Pexider’s equation and its extension. Quasi-extension of Cauchy’s equation. Determination of all generalized Hicks-neutral production functions

Abstract

We will deal first with the Pexider equation (an equation with three unknown functions) $$ h(xy) = f(x)g(y)\quad (x,y \in N) . $$ (1) (of which (1.5), (4.7) and others are special cases) for f,g,h:N→G in similar generality as with (1.19) in Proposition 1.3: We will suppose that (N,•) is a groupoid with a neutral element and (G,•) a group (so, unlike the situation for (1.19), G is supposed to have neutral and inverse elements but it will not be supposed in this general set-up that G is commutative). As in (l) we will continue to omit the •.’ sign; we will also denote, as usual, the inverse of c ∈ G by c-1.