2  Index number approach

This section presents an introduction to measuring productivity using the index number method. Subsection 2.1 discusses various productivity measures and subsection 2.2 presents several index number formulae often used in constructing productivity indices. The economic and axiomatic approaches to choosing an index number formula are discussed in subsection 2.3. Finally, subsection 2.4 presents numerical examples using the index formulae from subsection 2.2 to illustrate the construction of productivity indices and the differences between different index formulations. Readers interested in a more detailed review of the index number approach to productivity measurement should consult Diewert and Nakumara (2004).

2.1  Productivity index

Productivity measures attempt to capture the ability of inputs to produce output (usually over time). In general a productivity index is defined as the ratio of an output quantity index to an input quantity index, that is:

(1)    

for and where is a productivity index, is an output quantity index and is an input quantity index. Each index represents accumulated growth from period to period .

When comprises a single input, for example labour or physical capital, is a partial productivity index. The two well known partial productivity measures are labour and capital productivity. A limitation of partial productivity measures is that changes in productivity may reflect the impact of omitted inputs. For example, increases in labour productivity may be due to increases in the available amount of physical capital (one of the omitted inputs in the measurement of labour productivity) per worker, rather than increases in the underlying productivity of labour.

When comprises two or more inputs, is a multifactor productivity index.[2] Most often multifactor productivity is formed using labour and physical capital, although some productivity studies have included additional variables such as land and inventories (see, for example, Diewert and Lawrence, 1999).

Productivity indices are usually constructed using disaggregate prices and quantities of outputs and inputs. Because outputs and inputs are heterogenous it is simply not possible to add all outputs to form an output quantity index or, likewise, to add all inputs to form an input quantity index. Disaggregate data on the volumes of outputs and inputs need to be weighted to form output and input quantity indices. Output and input prices, or nominal output and input shares, are typically used as representative weights when forming output and input quantity indices.

2.2  Index number formulae

When constructing productivity indices it is not immediately apparent which weighting procedure should be used to weight output and input quantities when forming output quantity and input quantity indices and on what basis the weighting structure should be chosen. There are numerous index formulae that can be used to construct output and input indices. The Laspeyres, Paasche, Fisher and Törnqvist indexes are some of the more widely used index formulae.

Suppose information on the price and quantity of outputs is available for period . Denoting the output price and quantity vectors in period as and , the Laspeyres output quantity index () is defined as follows:

(2)    

where is output ’s nominal output share. Note that equation shows the Laspeyres output quantity index is the period share-weighted sum of quantity ratios.

The Paasche output quantity () index is defined as follows:

(3)    

The Paasche output quantity index uses period prices as the weights, in contrast to the Laspeyres output quantity index that uses period prices as weights.

The Fisher output quantity index () is found by taking the geometric average of the Laspeyres and Paasche output quantity indexes, that is:

(4)    

Finally, the Törnqvist output quantity () index is defined as follows:

(5)    

Input quantity indexes are defined in a similar manner using input prices () and input quantities ().