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

T_t = {(x_t,y_t) : x_t can produce y_t} ,                     (1)

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

A2: the set A(x_t) = {(u_t, y_t) ∈ T_t;u_t ≤ x_t} of dominating observations is bounded ∀x_t ∈ R₊^N ,
i.e., infinite outputs are not allowed with a finite input vector;

A3: T_tis closed;

A4: ∀(x_t, y_t) ∈ T_t,(x_t,-y_t) ≤ (u_t,-v_t) ⇒ (u_t,v_t) ∈ T_t , i.e., fewer outputs can always be produced
with more inputs, and inversely (strong disposal of inputs and outputs);

A5: T_t 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 D_t : R^n+p × R^n+p → R ∪ {-∞}∪ {+ ∞}defined by
is called directional distance function in the direction of g = (h, k) .

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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