<p>The utility function <span class="math-container">$u:\mathbb{R}^2_+\to\mathbb{R}$</span> given by <span class="math-container">$u(x,y)=(x+1)(y+1)$</span> is strictly quasiconcave and strictly monotone, but allows for boundary optima.</p>
<p>A sufficient condition for the existence of unique optima in the relative interior of the budget line is that the utility function <span class="math-container">$u$</span> is continuous, strictly increasing and strictly quasi-concave on <span class="math-container">$\mathbb{R}^l_{++}$</span> (the strictly positive orthant), and any commodity bundle that contains strictly positive amounts of every commodity is strictly preferred to any bundle containing zero of some commodity. Cobb-Douglas utility functions are a typical example. Note that they are neither strictly increasing nor strictly quasi-concave (because of boundary behavior).</p>

CC BY-SA 4.0
2023-12-06T12:31:00.030Z
2023-12-06T12:31:00.030Z
Michael Greinecker
<p>I was wondering under what set of conditions one is allowed to assume an interior solution to the Utility Maximisation Problem. In most of my classes and lecture notes, interior solutions are assumed from the outset.</p>
<p>Intuitively, it seems to me that if the utility function is:</p>
<ol>
<li>monotone (increasing in all its arguments)</li>
<li>strictly quasi-concave (implying strict convexity of indifference curves)</li>
</ol>
<p>we are guaranteed an interior optimum. Is this correct? Thank you in advance!</p>

CC BY-SA 4.0
2023-12-06T13:35:56.020Z
https://economics.stackexchange.com/a/56643/11824
Amit
<microeconomics><optimization>
Conditions for an interior solution to the UMP
TomUcl2003
# Conditions for an interior solution to the UMP

(Asked by: username_0 on 2023-12-06)

I was wondering under what set of conditions one is allowed to assume an interior solution to the Utility Maximisation Problem. In most of my classes and lecture notes, interior solutions are assumed from the outset.

Intuitively, it seems to me that if the utility function is:

1. monotone (increasing in all its arguments)
2. strictly quasi-concave (implying strict convexity of indifference curves)

we are guaranteed an interior optimum. Is this correct? Thank you in advance!

**Tags:** microeconomics, optimization
> **Comments:**
> **username_1**: https://economics.stackexchange.com/a/56643/11824  

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## 1 answers

### Answer by username_2 on 2023-12-06. Score: 2 | Accepted answer

The utility function $u:\mathbb{R}^2\_+\to\mathbb{R}$ given by $u(x,y)=(x+1)(y+1)$ is strictly quasiconcave and strictly monotone, but allows for boundary optima.

A sufficient condition for the existence of unique optima in the relative interior of the budget line is that the utility function $u$ is continuous, strictly increasing and strictly quasi-concave on $\mathbb{R}^l\_{++}$ (the strictly positive orthant), and any commodity bundle that contains strictly positive amounts of every commodity is strictly preferred to any bundle containing zero of some commodity. Cobb-Douglas utility functions are a typical example. Note that they are neither strictly increasing nor strictly quasi-concave (because of boundary behavior).


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