Can a function be homothetic if it's not homogeneous?

Question

I know that a homogeneous function of positive degree is homothetic, but can a function that is not homogeneous be homothetic?

Answer 1

I did not know the term in a non-geometric context, so got curious.

If we specialize to two variables, it seems that a function $f: \mathbb{R}^2 \to \mathbb{R}$ is called homothetic if the ratio of the partial derivatives $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial x}$ depends only on the ratio of $x$ and $y$.

If that is the case, there are simple examples that show that a homothetic function need not be homogeneous.

For instance, let $f(x,y)=xy +e^{xy}$. Then $$\frac{\partial f}{\partial y}=x +xe^{xy}$$ and $$\frac{\partial f}{\partial x}=y +ye^{xy}$$ So their ratio is $x/y$, but $f(x,y)$ is not homogeneous.

A more trivial sort of example is something like $x^7+y^7 +17$, but this could be ruled out by a minor change of definition.