I am working on some applications of measures of inequality (economic inequality). Upon reading a paper by Kolm (Kolm, S. C., Unequal Inequalities. I, Journal of Economic Theory, 12 pp 416-442, 1976) I found a result that I found intriguing and for which I would like to know if any one has a reference or a proof of.
The author introduces some axiomatization of measures of economic inequality, say $I:U\subset\mathbb{R}^n\rightarrow[0,\infty)$, for example "Impartiality" ($I$ is invariant under permutations); transfers principle ($\partial_iI-(x_i-x_j)\partial_jI\geq0$), among others.
The following properties are the ones that are intriguing to me: Let $\mathbf{x}=[x_1,\ldots,x_n]^\intercal\in U$ and denote $\bar{\mathbf{x}}=\frac1n\sum^n_{j=1}x_j$. The author defines welfare indices \begin{align} \hat{x}(\mathbf{x})&=\bar{\mathbf{x}}-I(\mathbf{x})\\ \tilde{x}(\mathbf{x})&=(1-I(\mathbf{x}))\bar{\mathbf{x}} \end{align} and introduces the following properties in his axiomatization:
(1) (absolute) For all $1\leq i,j\leq n$, the ration $\frac{\partial_i\hat{x}}{\partial_j\hat{x}}$ depends only on $x_i$ and $x_j$, that is $$\begin{align}\frac{\partial_i\hat{x}}{\partial_j\hat{x}}=\psi_{ij}(x_i,x_j)\tag{1}\label{one}\end{align}$$ for some function $\psi_{ij}:\mathbb{R}^2\rightarrow\mathbb{R}$.
(1') (relative) For all $1\leq i,j\leq n$, the ration $\frac{\partial_i\tilde{x}}{\partial_j\tilde{x}}$ depends only on $x_i$ and $x_j$, that is $$\begin{align}\frac{\partial_i\tilde{x}}{\partial_j\tilde{x}}=\phi_{ij}(x_i,x_j)\tag{1'}\label{onep}\end{align}$$ for some function $\phi_{ij}:\mathbb{R}^2\rightarrow\mathbb{R}$.
(1) and (1'), the author argues, may be called (or labeled) *welfare independence. There are no additional properties on $I$ except that presumably is smooth (as differentiable as one needs it). Under this general assumptions, he says:
Well-known results in economics show that (1) or (1') is equivalent to saying that there exists a function of this social index which can be written as a sum of funcyions of each of the $x_j$'s.
Mathematically, this is expressed by the author as saying that there is a (smooth) function $F:\mathbb{R}\rightarrow\mathbb{R}$ and (smooth) $V_j:\mathbb{R}^n\rightarrow\mathbb{R}$ ($j=1,\ldots,n$) such that $$\begin{align} \hat{x}(\mathbf{x})= F\big(\sum^n_{j=1}V_j(x_j)\big)\tag{2}\label{two} \end{align}$$ (similarly for $\tilde{x}$).
The problem: It is clear that if a function $\hat{x}$ is of the form \eqref{two}, then the ratios $\frac{\partial_i\hat{x}}{\partial_j\hat{x}}=\frac{V'_i(x_i)}{V'(x_j)}$ depend on $\mathbf{x}$ only through $x_i$ and $x_j$. It is the converse part is what it is not clear to me, that is, that of a (smooth) function $\hat{x}:U\subset \mathbb{R}^n\rightarrow\mathbb{R}$ satisfies a property as in \eqref{one}, then $\hat{x}$ must be of the form \eqref{two} ($n\geq3$).
Any reference (or additional assumptions pertaining to some other conditions that welfare indices must satisfy) will be appreciated.
