<p>If it is a lexicographic preference on <span class="math-container">$R^2$</span>, the question sounds weird and I don't know if it can be represented by a utility function from <span class="math-container">$N^2$</span> to <span class="math-container">$R$</span>. The domain of the utility should be the same as the domain of the preference relation, otherwise they cannot represent each other for sure.</p>
<p>Now suppose that, you mean, we have a preference on <span class="math-container">$R^2$</span>, but we are only taking some data from the <span class="math-container">$\mathbb R^2$</span>, for example only those data in <span class="math-container">$N^2$</span>, and see that if we could represent those <span class="math-container">$N^2$</span> data by a utility defined on <span class="math-container">$N^2$</span>. This way, I think the problem becomes more interesting and meaningful.  (Ok, so now the question is edited this way.)</p>
<p>If it a lexicographic preference on <span class="math-container">$N^2$</span>, I think it can be represented by a utility function from <span class="math-container">$N^2$</span> to <span class="math-container">$R$</span>.</p>
<p>The proof should be obvious in this case. A simple construction suffices.</p>
<p>If you have further questions on this case, please do let me know in comment.</p>
<p>Now let's think about the case of <span class="math-container">$u:N^2\to Z$</span>. This problem becomes more complicated. To solve this problem, you first need to get used to the proof that a lexicographic preference on <span class="math-container">$R^2$</span> cannot be represented by utility function with <span class="math-container">$R$</span> as codomain. By applying the same method, you'll see that a preference on <span class="math-container">$N^2$</span> cannot be represented with a utility with <span class="math-container">$N$</span> as codomain. Then going from <span class="math-container">$N$</span> to <span class="math-container">$Z$</span> will not take much efforts.</p>

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2020-10-12T21:30:57.863Z
2020-10-13T20:50:27.513Z
2020-10-13T20:50:27.513Z
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<p>I got this question on a homework:</p>
<p><strong>Could lexicographic preferences on <span class="math-container">$N^2$</span> be represented by a utility function u: <span class="math-container">$N^2$</span> to <span class="math-container">$Z$</span>?</strong></p>
<p>I've heard countless times that lexicographic preferences can't be represented by a utility function. But would anything change if u: <span class="math-container">$N^2$</span> to <span class="math-container">$Z$</span>?</p>

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2020-10-12T21:25:48.537Z
Assume $(0,0)$ maps to $0$. Now, what would $(1,0)$ map to....
201p
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2020-10-12T21:41:23.163Z
I think you have to show your efforts if it is a homework question
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2020-10-13T20:48:31.663Z
By the way, if you edited your question after someone answered it, you could add a note say that how the question was edited, so the readers wouldn't be confused. For example, you could say that you've changed "lexicographic preference on $R^2$" to "lexicographic preference on $N^2$".
High GPA
<microeconomics><preferences>
Could lexicographic preferences on N^2 be represented by a utility function u: $N^2$ to $Z$?
Walter Schmit
# Could lexicographic preferences on N^2 be represented by a utility function u: $N^2$ to $Z$?

(Asked by: username_0 on 2020-10-12)

I got this question on a homework:

**Could lexicographic preferences on $N^2$ be represented by a utility function u: $N^2$ to $Z$?**

I've heard countless times that lexicographic preferences can't be represented by a utility function. But would anything change if u: $N^2$ to $Z$?

**Tags:** microeconomics, preferences
> **Comments:**
> **username_1**: Assume $(0,0)$ maps to $0$. Now, what would $(1,0)$ map to....  
> **username_2**: I think you have to show your efforts if it is a homework question  
> **username_2**: By the way, if you edited your question after someone answered it, you could add a note say that how the question was edited, so the readers wouldn't be confused. For example, you could say that you've changed "lexicographic preference on $R^2$" to "lexicographic preference on $N^2$".  

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## 1 answers

### Answer by username_2 on 2020-10-12. Score: 2

If it is a lexicographic preference on $R^2$, the question sounds weird and I don't know if it can be represented by a utility function from $N^2$ to $R$. The domain of the utility should be the same as the domain of the preference relation, otherwise they cannot represent each other for sure.

Now suppose that, you mean, we have a preference on $R^2$, but we are only taking some data from the $\mathbb R^2$, for example only those data in $N^2$, and see that if we could represent those $N^2$ data by a utility defined on $N^2$. This way, I think the problem becomes more interesting and meaningful. (Ok, so now the question is edited this way.)

If it a lexicographic preference on $N^2$, I think it can be represented by a utility function from $N^2$ to $R$.

The proof should be obvious in this case. A simple construction suffices.

If you have further questions on this case, please do let me know in comment.

Now let's think about the case of $u:N^2\to Z$. This problem becomes more complicated. To solve this problem, you first need to get used to the proof that a lexicographic preference on $R^2$ cannot be represented by utility function with $R$ as codomain. By applying the same method, you'll see that a preference on $N^2$ cannot be represented with a utility with $N$ as codomain. Then going from $N$ to $Z$ will not take much efforts.


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