<p>Consider a locally non-satiated preference represented by the utility function
<span class="math-container">\begin{equation}
u(x,y)=-(x-4)^2(y-2)^2
\end{equation}</span>
It is easy to verify that <span class="math-container">$u(4,2)=0&gt;-9=u(1,3)$</span>. Therefore <span class="math-container">$B$</span> is strictly preferred to <span class="math-container">$A$</span>.</p>
<p>Likewise, you can construct a locally non-satiated preference, one represented by <span class="math-container">$$u(x,y)=-(x-1)^2(y-3)^2,$$</span> under which <span class="math-container">$A$</span> is strictly preferred to <span class="math-container">$B$</span>.</p>
<hr />
<p>What local non-satiation does is that it rules out indifference &quot;zones&quot;, so that we're left with indifference <em>curves</em>. Hence, with local non-satiation alone, it's impossible to tell whether one alternative is preferred to another.</p>
<p>The following figures from Jehle and Reny (2011) illustrate preferences with local non-satiation (Fig 1.3) and without it (Fig 1.2).</p>
<p><a href="https://i.sstatic.net/N8Hjl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/N8Hjl.png" alt="enter image description here" /></a></p>
<p><sub>Axioms 1 through 4' are completeness, transitivity, continuity, and local non-satiation, respectively.</sub></p>

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2019-03-09T20:40:38.223Z
2019-03-09T20:50:45.810Z
2020-06-17T09:40:54.643Z
Herr K.
<p>If a consumer has locally non-satiated preferences, which of these 2 bundles is preferred and why?</p>

<p>Bundle A: (1,3)</p>

<p>Bundle B: (4,2)</p>

<p>This is what I've reasoned from my very limited understanding of LNS preferences. </p>

<p>The distance between the two bundles has to be less than the radius of a ball around one of the bundles for the other bundle to be preferred. A ball around bundle A can't have a radius larger than 1, thus bundle B isn't preferred. However, a ball around bundle B can have a radius large enough so that Bundle A is within the ball. So bundle A is preferred.</p>

<p>I'm very unsure of this reasoning and I'd appreciate it if anyone could correct me where I'm wrong or help improve my answer.</p>

CC BY-SA 4.0
2019-03-09T17:57:00.810Z
Are there other assumptions/conditions you're not including here? With just the information given, both A and B could be preferred to the other.
Herr K.
CC BY-SA 4.0
2019-03-09T18:32:13.177Z
There were no other assumptions/conditions included in the question. How could B be preferred to A?
Aude
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2019-03-09T18:40:26.987Z
If A is at the point (1,3), it can't have a ball with a radius greater than 1. And for B to be preferred wouldn't it have to within the ball?
Aude
<microeconomics><consumer-theory><preferences><consumption>
Question about Locally non-satiated preferences
Aude
# Question about Locally non-satiated preferences

(Asked by: username_0 on 2019-03-09)

If a consumer has locally non-satiated preferences, which of these 2 bundles is preferred and why?

Bundle A: (1,3)

Bundle B: (4,2)

This is what I've reasoned from my very limited understanding of LNS preferences.

The distance between the two bundles has to be less than the radius of a ball around one of the bundles for the other bundle to be preferred. A ball around bundle A can't have a radius larger than 1, thus bundle B isn't preferred. However, a ball around bundle B can have a radius large enough so that Bundle A is within the ball. So bundle A is preferred.

I'm very unsure of this reasoning and I'd appreciate it if anyone could correct me where I'm wrong or help improve my answer.

**Tags:** microeconomics, consumer-theory, preferences, consumption

> **Comments:**
> **username_1**: Are there other assumptions/conditions you're not including here? With just the information given, both A and B could be preferred to the other.  
> **username_0**: There were no other assumptions/conditions included in the question. How could B be preferred to A?  
> **username_0**: If A is at the point (1,3), it can't have a ball with a radius greater than 1. And for B to be preferred wouldn't it have to within the ball?  

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## 1 answers

### Answer by username_1 on 2019-03-09. Score: 1

Consider a locally non-satiated preference represented by the utility function
\begin{equation}
u(x,y)=-(x-4)^2(y-2)^2
\end{equation}
It is easy to verify that $u(4,2)=0>-9=u(1,3)$. Therefore $B$ is strictly preferred to $A$.

Likewise, you can construct a locally non-satiated preference, one represented by $$u(x,y)=-(x-1)^2(y-3)^2,$$ under which $A$ is strictly preferred to $B$.

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What local non-satiation does is that it rules out indifference "zones", so that we're left with indifference *curves*. Hence, with local non-satiation alone, it's impossible to tell whether one alternative is preferred to another.

The following figures from Jehle and Reny (2011) illustrate preferences with local non-satiation (Fig 1.3) and without it (Fig 1.2).

[![enter image description here](https://i.sstatic.net/N8Hjl.png)](https://i.sstatic.net/N8Hjl.png)

Axioms 1 through 4' are completeness, transitivity, continuity, and local non-satiation, respectively.


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