Demand with Quasilinear Preferences

Exercise

A consumer has utility $u(x, y) = \ln x + y$, prices $p_x > 0, p_y=1$, and income $M > 0$. The quantity $y$ may be interpreted as money spent on all other goods, so $y \geq 0$ is required but $y$ need not be an integer.

1. Derive the demand functions for $x$ and $y$.
2. Show that, at an interior solution, the demand for $x$ is independent of income.