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?