<p>The proof of this is a bit involved but not too hard. You can find it in the beautiful paper of <a href="https://www.jstor.org/stable/1913043?seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">Goldman and Uzawa, (1964), A note on separability in demand analysis, Econometrica, 387-398</a>. It is basically Theorem 1.</p>

CC BY-SA 4.0
2022-02-01T11:21:46.553Z
Thanks! I think I will be able to derived the statement in my OP from the general theorems stated in the paper you referenced.
Oliver Díaz
CC BY-SA 4.0
2022-02-01T08:45:01.393Z
2022-02-01T08:45:01.393Z
tdm
<p>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.</p>
<p>The author introduces some axiomatization of measures of economic inequality, say <span class="math-container">$I:U\subset\mathbb{R}^n\rightarrow[0,\infty)$</span>, for example &quot;Impartiality&quot; (<span class="math-container">$I$</span> is invariant under permutations); transfers principle (<span class="math-container">$\partial_iI-(x_i-x_j)\partial_jI\geq0$</span>), among others.</p>
<p>The following properties are the ones that are intriguing to me: Let <span class="math-container">$\mathbf{x}=[x_1,\ldots,x_n]^\intercal\in U$</span> and denote  <span class="math-container">$\bar{\mathbf{x}}=\frac1n\sum^n_{j=1}x_j$</span>.
The author defines <em>welfare indices</em>
<span class="math-container">\begin{align}
\hat{x}(\mathbf{x})&amp;=\bar{\mathbf{x}}-I(\mathbf{x})\\
\tilde{x}(\mathbf{x})&amp;=(1-I(\mathbf{x}))\bar{\mathbf{x}}
\end{align}</span>
and introduces the following properties in his axiomatization:</p>
<p>(1) (absolute) For all <span class="math-container">$1\leq i,j\leq n$</span>, the ration <span class="math-container">$\frac{\partial_i\hat{x}}{\partial_j\hat{x}}$</span> depends only on <span class="math-container">$x_i$</span> and <span class="math-container">$x_j$</span>, that is
<span class="math-container">$$\begin{align}\frac{\partial_i\hat{x}}{\partial_j\hat{x}}=\psi_{ij}(x_i,x_j)\tag{1}\label{one}\end{align}$$</span>
for some function <span class="math-container">$\psi_{ij}:\mathbb{R}^2\rightarrow\mathbb{R}$</span>.</p>
<p>(1') (relative) For all <span class="math-container">$1\leq i,j\leq n$</span>, the ration <span class="math-container">$\frac{\partial_i\tilde{x}}{\partial_j\tilde{x}}$</span> depends only on <span class="math-container">$x_i$</span> and <span class="math-container">$x_j$</span>, that is
<span class="math-container">$$\begin{align}\frac{\partial_i\tilde{x}}{\partial_j\tilde{x}}=\phi_{ij}(x_i,x_j)\tag{1'}\label{onep}\end{align}$$</span>
for some function <span class="math-container">$\phi_{ij}:\mathbb{R}^2\rightarrow\mathbb{R}$</span>.</p>
<p>(1) and (1'), the author argues, may be called (or labeled) *<strong>welfare independence</strong>.
There are no additional properties on <span class="math-container">$I$</span> except that presumably is smooth (as differentiable as one needs it). Under this general assumptions, he says:</p>
<blockquote>
<p>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 <span class="math-container">$x_j$</span>'s.</p>
</blockquote>
<p>Mathematically, this is expressed by the author as saying that there is a (smooth) function <span class="math-container">$F:\mathbb{R}\rightarrow\mathbb{R}$</span> and (smooth) <span class="math-container">$V_j:\mathbb{R}^n\rightarrow\mathbb{R}$</span> (<span class="math-container">$j=1,\ldots,n$</span>) such that
<span class="math-container">$$\begin{align}
\hat{x}(\mathbf{x})= F\big(\sum^n_{j=1}V_j(x_j)\big)\tag{2}\label{two}
\end{align}$$</span>
(similarly for <span class="math-container">$\tilde{x}$</span>).</p>
<hr />
<p><strong>The problem:</strong> It is clear that if a function <span class="math-container">$\hat{x}$</span> is of the form \eqref{two}, then the ratios <span class="math-container">$\frac{\partial_i\hat{x}}{\partial_j\hat{x}}=\frac{V'_i(x_i)}{V'(x_j)}$</span> depend on <span class="math-container">$\mathbf{x}$</span> only through <span class="math-container">$x_i$</span> and <span class="math-container">$x_j$</span>. It is the converse part is what it is not clear to me, that is, that of a (smooth) function <span class="math-container">$\hat{x}:U\subset \mathbb{R}^n\rightarrow\mathbb{R}$</span> satisfies a property as in \eqref{one}, then <span class="math-container">$\hat{x}$</span> must be of the form \eqref{two} (<span class="math-container">$n\geq3$</span>).</p>
<p>Any reference (or additional assumptions pertaining to some other conditions that welfare indices must satisfy) will be appreciated.</p>

<mathematical-economics><inequality>
A question about "welfare independence"
Oliver Díaz
# A question about "welfare independence"

(Asked by: username_0 on 2022-01-31)

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.

**Tags:** mathematical-economics, inequality


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## 1 answers

### Answer by username_1 on 2022-02-01. Score: 3 | Accepted answer

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](https://www.jstor.org/stable/1913043?seq=1#metadata_info_tab_contents). It is basically Theorem 1.

> **Comments:**
> **username_0**: Thanks! I think I will be able to derived the statement in my OP from the general theorems stated in the paper you referenced.  

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