Based on the concept of translation elasticity we restate in this note the Fare and Grosskopf’s [1] conditions for additive separability of the profit function. We show that for the profit function to be additively separable, the technology must satisfy both simultaneous input-and-output translation homotheticity and graph translation homotheticity.
Färe and Grosskopf [
Let
their corresponding price vectors. The technology is defined in terms of
closed, allows for free disposability of inputs and outputs, and it contains
nology distance function, which is the negative of the shortage function introduced by Luenberger [
and has the following properties: first,
freely disposable; second, it is non-decreasing in
gree −1 in the direction vector
and it contains all other forms of directional functions as special cases. In particular,
Following Färe and Grosskopf [
where
On the other hand, additive separability of the profit function implies that [
In order to prove that (1) implies (2) and vice versa, Färe and Grosskopf [
that
at a first instance may be seen as a convenient normalization. Nevertheless, based on recent work by Balk, Färe and Karagiannis [
which gives the maximal number of times the output direction vector
with the corresponding first-order conditions being
translation elasticity is equal to the relative value of the input and the output direction vector. Then, constant re-
turns to translation in the direction of
that
[
Combining (1) and (4) results in the following form of the directional technology distance function:
We can thus replace the requirement of
lity of the profit function with that of the last two equalities in (5).
In this note we have restated the directional distance function characterization of the technology required for additive separability of the profit function based on the concept of translation elasticity. We have shown in particular that for the profit function to be additively separable, the technology must satisfy both simultaneous input-and-output translation homotheticity and graph translation homotheticity.