<p>One sufficient condition for interior solutions given strictly positive prices and income that sometimes works is that every bundle in the interior of the commodity space (a bundle that contains strictly positive amounts of all goods) is preferred to every bundle on the boundary of the commodity space (a bundle that contains zero of some commodity). Prime examples are <a href="https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function#Cobb%E2%80%93Douglas_utilities" rel="noreferrer">Cobb-Douglas utility functions</a>.</p>

<p>More generally (not really), to be sure that you get interior solutions for all positive prices, marginal utilities should become infinitely large when the amount of one commodity goes to zero. A typical example is <span class="math-container">$u:\mathbb{R}^2_+\to\mathbb{R}$</span> given by
<span class="math-container">$$u(x_1,x_2)=\sqrt{x_1}+\sqrt{x_2}.$$</span></p>

CC BY-SA 4.0
2018-09-27T05:46:13.420Z
Yes, though originally meant for production functions, the [Inada conditions](https://en.wikipedia.org/wiki/Inada_conditions) are useful here.
Giskard
CC BY-SA 4.0
2018-09-26T22:24:26.190Z
2018-09-26T22:24:26.190Z
Michael Greinecker
<p>No, it isn't.</p>

<p>A simple counterexample is
<span class="math-container">\begin{align*}
\max_x \hskip 10pt &amp; x_1 + x_2 \\
\\
\text{s.t.} \hskip 10pt &amp; x_1 + 2x_2 = 10.
\end{align*}</span>
The utility function is monotonic (strictly monotonic even), but the solution is a corner solution at <span class="math-container">$(x_1,x_2) = (10,0)$</span>.</p>

<hr>

<p>The above answers the question, but it is worthwhile to note that defining a sufficient condition (that is not too strong) seems to be difficult.</p>

<p>Not even convexity is sufficient. (The utility function above is convex, and could be made strictly convex with an extremely slight bend.)</p>

<p>Seems like you would have to compare the level curve gradients at all the possible corner solutions to the gradient of the budget set, but that does not really help you in higher dimensions.</p>

CC BY-SA 4.0
2018-09-26T20:50:31.863Z
2018-09-27T05:47:12.283Z
2018-09-27T05:47:12.283Z
Giskard
<p>Solving the basic Utility Maximization Problem, i.e.</p>

<p><span class="math-container">\begin{align}
max_{x\geq 0} u(x) \\
s.t. \,\,\, p^Tx\leq w
\end{align}</span></p>

<p>we get the Kuhn-Tucker first order condition</p>

<p><span class="math-container">\begin{gather}
\frac{\partial u(x)}{\partial x_l}\leq \lambda p_l
\end{gather}</span></p>

<p>which holds with equality when <span class="math-container">$x_l &gt; 0$</span>, i.e. when solutions are interior. </p>

<p>Is it enough to say that since preferences are assumed to be monotonous, we can rule out the possibility of boundary solutions and thus we are safe in setting the FOC with the equal sign?</p>

<microeconomics><utility>
Ruling out boundary solutions in Utility Maximization
PostDocing
# Ruling out boundary solutions in Utility Maximization

(Asked by: username_0 on 2018-09-26)

Solving the basic Utility Maximization Problem, i.e.

\begin{align}
max\_{x\geq 0} u(x) \\
s.t. \,\,\, p^Tx\leq w
\end{align}

we get the Kuhn-Tucker first order condition

\begin{gather}
\frac{\partial u(x)}{\partial x\_l}\leq \lambda p\_l
\end{gather}

which holds with equality when $x\_l > 0$, i.e. when solutions are interior.

Is it enough to say that since preferences are assumed to be monotonous, we can rule out the possibility of boundary solutions and thus we are safe in setting the FOC with the equal sign?

**Tags:** microeconomics, utility

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## 2 answers

### Answer by username_2 on 2018-09-26. Score: 6 | Accepted answer

No, it isn't.

A simple counterexample is
\begin{align\*}
\max\_x \hskip 10pt & x\_1 + x\_2 \\
\\
\text{s.t.} \hskip 10pt & x\_1 + 2x\_2 = 10.
\end{align\*}
The utility function is monotonic (strictly monotonic even), but the solution is a corner solution at $(x\_1,x\_2) = (10,0)$.

---

The above answers the question, but it is worthwhile to note that defining a sufficient condition (that is not too strong) seems to be difficult.

Not even convexity is sufficient. (The utility function above is convex, and could be made strictly convex with an extremely slight bend.)

Seems like you would have to compare the level curve gradients at all the possible corner solutions to the gradient of the budget set, but that does not really help you in higher dimensions.


---

### Answer by username_1 on 2018-09-26. Score: 6

One sufficient condition for interior solutions given strictly positive prices and income that sometimes works is that every bundle in the interior of the commodity space (a bundle that contains strictly positive amounts of all goods) is preferred to every bundle on the boundary of the commodity space (a bundle that contains zero of some commodity). Prime examples are [Cobb-Douglas utility functions](https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function#Cobb%E2%80%93Douglas_utilities).

More generally (not really), to be sure that you get interior solutions for all positive prices, marginal utilities should become infinitely large when the amount of one commodity goes to zero. A typical example is $u:\mathbb{R}^2\_+\to\mathbb{R}$ given by
$$u(x\_1,x\_2)=\sqrt{x\_1}+\sqrt{x\_2}.$$

> **Comments:**
> **username_2**: Yes, though originally meant for production functions, the [Inada conditions](https://en.wikipedia.org/wiki/Inada_conditions) are useful here.  

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