Example of a utility maximization problem with a non-binding budget constraint

Question

Given a utility function $U(x,y): \mathbb{R}^{2} \to \mathbb{R}$, the general utility maximization can be stated as follows: $$ \max_{x, y} U(x,y) \text{ s.t. } p_{x}x + p_{y}y \leq m $$ where the $p_{i}$ are prices, and $m$ is total income. In the Cobb-Douglas case, for example, $$ U(x,y) = x^{\alpha} y^{1 - \alpha} $$ for some $\alpha \in (0,1)$. We notice that in Cobb-Douglas, the more you buy, the more utility you get. Thus your budget constraint must hold with equality, because the more you spend, the more utility you get.

What are examples of utility functions that don't have positive marginal utility of income after some point? I can think of functions that have bliss points. For instance, if $U(x,y) = -(x-25)^{2} - (y-20)^{2}$, then $(25,20)$ is the best you can get, even if your income allows you to spend more. What are other examples of this?

Answer 1

Assuming positive prices, any utility function that doesn't satisfy strong monotonicity of at least one good would have a non-binding budget constraint. Since you asked for some examples:

• $U(x, y) = 0$,
• $U(x, y) = -x$.

Re: Giskard's comment.

I'm not sure I used the correct term here when I said "strong monotonicity of at least one good." To be precise, what I meant to say was:

Let $U(x_1, x_2)$ be the utility function. The function would have non-binding budget constraint if there are some $(x_1, x_2)$ such that $dU/dx_1 \leq 0$ AND $dU/dx_2 \leq 0$.