In practice, Törnqvist index values are calculated for consecutive periods, then these are strung together, or "chained".
A Törnqvist or Törnqvist-Theil price index is the weighted geometric mean of the price relatives using arithmetic averages of the value shares in the two periods as weights.
to be the quantity purchased of item i at time t, then, the Törnqvist price index
at time t can be calculated as follows:[2] The denominators in the exponent are the sums of total expenditure in each of the two periods.
[3] A Törnqvist quantity index can be calculated analogously using prices for weights.
A Divisia index is a theoretical construct, a continuous-time weighted sum of the growth rates of the various components, where the weights are the component's shares in total value.
For a Törnqvist index, the growth rates are defined to be the difference in natural logarithms of successive observations of the components (i.e. their log-change) and the weights are equal to the mean of the factor shares of the components in the corresponding pair of periods (usually years).
Divisia-type indexes have advantages over constant-base-year weighted indexes, because as relative prices of inputs change, they incorporate changes both in quantities purchased and relative prices.
For example, a Törnqvist index summarizing labor input may weigh the growth rate of the hours of each group of workers by the share of labor compensation they receive.
To express that thought, Diewert (1978) uses this phrasing which other economists now recognize: the Törnqvist index procedure "is exact for" the translog production or utility function.
For some purposes (like large annual aggregates), this is treated as consistent enough, and for others (like monthly price changes) it is not.
[8][9] Törnqvist indexes are used in a variety of official price and productivity statistics.
[10][11][12][13] The time periods can be years, as in multifactor productivity statistics, or months, as in the U.S.'s Chained CPI.