Appendix B
Methodology with the Tornqvist Index
Multifactor productivity is the ratio of the output index to a weighted average of 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 various 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) ![uppercase delta of uppercase L lowercase n of uppercase A = upeprcase L lowercase n( uppercase A subscript{lowercase t} over uppercase A subscript{lowercase t minus one}) = uppercase L lowercase n( upeprcase Q subscript{t} over uppercase Q subscript{lowercase t minus one}) minus [uppercase W lowercase k( uppercase L lowercase n of uppercase K subscript{lowercase t} over upercase K subscript{lowercase t minus one}) plus uppercase W subscript{lowercase l}) plus uppercase W subscript{lowercase l}(uppercase L lowercase n of uppercase L subscript{lowercase t} over uppercase L subscript{lowercase t minus one}) plus uppercase W lowercase m(uppercase L lowercase n of uppercase M subscript{lowercase t} over uppercase M subcript{lowercase t minus one})].](https://webarchive.library.unt.edu/eot2008/20090510095214im_/https://www.bts.gov/publications/research_papers/estimating_multifactor_productivity_in_truck_transportation/images/equation_01.gif)
OR
2) 
A multifactor productivity index can be further developed by calculating the antilog of ΔLnA, 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  . Again,  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 defl ated separately by more detailed price indexes. In that case, the total output quantity index can be derived through a Tornqvist aggregation such as:
3) 
where Qi is the output of the ith product, and
wi is the average share of the ith product in the total output.
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