# Existence of maximum utility with two bads

I am working with a consumption set $$X = R_+^2$$ and preferences that are complete, transitive, continuous and strongly monotonically decreasing. The economy is characterized by the presence of two economic bads. I need to show that "there always exist a solution to the consumer's utility maximization problem".

My point is that if I had a continuous utility function over a compact budget set, using Weierstrass theorem I have the certainty of the function reaching its extrema. The problem here, if I am right, is that the budget set is no longer bounded and thus neither compact.

Are there other results I can use to show the statement?

• I apologize if this is naive: But if the budget set B is a subset of $R^2_+$ and the utility function U is continuous in $R^2_+$, then the function $U:B->R^2_+$ is bounded. So the sup and inf of U[.] in B exist. Which means that in the budget set the utility function has a solution. I do not understand the problem with this logic. Again, sorry if this is naive.
– erik
Oct 6, 2018 at 5:12
• @erik: Note that the consumption set is the entire $\mathbb R^2_+$, not a subset of it. And since $\mathbb R^2_+$ is not bounded and thus not compact, we can't directly apply Weierstrass theorem. Oct 6, 2018 at 6:56
• @ Herr K: I saw X = $R^2_+$ and assumed that this meant that all goods are measured on the positive quadrant...I glossed over the consumption set before it. Thank you for you clarification.
– erik
Oct 6, 2018 at 6:58

I assume, by a strongly monotonically decreasing preference $$\succsim$$, you mean: for all $$x,y\in\mathbb R^2_+$$, $$\begin{equation} x\succ y \text{ only if x\le y and x\ne y}. \end{equation}$$ Then it is clear that $$(0,0)$$ is the most preferred bundle in $$\mathbb R^2_+$$, i.e. $$(0,0)\succ x$$ for all $$x\in\mathbb R^2$$ such that $$x\ne (0,0)$$.
Therefore, any utility function representing $$\succsim$$ must also satisfy $$u(0,0)\ge u(x)$$ for all $$x\in\mathbb R^2_+$$. Hence, $$\begin{equation} (0,0)=\arg\max_{x} u(x) \end{equation}$$
• Nice one! Thanks! So you mean that $(0,0)$ would always be a solution but not necessarily the unique one? Oct 6, 2018 at 16:09
• But what if $w>0$? I mean $(0,0)$ can be not included in the budget set Oct 6, 2018 at 18:29
• @Alessandro: By specifying $\mathbb R^2_+$ as the consumption set, you're already assuming implicitly that all bundles in $\mathbb R^2_+$ are feasible. Plus, why would anyone spend a positive amount on a "bad"? Oct 6, 2018 at 19:20