With quasilinear utiliy, I've seen that often there is no non-negativity restriction imposed on the numeraire good. MWG justify it as "to avoid dealing with boudary problems", but I cannot see the - probably trivial - step.
Moreover, are there situations when the nonnegativity constraint on the numeraire can be applied without adding restrictions?
Consider $u(x,y)=2\sqrt x+y$, and the budget constraint is $x+y\le 0.5$.
If non-negativity is imposed, i.e. $x,y\ge0$, then optimal solution to the utility maximization is $\bar x=0.5$ and $\bar y=0$. Note that under this solution, the FOC holds with strict inequality. So we need to discuss various complementary slackness conditions before arriving at the solution.
If we allow $y$ to be negative, then the optimal solution is $x^*=1$ and $y^*=-0.5$. Under this solution, FOC holds with equality, and both $x^*$ and $y^*$ can be solved directly from the FOC (and the budget constraint).
From the perspective of characterizing solutions, it is thus more convenient not to impose the non-negativity constraint(s).
