Appendix B. Methodology with the Tornqvist Index

Appendix B.Methodology with the Tornqvist Index

Multifactor productivity is the ratio of the output index to a weighted average of the input indexes. A Tornqvist formula expresses the change in multifactor productivity as the difference between the rate of change in output and the weighted average of the rates of change in the inputs. Let:

Ln = the natural logarithm of a variable

A = multifactor productivity

Q = output

I = combined input

K = capital input

L = labor input

M = intermediate input

Wk = the average share of capital cost in total cost in two adjacent periods

Wl = the average share of labor cost in total cost in two adjacent periods

Wm = the average share of intermediate input cost in total cost in two adjacent periods,

The change in the multifactor productivity is then:

(1) delta lon uppercase a is equal to lon (uppercase a subscript {lowercase t} over uppercase a subscript {lowercase t minus 1}) is equal to lon (uppercase q subscript {lowercase t} over uppercase q subscript {lowercase t minus 1}) minus [uppercase w subscript {lowercase k} (lon (uppercase k subscript {lowercase t} over uppercase k subscript {lowercase t minus 1})) plus uppercase w subscript {lowercase l} (lon (uppercase l subscript {lowercase t} over uppercase l subscript {lowercase t minus 1})) plus uppercase w subscript {lowercase m} (lon (uppercase m subscript {lowercase t} over uppercase m subscript {lowercase t minus 1}))]

Or

(2) delta lon uppercase a is equal to lon (uppercase a subscript {lowercase t} over uppercase a subscript {lowercase t minus 1}) is equal to lon (uppercase q subscript {lowercase t} over uppercase q subscript {lowercase t minus 1}) minus lon (uppercase i subscript {lowercase t} over uppercase subscript {lowercase t minus 1})


A multifactor productivity index can be further developed by calculating the antilogs of ∆LnA, chaining up the resulting annual rates of change, and expressing the resulting series as a percentage of a selected base year. Equivalently, the change in the multifactor productivity can be directly expressed as At / At-1 = (Qt / Qt-1) / (It / It-1). Again, At / At-1 can be chained over time and converted into an index number.

All variables, except for cost shares, are in the form of a constant dollar quantity index. The output quantity index is usually derived by deflating the industry output in current dollars by an appropriate price index when the industry output is a single measure. When an industry produces multiple products and the output measure of each individual product is available, such individual outputs may be deflated separately by more detailed price indexes. In that case, the total output quantity index can be derived through a Tornqvist aggregation such as:

(3) summation from 1 to lowercase n of (lowercase w subscript {lowercase i} delta lon uppercase q subscript {lowercase i}) ,

where Qi is the output of the ith product, and wi is the average share of the ith product in the total output.