<p>MWG defines two types of montone preferences (definition 3.B.2).</p>
<p>For a first they define preferences to be <strong>montone</strong> if <span class="math-container">$y \gg x$</span> implies that <span class="math-container">$y \succ x$</span>. Here <span class="math-container">$y \gg x$</span> means that every component of the vector <span class="math-container">$y$</span> is strictly larger than every corresponding component of the vector <span class="math-container">$x$</span>. (e.g. <span class="math-container">$y = [1,1]$</span> and <span class="math-container">$x = [0.5, 0]$</span>)</p>
<p>For the second, they define preferences to be <strong>strongly monotone</strong> if <span class="math-container">$y \ge x$</span> and <span class="math-container">$x \ne y$</span> then <span class="math-container">$y \succ x$</span>. This means that <span class="math-container">$y$</span> contains at least as much of every component as <span class="math-container">$x$</span> but is not equal to <span class="math-container">$x$</span> (so one component should be strictly larger. For example if <span class="math-container">$y = [1,1]$</span> and <span class="math-container">$x = [1,0]$</span>).</p>
<p>From your question, it is unclear what you mean by <span class="math-container">$y &gt; x$</span>.</p>
<ul>
<li>Does you mean that <span class="math-container">$y \ge x$</span> and <span class="math-container">$x \ne y$</span>, then both your definitions are basically the definition of strict monotonicity as in MWG (as <span class="math-container">$x \ge y$</span> and not <span class="math-container">$x &gt; y$</span> is the same as <span class="math-container">$x = y$</span>).</li>
<li>Does it mean that <span class="math-container">$y \gg x$</span>, then your second definition boils down to the definition of monotonicity as in MWG.</li>
</ul>

CC BY-SA 4.0
2023-08-16T15:05:54.893Z
$y>x$ means $y>>x$. I am interested in the reference for the first definition in the question.
High GPA
CC BY-SA 4.0
2023-08-17T07:13:13.623Z
@HighGPA The handbook of Jehle and Reny (Advanced Microeconomic Theory) uses your first definition (axiom 4 on page 10, I have the 3rd edition) and call it Strict Monotonicity. Maybe you can use this as a reference.
tdm
CC BY-SA 4.0
2023-08-17T07:46:54.997Z
This is great!!
High GPA
CC BY-SA 4.0
2023-08-16T13:22:56.640Z
2023-08-16T13:22:56.640Z
tdm
<p>I've seen this definition for monotonicity many times on different papers and on this site:</p>
<blockquote>
<p><span class="math-container">$x\geq y\implies x\succsim y$</span> and <span class="math-container">$x&gt;&gt;y\implies x\succ y$</span>.</p>
</blockquote>
<p>However, what I read on MWG's microeconomic theory textbook is:</p>
<blockquote>
<p>Definition of monotonicity: <span class="math-container">$x&gt;&gt;y\implies x\succ y$</span>.</p>
</blockquote>
<p>Here <span class="math-container">$x&gt;&gt;y$</span> means <span class="math-container">$x_i&gt;y_i$</span> for all coordinates <span class="math-container">$i$</span>.</p>
<p>My question: what some famous <strong>references</strong> for the first definition of monotonicity?</p>

<microeconomics><reference-request><preferences><theory>
Reference for monotonicity: $x\geq y\implies x\succsim y$ and $x>y\implies x\succ y$
High GPA
# Reference for monotonicity: $x\geq y\implies x\succsim y$ and $x>y\implies x\succ y$

(Asked by: username_0 on 2023-08-16)

I've seen this definition for monotonicity many times on different papers and on this site:

> $x\geq y\implies x\succsim y$ and $x>>y\implies x\succ y$.

However, what I read on MWG's microeconomic theory textbook is:

> Definition of monotonicity: $x>>y\implies x\succ y$.

Here $x>>y$ means $x\_i>y\_i$ for all coordinates $i$.

My question: what some famous **references** for the first definition of monotonicity?

**Tags:** microeconomics, reference-request, preferences, theory


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## 1 answers

### Answer by username_1 on 2023-08-16. Score: 2

MWG defines two types of montone preferences (definition 3.B.2).

For a first they define preferences to be **montone** if $y \gg x$ implies that $y \succ x$. Here $y \gg x$ means that every component of the vector $y$ is strictly larger than every corresponding component of the vector $x$. (e.g. $y = [1,1]$ and $x = [0.5, 0]$)

For the second, they define preferences to be **strongly monotone** if $y \ge x$ and $x \ne y$ then $y \succ x$. This means that $y$ contains at least as much of every component as $x$ but is not equal to $x$ (so one component should be strictly larger. For example if $y = [1,1]$ and $x = [1,0]$).

From your question, it is unclear what you mean by $y > x$.

* Does you mean that $y \ge x$ and $x \ne y$, then both your definitions are basically the definition of strict monotonicity as in MWG (as $x \ge y$ and not $x > y$ is the same as $x = y$).
* Does it mean that $y \gg x$, then your second definition boils down to the definition of monotonicity as in MWG.

> **Comments:**
> **username_0**: $y>x$ means $y>>x$. I am interested in the reference for the first definition in the question.  
> **username_1**: @HighGPA The handbook of Jehle and Reny (Advanced Microeconomic Theory) uses your first definition (axiom 4 on page 10, I have the 3rd edition) and call it Strict Monotonicity. Maybe you can use this as a reference.  
> **username_0**: This is great!!  

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