I can prove that if $f(x_1, \ldots, x_n)$ is a homogeneous function of degree $k$, then each of its partial derivatives must be a homogeneous function of degree $k-1$;
but I'm not sure if the converse is true: if we know that each and all of its partial derivatives are homogeneous functions of degree $k$, do we know that the function itself must be a homogeneous function of degree $k+1$?
(This is not homework, I'm trying to find a link between homogeneity of utility function and demand function which is not specified in textbook)
Many thanks
Best Answer
In general. homogeneity doesn't run upstream. Consider the function $g(x_1, \ldots, x_n) = f(x_1, \ldots, x_n) + A\;, A\in \mathbb R$ a constant, with $f$ homogeneous of degree $k$. Then
$$\frac {\partial g}{\partial x_i} = \frac {\partial f}{\partial x_i}\; \; \forall i=1,...,n$$
The partials of both function are homogeneous of degree $k-1$ but $g$ is not homogeneous.
But this depends on the existence of an additive constant term. In your particular case, upstream homogeneity may hold, depending on the functional form of the utility function (which usually does not contain additive constants).