# Conditions for an interior solution to the UMP

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!

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).