2.1.4.1 Input-orientated
measure of technical efficiency
The input-orientated measure of technical efficiency seeks to
answer the question: «By how much can input quantities be proportionally
reduced without changing the output quantities produced?» Farrell (1957)
illustrated the definition of the input-orientated measure of technical
efficiency using a simple example involving firms, which use 2 inputs (x1 and
x2) to produce a single output (y), under the assumption of constant returns to
scale (to enable the representation of the production technology on a single
isoquant). Knowledge of the unit isoquant (represented by the line SS' in
figure 2.1) of the fully efficient firm permits the measurement of technical
efficiency.
If a given firm uses quantities of inputs defined by point P
to produce a unit of output, the technical inefficiency of that firm could be
represented by the distance QP which is the amount by which all inputs could be
proportionally reduced without a reduction in output. This is usually expressed
in percentage terms by the ratio which represents the percentage by which all
inputs could be reduced, keeping output constant. The technical efficiency of a
firm is most commonly measured by the ratio which is equal to . Technical
efficiency takes on either 1 or 0 or values between 1 and 0 and hence provides
an indicator of the degree of the technical inefficiency of a firm. A value of
1 indicates that the firm is fully technically efficient while the value of 0
indicates that the firm is fully technically inefficient. For example the point
Q is technically efficient because it lies on the efficient isoquant.
A S P
Q
R
Q'
S'
0 A'
x1/y
Figure .1:
Technical efficiency and allocative efficiency under input-orientated
measure
If the input price ratio represented by the line AA' is also
known, allocative efficiency of the firm can also be calculated. Allocative
efficiency of the firm operating at point P is defined as
. The distance RQ represents the reduction in production costs
that would occur if production were to occur at the allocatively (and
technically) efficient point Q' instead of the technically efficient but
allocatively inefficient point, Q. The total economic efficiency is defined to
be the ratio where the distance RP can also be interpreted in terms of a cost
reduction. Note that the product of technical efficiency (TE) and allocative
efficiency (AE) provides the overall economic efficiency (EE). That is EEI= TEI
OR/OQ = OR/OP. Note also that TE, AE and EE are bounded by 0 and 1.
The efficiency measures explained above assume that the
production function of a fully efficient firm is known. In practice this is not
the case, and the efficient isoquant must be estimated from the sample data.
Farrell (1957) suggested the use of (a) a non-parametric piecewise-linear
convex isoquant constructed such that no observed point lies to the left or
below it as shown in figure 2.2, and (b) a parametric function such as the
Cobb-Douglas production function, fitted to the data, again such that no
observed point should lie to the left or below it.
x2/y S
.
.
.
S'
O
x1/y
Figure 2.2: Piece wise linear convex
isoquant
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