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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?

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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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