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enter image description here

my current thinking is i have to dis/prove two things

  1. cardinality
  2. continuity but im not sure about how it would apply since the above is a natural X natural choice space

I know cardinality of natural choice space = cardinality of rational numbers, but im not sure how I can relate that to representation in a utility function

Im not sure about continuity because epsilon balls around numbers in a natural space doesnt include anything else

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    $\begingroup$ Have you tried writing down a few bundles and seeing if there is a way to compare them? Try out out some examples and it should be clear if such a representation is possible. $\endgroup$ Jan 25, 2022 at 18:58
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    $\begingroup$ No it doesn't. The link you provide shows an example where they are comparing real to rational as a proof, whereas my question is set in the natural choice space @Giskard $\endgroup$ Jan 25, 2022 at 19:03
  • $\begingroup$ @WalrasianAuctioneer yeah, the lexicographic preference can be compared by drawing but im not sure how i can prove that $\endgroup$ Jan 25, 2022 at 19:05
  • $\begingroup$ Sorry, mistake! But why are you bringing continuity into this? It is not required that $U$ is continuous, is it? $\endgroup$
    – Giskard
    Jan 25, 2022 at 19:06
  • $\begingroup$ @Giskard my lecture notes state that discontinuous preferences cannot be represented by a utility function, hence the continuity argument $\endgroup$ Jan 25, 2022 at 19:10

1 Answer 1

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Take a strictly increasing mapping $f:\mathbb{N} \to [0,1)$, such as $$ f(y) = 1 - \frac{1}{y+1}. $$ Then $$ U(x,y) = x + f(y) $$ represents the Lexicographic preference in the $\mathbb{N}^2$ choice space.

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