Measuring scale efficiency from a parametric hyperbolic distance function

This paper shows how scale efficiency can be measured from an arbitrary parametric hyperbolic distance function with multiple outputs and multiple inputs. It extends the methods introduced by Ray (J Product Anal 11:183-194, 1998), and Balk (J Product Anal 15:159-183, 2001) and Ray (2003) that measure scale efficiency from a single-output multi-input distance function and from a multi-output and multi-input distance function, respectively. The method developed in the present paper is different from Ray's and Balk's in that it allows for simultaneous contraction of inputs and expansion of outputs. Theorems applicable to an arbitrary parametric hyperbolic distance function are introduced first, and then their uses in measuring scale efficiency are illustrated with the translog functional form.