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Suppose we define $\Omega$ = $\mathbb R_+$. The preference relation $\succeq $ is defined as $$(x_1, x_2)\succeq(y_1,y_2)\iff x_1>y_1 \text{ or } [x_1=y_1 \text{ and }x_2\geq y_2]$$ where $x_1, x_2,y_1, y_2\in \mathbb R_+$

How to prove that there does not exist utility function that represents this preference?

Thanks in advance for the help!

Luka
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  • @Giskard. I don't think that this question is a duplicate, because it points out the relevance of the uncountability of the sets on which the preference relation is defined. Lexicografic preferences can be represented by a utlity function if they are defined on $\mathbb{Q} \times \mathbb{R}$, as $\mathbb{Q}$ is countable. See https://economics.stackexchange.com/questions/24415/lexicographic-preference-relation-on-the-qxr?rq=1e. This is ia subtle but important issue, because it is not true that lexicografic preferences cannot be represented by a utility function in general. – BakerStreet Oct 19 '23 at 11:42
  • I didn't find a question on SE that addresses specifically this issue. – BakerStreet Oct 19 '23 at 11:47
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    @BakerStreet Thanks for the clarification. I guess in the example case it provides a counterexample for uncountable outcomes. The proof on the other thread sounds good to me. – Luka Oct 19 '23 at 11:48
  • You are welcome. I wanted to point out tha there are proofs for the case of uncountable sets that fail if applied to countable sets. – BakerStreet Oct 19 '23 at 11:50

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