<p>Using <a href="https://en.wikipedia.org/wiki/Monotone_preferences" rel="nofollow noreferrer">this</a> definition of monotone preferences (Thanks to User <a href="https://economics.stackexchange.com/users/5145/art">Art</a>). Let <span class="math-container">$X\subseteq \mathbb{R}^n$</span> be the set of possible consumption bundles, and <span class="math-container">$x,y\in X$</span>. We have weak monotone preferences if <span class="math-container">$y&gt;&gt;x \implies y\succ x $</span>, and strict preferences if <span class="math-container">$y\geq x \implies y\succ x$</span>. If we have a utility function <span class="math-container">$u: X\to \mathbb{R}$</span>, then <span class="math-container">$y\succ x$</span> is the same as <span class="math-container">$u(y)&gt; u(x)$</span>. </p>

<p>Regarding notation for two bundles <span class="math-container">$x,y\in X, x=(x_1,x_2,...,x_n), y=(y_1,y_2,...,y_n)$</span>, we say <span class="math-container">$y&gt;&gt;x$</span> if <span class="math-container">$y_i&gt;x_i$</span> forall <span class="math-container">$i=1,2,...,n$</span> and we say <span class="math-container">$y\geq x$</span> if <span class="math-container">$y_i\geq x_i$</span> forall <span class="math-container">$i=1,2,...,n$</span> and there is a <span class="math-container">$j$</span>, such that <span class="math-container">$y_j&gt;x_j$</span>. (This notation is not universal) </p>

<p>Now, see if you can use this to answer your question</p>

CC BY-SA 4.0
2019-10-22T09:15:43.427Z
2019-10-22T09:15:43.427Z
Nikolaj Pedersen
<p>Given a function <span class="math-container">$u(x_1, x_2) = x_1 +x_2 + \min(2x_1, x_2)$</span>, how do we mathematically prove that it monotonic or not? </p>

<p>Is there is a general algebraic technique to show monotonicity of suchlike functions?</p>

CC BY-SA 4.0
2019-10-19T04:19:09.310Z
For problems like this, you always start with a definition. What's the definition of a monotonic transformation?
Art
CC BY-SA 4.0
2019-10-19T09:05:01.283Z
@Art I don't think monotonic transformation is what you meant. There is no need to transform anything.
Giskard
CC BY-SA 4.0
2019-10-19T09:52:47.327Z
@Giskard Oops my bad. [This](https://en.wikipedia.org/wiki/Monotone_preferences) is the definition I meant. The question implies that the function $u(x_1, x_2)$ could be said to be monotonic or not. My understanding is that a "monotonic function" applies to $f: \mathbb{R} \to \mathbb{R}$ only. So I guess what I'm suggesting is to change title to "Check if a utility function represents a monotone preference."
Art
<microeconomics><expected-utility>
Check if a utility function represents a monotone preference
Frodo Baggins
# Check if a utility function represents a monotone preference

(Asked by: username_0 on 2019-10-19)

Given a function $u(x\_1, x\_2) = x\_1 +x\_2 + \min(2x\_1, x\_2)$, how do we mathematically prove that it monotonic or not?

Is there is a general algebraic technique to show monotonicity of suchlike functions?

**Tags:** microeconomics, expected-utility
> **Comments:**
> **username_1**: For problems like this, you always start with a definition. What's the definition of a monotonic transformation?  
> **username_2**: @Art I don't think monotonic transformation is what you meant. There is no need to transform anything.  
> **username_1**: @Giskard Oops my bad. [This](https://en.wikipedia.org/wiki/Monotone_preferences) is the definition I meant. The question implies that the function $u(x_1, x_2)$ could be said to be monotonic or not. My understanding is that a "monotonic function" applies to $f: \mathbb{R} \to \mathbb{R}$ only. So I guess what I'm suggesting is to change title to "Check if a utility function represents a monotone preference."  

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## 1 answers

### Answer by username_3 on 2019-10-22. Score: 1

Using [this](https://en.wikipedia.org/wiki/Monotone_preferences) definition of monotone preferences (Thanks to User [Art](https://economics.stackexchange.com/users/5145/art)). Let $X\subseteq \mathbb{R}^n$ be the set of possible consumption bundles, and $x,y\in X$. We have weak monotone preferences if $y>>x \implies y\succ x $, and strict preferences if $y\geq x \implies y\succ x$. If we have a utility function $u: X\to \mathbb{R}$, then $y\succ x$ is the same as $u(y)> u(x)$.

Regarding notation for two bundles $x,y\in X, x=(x\_1,x\_2,...,x\_n), y=(y\_1,y\_2,...,y\_n)$, we say $y>>x$ if $y\_i>x\_i$ forall $i=1,2,...,n$ and we say $y\geq x$ if $y\_i\geq x\_i$ forall $i=1,2,...,n$ and there is a $j$, such that $y\_j>x\_j$. (This notation is not universal)

Now, see if you can use this to answer your question


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