<p>The second to top &quot;Related&quot; question (<a href="https://economics.stackexchange.com/questions/6908/what-utility-functions-are-equivalent-to-additive-functions?rq=1">What utility functions are equivalent to additive functions?</a>) contained a link to Ted Bergstrom's <a href="http://www.econ.ucsb.edu/%7Etedb/Courses/GraduateTheoryUCSB/separabilitynotes.pdf" rel="nofollow noreferrer">Lecture Notes on Separable Preferences</a>, which answered this question in the affirmative. The full proof is given in &quot;Donald W. Katzner. Static Demand Theory. Macmillan, New York, 1970&quot;.</p>

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2023-03-01T12:41:35.710Z
2023-03-01T12:46:40.373Z
2023-03-01T12:46:40.373Z
cfp
<p>Are homothetic additively separable preferences always a monotonic transformation of CES preferences?</p>
<p>In technical language, the question is the following:</p>
<p>Let <span class="math-container">$n&gt;1$</span>, and let <span class="math-container">$f:\mathbb{R}^n_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$</span> be continuously differentiable, concave, and homogeneous of degree one. Here, homogeneity of degree one means that for all <span class="math-container">$s\in\mathbb{R}_{\ge 0}$</span>, and <span class="math-container">$x\in\mathbb{R}^n_{\ge 0}$</span>, <span class="math-container">$f(sx)=sf(x)$</span>.</p>
<p>And suppose that there exist continuously differentiable monotonic increasing functions <span class="math-container">$g:\mathbb{R}\rightarrow\mathbb{R}_{\ge 0}$</span> and <span class="math-container">$h_1,\dots,h_n:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}$</span> such that for all <span class="math-container">$x\in\mathbb{R}^n_{\ge 0}$</span>:
<span class="math-container">$$f(x)=g(h_1(x_1)+\cdots+h_n(x_n)).$$</span></p>
<p>Must it be the case that there exists <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}$</span> and <span class="math-container">$\rho,b_1,\dots,b_n\in\mathbb{R}_{\ge 0}$</span> such that for all <span class="math-container">$i\in\{1,\dots,n\}$</span>, <span class="math-container">$h_i(x)=a_i+b_i \frac{x^{1-\rho}-1}{1-\rho}$</span>? (Where, when <span class="math-container">$\rho=1$</span>, we understand this as stating <span class="math-container">$h_i(x)=a_i+b_i\log(x)$</span>.)</p>
<hr />
<p>Note: This was first posted to math.se here <a href="https://math.stackexchange.com/questions/4648187/homogeneous-of-degree-one-functions-that-are-a-monotonic-transformation-of-an-ad">https://math.stackexchange.com/questions/4648187/homogeneous-of-degree-one-functions-that-are-a-monotonic-transformation-of-an-ad</a> but I think I have a higher chance of getting an answer here.</p>

<utility><preferences><production-function><ces-function><ces>
Are homothetic additively separable preferences always equivalent to CES?
cfp
# Are homothetic additively separable preferences always equivalent to CES?

(Asked by: username_0 on 2023-03-01)

Are homothetic additively separable preferences always a monotonic transformation of CES preferences?

In technical language, the question is the following:

Let $n>1$, and let $f:\mathbb{R}^n\_{\ge 0}\rightarrow\mathbb{R}\_{\ge 0}$ be continuously differentiable, concave, and homogeneous of degree one. Here, homogeneity of degree one means that for all $s\in\mathbb{R}\_{\ge 0}$, and $x\in\mathbb{R}^n\_{\ge 0}$, $f(sx)=sf(x)$.

And suppose that there exist continuously differentiable monotonic increasing functions $g:\mathbb{R}\rightarrow\mathbb{R}\_{\ge 0}$ and $h\_1,\dots,h\_n:\mathbb{R}\_{\ge 0}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}^n\_{\ge 0}$:
$$f(x)=g(h\_1(x\_1)+\cdots+h\_n(x\_n)).$$

Must it be the case that there exists $a\_1,\dots,a\_n\in\mathbb{R}$ and $\rho,b\_1,\dots,b\_n\in\mathbb{R}\_{\ge 0}$ such that for all $i\in\{1,\dots,n\}$, $h\_i(x)=a\_i+b\_i \frac{x^{1-\rho}-1}{1-\rho}$? (Where, when $\rho=1$, we understand this as stating $h\_i(x)=a\_i+b\_i\log(x)$.)

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Note: This was first posted to math.se here <https://math.stackexchange.com/questions/4648187/homogeneous-of-degree-one-functions-that-are-a-monotonic-transformation-of-an-ad> but I think I have a higher chance of getting an answer here.

**Tags:** utility, preferences, production-function, ces-function, ces

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## 1 answers

### Answer by username_0 on 2023-03-01. Score: 1

The second to top "Related" question ([What utility functions are equivalent to additive functions?](https://economics.stackexchange.com/questions/6908/what-utility-functions-are-equivalent-to-additive-functions?rq=1)) contained a link to Ted Bergstrom's [Lecture Notes on Separable Preferences](http://www.econ.ucsb.edu/%7Etedb/Courses/GraduateTheoryUCSB/separabilitynotes.pdf), which answered this question in the affirmative. The full proof is given in "Donald W. Katzner. Static Demand Theory. Macmillan, New York, 1970".


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