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

based on simultaneous proportional modifications of inputs and outputs; it generalizes
Debreu’s and Farrell’s measure and is equally straightforward to interpret.

To estimate the proportional distance function, we use a non-parametric approach (see
Banker and Maindiratta, 1988; Varian, 1984). The technology can be written as:

T = ](x_t,y_t),x_t ≥∑θ_jx^jj,y_t ≤∑θ_jy^jt,∑θ_j = 1,θ_j ≥ 0, j = ι,∙

The linear program that calculates the values of the directional distance function is given by⁷:

D_t(x_t, y_t) = maxδ_t

s.t. x_t -δ_tx_t ≥ ∑θ_j x_t^j ,                           (4)

^j

y_t +δ_ty_t ≤ ∑θ_j y_t^j ,
^j

∑θj =1, j=1KJ.

^j

Suppose that an individual bank is represented by a production vector(x_t, y_t) with
corresponding technologyT_t, and then the production vector is changed to(x_t+1, y_t+1) with
corresponding technology T_t+1 . In order to assign a cardinal measure to the productivity
change we can use the directional distance function in one of two ways; corresponding to
using either the initial technology at t or the final technology at t+1 as reference. In this case,
the Luenberger productivity indicator proposed by Chambers (1996) can be employed to
evaluate productivity change. The productivity indicator is constructed as the arithmetic mean
of the productivity change measured by the technology at T_t+1 and the productivity change
measured by the technology at T_t .

The Luenberger productivity indicator is defined as⁸:

L(zt^, zt+ι ) = 2 [Dt+ι(z_t ; g)^- D+ι (zt+ι ; g)⁺D_t (zt ; g)^- Dt (zt+ɪ ; g^)].       ⁽⁵⁾

⁷ All the computations are programmed in Mathematica language with the mathematica 5.0 software.

⁸ We simplify the notations by posing z_t = (x_t, y_t) .

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