3.2  Index number approaches to measuring productivity

The majority of statistical agencies that produce regular productivity statistics use the index number approach. For example, the Australian Bureau of Statistics calculates market sector multifactor productivity using the index number approach based on a Törnqvist index, as does the US Bureau of Labor Statistics.

The index number approach to calculating productivity involves dividing an output quantity index by an input quantity index to give a productivity index. Therefore:

(9)     

Where is TFP, is an index of output quantities (a real output index) and is an index of input quantities. The subscript t indicates the time period. Equation (9) can be viewed as a re-arrangement of (3), but in (9) no production function has been specified meaning that the TFP level (measured by ) and subsequent growth rates may not be the same as that which would result from using the growth accounting procedure.

After obtaining an output and input quantity indexes the calculation of a TFP index is straightforward. From this TFP index productivity growth rates can be easily calculated. The difficulty is in determining what type of index to use and then obtaining the necessary price and quantity data to construct them.

To construct an output (input) quantity index it is necessary to determine an appropriate way to aggregate the different outputs (inputs) produced in an economy. For example, it would not make sense to add the number of car tyres produced to the number of woollen jerseys. There are a number of different index formulations that try to overcome this problem by using prices (or output shares) to weight the various different kinds of outputs.

With vectors of prices and quantities for the different outputs produced in an economy at time , a Laspeyres index is calculated as:

(10)    

where is quantity nominal output share. Equation (10) shows that the Laspeyres index is a nominal share-weighted sum of quantity ratios.

The Paasche quantity index is obtained by using period 0 prices rather period 1 prices:

(11)     

The geometric average of the Laspeyres and Paasche indexes is known as a Fisher index.

Finally, the Törnqvist output index is defined as follows:

(12)     

Similarly, input indexes are defined when using data on input quantities and prices.

Calculating a TFP index using the index number approach requires a decision regarding the type of index formulation to be used in constructing the output quantity and input quantity indexes. Two approaches to choosing between different index number formulations are the economic and axiomatic approaches.

Diewert and Lawrence stated:

The economic approach selects index number formulations on the basis of an assumed underlying production function and assuming price taking, profit maximising behaviour on the part of producers. For example, the Törnqvist index used extensively in past TFP studies can be derived assuming the underlying production function has the translog form and assuming producers are price taking revenue maximisers and price taking cost minimisers.

Diewert and Lawrence, 1999: 9

The axiomatic approach involves comparing the properties of the different index number formulations with a number of desirable mathematical properties. The index that passes the most tests is the “preferred” index formulation. Diewert and Lawrence (1999) used this axiomatic approach in determining which index number formulation to use. The axioms (or desirable properties) were:

1. Constant quantities test: If quantities are the same in two periods, then the output index should be the same in both periods irrespective of the price of the goods in both periods;
2. Constant basket test: If prices are constant over two periods, then the level of output in period 1 compared to period 0 is equal to the value of output in period 1 divided by the value of output in period 0;
3. Proportional increase in quantity test: If all quantities in period t are multiplied by a common factor, λ, then the quantity index in period t compared to period 0 should increase by λ also; and
4. Time reversal test: If the prices and quantities in period 0 and t are interchanged, then the resulting output index should be the reciprocal of the original index.

Diewert and Lawrence note that of the four index formulations defined above, only the Fisher index has all four desirable properties. Both the Laspeyres and Paasche indexes are inconsistent with time reversal test and the Törnqvist does not pass the constant basket test. For this reason Diewert and Lawrence use a chained Fisher index for constructing output and input quantity indices in their construction of a TFP index. They also note that when a more extensive list of axiomatic tests is used, the Fisher index continues to satisfy more tests than other index formulations.