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.



More intriguing information

1. Fiscal Reform and Monetary Union in West Africa
2. Financial Development and Sectoral Output Growth in 19th Century Germany
3. Second Order Filter Distribution Approximations for Financial Time Series with Extreme Outlier
4. Weak and strong sustainability indicators, and regional environmental resources
5. The name is absent
6. The Formation of Wenzhou Footwear Clusters: How Were the Entry Barriers Overcome?
7. Großhandel: Steigende Umsätze und schwungvolle Investitionsdynamik
8. The name is absent
9. Testing the Information Matrix Equality with Robust Estimators
10. Multifunctionality of Agriculture: An Inquiry Into the Complementarity Between Landscape Preservation and Food Security