# A question about "welfare independence"

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.

The proof of this is a bit involved but not too hard. You can find it in the beautiful paper of Goldman and Uzawa, (1964), A note on separability in demand analysis, Econometrica, 387-398. It is basically Theorem 1.

• Thanks! I think I will be able to derived the statement in my OP from the general theorems stated in the paper you referenced. Feb 1, 2022 at 11:21