Ruling out boundary solutions in Utility Maximization

Question

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?

Answer 1

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 $(x_1,x_2) = (10,0)$.

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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 2

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.

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 $$u(x_1,x_2)=\sqrt{x_1}+\sqrt{x_2}.$$