# Revisiting Transformation and Directional Technology Distance Functions

## TL;DR

This paper clarifies the theoretical foundations of transformation and directional technology distance functions, proving equivalences and highlighting issues with common quadratic forms, and proposes methods for deriving valid functional forms.

## Contribution

It establishes the equivalence between unsymmetric transformation functions and efficient joint production functions under certain conditions, and corrects misconceptions about the quadratic DTDF form.

## Key findings

- Quadratic DTDF does not satisfy homogeneity of degree -1 in the direction vector.
- Symmetric transformation functions can be used to derive valid DTDFs with desired properties.
- The paper provides a method to obtain functional forms satisfying translation and homogeneity properties.

## Abstract

In the first part of the paper, we prove the equivalence of the unsymmetric transformation function and an efficient joint production function (JPF) under strong monotonicity conditions imposed on input and output correspondences. Monotonicity, continuity, and convexity properties sufficient for a symmetric transformation function to be an efficient JPF are also stated. In the second part, we show that the most frequently used functional form for the directional technology distance function (DTDF), the quadratic, does not satisfy homogeneity of degree $-1$ in the direction vector. This implies that the quadratic function is not the directional technology distance function. We provide derivation of the DTDF from a symmetric transformation function and show how this approach can be used to obtain functional forms that satisfy both translation property and homogeneity of degree $-1$ in the direction vector if the optimal solution of an underlying optimization problem can be expressed in closed form.