12.1 The Basic Concepts
The activity of production units such as banks, universities,
governments, administrations, and hospitals may be described and formalized by the
production set:
where
is a vector of inputs and
is a vector of outputs.
This set is usually assumed to be free disposable, i.e. if
for given
all
with
and
belong to
, where the inequalities between vectors are
understood componentwise. When
is one-dimensional,
can
be characterized by a function
called the frontier
function or the production function:
Under free disposability condition the frontier function
is
monotone nondecreasing in
. See Figure 12.1 for an
illustration of the production set and the frontier function in
the case of
. The black curve represents the frontier
function, and the production set is the region below the curve.
Suppose the point
represent the input and output pair of a
production unit. The performance of the unit can be evaluated by
referring to the points
and
on the frontier. One sees that with
less input
one could have produced the same output
(point
).
One also sees that with the input of
one could have produced
. In the
following we describe a systematic way to measure the efficiency of any
production unit compared to the peers of the production set in a
multi-dimensional setup.
The production set
can be described by its sections. The
input (requirement) set
is defined by:
which is the set of all input vectors
that
yield at least the output vector
. See Figure 12.2 for a
graphical illustration for the case of
. The region over the
smooth curve represents
for a given level
. On the other
hand, the output (correspondence) set
is defined by:
the set of all output vectors
that is
obtainable from the input vector
. Figure 12.3
illustrates
for the case of
. The region below the
smooth curve is
for a given input level
.
Figure 12.1:
The production set and the frontier
function,
.
|
Figure 12.2:
Input requirement set,
.
|
Figure 12.3:
Output corresponding set,
.
|
In productivity analysis one is interested in the input and output
isoquants or efficient boundaries, denoted by
and
respectively. They consist of the attainable
boundary in a radial sense:
and
Given a production set
with the scalar output
, the production function
can also be defined for
:
It may be defined via the input set and the output set as well:
For a given input-output point
, its input efficiency
is defined as
The efficient level of input corresponding to the output level
is then given by
 |
(12.1) |
Note that
is the intersection of
and the ray
, see Figure
12.2. Suppose that the point
in Figure 12.2
represent the input used by a production unit. The point
is
its efficient input level and the input efficient score of the
unit is given by
. The output efficiency score
can be defined similarly:
 |
(12.2) |
The efficient level of output corresponding to the input level
is given by
In Figure 12.3, let the point
be the output produced by
a unit. Then the point
is the efficient output level and the
output efficient score of the unit is given by
. Note that,
by definition,
|
|
 |
(12.3) |
|
|
 |
|
Returns to scale is a characteristic of the surface of the
production set. The production set exhibits
constant returns to scale (CRS)
if, for
and
,
; it exhibits non-increasing returns to scale (NIRS) if,
for
and
,
; it
exhibits non-decreasing returns to scale (NDRS) if, for
and
,
. In particular, a convex
production set exhibits non-increasing returns to scale. Note,
however, that the converse is not true.
For more details on the theory and method for productivity
analysis, see Shephard (1970), Färe, Grosskopf, and Lovell (1985),
and Färe, Grosskopf, and Lovell (1994).