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220
The initial condition is given by
i
y and may be thought of as the stocks
of “capital” initially available. The transversality condition could include
the number of periods, say
1tT+= , the state of the system at
1
,
i
t
Ty
+
, and
the final output vector from the last period,
1
f
t
y
+
. The chosen conditions are
specific to the research problem to be analyzed.
Thus far we have not introduced discounting. However, if outputs are
given in value terms, a discount factor of
,0 1
ττ
δδ
≤
≤ , may be introduced
to account for the lesser of future income compared to present income. For
example,
tf
t
y
δ
are the discounted values of the final output in period t.
Figure 12-5. A dynamic network model
The dynamic network DEA model consists of the interaction of a finite
number of static models. Therefore let us start by studying the sub-
technology
t
P from above. This technology uses inputs
t
x
and intermediate
inputs
1
i
t
y
−
to produce outputs
f
i
tt
yy+ , where
f
t
y is the final output and
i
t
y
is the intermediate output that is used as an input in the next period. Thus
we may write
()( )
{
()( )
}
1
1
11
1
1
,:
1, , ,
,1,,,
,1,,,
0, 1, , ,
t
t
t
tti f i
ttt
K
fi tf i
tm tm k tkm tkm
k
K
ti i
kt km t km
k
K
tt
kkm kn
k
tt
k
Px y y y
yy zy y m M
zy y m M
zx x n N
zkK
−
=
′
−−
=
′
=
=+
+≤ + =
∑
≤=
∑
≤=
∑
≥=
K
K
K
K
(12.9)
where
f
tkm
y ,
i
tkm
y ,
1
i
tkm
y
−
, and
kn
x
are observed inputs and outputs. We allow
the number of observations to differ between periods, hence the notation
t
K .
Chapter 12
221
The output vector
(
)
fi
tt
yy+ is specified so that one can include
t
M
for all
t, i.e.,
1tt
M
MM
−
=++K by including appropriate zeroes.
Recall from the static model, that if the sub-technologies have the
properties A.1-A.6, then the whole network model also has those properties.
This observation also applies to the dynamic model below:
()( )
{
()( )
1
1
1
11
111
1
11 1 1
1
1
1
11 1
1
11
,, , , ,( ):
1, , ,
,1,,,
,1,,,
0, 1, , ,
t
t
t
tt tti f f f i
ttt t
K
fi tf i
tmtm ktkmtkm
k
K
ti i
kkm km
k
K
tt t
kkn n
k
tt
k
Px xx y y y y y
yy zy y m M
zy y m M
zx x n N
zkK
−
−
−
−+
−++
−
−− − −
=
−
=
−− −
=
−−
=+
+≤ + =
∑
≤=
∑
≤=
∑
≥=
K
K
K
K
(12.10)
()( )
()( )
1
1
1
11
1
1
1
11 1 1
1
1
1
1
1, , ,
,1,,,
,1,,,
0, 1, , ,
1, , ,
,1,,,
t
t
t
t
t
K
fi tf i
tm tm k tkm tkm
k
K
ti i
kt km t m
k
K
tt t
kkn n
k
tt
k
K
fi tf i
tmtm ktkmtkm
k
K
ti i
ktkm tkm
k
t
k
k
yy zy y m M
zy y m M
zx x n N
zkK
yy zy y m M
zy y m M
z
+
+
=
−−
=
=
+
++ + +
=
+
=
+
+≤ + =
∑
≤=
∑
≤=
∑
≥=
+≤ + =
∑
≤=
∑
K
K
K
K
K
K
}
1
11
1
11
,1,,,
0, 1, , ,
t
K
tt
kn n
tt
k
xx n N
zkK
+
++
=
++
≤=
∑
≥=
K
K
Note that each sub-technology has its own intensity vector
,1,,1zttt
τ
τ
=− +, and that the interaction between time periods comes
through the intermediate outputs.
Färe and Grosskopf (1997) use this model to study the inefficiency of
APEC countries due to dynamic misallocation of resources. They used the
sum of Shephard (1970) sub-technology distance functions as their
optimization criterion. Nemota and Gota (2003) applied the dynamic
network model to study Japanese electricity production over time. They
used cost minimization for the optimization criterion. Jaenicke (2000)
applied the dynamic model in the analysis of the yield effects of crop
rotation. Nemota and Gota (1999) applied the dual linear programming
Färe, Grosskopf & Whittaker, Network DEA
222
problem formulation to the cost minimization and derived the fundamental
equation (Hamilton-Jacobi-Bellman) of dynamic programming.
4. TECHNOLOGY ADOPTION
We now turn to our simplest, yet quite powerful network model, the
technology adoption model. Here we give two applications, one to
“embodied technical change” and the other to permit allocation. Although
the applications appear different, the basic approach is the same; allocation
of one or more resources among two or more technologies. Figure 12-6
illustrates a simple version of the problem.
xx
12
+≤
y
1
y
2
P
1
P
2
x
Figure 12-6. Distribution of resources among technologies
The two technologies
1
P
and
2
P
use inputs
1
x
and
2
x
respectively to
produce outputs
1
y
and
2
y
. The total amount of inputs that can be allocated
among the technologies is x. In this illustration, to analyze permit allocation,
for example, we would define one of the components of x as a permit for
pollution emission.
Chapter 12
223
Let us start with the case of technology adoption, where one of the
technologies is vintage. The original vintage models due to Johansen (1959)
and Solow (1960) were based on continuous time. Here we follow Färe and
Grosskopf (1996) and discuss the discrete time version. We distinguish
between two forms of inputs, durable and instantaneous. The durable inputs
are vintage specific, and they form sub-technologies,
1
P and
2
P above. The
instantaneous inputs are allocated between the vintage sub-technologies,
1
x
and
2
x
above.
The adoption of technology is generated by the reallocation of
instantaneous inputs away from “old” vintages to new ones. Hence making
the old obsolete and replaced by the newer vintages.
Assume there are
1, ,vV= K vintages and
v
K observations of outputs
,kv M
y
+
∈ℜ , durable inputs
,kv V
X
+
∈ℜ and instantaneous inputs
,kv N
x
+
∈
ℜ
7
.
The vintage technology v at time
,tt v≥ has the following DEA
formulation:
() ()
()
()
{
()
()
()
}
1
1
1
,:
,1,,,
,1,,,
,1,,,
0, 1, , .
v
v
v
vv v v M
K
vvv
mkkm
k
K
vvv
lkkl
k
K
vlv
nkkn
k
vV
k
PXtxt yt
yt zy m M
Xt zX l L
xt zx n N
zkK
+
=
=
=
=∈ℜ
≤=
∑
≥=
∑
≥=
∑
≥=
K
K
K
K
(12.11)
This vintage model has outputs, inputs both durable and instantaneous,
strong disposability, and satisfies constant returns to scale (CRS). Other
conditions may be substituted in the usual manner.
The technology is unchanged over time, ,
vv
km kl
yXand
v
kn
x
are independent
of time. However, the durable inputs age or degrade over time and may lose
their capabilities. We model this observation with an aging
function
()
l
At v− so that
() ( )
,1,,.
VV
ll l
X
tAtvX l L=− =K
Thus, as t increases, the effectiveness of
()
V
l
X
t may decline as
(
)
l
At v
−
decreases. Characteristics of aging are specific to each application, and we
leave the properties of
()
l
At v− to the researcher to address.
If there are V vintages, the multi-vintage model at time t is:
7
This part is based on Färe and Grosskopf (1996).
Färe, Grosskopf & Whittaker, Network DEA
224
() () ()
()
()
{
()
()
()
() () }
1
1
1
1
1
1
,, , :
,1,,,
,1,,,
,1,,,
0, 1, , , 1, ,
,1,,.
v
v
v
V
VV
v
K
vvv
mkkm
k
K
vvv
lkkl
k
K
vvv
lkkn
k
vV
k
V
v
nn
v
PX t X t xt y t
yt zy m M
Xt zX l L
xt zx n N
zkKvV
xt xt n N
=
=
=
=
=
=
∑
≤=
∑
≥=
∑
≥=
∑
≥==
≤=
∑
K
K
K
K
KK
K
(12.12)
Equation (12.12) may also be written as
() () ()
()
() ()
()
{
() ()
}
1
1
1
,, , , :
.
V
Vvvv
v
V
v
v
PX t X t xt P X t x t
xt xt
=
=
=
∑
≤
∑
K
(12.13)
Equation (12.13) states that the multi-vintage model is the sum of the
single-vintage models, with the instantaneous input allocated over all
vintages. This model can be used in combination with any of the objective
functions discussed in section 2.
Some prior information is needed to specify the permit allocation model.
The technologies in question are assumed to produce desirable and
undesirable outputs, as in our network model with externalities above.
Therefore we assume that the output vector of “goods”,
M
y
+
∈
ℜ , is nulljoint
with the “bads” output vector u
ℑ
+
∈ℜ , i.e.
()()
,0,0,if y u P x and u then y∈==
where
()
Pxis the output set and
N
x
+
∈ℜ is an input vector. In addition we
assume that y and u are weakly disposable, i.e.
()()
(
)
(
)
,01,,.if y u P x and then y u P x
θθθ
∈≤≤ ∈
A typical output set of a polluting technology may then be illustrated for
one good and one bad as in Figure 12-7.
Chapter 12
225
a
y
u
0
.
Px()
Figure 12-7. Polluting technology
The output set defined above is such that if 0u
=
, then for
(
)
,yu to
belong to it, y must also equal zero. That is y is nulljoint with u. In addition,
for any element, say
()
aPx∈ , its proportional contraction
()
,0 1a
θθ
≤≤also belongs to
()
Px. Thus the technology has weakly
disposable outputs
(
)
,yu .
To regulate the bad outputs u, we introduce a restriction on the bads,
namely,
(
)
(
)
{
}
,: 0 .
B
uyuyanduu=≥≤
The addition of these regulatory constraints to the polluting technology in
Figure 12-7 is illustrated in Figure 12-8.
y
u
0
u
Px()
Bu()
Figure 12-8. Regulated polluting technology
This diagram shows that the regulated technology is constrained by both
the technology
()
Px and the regulation
(
)
B
u . Thus, for feasibility
()() ()
,yu P x B u∈∩ .
To study permit allocation, we specify a number of firms 1, ,
f
F
=
K ,
and a total amount of pollutant that society will allow,
*
u . Then our model
Färe, Grosskopf & Whittaker, Network DEA
226
should allocate
*
u optimally among the F firms so that the sum does not
exceed
*
u ,
1
,, .
F
f
fJ
f
uu uu
+
=
≤∈ℜ
∑
Here we follow Brännlund, Färe and Grosskopf (Brännlund et al., 1995)
and assume that each firm is a profit maximizer. The first question we
address is the calculation of the optimal allocation of
*
u under profit
maximization,
()
*1 *
,,
F
uuK . To find this allocation we solve
(
)
(,,)
11 1
1
1
1
1
1
max
.. , 1, , ,
,1,,,
,1,,,
1, 0, 1, ,
,1,,,
1, , .
f
f
f
f
FM N
ff
mm nn
yux
fm n
K
fff
mkkm
k
K
fff
jkkj
k
K
fff
nkkn
k
K
f
ff
kk
k
F
f
jj
f
py wx
s
tyzy m M
uzu j J
x
zx n N
zz kK
uu j J
fF
== =
=
=
=
=
=
Π= −
∑∑ ∑
≤=
∑
==
∑
==
∑
≤≥=
∑
≤=
∑
=
K
K
K
K
K
K
(12.14)
Equation (12.14) consists of F sub-problems, one for each firm. These
problems are connected by the pollution allocation expressions
1
,1,,.
f
K
f
jj
k
uu j J
=
==
∑
K
Each firm is here assumed to satisfy non-increasing returns to scale
1
1.
f
K
f
n
k
z
=
≤
∑
Thus, we have imposed the condition of no negative profits. This condition
can be relaxed by imposing variable returns to scale, if required by the
application.
The solution to this problem yields the total profit
*
Π
, the firm’s profit
*
f
Π and the optimal allocation of undesirables
*
f
j
u ,
1, , and 1, , .
f
Fj J==KK Thus we may compare
*
f
j
u
to the permit
Chapter 12
227
amount allocated to each firm, denoted by
f
j
u . This comparison measures if
there is an opportunity for permit trading. We compute
f
j
u as
()
*
11
1
1
1
1
max
.. , 1, , ,
,1,,,
,1,,,
1,
0, 1, , .
f
f
f
MN
ff f f
mm nn
mn
K
fff
mkkm
k
K
fff
jkkj
k
K
fff
nkkn
k
F
f
k
f
f
f
k
upywx
s
tyzy m M
uzu j J
x
zx n N
z
zkK
==
=
=
=
=
Π= −
∑∑
≤=
∑
==
∑
==
∑
≤
∑
≥=
K
K
K
K
(12.15)
If
()
***
11
FF
ff f
ff
u
==
Π<Π=Π
∑∑
then there is a possibility for permit trading, although the trading cost must
also considered. If firm f has
()
**
f
ff
uΠ<Π
then there is a possibility of adjustment from
f
u to
*
f
u .
5. EPILOG
We end this chapter by referring the reader to some other relevant literature.
The classic book, Linear Programming & Economic Analysis, Dorfman,
Samuelson and Solow (1958) (often referred to as “DOSSO”) provides a
basis for further study for someone new to this area of study. The book
covers both static and dynamic linear models, including the Leontief system.
For the German speaker, May’s book, Dynamische Produktionstheorie auf
Basis der Aktivitätsanalys (1992) , connects the models discussed here to
production models developed in Europe.
We also refer the reader to the work of J.K. Sengupta. A search for “J.K.
Sengupta dynamic models” using the Google search engine shows examples
of the large number of dynamic non-parametric models he has analyzed. See
also the paper “Nonparametric approaches to dynamic efficiency” (Sengupta,
Färe, Grosskopf & Whittaker, Network DEA
228
1992). Last, we would like to direct attention to an interesting paper by
Silva and Stefanou, “Dynamic Efficiency Measurement: Theory and
Application” (in press).
REFERENCES
1. Brännlund, R., Färe, R., and Grosskopf, S. (1995). “Environmental
Regulation and Profitability: An Application to Swedish Pulp and Paper
Mills.” Environmental and Resource Economics, 6(1), 23-36.
2.
Dorfman, R., Samuelson, P., and Solow, R. (1958). Linear
Programming & Economic Analysis, McGraw-Hill, New York.
3.
Färe, R., and Grosskopf, S. (1996). Intertemporal production frontiers: With
dynamic DEA, Kluwer Academic, In collaboration with R. Brännlund et al.
Boston; London and Dordrecht.
4.
Färe, R., and Grosskopf, S. (1997). “Efficiency and Productivity in Rich
and Poor Countries.” Dynamics, economic growth, and international
trade, B. S. Jensen and K. Wong, eds., University of Michigan Press,
Studies in International Economics. Ann Arbor, 243-63.
5.
Färe, R., Grosskopf, S., and Lee, W. F. (2004). “Property rights and
profitability.” New Directions: Efficiency and Productivity, R. Färe and
S. Grosskopf, eds., Kluwer Academic Publishers, Boston, 65-77.
6.
Färe, R., and Whittaker, G. (1996). “Dynamic measurement of efficiency:
an application to western public grazing.” Intertemporal production
frontiers: With dynamic DEA, R. Fare and S. Grosskopf, eds., Kluwer
Academic, Boston; London and Dordrecht, 168-186.
7.
Jaenicke, E. C. (2000). “Testing for Intermediate Outputs in Dynamic
DEA Models: Accounting for Soil Capital in Rotational Crop Production
and Productivity Measures.” Journal of Productivity Analysis, 14(3),
247-266.
8.
Johansen, L. (1959). “Substitution versus fixed production coefficient
in the theory of economic growth: a synthesis.” Econometrica, 27(2),
157-176.
9.
Kemeny, J. G., Morgenstern, O., and Thompson, G. L. (1956). “A
generalization of the von Neumann model of an expanding economy.”
Econometrica, 24, 115-135.
10.
Lewis, H. F., and Sexton, T. R. (2004). “Network DEA: Efficiency
analysis of organizations with complex internal structure.” Computers
and Operations Research, 31(9), 1365.
11.
May, E. (1992). Dynamische Produktionstheorie auf Basis der
Aktivitätsanalyse, Physica-Verlag, Heidelberg.
12.
Nemota, J., and Gota, M. (1999). “Dynamic data envelopment analysis:
modeling intertemporal behavior of a firm in the presence of productive
inefficiencies.” Economic Letters, 64, 51-56.
Chapter 12
229
13.
Nemota, J., and Gota, M. (2003). “Measuring dynamic efficiency in
production: An application of data envelopment analysis to Japanese
electric utilities.” Journal of Productivity Analysis, 19, 191-210.
14.
Ramsey, R. (1928). “A Mathematical Theory of Saving.” Economic
Journal, 38, 543-559.
15.
Sengupta, J. K. (1992). “Non-parametric approaches to dynamic
efficiency: a non-parametric application of cointegration fo production
frontiers.” Applied Economics, 24, 153-159.
16.
Shephard, R. W. (1970). Theory of Cost and Production Functions,
Princeton University Press, Princeton.
17.
Shephard, R. W. and Färe, R (1975). “A dynamic theory of production
correspondences,” ORC 75-13, OR Center, U.C. Berkeley.
18.
Silva, E., and Stefanou, S. E. (in press). “Dynamic efficiency
measurement: theory and applications.” American Journal of
Agricultural Economics.
19.
Solow, R. M. (1960). “Investment and technical progress.” Mathematical
methods in the social sciences, K. J. Arrow, S. Karlin, and P. Suppes,
eds., Stanford University Press, Stanford, CA.
Appendix I
This appendix contains a program listing and output using the General
Algebraic Modeling System (GAMS) for the network shown in Figure 12-2.
The example data are called from separate file, which is listed first. The
variables are named to follow the nomenclature in Figure 12-2, e.g.,
1
0
x
is
X10 in the program. The page numbers and equation names refer to Färe
and Grosskopf (1996).
Listing of fig2.dat (example data):
Listing of GAMS program:
* NETWORK MODEL FOR FIGURE 12-2
*
*
$TITLE NETWORK FIGURE 12-2
$OFFSYMXREF
$OFFSYMLIST
x0 x10 x20 x30 y31 y32 y43 y41 y42 y0
FM1 10352423249
FM2 7213112114
FM3 11424222226
TABLE D(FM,R)
Färe, Grosskopf & Whittaker, Network DEA