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

  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.