Numeraire negativity to avoid boundary solution

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?

• Do you realize that with non-negativity constraints, quasilinear utility function may produce corner solutions? – Herr K. Jul 26 '17 at 17:20
• Sure, but I don't see how this could be avoided with non-negativity constraint – Mino Jul 26 '17 at 21:17
• Without non-negativity constraint, corner solution is not an issue. – Herr K. Jul 26 '17 at 22:14
• Sorry, I meant what you wrote. I can't see how, could you help me/give me a resource which explains it? – Mino Jul 26 '17 at 22:38

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).