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

CC BY-SA 4.0
2022-01-25T19:09:08.960Z
2022-01-25T19:17:38.823Z
2022-01-25T19:17:38.823Z
Giskard
<p><a href="https://i.sstatic.net/QOODQ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QOODQ.png" alt="enter image description here" /></a></p>
<p>my current thinking is i have to dis/prove two things</p>
<ol>
<li>cardinality</li>
<li>continuity
but im not sure about how it would apply since the above is a natural X natural choice space</li>
</ol>
<p>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</p>
<p>Im not sure about continuity because epsilon balls around numbers in a natural space doesnt include anything else</p>

CC BY-SA 4.0
2022-01-25T18:58:02.003Z
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.
Walrasian Auctioneer
CC BY-SA 4.0
2022-01-25T19:03:57.403Z
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
theshadowers
CC BY-SA 4.0
2022-01-25T19:05:28.740Z
@WalrasianAuctioneer yeah, the lexicographic preference can be compared by drawing but im not sure how i can prove that
theshadowers
CC BY-SA 4.0
2022-01-25T19:06:07.010Z
Sorry, mistake! But why are you bringing continuity into this? It is not required that $U$ is continuous, is it?
Giskard
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2022-01-25T19:10:45.100Z
@Giskard my lecture notes state that discontinuous preferences cannot be represented by a utility function, hence the continuity argument
theshadowers
CC BY-SA 4.0
2022-01-25T19:11:30.977Z
I am afraid your lecture notes are mistaken in this instance.
Giskard
CC BY-SA 4.0
2022-01-25T19:16:50.903Z
E.g., the *sign* function defines discontinuous preferences over $\mathbb{R}$.
Giskard
CC BY-SA 4.0
2022-01-25T19:20:13.023Z
@Giskard okay, if we assume to ignore the continuity statement, do you have any suggestions as to how I would proceed with the first point?
theshadowers
CC BY-SA 4.0
2022-01-25T19:20:59.377Z
I posted an answer 10 minutes ago (:
Giskard
CC BY-SA 4.0
2022-01-25T19:21:07.900Z
At first I linked the wrong question, but here is a true duplicate: [Lexicographic Preference Relation on the QxR](https://economics.stackexchange.com/q/24415/1601)
Giskard
CC BY-SA 4.0
2022-01-25T19:21:13.020Z
@Giskard Can I ask about the significance of the first line in your answer below? I don't really understand how we can map the entire natural plane in [0,1)
theshadowers
CC BY-SA 4.0
2022-01-25T19:21:49.923Z
You don't understand the function that assigns $1 - 1/(y+1)$ to $y$? What do you not understand about it? Do you think this can go under 0 or above 1? Note that it is not about the natural *plane*, just natural numbers.
Giskard
CC BY-SA 4.0
2022-01-25T19:24:06.437Z
@Giskard Sorry, not the function, but I dont think I understand the significance or implications of Take a strictly increasing mapping f:N→[0,1)
What are we doing here and why can we map the natural plane as [0,1)?
theshadowers
CC BY-SA 4.0
2022-01-25T19:25:25.803Z
It does not map the natural plane, just natural numbers. I suggest you take  
WalrasianAuctioneer's advice and experiment by plugging in some numbers, e.g., is (2,0) preferred to (1,99) according to this function?
Giskard
CC BY-SA 4.0
2022-01-25T20:11:56.030Z
@Giskard okay after prolonged thinking I believe I understand most of it. do you think it is necessary for me to state the cardinality stuff? as the question asks to show all details of your claim and proof
theshadowers
CC BY-SA 4.0
2022-01-25T20:28:02.017Z
I cannot speak for your university program, but I am sure you can talk about the exact requirements with your professor.
Giskard
CC BY-SA 4.0
2022-01-25T21:47:36.900Z
@Giskard thank you so much for your help! I understand it now
theshadowers
<mathematical-economics><utility><preferences><proof><discrete-preferences>
Representing a Lexicographic Preference in a Natural X Natural Choice Space With Utility Function
theshadowers
# Representing a Lexicographic Preference in a Natural X Natural Choice Space With Utility Function

(Asked by: username_0 on 2022-01-25)

[![enter image description here](https://i.sstatic.net/QOODQ.png)](https://i.sstatic.net/QOODQ.png)

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

**Tags:** mathematical-economics, utility, preferences, proof, discrete-preferences
> **Comments:**
> **username_1**: 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.  
> **username_0**: 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  
> **username_0**: @WalrasianAuctioneer yeah, the lexicographic preference can be compared by drawing but im not sure how i can prove that  
> **username_2**: Sorry, mistake! But why are you bringing continuity into this? It is not required that $U$ is continuous, is it?  
> **username_0**: @Giskard my lecture notes state that discontinuous preferences cannot be represented by a utility function, hence the continuity argument  
> **username_2**: I am afraid your lecture notes are mistaken in this instance.  
> **username_2**: E.g., the *sign* function defines discontinuous preferences over $\mathbb{R}$.  
> **username_0**: @Giskard okay, if we assume to ignore the continuity statement, do you have any suggestions as to how I would proceed with the first point?  
> **username_2**: I posted an answer 10 minutes ago (:  
> **username_2**: At first I linked the wrong question, but here is a true duplicate: [Lexicographic Preference Relation on the QxR](https://economics.stackexchange.com/q/24415/1601)  
> **username_0**: @Giskard Can I ask about the significance of the first line in your answer below? I don't really understand how we can map the entire natural plane in [0,1)  
> **username_2**: You don't understand the function that assigns $1 - 1/(y+1)$ to $y$? What do you not understand about it? Do you think this can go under 0 or above 1? Note that it is not about the natural *plane*, just natural numbers.  
> **username_0**: @Giskard Sorry, not the function, but I dont think I understand the significance or implications of Take a strictly increasing mapping f:N→[0,1) What are we doing here and why can we map the natural plane as [0,1)?  
> **username_2**: It does not map the natural plane, just natural numbers. I suggest you take   WalrasianAuctioneer's advice and experiment by plugging in some numbers, e.g., is (2,0) preferred to (1,99) according to this function?  
> **username_0**: @Giskard okay after prolonged thinking I believe I understand most of it. do you think it is necessary for me to state the cardinality stuff? as the question asks to show all details of your claim and proof  
> **username_2**: I cannot speak for your university program, but I am sure you can talk about the exact requirements with your professor.  
> **username_0**: @Giskard thank you so much for your help! I understand it now  

---

## 1 answers

### Answer by username_2 on 2022-01-25. Score: 4

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.


---

