<p>Consider $u(x,y)=2\sqrt x+y$, and the budget constraint is $x+y\le 0.5$. </p>

<p>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. </p>

<p>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).</p>

<p>From the perspective of characterizing solutions, it is thus more convenient not to impose the non-negativity constraint(s). </p>

CC BY-SA 3.0
2017-07-27T02:51:52.117Z
2017-07-27T02:51:52.117Z
Herr K.
<p>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.</p>

<p>Moreover, are there situations when the nonnegativity constraint on the numeraire can be applied without adding restrictions?</p>

CC BY-SA 3.0
2017-07-26T17:20:59.370Z
Do you realize that with non-negativity constraints, quasilinear utility function may produce [**corner solutions**](https://en.wikipedia.org/wiki/Corner_solution)?
Herr K.
CC BY-SA 3.0
2017-07-26T21:17:57.703Z
Sure, but I don't see how this could be avoided with non-negativity constraint
Mino
CC BY-SA 3.0
2017-07-26T22:14:53.657Z
*Without* non-negativity constraint, corner solution is not an issue.
Herr K.
CC BY-SA 3.0
2017-07-26T22:38:06.280Z
Sorry, I meant what you wrote. I can't see how, could you help me/give me a resource which explains it?
Mino
<microeconomics>
Numeraire negativity to avoid boundary solution
Mino
# Numeraire negativity to avoid boundary solution

(Asked by: username_0 on 2017-07-26)

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?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Do you realize that with non-negativity constraints, quasilinear utility function may produce [**corner solutions**](https://en.wikipedia.org/wiki/Corner_solution)?  
> **username_0**: Sure, but I don't see how this could be avoided with non-negativity constraint  
> **username_1**: *Without* non-negativity constraint, corner solution is not an issue.  
> **username_0**: Sorry, I meant what you wrote. I can't see how, could you help me/give me a resource which explains it?  

---

## 1 answers

### Answer by username_1 on 2017-07-27. Score: 4 | Accepted answer

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).


---

