CHAPTER 5 ANALYTICAL FOUNDATIONS AND MEASUREMENT OF FRONTIER
5.2 Conceptual framework of frontier efficiency
5.2.3 Distance functions and measures of technical efficiency
Efficiency can be measured as the distance from the efficient frontier by means of a distance function. Both input distance function (from input minimisation perspective) and output distance function (from output maximisation perspectives are discussed in detail below.
(a) Input distance function and measures of input-oriented technical efficiency
Formally, given the input set W(X) defined in Section 5.2, we define an input distance function as the function:
}~(X, ') maxd8e•‚ ∈ W(X)}where 1. (5.7) An input distance function (uses an input-conserving approach) examines the production technology by looking at minimal proportional contraction of the input vector, given an output vector. It therefore measures the maximum amount by which input usage can be radially reduced but remains feasible to produce a given vector of outputs (Kumbhakar and Lovell, 2003).
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Distance functions were introduced by Shephard (1953), who used it as a tool to establish duality between production technology and cost functions. In terms of empirical estimation of efficiency, distance functions are less frequently used. Similar to production frontiers, distant functions do not require price data and imposition of economic behavioural objectives, and thus are used to measure only technical efficiency.
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We illustrate the input distance function in the 2-input space ' and ' – used in producing output vectorX in Figure 5.4a, using the input set, W(X) and input isoquantƒsl* W(X). We observe that the input vector A is feasible for output vectorX, but X can be produced with the radially contracted input vector •
‚D
.
The distance function for the production point A where the firm uses ' „ of input ' and ' „ of input' to produce the vector outputX is equal to the ratio6{;6 .Figure 5.4a Input distance function
' ' „ A • ‚D W(X) B ƒsl* W(X) 0 ' „ '
Since distance functions are defined in terms of input sets, which have to satisfy certain properties, discussed in the context of the economic feasible region of the production frontier in Section 5.3, the input distance function (DI) must satisfy the following properties summarised below (Kumbhakar et al., 2015):
1. DI (Y, X) is decreasing in each output level; 2. DI (Y, X) is increasing in each input level;
3. DI (Y, X) is homogenous of degree 1 in the feasible input vector X; 4. DI (Y, X) is concave in X.
It is clear from the above and earlier discussions under Section 5.2 thatW(X) d'e …~(† ‡ y 7a and that b\ c W X d'e …~ † ‡ 7ax Thus, the isoquant which serves as a standard against which to measure technical efficiency corresponds to the set of input vectors having an input distance function value of unity.
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Input distance functions therefore also serve as a standard against which we measure input- oriented measure of technical efficiency, and it is given as the reciprocal of the input distance function, that is,
TE = 1/DI(Y, X) (5.8)
A firm is technically efficient if it is on the frontier, in which case TE=1 and DI(Y, X) is also equal to 1. From Figure 5.4a, we define the input-oriented technical efficiency by the function
TEI (Y, X) = min { : DI(Y, X), 1}-1 (5.9)
(b) Output distance function and measures of output -oriented technical efficiency
Similarly, an output distance function adopts an output-expanding approach and considers a maximal proportional expansion of the output vector, given an input vector. It gives the minimum amount by which an output vector can be deflated and still remain producible with a given input vector.
Based on the output sets ofY R , we define an output distance function as:
}ˆ ', X) mind Š: Œ‹∈ Y(R)}, (5.10)
where, as defined earlier, Y(R) is the set of output vectors that are feasible for each input vector X.
We illustrate the output distance function in a 2-commodity output case, X and X produced by a vector of inputs', in figure 5.4b. The production possibility set, P(X), is the area bounded by the production possibility frontier, PPC – P(X), and X and X axes.
We note that output X„ can be produced with input ' but so can a large output ‹
Œ∗so
that}ˆ(', X) < 1. The value of the distance function for the firm using input level X to produce the outputs, defined by the point A, is equal to the ratio 0A/0B.
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Figure 5.4b Output distance function Y2
B
‹ ŒDY2A
A
PPC–P(x)
0
Y1 Y1ASimilar to the properties of the input distance function, we summarise the properties of the output distance function (DO) below (Kumbhakar et al., 2015):
1. DO (X, Y) is decreasing in each input level; 2. DO (X, Y) is increasing in each output level;
3. DO (X, Y) is homogenous of degree 1 in the feasible input vector Y; 4. DO (X, Y) is concave in Y.
The output-oriented measure of technical efficiency coincides with the output distance function, that is,
TEO = DO(x, y) (5.11)
In other words, the output-oriented measure of technical efficiency is given by the function
TEO (x, y) = [max { : D0 (x, y) 1}]-1 (5.12)
The measure of technical inefficiency using production frontiers or distance functions is however limited as it does not address whether the observed combination of inputs are allocatively optimal. Allocative efficiency relates to the combination of inputs and outputs that meets some behavioural objective such as cost minimization, revenue or profit maximization and therefore takes into account input and/or output prices. Thus a firm may achieve technical efficiency but not allocative efficiency. Where a behavioural criterion is assumed, efficiency measures can be altered to accommodate allocative efficiency. Allocative
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and technical efficiencies together provide a measure of economic efficiency, which we discuss in the next section.