Homogeneous Utilities: Anything other than CES? 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?
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.