<p>There is an infinity of such functions. You can for instance construct a linear homogeneous function <span class="math-container">$u$</span> from any utility function <span class="math-container">$U$</span> by using a linear homogeneous function <span class="math-container">$h: \mathbb{R}^J \rightarrow \mathbb{R} $</span> as follow:
<span class="math-container">$$ u(x) = h(x)U(x/h(x)). $$</span></p>
<p>Example:<br />
<span class="math-container">$U(x)= \alpha x_1^2 + \beta x_1x_2 + \gamma x_2^5 $</span>,<br />
<span class="math-container">$h(x)=x_1+x_2$</span><br />
yields
<span class="math-container">$$u(x) = (x_1+x_2) \left(  \alpha(\frac{x_1}{x_1+x_2})^2 + \beta\frac{x_1x_2}{(x_1+x_2)^2} + \gamma(\frac{x_2}{x_1+x_2})^5 \right). $$</span></p>
<p>If you also want to ensure that <span class="math-container">$u$</span> is a utility function (increasing in <span class="math-container">$x$</span>) it is a bit more tricky, however.</p>

CC BY-SA 4.0
2020-11-05T22:29:37.030Z
2020-11-05T22:29:37.030Z
Bertrand
<p>Does there exist any homogeneous utility function, i.e., <span class="math-container">$u(\lambda \mathbf{x}) = \lambda u(\mathbf{x})$</span>, that is not a special case of the CES (or nested CES) family of utility functions or its limits (i.e., including Leontief and Cobb-Douglas)? I have been thinking a lot about this but cannot think of any example. If so, does homogeneity imply constant elasticity of substitution?</p>

<mathematical-economics><utility><ces-function>
Homogeneous Utilities: Anything other than CES?
Denizalp
# Homogeneous Utilities: Anything other than CES?

(Asked by: username_0 on 2020-11-05)

Does there exist any homogeneous utility function, i.e., $u(\lambda \mathbf{x}) = \lambda u(\mathbf{x})$, that is not a special case of the CES (or nested CES) family of utility functions or its limits (i.e., including Leontief and Cobb-Douglas)? I have been thinking a lot about this but cannot think of any example. If so, does homogeneity imply constant elasticity of substitution?

**Tags:** mathematical-economics, utility, ces-function

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## 1 answers

### Answer by username_1 on 2020-11-05. Score: 2 | Accepted answer

There is an infinity of such functions. You can for instance construct a linear homogeneous function $u$ from any utility function $U$ by using a linear homogeneous function $h: \mathbb{R}^J \rightarrow \mathbb{R} $ as follow:
$$ u(x) = h(x)U(x/h(x)). $$

Example:  
$U(x)= \alpha x\_1^2 + \beta x\_1x\_2 + \gamma x\_2^5 $,  
$h(x)=x\_1+x\_2$  
yields
$$u(x) = (x\_1+x\_2) \left( \alpha(\frac{x\_1}{x\_1+x\_2})^2 + \beta\frac{x\_1x\_2}{(x\_1+x\_2)^2} + \gamma(\frac{x\_2}{x\_1+x\_2})^5 \right). $$

If you also want to ensure that $u$ is a utility function (increasing in $x$) it is a bit more tricky, however.


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