<p>The Wikipedia article shows it graphically.  But here, you're leaving out the important bit:  "The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it." This is the statement that you want to connect with the math.</p>

<p>So how does this relate? Let's go bit-by-bit. Refer to the graph on the Wikipedia page as needed.</p>

<p>-"for any bundle of goods" means "choose any bundle <span class="math-container">$x \in X$</span>, where X is the box in the Wikipedia graph.</p>

<p>-"arbitrarily close" means "choose any <span class="math-container">$\epsilon &gt; 0, \epsilon \in \mathbb{R}$</span>".  So, pick any non-zero real number as small or as large as you like.</p>

<p>-"there is always another bundle of goods arbitrarily close" means "given <span class="math-container">$x$</span> and <span class="math-container">$\epsilon$</span>, you can always find a bundle <span class="math-container">$y \in X$</span> that is within a distance <span class="math-container">$\epsilon$</span> from <span class="math-container">$x%$</span>." This is expressed as <span class="math-container">$||y-x|| \le \epsilon$</span>.</p>

<p>-"is preferred to it" means that for any utility function <span class="math-container">$U(x)$</span>, we have <span class="math-container">$U(y) &gt; U(x)$</span>.</p>

<p>Putting this all together, "non-satiation" means that (without a budget constraint) an agent can "never be satisfied" because no matter what bundle of goods they have (<span class="math-container">$x$</span>) there is always a <span class="math-container">$y$</span> that yields greater utility. "Local" means that <span class="math-container">$y$</span> and <span class="math-container">$x$</span> can be very similar (ie arbitrarily close to each other in <span class="math-container">$X$</span>).</p>

CC BY-SA 4.0
2019-10-21T15:02:58.740Z
2019-10-21T15:02:58.740Z
heh
<p>I am having trouble completely understanding the mathematical definition of non-satiation. I have stated the definition from Wikipedia below. It would be great if someone can graphically explain.</p>

<blockquote>
  <p>Formally, if <span class="math-container">$X$</span> is the consumption set, then for any <span class="math-container">$x \in X$</span> and every <span class="math-container">$\varepsilon &gt; 0$</span>, there exists <span class="math-container">$y \in X$</span> such that <span class="math-container">$||y-x|| \leq \varepsilon$</span> and <span class="math-container">$y$</span> is preferred to <span class="math-container">$x$</span>.</p>
</blockquote>

CC BY-SA 4.0
2019-10-18T06:05:57.867Z
Which part exactly is causing you difficulties? Any explanation is bound to repeat the definition in some way.
Giskard
CC BY-SA 4.0
2019-10-18T14:16:06.683Z
I am not able to fully understand how he part that states |y-x|< epsilon is related to the definition. Also, if are given a function, say, F(u,v) = uv. How do we mathematically show if it is locally non-satiated or not.
Frodo Baggins
<microeconomics><expected-utility>
Local non-satiation in economics
Frodo Baggins
# Local non-satiation in economics

(Asked by: username_0 on 2019-10-18)

I am having trouble completely understanding the mathematical definition of non-satiation. I have stated the definition from Wikipedia below. It would be great if someone can graphically explain.

> Formally, if $X$ is the consumption set, then for any $x \in X$ and every $\varepsilon > 0$, there exists $y \in X$ such that $||y-x|| \leq \varepsilon$ and $y$ is preferred to $x$.

**Tags:** microeconomics, expected-utility
> **Comments:**
> **username_1**: Which part exactly is causing you difficulties? Any explanation is bound to repeat the definition in some way.  
> **username_0**: I am not able to fully understand how he part that states |y-x|< epsilon is related to the definition. Also, if are given a function, say, F(u,v) = uv. How do we mathematically show if it is locally non-satiated or not.  

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## 1 answers

### Answer by username_2 on 2019-10-21. Score: 2 | Accepted answer

The Wikipedia article shows it graphically. But here, you're leaving out the important bit: "The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it." This is the statement that you want to connect with the math.

So how does this relate? Let's go bit-by-bit. Refer to the graph on the Wikipedia page as needed.

-"for any bundle of goods" means "choose any bundle $x \in X$, where X is the box in the Wikipedia graph.

-"arbitrarily close" means "choose any $\epsilon > 0, \epsilon \in \mathbb{R}$". So, pick any non-zero real number as small or as large as you like.

-"there is always another bundle of goods arbitrarily close" means "given $x$ and $\epsilon$, you can always find a bundle $y \in X$ that is within a distance $\epsilon$ from $x%$." This is expressed as $||y-x|| \le \epsilon$.

-"is preferred to it" means that for any utility function $U(x)$, we have $U(y) > U(x)$.

Putting this all together, "non-satiation" means that (without a budget constraint) an agent can "never be satisfied" because no matter what bundle of goods they have ($x$) there is always a $y$ that yields greater utility. "Local" means that $y$ and $x$ can be very similar (ie arbitrarily close to each other in $X$).


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