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?

  • $\begingroup$ What about a non-satiated utility function with a non-budget constraint? For example, the ability to carry a maximum weight or volume home from the store. This can cause effective satiation but the satiation isn't baked into the utility function. $\endgroup$ – BKay Jan 28 at 21:24

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 $dU/dx_2 \leq 0$.

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    $\begingroup$ Would the second utility function not result in the agent spending all their income on $x$ such that $x=\frac{m}{p_x}$? $\endgroup$ – Brennan Jan 28 at 2:15
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    $\begingroup$ @Brennan Oops my bad... fixed. Thanks a lot! $\endgroup$ – Art Jan 28 at 2:28
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    $\begingroup$ Not sure what you mean by "strong monotonicity of at least one good", but it seems like Leontief preferences such as $\min(x,y)$ do not have them, yet the budget constraint is binding due to local non-satiation. $\endgroup$ – Giskard Jan 28 at 5:53
  • $\begingroup$ @Giskard Thanks. What I meant to say is that at any point, the preference should have at least one parameter that has positive derivative. Not sure what that's called though... Can just write out the math expression but not sure if that's helpful for the OP. $\endgroup$ – Art Jan 28 at 6:16
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    $\begingroup$ What about the utility function $U(x_1,x_2) = x_1 - (x_1-x_2)^2$? This seems to fulfill your conditions, yet the budget constraint will always be binding. I don't think you can easily do better than negating local non-satiation. $\endgroup$ – Giskard Jan 28 at 7:49

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