A Note on Productivity Change in European Co-operative Banks: The Luenberger Indicator Approach



measures the smallest changes in inputs and outputs in a given direction which are necessary
for a firm to reach the production frontier, rendering it an indicator of firm performance.

Let the technology be described by a set, T R+N × R+M , defined by

Tt = {(xt,yt) : xt can produce yt} ,                     (1)

where xt R+N is a vector of inputs and yt R+M is a vector of outputs at the time period t.
Throughout this paper, technology satisfies the following conventional assumptions6:
A1: (0,0)
Tt,(0,yt) Tt yt = 0 i.e., no fixed costs and no free lunch;

A2: the set A(xt) = {(ut, yt) Tt;utxt} of dominating observations is bounded xt R+N ,
i.e., infinite outputs are not allowed with a finite input vector;

A3: Ttis closed;

A4: (xt, yt) Tt,(xt,-yt) (ut,-vt) (ut,vt) Tt , i.e., fewer outputs can always be produced
with more inputs, and inversely (strong disposal of inputs and outputs);

A5: Tt is convex.

The directional distance function generalizes the traditional Shephard distance
function (1970). Directional distance functions project input and/or output vector from itself
to the technology frontier in a preassigned direction. In the case of a radial direction out of the
origin, we retrieve the classical Shephard distance function. The directional distance function
is defined as follows.

The function Dt : Rn+p × Rn+pR {-∞} {+ ∞}defined by
is called directional distance function in the direction of
g = (h, k) .

Dt(xt,yt;g)


sup p : (xt

-∞


-δh; yt +δk) Tt } if (xt -δh; yt +δk) Tt,δ R
otherwise


(2)


To operate the approach, it is necessary to take an appropriate direction. We do this by
considering the direction
g = (x, y) . Then, the directional distance function is similar to the
proportional distance function introduced by Briec (1995, 1997). This distance function is

6 See Shephard (1970) and Fare et al. (1985) for thorough analysis of their implications on technology.



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