 percentage points by 1991, or an annual rate of 0.27 percentage points, compared with 2.6 index points using the fixed-weight index, or an annual rate of 0.2.