# 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: $$u(x) = h(x)U(x/h(x)).$$
$$U(x)= \alpha x_1^2 + \beta x_1x_2 + \gamma x_2^5$$,
$$h(x)=x_1+x_2$$
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.