<p>Monotonicity is in economics defined the same way as in mathematics. According to Pemberton and Rau, Mathematics for Economists, pp 130;</p>
<blockquote>
<p>function <span class="math-container">$f$</span> is said to be non-decreasing if <span class="math-container">$f(x_1) \leq f(x_2)$</span> whenever <span class="math-container">$x_1&lt;x_2$</span>. ...  <span class="math-container">$f$</span> is said to be <strong>non-increasing</strong> if <span class="math-container">$f(x_1)\geq f(x_2)$</span> whenever <span class="math-container">$x_1&gt;x_2$</span>. A non-increasing or non-decreasing function is said to be <strong>weakly monotonic</strong>.</p>
</blockquote>
<p>The definition is thus exactly the same (the wolfram alpha also talks about weak monotonicity). So in economics the definition of monotonic function exactly matches the definition in mathematics.</p>
<p>What might confuse you that in economics some functions already have restrictions put at them. For example, in most branches of economics, by default, it is assumed that <span class="math-container">$U'&gt;0$</span>. Hence any monotonic utility function is by definition increasing monotonic function because decreasing utility is not allowed by assumption.  There are some subfields of economics where these assumptions might be relaxed, but by default any utility function should be assumed to be increasing function.</p>
<p>So monotonic utility function is not non-decreasing because in economics monotonicity is redefined. Monotonic utility is non-decreasing, because it is a utility function.</p>
<hr />
<p>PS: Random working paper is also not good evidence for something applying to a whole field.</p>

CC BY-SA 4.0
2023-10-24T21:06:36.120Z
MAke sense!!!!!
dodo
CC BY-SA 4.0
2023-10-24T23:20:57.170Z
I think the P.S. part is uncalled for.
Michael Greinecker
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2023-10-24T23:31:53.813Z
@MichaelGreinecker I toned it down
1muflon1
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2023-10-24T20:51:17.467Z
2023-10-24T23:31:46.177Z
2023-10-24T23:31:46.177Z
1muflon1
<p>Obviously a monotonic function can be either nondecreasing or nonincreasing:<a href="https://i.sstatic.net/O3nBy.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/O3nBy.png" alt="enter image description here" /></a></p>
<p>However, in Economics, a quick google search gives:</p>
<p><a href="https://i.sstatic.net/DVkVt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/DVkVt.png" alt="enter image description here" /></a></p>
<p>I am interested in the history or the motivation behind the econ definition that deviate from the definition in math. Why economists invent their own math terms?</p>

CC BY-SA 4.0
2023-10-24T14:53:58.267Z
What's the invented term that deviates from the mathematical definition? You haven't presented any other definition of monotonic here beside than the usual mathematical meaning of "nonincreasing or nondecreasing".
Nuclear Hoagie
CC BY-SA 4.0
2023-10-24T15:15:18.413Z
Hi @Dodo. Regarding monotonicity of preferences, you must recognise that this is a choice-theoretical term. In other words, monotonic preferences draws on the fact that when people make choices about economic "goods." Since there economic goods (and not bads), we assume that welfare is at least increasing when any of these goods is increased. https://felixmunozgarcia.files.wordpress.com/2020/08/intermediate-micro-chapter2.pdf
EB3112
CC BY-SA 4.0
2023-10-24T15:29:13.477Z
@NuclearHoagie In econ, a monotonic utility function can only be nondecreasing, not nonincreasing.
dodo
CC BY-SA 4.0
2023-10-24T15:44:05.137Z
@dodo If the utility function is nondecreasing, it is absolutely correct to describe it as monotonic. Monotonic alone doesn't imply directionality, it must be the domain knowledge of the economic utility function that makes it nondecreasing. I don't think this is a re-definition of "monotonic", it's an additional specification of directionality by including "utility function" in a particular context. It would defy the common notion of "utility" to have more of something be less useful.
Nuclear Hoagie
CC BY-SA 4.0
2023-10-24T16:23:47.343Z
I usually read about increasing utility functions and monotonic preferences.
Michael Greinecker
CC BY-SA 4.0
2023-10-24T16:55:01.060Z
Could you please provide examples where economic definition deviates from the mathematical one?
1muflon1
CC BY-SA 4.0
2023-10-24T19:38:49.527Z
@MichaelGreinecker You are right. That's true. After further research, it is true that more than 50% of economists use "increasing utility" rather than "monotonic utility".
dodo
<microeconomics><utility><preferences>
Why is monotonic preference and monotonic utility function non-decreasing?
dodo
# Why is monotonic preference and monotonic utility function non-decreasing?

(Asked by: username_0 on 2023-10-24)

Obviously a monotonic function can be either nondecreasing or nonincreasing:[![enter image description here](https://i.sstatic.net/O3nBy.png)](https://i.sstatic.net/O3nBy.png)

However, in Economics, a quick google search gives:

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

I am interested in the history or the motivation behind the econ definition that deviate from the definition in math. Why economists invent their own math terms?

**Tags:** microeconomics, utility, preferences

> **Comments:**
> **username_1**: What's the invented term that deviates from the mathematical definition? You haven't presented any other definition of monotonic here beside than the usual mathematical meaning of "nonincreasing or nondecreasing".  
> **username_2**: Hi @username_0. Regarding monotonicity of preferences, you must recognise that this is a choice-theoretical term. In other words, monotonic preferences draws on the fact that when people make choices about economic "goods." Since there economic goods (and not bads), we assume that welfare is at least increasing when any of these goods is increased. https://felixmunozgarcia.files.wordpress.com/2020/08/intermediate-micro-chapter2.pdf  
> **username_0**: @username_1 In econ, a monotonic utility function can only be nondecreasing, not nonincreasing.  
> **username_1**: @username_0 If the utility function is nondecreasing, it is absolutely correct to describe it as monotonic. Monotonic alone doesn't imply directionality, it must be the domain knowledge of the economic utility function that makes it nondecreasing. I don't think this is a re-definition of "monotonic", it's an additional specification of directionality by including "utility function" in a particular context. It would defy the common notion of "utility" to have more of something be less useful.  
> **username_3**: I usually read about increasing utility functions and monotonic preferences.  
> **username_4**: Could you please provide examples where economic definition deviates from the mathematical one?  
> **username_0**: @username_3 You are right. That's true. After further research, it is true that more than 50% of economists use "increasing utility" rather than "monotonic utility".  

---

## 1 answers

### Answer by username_4 on 2023-10-24. Score: 2 | Accepted answer

Monotonicity is in economics defined the same way as in mathematics. According to Pemberton and Rau, Mathematics for Economists, pp 130;

> function $f$ is said to be non-decreasing if $f(x\_1) \leq f(x\_2)$ whenever $x\_1<x\_2$. ... $f$ is said to be **non-increasing** if $f(x\_1)\geq f(x\_2)$ whenever $x\_1>x\_2$. A non-increasing or non-decreasing function is said to be **weakly monotonic**.

The definition is thus exactly the same (the wolfram alpha also talks about weak monotonicity). So in economics the definition of monotonic function exactly matches the definition in mathematics.

What might confuse you that in economics some functions already have restrictions put at them. For example, in most branches of economics, by default, it is assumed that $U'>0$. Hence any monotonic utility function is by definition increasing monotonic function because decreasing utility is not allowed by assumption. There are some subfields of economics where these assumptions might be relaxed, but by default any utility function should be assumed to be increasing function.

So monotonic utility function is not non-decreasing because in economics monotonicity is redefined. Monotonic utility is non-decreasing, because it is a utility function.

---

PS: Random working paper is also not good evidence for something applying to a whole field.

> **Comments:**
> **username_0**: MAke sense!!!!!  
> **username_3**: I think the P.S. part is uncalled for.  
> **username_4**: @username_3 I toned it down  

---

