Existence of maximum utility with two bads

Question

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?

Answer 1

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}