Activity Analysis (OnFront-Faire and Grosskopf)
1.
The Best Practice Frontier
We first show how to identify benchmark firms or observations and use them to
construct the best practice frontier, or reference technology, production frontier,
or simply technology. Technologies can be constructed for airlines, banks,
government agencies or an decision-making-unit (DMU) that use recourses
(inputs) to produce outcomes (outputs). The frontier is constructed from
observations of inputs and outputs.
Let there be N inputs, each of which is xn , n  1,...N . The vector of inputs is then
x  ( x1 ,..., xN ). For each of the inputs we have data that is nonnegative. These
inputs should be as inclusive as possible and should be measured as accurately as
possible. Activity analysis is especially sensitive to ‘outliers,’ so box plots, etc.
are a useful diagnostic to screen your data.
As for outputs, we can have many types for any DMU. Let there be M outputs,
each of which is ym , m  1,..., M with the corresponding (nonnegative) output
vector y  ( y1 ,..., yM ) . In constructing the best practice frontier and identifying
benchmark firms we have available a number of observations (DMU’s). Let there
be K DMU’s. Then each observation on all inputs and on all outputs is
x k  ( xk1 ,..., xkN ) and y k  ( yk1 ,..., ykM ).
2.
The Input Requirement Set
The input requirement set L ( y ) shows all the combinations of inputs that can be
used to produce the output vector and it fully describes the reference technology
or best practice technology.
Numerical Example- L ( y ) can be constructed by using what is referred to as
Activity Analysis. It can be written in the following way:
L( y )  {( x1 ,..., xN ) :
K
z
k 1
k
ykm  ym , m  1,..., M ,
K
z x
k 1
k
kn
 xn , n  1,..., N ,
zk  0, k  1,..., K }.
That is, an output is no bigger and an input is no smaller than any convex
combination of output and input used by particular DMU’s (or any arbitrary
output and input). Let’s construct the reference technology using real numbers
based on the information below:
Firm(DMU)
1
2
3
Data for a Best Practice Technology
Input 1 ( x1 )
Input 2 ( x2 )
1
2
2
1
2
2
Output ( y )
1
1
1
There are three firms (DMU’s) that use two inputs x1 and x2 to produce one
output ( y ). For example, the amount of input one used by firm two is x21  2 .
The best practice technology can be constructed from the data using a graphical
approach or by the equations. In equation form, with the data above, we get
L( y )  {( x1 , x2 ) :
z11  z21  z31  y, (a)
z11  z2 2  z3 2  x1 , (b)
z1 2  z21  z3 2  x2 , (c)
z1 , z2 , z3  0},
and the diagram is given as
In the figure the three firms are labeled as 1,2,and 3, where, for example,
x1  (1, 2) and x 2  (2,1). Note that they all produce the same amount of output
y=1 (alternatively, since we assuming CRS when could have had outputs vary
from firm to firm but the inputs would have been in terms of input/output ratios).
If the intensity variables take the values z1  1, z2  z3  0, we see from the
equations that we will get back x1  (1, 2) as a feasible member of the input
requirement set L (1) . If we set the intensity variables to z2  1, z1  z2  0, we get
back the observed values of x for firm 2. ‘Hypothetical’ members (including
benchmark members) can be represented by varying the values of the intensity
variables. If z1  z2  1/ 2, z3  0, then we will have a firm that will be halfway
between firm 1 and 2. In fact, all convex combinations of 1,2,and 3 are feasible
(hypothetical) members of the input requirement set. Also, because of the
inequalities in (b) and (c), all points northeast of the line segment between 1 and 2
are also feasible members of the input requirement set. The benchmarks or
frontier of the input requirement set are clearly the points that form the ‘lower’
boundary of the set, that is, they represent real or hypothetical firms that use the
fewest possible inputs to produce the same outputs.
3.
Technical Efficiency
Technical efficiency can be understood best with help of a diagram.
In the figure it is clear that firms 1 and 2 are on the boundary of the input
requirement set and firm 3 is not. We say firms 1 and 2 are technically efficient
and firm 3 is technically inefficient. How inefficient is firm 3? With the
reference technology we can compare firm 3 to a (in this case hypothetical)
benchmark firm that has exactly the same mix of inputs as firm 3. This is the
hypothetical firm at point b. Point b is on the lower boundary of the input
requirement set L (1) and thus is on the best practice frontier.
When we choose point b as the benchmark, we choose to measure inefficiency
proportionally. The mix of inputs is the same at point a and at point b. In
principle then firm 3 should be able to scale down its input use to point b and still
produce the same amount of output. Technical efficiency is then measured as the
greatest proportion that inputs could be reduced and still produce the same output.
Alternatively, it is the ratio of (the size of) minimal feasible input usage to (the
size of) current input usage. In the figure it is ob/oa and is usually referred to as
the Farrell Input-Saving Measure of Technical Efficiency ( Fi ) and is formally
defined as
Fi ( y, x)  min{ :  x  L( y)}
To see how this can be calculated use the data from the table above. First, we will
set up the problem for firm 3, which uses x1  2, x2  2 to produce y  1 . For
firm 3 we then have
Fi (1, 2, 2)  min 
s.t.
z11  z21  z31  1
z11  z2 2  z3 2   2,
z1 2  z21  z3 2   2,
z1 , z2 , z3  0,
which has a solution value of Fi (1, 2, 2)  .75 . This indicates that firm 3’s inputs
could be scaled back by multiplying them by 0.75, which is equivalent to a 25%
reduction. That would give us input usage of x1  x2  1.5 , which is exactly the
amount at point b, the benchmark firm.
Looking at the measure of efficiency for firm’s 1 and 2 we find that
Fi (1,1, 2)  Fi (1, 2,1)  1.0
In general 0  Fi ( y, x)  1, and we refer to a firm k as technically input efficient if
Fi ( y k , x k )  1,
and inefficient if the value is less than one.