3.3  A distance function based approach

The distance function based approach to measuring TFP seeks to separate TFP into two components.[13] This is done using an output distance function that measures the distance of an economy from its production function. In principle, this technique enables a change in TFP to be decomposed into changes resulting from a movement towards the production frontier and shifts in the frontier. The output distance function measures how close a particular level of output is to the maximum attainable level of output that could be obtained from the same level of inputs if production is technically efficient. In other words, it represents how close a particular output vector is to the production frontier given a particular input vector. Fare, Grosskopf, Norris and Zhang (1994) define an output distance function at time t as

(13)     

where is a vector of input quantities at time t and is a vector of output quantities at time t. describes a production technology or production possibility set that is feasible using the technology available at time t.

The term in equation (13) states that of the set of real numbers, θ, where θ is such that the input/output combination is part of the production possibility set that is technically feasible given time t technology, we need to find the infimum or greatest lowest bound of θ. The infimum of θ is the biggest real number that is less than or equal to every number in θ. The last part of equation states that finding this infimum is equivalent to finding the reciprocal of . That is we want to find the reciprocal of the supremum of the set of real numbers θ, where this time θ is the set of real numbers such that for a given input vector the input/output combination is part of the production possibility set that is technically feasible given time t technology. The supremum (sup) of θ is the smallest real number that is greater than or equal to every number in θ.

Figure 1 presents two possible production frontiers in the case where there is one output and one input (ie, and are scalars). In panel A the production frontier exhibits constant returns to scale. That is, if we double the level of input we double the level of output. In panel B the production frontier exhibits diminishing returns to scale implying that a doubling in the level of input results in an increase in output that is less than double. The grey shaded area (including the production frontier) on each of the two panels represents the production possibility set, given the production technology shown. It is technically feasible for the economy to be at any point in this set, with the determining factors being the level of input available (ie, how far to the right in Figure 1 the economy is operating at) and how efficiently the level of input is converted into output (ie, given the level of input how close is the level of output to the production frontier) Given the technology shown, it is not possible for the economy to be operating at a point above the production frontier. The point shown in panel A is feasible if the level of input is available for production in the economy. Given that level of input, the economy could produce anywhere along the line .

Figure 1: Production frontiers

The term in equation is the output distance function based on the input and output vectors at time t. The subscript “O” signals that the distance function is an output distance function. The superscript "t" on the D is important as it signals which period’s reference technology (or production possibility frontier) the distance is being measured from.

To calculate , it is necessary to find the largest factor by which all the outputs in the output vector could be increased when making production as technically as efficient as possible, based on the input vector . is then the reciprocal of this value. The closer the economy is to the production frontier the smaller the factor increase will be and consequently the larger the value of . If the economy is operating on the frontier then will take a value of 1. In contrast, when the economy is below the frontier will be less than 1.

In terms of Figure 1 (panel A), if the economy is operating at point then it is producing output quantity "a". In this case, is technically inefficient. If production were on the frontier, then the output quantity could be produced. , ie, is times as large as a. Therefore and . Therefore the distance function is actual output divided by the frontier level of output.

Caves, Christensen and Diewert (1982a and b) define a Malmquist productivity index as:

(14)    

ie, they define their productivity index as the ratio of two output distance functions which both utilise technology at time t as a reference technology. The numerator is the output distance function at time based on period t technology. The denominator is the output distance function at time t based on period t technology.

Instead of using period technology as the reference technology it is possible to construct output distance functions based on period technology and consequently we may construct a Malmquist productivity index as:

(15)     

Fare et al (1994) avoid choosing an arbitrary benchmark technology by specifying their Malmquist productivity change index as the geometric mean of the indexes shown in equations and . That is:

(16)     

Equation can also be written as:

(17)     

Fare et al (1994) give the following interpretation to the two terms on the right hand side of equation (17):

Efficiency change =

Technical change =

Hence the Malmquist productivity index they derive is simply the product of the change in relative efficiency that occurred between periods t and , and the change in technology that occurred between periods t and .

Figure 2: A Malmquist productivity index