Is it known when a marginal-rate-of-substitution function can be rationalized by some utility function?
More precisely, and focusing on the case of two goods, what conditions are required on $M: (\mathbb R_{\geq 0})^2 \to \mathbb R$ in order for there to exist $u: (\mathbb R_{\geq 0})^2 \to \mathbb R$ such that for all $x,y \geq 0$, $$ M(x,y)=\frac{u_x(x,y)}{u_y(x,y)} $$
This is a rather indirect way.
For $\omega, z \in \mathbb{R}_{++}$, define the (demand) correspondence:
$$ D(\omega, z) = \left\{(x,y) \in \mathbb{R}^2_+| MRS(x,y) = \omega \text{ and } \omega x + y = z.\right\} $$ The idea is that $D(\omega, z)$ gives the demand correspondence for (relative prices $\omega = MRS(x,y)$ and (normalized) income $z = \omega x + y$.
If $D(\omega, z)$ would turn out to be a (smooth) function, one could then check if it satisfies the usual integrability conditions: Slutsky symmetry and negative definiteness. (Alternatively, one could check if $D(\omega, z)$ satisfies the SARP (strong axiom of revealed preference).)
