3.3 A distance function based approach (continued)

Consider the following example based on Figure 2, which gives the necessary pieces of information to construct a Malmquist output based productivity index. Given two constant returns to scale production frontiers ( and ) that describe the technology available in periods t and and also information on the input/output combinations that were realised by the economy during periods t and . In period t, the economy used inputs to produce outputs. In period the economy used inputs to produce outputs.

To calculate the efficiency change component we need to construct two distance functions, and . These two functions tell us for each period, how close the level of output actually produced was to the frontier level of output for the same level of inputs based on the technology available in the period under consideration. These two measures describe the relative efficiency of production in each period and both will take values between 0 and 1. Based on Figure 2, as the level of output was , whereas had production be technically efficient, output would have been b. and thus efficiency change equals .

To calculate the technical change component we need to use the two distance functions calculated above as well as and . measures how close the level of output actually produced in period is to the maximum level of output that could have been produced with inputs had period technology been available. In Figure 2, . Note that can exceed 1 (and does in this case) if technological advancement has occurred. measures how close the level of output actually produced in period is to the maximum level of output that could have been produced from period inputs, had period technology been available. . Thus in this example technical change equals

To compute distance functions the researcher needs to know the production technology of or production frontier at all time periods under consideration. Fare et al (1994) who looked at productivity growth in 17 OECD countries, produced a world production frontier nonparametrically using an approach known as activity analysis or Data Envelopment Analysis (DEA). They then compared their 17 countries to the “world” production frontier calculated from data for the 17 countries. Fare, Grosskopf and Margaritis (1996) (hereafter, FGM) followed this approach but used sector level input and output data for New Zealand to produce an aggregate production frontier for the New Zealand market sector. Individual sectors are then compared to this frontier.

The basic approach used in both these studies involved having data for countries (or sectors in the case of FGM). Each of these countries (or sectors) uses inputs at each time period , to produce outputs (in FGM each sector produced one output so ). The production frontier or technology can be constructed under the assumption of constant returns to scale:

(18)    

where is an intensity variable.

It is possible to allow for different returns to scale by adding restrictions involving the intensity variable. For example it is possible to relax the assumption of constant returns to scale by adding the restriction shown in equation (19) which allows for non-increasing returns to scale:

(19)     

Figure 3 illustrates the construction of a production frontier, for a particular period, based on equation , in the simple case where we have one input and one output.[14]

Figure 3: Construction of a production frontier at time t

In Figure 3 there are three different output-input combinations (, and ), which could relate to three countries or three sectors depending on the level at which the analysis is being undertaken. If the restriction shown in equation is imposed, that the sum of the intensity variables must be less than or equal to 1, then the production frontier is the line and the horizontal extension from . If instead constant returns to scale is imposed (ie, the formulation given in equation ), the production frontier is given by the ray from the origin passing through . In the variable returns to scale case, the production frontier will consist of the line and the horizontal extension from .

A number of economists have expressed scepticism about the practical merit of the DEA approach to identifying an economy-wide or world production frontier. For example, Diewert and Lawrence (1999) stated in relation to FGM’s study:

Evidently, they just used a variant of DEA analysis, assuming that the value added outputs of each industry can be produced by every other industry. This seems to be a rather untenable assumption to say the least and hence we suspect that their measures of efficiency change and technical progress are essentially worthless.

Diewert and Lawrence, 1999: 98

Carlaw and Lipsey (2003) had similar concerns noting that the underlying assumption when using this technique to construct an economy’s production frontier from industry level data is that all the industries have the same aggregate production function.[15]

Given what we know about technological complementarities and the need to adapt technologies for specific uses, identical production functions across industries is not an acceptable assumption. For example, it is difficult to believe that the application of electricity to communications technologies can be considered to be the same production technology as the application of electricity to mining or machining?

Carlaw and Lipsey, 2003: 10

3.4  The econometric approach to productivity measurement

The econometric approach to productivity measurement involves estimating the parameters of a specified production function (or cost, revenue, or profit function, etc). Examples of the approach using New Zealand data include Grimes (1983), Szeto (2001) and Razzak (2002). Often the production function is expressed in growth rate form and then estimated to yield an estimate of the parameter that reflects the growth in technological progress, which is typically interpreted as a measure of productivity growth.

One advantage of the econometric approach is the ability to gain information on the full representation of the specified production technology. In addition to estimates for productivity, information is also gained on other parameters of the production technology. It is not possible to generate this additional information using the growth accounting or index number approaches. Moreover, because the econometric approach is based on using information on outputs and inputs, there is greater flexibility in specifying the production technology. For example, it is possible to introduce other forms of factor-augmenting technological change other than the Hicks-neutral formulation implied by the growth accounting and index number approaches, and to make allowance for adjustment costs and variation in input utilisation.

Within the econometric framework it is possible to test the validity of assumptions that underpin the growth accounting and index number approaches because of the sampling properties of the production technology. For example, it is possible to test the assumption of constant returns to scale that is often used in the growth accounting approach to productivity measurement.

The increased flexibility and the ability to test the validity of different assumptions of the econometric approach do not come without cost. The use of the econometric approach raises estimation issues that may call in to question the robustness of parameter estimates. For example, the parameter estimates for a specified technology may appear implausible (such as negative factor income shares for a Cobb Douglas production function), causing researchers to impose a priori restrictions on parameter values. Furthermore, when data samples are small, restrictions such as constant returns to scale are often imposed to preserve degrees of freedom. The use of more flexible function forms for the production technology often means that non-linear estimation techniques are required, which bring their own complications. Szeto (2001) provides a good example of the non-linear estimation issues that are often faced by researchers when using the econometric approach to estimate productivity.

A further drawback of the econometric approach is the greater difficulty of explaining the econometric methodology to a range of users, as well as the difficulty in replicating and producing productivity estimates on an ongoing basis. This is why measuring productivity using the econometric approach is best suited to single, one-off studies.

Hulten (2000) has pointed out that the econometric approach to productivity measurement should be viewed as complimentary to the growth accounting and index number approaches for several key reasons. First, index number techniques are often used to construct the output and input series that are used as variables in estimating productivity using the econometric approach, thus “the question of whether or when to use econometrics to measure productivity change is really a question of which stage of the analysis index number procedures should be abandoned” (Hulten, 2000: 23). Second, the relative simplicity of the growth accounting and index number approaches can be used to help interpret the richer results of the econometric approach. Finally, by merging the different approaches, econometrics can be used to help explain TFP. In summary the “potential richness and testable set-up [of the econometric approach] make them a valuable complement to the non-parametric, index number methods that are the normal tool for productivity statistics” (Schreyer and Pilat, 2001: 133).