<p>Preferences represented by <span class="math-container">$u:X\rightarrow\mathbb{R}$</span>, where <span class="math-container">$X$</span> is a non-empty convex subset of <span class="math-container">$\mathbb{R}^2_+$</span>, satisfy LNS if</p>
<p><span class="math-container">$(\forall (x, y)\in X)(\forall \epsilon &gt; 0)(\exists \ (x',y')\in X)(\sqrt{(x-x')^2+(y-y')^2} &lt;\epsilon \ \wedge u(x',y') &gt; u(x,y)) $</span></p>
<p>Observe that for any given <span class="math-container">$y\in [0,2)$</span>, <span class="math-container">$u(x, y) = \dfrac{x-1}{(2-y)^2}$</span> is strictly increasing in <span class="math-container">$x$</span> over <span class="math-container">$(1, \infty)$</span>. So it satisfy LNS.</p>

CC BY-SA 4.0
2022-06-12T14:27:29.293Z
2022-06-12T14:27:29.293Z
Amit
<p>I got this utility function  <a href="https://i.sstatic.net/Je5Q7.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Je5Q7.png" alt="enter image description here" /></a> representing certain preferences.
Are these preferences locally non satiated?
Can somebody please explain me with the exact definition of local non satiation why these preferences are locally non satiated or not?</p>
<p>Furthermore, consider that the choice set X= {(x1,x2) element of R2+ : x1&gt;1, x2&lt;2}</p>
<p>Thank you</p>

<utility><preferences>
Are these preferences locally non satiated? U(x1,x2)=(x1-1)/(2-x2)^2
dewewdew
# Are these preferences locally non satiated? U(x1,x2)=(x1-1)/(2-x2)^2

(Asked by: username_0 on 2022-06-12)

I got this utility function [![enter image description here](https://i.sstatic.net/Je5Q7.png)](https://i.sstatic.net/Je5Q7.png) representing certain preferences.
Are these preferences locally non satiated?
Can somebody please explain me with the exact definition of local non satiation why these preferences are locally non satiated or not?

Furthermore, consider that the choice set X= {(x1,x2) element of R2+ : x1>1, x2<2}

Thank you

**Tags:** utility, preferences

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## 1 answers

### Answer by username_1 on 2022-06-12. Score: 3 | Accepted answer

Preferences represented by $u:X\rightarrow\mathbb{R}$, where $X$ is a non-empty convex subset of $\mathbb{R}^2\_+$, satisfy LNS if

$(\forall (x, y)\in X)(\forall \epsilon > 0)(\exists \ (x',y')\in X)(\sqrt{(x-x')^2+(y-y')^2} <\epsilon \ \wedge u(x',y') > u(x,y)) $

Observe that for any given $y\in [0,2)$, $u(x, y) = \dfrac{x-1}{(2-y)^2}$ is strictly increasing in $x$ over $(1, \infty)$. So it satisfy LNS.


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