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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?

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  • $\begingroup$ 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. $\endgroup$ – erik Oct 6 '18 at 5:12
  • $\begingroup$ @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. $\endgroup$ – Herr K. Oct 6 '18 at 6:56
  • $\begingroup$ @ 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. $\endgroup$ – erik Oct 6 '18 at 6:58
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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}

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  • $\begingroup$ Nice one! Thanks! So you mean that $(0,0)$ would always be a solution but not necessarily the unique one? $\endgroup$ – PhDing Oct 6 '18 at 16:09
  • $\begingroup$ @Alessandro if preference is strongly monotone then solution must be unique. $\endgroup$ – Herr K. Oct 6 '18 at 16:29
  • $\begingroup$ But what if $w>0$? I mean $(0,0)$ can be not included in the budget set $\endgroup$ – PhDing Oct 6 '18 at 18:29
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    $\begingroup$ @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"? $\endgroup$ – Herr K. Oct 6 '18 at 19:20

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