A Note on Separability of the Profit Function

Rolf Färe,Giannis Karagiannis

doi:10.4236/tel.2014.48089

Abstract

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.

Highlights

Färe and Grosskopf [1] derived conditions on production technology which are required for the profit function to be additively separable into a revenue function component depending only on output prices and a cost function component depending only on input prices. They showed that simultaneous input-and-output translation homotheticity of production technology implies additive separability of the profit function and vice versa, for some input and output direction vectors such as that the inner product of output prices and the output direction vector is equal to the inner product of inputs prices and the input direction vector
In order to prove that (1) implies (2) and vice versa, Färe and Grosskopf [1] had to chose gx and g y such that w′gx = p′g y, i.e., the value of output direction vector is equal to the value of input direction vector, which at a first instance may be seen as a convenient normalization
By substituting them into (3) one can verify that θ = w′gx p′g y, namely that the translation elasticity is equal to the relative value of the input and the output direction vector

Summary

Introduction

Färe and Grosskopf [1] derived conditions on production technology which are required for the profit function to be additively separable into a revenue function component depending only on output prices and a cost function component depending only on input prices. They showed that simultaneous input-and-output translation homotheticity of production technology implies additive separability of the profit function and vice versa, for some input and output direction vectors such as that the inner product of output prices and the output direction vector is equal to the inner product of inputs prices and the input direction vector.

Results

Conclusion