<p>There must be something wrong with your copy of Varian's book, here strong monotonicity is correctly written with a strict preference symbol:</p>
<p><a href="https://i.sstatic.net/mnTWg.png" rel="noreferrer"><img src="https://i.sstatic.net/mnTWg.png" alt="copy of Varian's definitions of weak and strong monotonicity" /></a></p>
<p>The term <em>non-satiation</em> is usually used differently than in your coursepack; the usual interpretation is that there is no bundle <span class="math-container">$x^*$</span> such that <span class="math-container">$x^*\succeq x$</span> for all <span class="math-container">$x$</span>; there is no &quot;bliss-point.&quot;</p>
<p>The way your coursepack uses the term, it is weaker than Varian's strong monotonicity (because the assumption <span class="math-container">$x&gt;y$</span> is weaker than the assumption <span class="math-container">$x\gg y$</span>).</p>
<p>Strong monotonicity always implies weak monotonicity. If preferences are continuous and satisfy the coursepack version of non-satiation, they also satisfy weak monotonicity. To see this, let <span class="math-container">$e=(1,1,\ldots,1)\in\mathbb{R}^n$</span>. If <span class="math-container">$x\geq y$</span>,
then <span class="math-container">$x+(1/n) e\gg y$</span>, so <span class="math-container">$x+(1/n) e\succeq y$</span>. Taking limits and using the continuity of <span class="math-container">$\succeq$</span>, we get <span class="math-container">$x\geq y$</span>.</p>
<p>It is possible to construct preferences that satisfy the coursepack version of non-satiation but not weak monotonicity. They can, of course, not be continuous. Here is an example with <span class="math-container">$n=2$</span>, a slight twist on <a href="https://en.wikipedia.org/wiki/Lexicographic_preferences" rel="noreferrer">lexicographic preferences</a>. We let <span class="math-container">$\succeq$</span> be defined by <span class="math-container">$(x_1,x_2)\succeq (y_1,y_2)$</span> if either <span class="math-container">$x_1&gt;y_1$</span> or <span class="math-container">$x_1=y_1$</span> and <span class="math-container">$x_2\leq y_2$</span>. Intuitively, more of good 1 is better, and more of good 2 is worse, but the amount of good 1 matters infinitely much more than good 2. These preferences satisfy the coursepack version of non-satiation but do not satisfy weak monotonicity.</p>

CC BY-SA 4.0
2023-05-11T14:28:12.840Z
Hi Michael, again another superb answer and so helpful:
1) Did you mean this the other way round? "because the assumption > is weaker than the assumption ≫". I would have thought the double arrow was a healer condition because it's more demanding? Also the context of what you say seems to imply you think my course-pack is weaker? And my course pack is the double arrow >>?
CormJack
CC BY-SA 4.0
2023-05-11T14:29:39.443Z
2) Also given that Varian doesn't actually state x > y anywhere. Are you taking the condition. x > y to mean   ≥  and  ≠ then >
CormJack
CC BY-SA 4.0
2023-05-11T14:40:47.187Z
^^As stated by my course pack. As in, I understand that out of the two definitions for non-satiation my course pack gives. This one is identical in meaning to to Varian's definition of strong montonoicty. It's just that Varian doesn't explicitly state that strong Monotonocity can be described as x > y.

Where as my course pack basically says x > y means [varian's definition of strong montonocoity] and this implies strict preference $x \succ y$. 

I'm just checking we are on the same page of using x > y for strong monotnocity. Which both sources agree implies strict preference $x \succ y$
CormJack
CC BY-SA 4.0
2023-05-11T14:53:57.077Z
3) Also could I get your thoughts on the fact that the course pack provides two different definitions of non-satiation.

(1) "The non-satiation assumption on preferences is that for any $x,y \in \mathbb{R^{n+}}$, if $x > y$ that is $x_i \ge y_i \; \forall i$  and $x_i > y_i$ for some $i$ then $x \succ y$"....Which I believe is equivalent to Varian Strong Montonicity. 

(2)"$x,y \in \mathbb{R^{n+}} \;\; x >>y$ that is $x_i > y_i \; \forall i\in [1,n]$ implies $x \succ y$"... which is  a stricter condition?
CormJack
CC BY-SA 4.0
2023-05-11T16:26:23.320Z
The condition $x\geq y$ and $x\neq y$ (which I indeed wrote as $x>y$) means that every coordinate of $x$ is at least as large as the corresponding coordinate of $y$ and at least one coordinate must be strictly larger. The condition $x\gg y$ means that every coordinate of $x$ is strictly larger than the corresponding coordinate of $y$, a stronger condition. Since it requires a stronger assumption, the coursepack condition is weaker. And yes, $\gg$ is a typographically nicer way to write $>>$.
Michael Greinecker
CC BY-SA 4.0
2023-05-11T16:28:20.337Z
And, yes, the first coursepack definition you cite in the comment is the same as Varian's strong monotonicity.  And the second one is weaker.
Michael Greinecker
CC BY-SA 4.0
2023-05-11T16:33:39.840Z
Hi Michael this is great, I think we are on the same page! And sorry I wasn't commenting on the way you typed >>, that must be miscommunication on my part. So to clarify. 1) The line in your answer "The way your course pack uses the term, it is weaker than Varian's strong monotonicity (because the assumption > is weaker than the assumption ≫)." Is stating that the assumption >> is stronger, hence the condition is weaker (because it's harder to satisfy).
CormJack
CC BY-SA 4.0
2023-05-11T16:41:47.137Z
Exactly, that's what I meant.
Michael Greinecker
CC BY-SA 4.0
2023-05-11T17:04:38.690Z
Fantastic. If you have any thoughts on  why / the value of providing two definitions of non-satiation in the course pack. Do let me know! Thanks again!
CormJack
CC BY-SA 4.0
2023-05-11T09:20:44.633Z
2023-05-11T09:30:09.273Z
2023-05-11T09:30:09.273Z
Michael Greinecker
<p>I understand that there is no universally agreed terminology for preference relations. However I need to pin down a definitive way to think about them (both for my exam, and my own sanity). Please can we help me decipher the following.</p>
<ul>
<li><em>Note: I apologise for the extended nature of this question. Hours of internet research, book reading, and past papers has still not given ultimate clarity. Please just answer individual components as and when you have time. Thanks!</em></li>
</ul>
<p>Using <em>Varian Microeconomic Analysis 3rd edition</em> as follows:</p>
<blockquote>
<p>p.96 states:</p>
<ul>
<li><strong>Weak Monotonicity:</strong> If <span class="math-container">$x \ge y$</span> then <span class="math-container">$x \succeq y$</span></li>
<li><strong>Strong Monotonicity:</strong> if <span class="math-container">$x \ge y$</span> and <span class="math-container">$x ≠ y$</span> then <span class="math-container">$x &gt; y$</span></li>
</ul>
</blockquote>
<ul>
<li><strong>Questions</strong></li>
<li>Why does <em>weak monotonicity</em> give me an inequality that implies a preference relation <span class="math-container">$x \succeq y$</span>,  but <em>Strong Monotonicity</em> gives me an inequality that just implies another inequality <span class="math-container">$x &gt; y$</span>.</li>
</ul>
<p>My course pack supposedly based of Varian states that:</p>
<blockquote>
<p><em>&quot;The non-satiation assumption on preferences is that for any <span class="math-container">$x,y \in \mathbb{R^{n+}}$</span>, if <span class="math-container">$x &gt; y$</span> that is <span class="math-container">$x_i \ge y_i \; \forall i$</span>  and <span class="math-container">$x_i &gt; y_i$</span> for some <span class="math-container">$i$</span> then <span class="math-container">$x \succ y$</span>&quot;</em></p>
</blockquote>
<ul>
<li><strong>Questions</strong></li>
<li>I understand how this definition is equivalent to Varian's Strong Monotonicity. However Varian makes no reference of the <span class="math-container">$\succ$</span> sign used here. Does Varian mean to use this? Why is one using <em>strict preference</em> and the other using <em>weak preference</em>?</li>
</ul>
<p>My course pack also gives another definition of Non-satiation at the very start that Varian makes no mention of:</p>
<blockquote>
<p><em>Preferences satisfy non-satiation if for any <span class="math-container">$x,y \in \mathbb{R^{n+}} \;\; x &gt;&gt;y$</span> that is <span class="math-container">$x_i &gt; y_i \; \forall i\in [1,n]$</span> implies <span class="math-container">$x \succ y$</span></em></p>
</blockquote>
<ul>
<li><strong>Questions</strong></li>
</ul>
<ol>
<li>How do these two definitions of non-satiation match up. Is one implicitly referring to  strong monotonicity and the second one weak monotonicity? How should i interpret this? Both course pack definitions claim to imply <span class="math-container">$\succ$</span> so how/why should I differentiate the two?</li>
<li>Both non-satiation definitions from my course pack end up with <span class="math-container">$x \succ y$</span> is there a condition which specifically references <span class="math-container">$x \succeq y$</span> or do i rely on something like  <span class="math-container">$x \succ y \implies x \succeq y$</span>. In general what's a guide for using <span class="math-container">$\succ$</span> vs <span class="math-container">$\succeq$</span> in the context of Varian and what i've shown here from my course pack.</li>
</ol>
<p><strong>Monotonicity, preference, and non-satiation</strong></p>
<blockquote>
<p>My course pack says <em>&quot;some books refer to <strong>non-satiation</strong> as 'more is better,' others use the mathematical term '<strong>monotonicity</strong>'&quot;</em></p>
</blockquote>
<p>I understand <strong>monotonicity</strong> to just be a bit like not changing direction/order i.e.</p>
<ul>
<li>Monotonically increasing - is a non decreasing function.</li>
</ul>
<p>And a <strong>Monotonic transformation</strong> is a transformation that preserves the order of the function it's transforming.</p>
<p>So i have 5 terms:</p>
<ul>
<li>Non-Satiation</li>
<li>Weak Monotonicity</li>
<li>Strong Monotonicity</li>
<li>Weak preference</li>
<li>Strict preference.</li>
</ul>
<p>Almost like a game of snap, Can you help me match them up, with their appropriate symbols? E.g. <em>Strong Monotonicity, implies weak preference, implies satiation??</em></p>
<p>Thanks!</p>

<mathematical-economics><consumer-theory><preferences>
Preference relations based on Varian
CormJack
# Preference relations based on Varian

(Asked by: username_0 on 2023-05-11)

I understand that there is no universally agreed terminology for preference relations. However I need to pin down a definitive way to think about them (both for my exam, and my own sanity). Please can we help me decipher the following.

* *Note: I apologise for the extended nature of this question. Hours of internet research, book reading, and past papers has still not given ultimate clarity. Please just answer individual components as and when you have time. Thanks!*

Using *Varian Microeconomic Analysis 3rd edition* as follows:

> p.96 states:
>
> * **Weak Monotonicity:** If $x \ge y$ then $x \succeq y$
> * **Strong Monotonicity:** if $x \ge y$ and $x ≠ y$ then $x > y$

* **Questions**
* Why does *weak monotonicity* give me an inequality that implies a preference relation $x \succeq y$, but *Strong Monotonicity* gives me an inequality that just implies another inequality $x > y$.

My course pack supposedly based of Varian states that:

> *"The non-satiation assumption on preferences is that for any $x,y \in \mathbb{R^{n+}}$, if $x > y$ that is $x\_i \ge y\_i \; \forall i$ and $x\_i > y\_i$ for some $i$ then $x \succ y$"*

* **Questions**
* I understand how this definition is equivalent to Varian's Strong Monotonicity. However Varian makes no reference of the $\succ$ sign used here. Does Varian mean to use this? Why is one using *strict preference* and the other using *weak preference*?

My course pack also gives another definition of Non-satiation at the very start that Varian makes no mention of:

> *Preferences satisfy non-satiation if for any $x,y \in \mathbb{R^{n+}} \;\; x >>y$ that is $x\_i > y\_i \; \forall i\in [1,n]$ implies $x \succ y$*

* **Questions**

1. How do these two definitions of non-satiation match up. Is one implicitly referring to strong monotonicity and the second one weak monotonicity? How should i interpret this? Both course pack definitions claim to imply $\succ$ so how/why should I differentiate the two?
2. Both non-satiation definitions from my course pack end up with $x \succ y$ is there a condition which specifically references $x \succeq y$ or do i rely on something like $x \succ y \implies x \succeq y$. In general what's a guide for using $\succ$ vs $\succeq$ in the context of Varian and what i've shown here from my course pack.

**Monotonicity, preference, and non-satiation**

> My course pack says *"some books refer to **non-satiation** as 'more is better,' others use the mathematical term '**monotonicity**'"*

I understand **monotonicity** to just be a bit like not changing direction/order i.e.

* Monotonically increasing - is a non decreasing function.

And a **Monotonic transformation** is a transformation that preserves the order of the function it's transforming.

So i have 5 terms:

* Non-Satiation
* Weak Monotonicity
* Strong Monotonicity
* Weak preference
* Strict preference.

Almost like a game of snap, Can you help me match them up, with their appropriate symbols? E.g. *Strong Monotonicity, implies weak preference, implies satiation??*

Thanks!

**Tags:** mathematical-economics, consumer-theory, preferences

---

## 1 answers

### Answer by username_1 on 2023-05-11. Score: 5 | Accepted answer

There must be something wrong with your copy of Varian's book, here strong monotonicity is correctly written with a strict preference symbol:

[![copy of Varian's definitions of weak and strong monotonicity](https://i.sstatic.net/mnTWg.png)](https://i.sstatic.net/mnTWg.png)

The term *non-satiation* is usually used differently than in your coursepack; the usual interpretation is that there is no bundle $x^\*$ such that $x^\*\succeq x$ for all $x$; there is no "bliss-point."

The way your coursepack uses the term, it is weaker than Varian's strong monotonicity (because the assumption $x>y$ is weaker than the assumption $x\gg y$).

Strong monotonicity always implies weak monotonicity. If preferences are continuous and satisfy the coursepack version of non-satiation, they also satisfy weak monotonicity. To see this, let $e=(1,1,\ldots,1)\in\mathbb{R}^n$. If $x\geq y$,
then $x+(1/n) e\gg y$, so $x+(1/n) e\succeq y$. Taking limits and using the continuity of $\succeq$, we get $x\geq y$.

It is possible to construct preferences that satisfy the coursepack version of non-satiation but not weak monotonicity. They can, of course, not be continuous. Here is an example with $n=2$, a slight twist on [lexicographic preferences](https://en.wikipedia.org/wiki/Lexicographic_preferences). We let $\succeq$ be defined by $(x\_1,x\_2)\succeq (y\_1,y\_2)$ if either $x\_1>y\_1$ or $x\_1=y\_1$ and $x\_2\leq y\_2$. Intuitively, more of good 1 is better, and more of good 2 is worse, but the amount of good 1 matters infinitely much more than good 2. These preferences satisfy the coursepack version of non-satiation but do not satisfy weak monotonicity.

> **Comments:**
> **username_0**: Hi Michael, again another superb answer and so helpful: 1) Did you mean this the other way round? "because the assumption > is weaker than the assumption ≫". I would have thought the double arrow was a healer condition because it's more demanding? Also the context of what you say seems to imply you think my course-pack is weaker? And my course pack is the double arrow >>?  
> **username_0**: 2) Also given that Varian doesn't actually state x > y anywhere. Are you taking the condition. x > y to mean   ≥  and  ≠ then >  
> **username_0**: ^^As stated by my course pack. As in, I understand that out of the two definitions for non-satiation my course pack gives. This one is identical in meaning to to Varian's definition of strong montonoicty. It's just that Varian doesn't explicitly state that strong Monotonocity can be described as x > y.  Where as my course pack basically says x > y means [varian's definition of strong montonocoity] and this implies strict preference $x \succ y$.   I'm just checking we are on the same page of using x > y for strong monotnocity. Which both sources agree implies strict preference $x \succ y$  
> **username_0**: 3) Also could I get your thoughts on the fact that the course pack provides two different definitions of non-satiation.  (1) "The non-satiation assumption on preferences is that for any $x,y \in \mathbb{R^{n+}}$, if $x > y$ that is $x_i \ge y_i \; \forall i$  and $x_i > y_i$ for some $i$ then $x \succ y$"....Which I believe is equivalent to Varian Strong Montonicity.   (2)"$x,y \in \mathbb{R^{n+}} \;\; x >>y$ that is $x_i > y_i \; \forall i\in [1,n]$ implies $x \succ y$"... which is  a stricter condition?  
> **username_1**: The condition $x\geq y$ and $x\neq y$ (which I indeed wrote as $x>y$) means that every coordinate of $x$ is at least as large as the corresponding coordinate of $y$ and at least one coordinate must be strictly larger. The condition $x\gg y$ means that every coordinate of $x$ is strictly larger than the corresponding coordinate of $y$, a stronger condition. Since it requires a stronger assumption, the coursepack condition is weaker. And yes, $\gg$ is a typographically nicer way to write $>>$.  
> **username_1**: And, yes, the first coursepack definition you cite in the comment is the same as Varian's strong monotonicity.  And the second one is weaker.  
> **username_0**: Hi Michael this is great, I think we are on the same page! And sorry I wasn't commenting on the way you typed >>, that must be miscommunication on my part. So to clarify. 1) The line in your answer "The way your course pack uses the term, it is weaker than Varian's strong monotonicity (because the assumption > is weaker than the assumption ≫)." Is stating that the assumption >> is stronger, hence the condition is weaker (because it's harder to satisfy).  
> **username_1**: Exactly, that's what I meant.  
> **username_0**: Fantastic. If you have any thoughts on  why / the value of providing two definitions of non-satiation in the course pack. Do let me know! Thanks again!  

---

