On scale-invariant parametric families of functions

In this paper, we attempt to characterize parametric families of functions such that the statement “a function is an element of the parametric family” is meaningful with respect to a given scale of measurement (a statement is said to be meaningful if its truth or falsity is unchanged when admissible transformations are applied to all of the scales in the statement). A few special cases of the problem are solved for nominal, ordinal, and some quantitative scales. As economic applications, axiomatizations of homothetic production functions and the Cobb–Douglas production function are given.