The Performance of German Water Utilities: A First Non-Parametric ...

3. Methodology
3.1 DEA approaches
For our efficiency analysis, the method of Data Envelopment Analysis is chosen because of
the following two reasons: first, the absence of panel data restricts the applicability of
stochastic frontier models and second, Data Envelopment Analysis is able to discriminate
best between different outputs like in our case between household and industrial demand,
which are not under the control of the management. DEA puts individual weights on the
outputs of each firm. Hence firms with e.g. low industrial demand are not punished in the
model.
Data Envelopment Analysis uses linear programming methods to obtain measures of
technical efficiency. A piece-wise surface (frontier) over the data, consisting of input and
output variables, for a sample of firms can be constructed. The efficiency of each firm is
measured through calculating the distance between each data point and the point on the
frontier, and lies between 0 and 1. The frontier represents the most efficient firms with
technical efficiency equal to one, the so-called peer firms. Under input orientation, those
firms produce the same output with fewer inputs. The Banker, Charnes and Cooper (BCC)
formulation of Data Envelopment Analysis can be expressed by the following linear
programming problem (see Banker et al., 1984):
min θ ,
s.
t.
λ θ
− qi
+ Qλ
≥0
θxi
− Xλ
≥0
I1'
λ = 1
λ ≥ 0
with θ as a scalar, X as N ∗ I input matrix for N inputs and I firms, Q as M ∗ I output
matrix for M outputs and I1 as a I ∗1
vector of ones. Inputs and outputs for the i-th firm are
represented by the column vectors i x and q i , λ represents a 1 ∗ I vector of constants. Using
the BCC formulation, a convex hull enveloping the data points is constructed. The BCC
8