2  Index number methodology

This paper uses the index number approach to measure aggregate and industry productivity.[1] Construction of aggregate and industry productivity series using index number techniques is common internationally, especially by statistical agencies. For example, the Australian Bureau of Statistics (ABS) publishes productivity series for the Australian economy using the index number approach. What follows is a brief introduction to productivity measurement using the index number methodology. A more detailed review of the index number approach to productivity measurement is available in McLellan (2003).

In general, a productivity index is defined as the ratio of an output index to an input index, that is:

(1)      ;

where is a productivity index, is an output index and is an input index. Each index represents accumulated growth from period 0 to period t.

When is comprised of a single type of input, say labour or capital, is a partial productivity index. The two most common partial productivity measures are labour productivity and capital productivity. Labour and capital productivity indexes measure changes in the ability of the labour and capital inputs, respectively, to produce output over time. Caution should be exercised when using partial productivity measures as changes in the mix of inputs can influence these measures. For example, substitution of physical capital for labour, owing to a relative change in the price of labour to physical capital, may raise labour productivity. In this case, “…labour productivity statistics do not always represent true changes in the underlying productivity of labour…” (Dixon, 1990, p. 6).

When is a composite index of two or more inputs, is a multifactor productivity index. Most often is formed using labour and capital inputs, although researchers have also included other inputs, such as land, in addition to labour and physical capital inputs.[2] This paper presents a suite of productivity measures, including labour and capital productivity. However, owing to limitations with the partial productivity measures, more emphasis is given to multifactor productivity series throughout this paper.

Calculating productivity at the aggregate and industry level requires the construction of both output and input indices. Because outputs and inputs are heterogenous it is not possible to simply add all the outputs to get an output index and, likewise, to add all the inputs to get an input index. Both outputs and inputs need to be weighted to form aggregate and sub-aggregate output and input indices. Output prices and input costs are often used as respective weights to form output and input indices.

When constructing productivity indices, it is not immediately apparent which weighting procedure should be used to form output and input series and on what basis this weighting procedure should be chosen. There is a multiplicity of index number formulae available to users when constructing output and input indices. Some of the better known indexes include the Laspeyres, Paasche, Fisher and Törnqvist.

Suppose information on the price and quantity of I outputs is available for period . Denoting the price and quantity vectors in period t as and , the Laspeyres , Paasche, Fisher and Törnqvist quantity indexes are defined as follows:

(2)     

(3)     

(4)     

(5)     

for and , and where .

There are two approaches to choosing an index number formula: the economic approach and the axiomatic approach. The economic approach bases the choice of an index number formula on an assumed underlying aggregator function (ie, production, cost, revenue or profit function). This approach assumes competitive optimising behaviour and embodies production technology. For example, firms are assumed to maximise profit for a given production technology. The axiomatic (or test) approach bases the choice of an index number formula on properties the index should exhibit, with these properties being embodied in axioms. One of the appealing features of this approach is that it does not make any assumptions about competitive optimising behaviour. A strong case can be made in favour of using the Törnqvist and Fisher index formulae as both possess properties that are desired under the economic and axiomatic approaches.[3]

In addition to deciding on an index formula, a decision needs to be made whether to construct direct or chained productivity indices. A direct quantity index compares quantities in period t relative to some fixed base period (which is why a direct index is also known as fixed-weight index or a fixed-base index). Information on price movements and therefore weighting changes in the intervening period is ignored. In contrast, a chained quantity index compares quantities between two periods taking into account information on weighting changes in the intervening period. Put another way, a chained index uses price information that is more representative of that faced by economic agents in each period than does a direct index.

When relative prices change, relative quantities also tend to change. For example, if the price of a particular good rises relative to all other goods in the economy due to a demand increase, then price taking producers will tend to produce more of this good relative to other goods. Using a direct quantity index to measure quantity changes in the face of relative price changes will introduce substitution bias into the quantity index. Biases arise because changes in producer behaviour in response to relative price changes are not taken into account when using direct indexes. Moreover, the substitution bias usually becomes cumulatively larger with the passage of time, as historical fixed-weights become increasingly unrepresentative. Chaining direct indexes usually reduces substitution bias.

The chained quantity index is formed by linking direct quantity indexes. Generally, a chained index is constructed as follows:

(6)     ;

where denotes the chained index between time 0 and time t and the direct index. Chaining can be applied to any of the index number formulae outlined in equations (2) to (5). One of the consequences of chaining is that it usually reduces the index number spread (the range between the Laspeyres and Paasche indices).[4] The one situation where it is inappropriate to adopt chaining is when price and quantities exhibit large fluctuations. In this situation, the chained index and counterpart direct index will diverge and the index number spread will be accentuated.

As discussed, under either the axiomatic or economic approaches, a strong case can be made for using either the Fisher or Törnqvist indexes. This paper uses the Fisher index to construct market sector and industry productivity series to provide methodological continuity with Diewert and Lawrence (1999), who also used the Fisher index when constructing productivity series for New Zealand. Furthermore, chaining is also employed throughout. Nonetheless, the paper also constructs productivity series using alternative indexes to test the sensitivity of New Zealand productivity series to different index number formulae.