Econometric approaches to cost efficiency estimate a parametric cost
function and then identify cost efficiency via differences between observed
cost and the minimum predicted cost. Consider carrier k in year t. It
has a V-vector zkt=(zkt1,... ,zktV) of system
characteristics and produces the M-vector of outputs ykt=(ykt1,... ,yktM) from the N-vector of inputs xkt=(xkt1,... ,xktN) according to the transformation function F(ykt,xkt;zkt)=0. It is assumed to face
the N factor prices wkt=(wkt1,... ,wktN). Its
(minimum) cost function C* (ykt,wkt;zkt) is the indirect objective function of the problem: choose inputs xkt to
minimize
å
n
wktnxktn
subject to:
F(ykt,xkt;zkt)=0.
If Ckt is the observed cost of carrier k in year t, we allow
for a failure to attain the minimum cost by writing
Ckt=C
*
(ykt,wkt;z
kt)e
e kt
(1)
where e kt is a random variable, so that
ln Ckt=ln
C
*
(ykt,wkt;zkt)+e kt
(2)
Because the observed cost Ckt cannot exceed the theoretical minimum
cost Ckt* , e kt is necessarily non-negative. We
obtain a parametric cost function by specifying a functional form for C* (ykt,wkt;zkt). For the
remainder of this section, we shall assume that the cost function is
translog in inputs and outputs; but that system characteristics enter
linearly.1
Then we may write:
ln
C
*
(ykt,xkt;zkt)
=
a
0+
V
å
i=1
a izkti+
N
å
n=1
b nln
yktn+
M
å
m=1
g mln wktm
(3)
+
1
2
N
å
n=1
N
å
p=1
d npln yktnln yktp+
1
2
M
å
m=1
M
å
j=1
q mjln wktmln wktj
+
1
2
N
å
n=1
M
å
m=1
x nmln wktnln wktm
The theory of cost minimization implies that C* (ykt,wkt;zkt) is homogeneous of degree zero in input
prices, which places restrictions on the parameters of the cost function
(see, eg, [spady-friedlaender:78]); in addition the parameters d
,q and x are symmetric, so that, for example d
np=d pn.
2 The DEA Approach
Data Envelopment Analysis, or DEA, see, eg [bcc:84], or [fgl:94],
begins by constructing the set of feasible input-output combinations. It
then finds, with reference to that set, the input bundle that minimizes the
cost of producing the observed output. DEA is non-parametric in that it
makes no assumption about the structure of the cost function ---
indeed it does not explicitly obtain the cost function at all ---
and, being non-statistical, it has no need for parametric distributional
assumptions.2 To construct the feasible set it begins by
assuming that the observed input-output combinations (the data) are feasible
and then adds unobserved combinations by considerations of additivity
(scale) and disposal.3
The upshot is that cost-minimizing carrier selects an input bundle x~kt to minimize total cost, ånx~ktnwktn=x~kt' wkt subject to feasibility;
and this is the solution to the non-linear program
minimize
å
n
~
x
ktnwktn
subject to
:
yktm
=
µ
å
l,s
rlsylsm " m=1,2,... ,M
l
~
x
ktn
=
å
l,st
rlsxlsn " n=1,2,... ,N
å
l,s
rls
=
1
0
<
l £ 1
0
<
µ £ 1
and we define the cost-efficiency of carrier k at period t by
CEkt=
w
'
kt
~
x
kt
w
'
kt
xkt
the ratio of observed to minimal cost.
References
[bcc:84] Banker, R., A. Charnes, and W. W. Cooper (1984), ``Some models for estimating
technical and scale inefficiencies in data envelopment analysis,'' Mangement Science, 1984, 30 (9), 1078--1092.
[banker:93] Banker, Rajiv D. (1993), ``Maximum Likelihood, Consistency and Data Envelopment
Analysis: A Statistical Foundation,'' Management Science, 1993, 39 (10), 1265--1273.
[fgl:94] Färe, R., S. Grosskopf, and C. A. K. Lovell (1994), Production Frontiers,
Cambridge: Cambridge University Press, 1994.
[spady-friedlaender:78] Spady, R. H. and A. F. Friedlaender (1978), ``Hedonic cost functions for the
regulated trucking industry,'' Bell Journal of Economics, 1978, 9
(1), 159--179.
This is not to say that DEA cannot be given a statistical foundation: for
example, [banker:93] shows that the DEA solution can be considered the
maximum-likelihood estimator of a non-parametric monotone increasing and
concave production frontier.
Feasibility of the observed data is not a completely innocuous assumption.
If, for example, there are measurement errors, then the observed data may
not in fact be feasible.