<p>It's not necessarily a contradiction, but more like Varian's definition is more permissive, while Jehle's is stricter. (These 2 are indeed different definitions)</p>
<p>For Varian:</p>
<p>the preference relation ≻ is strictly monotonic for bundles  and , if  ≥  and and  ≠ , <em>then</em>  ≻ .</p>
<p>For Jehle:</p>
<p>if  ≥  then,  ≳  (weak preference). However, for strict preference  ≻ , Jehle requires  &gt;&gt; , which means that  has strictly more of every good than .</p>
<p>Random example:</p>
<p>If  = (2,1) and  = (1,1), Varian says  ≻  because  has more of the first group, even though its a tie for the second. However, Jehle wouldn't agree with  ≻  since  doesn't dominate  in the second good. I.e.  ≻  would only hold if  was (2,2) instead.</p>
<p>So yes your intuition is correct.</p>

CC BY-SA 4.0
2024-10-16T07:15:54.000Z
2024-10-16T07:15:54.000Z
Economic-Theory
<p>While studying microeconomics using many books at the same time (Mas-Colell, Kreps, Varian and Jehle) I have encountered a problem that I don't seem to understand.</p>
<p>Varian defines strong(or strict) monotonicity of a preference relation as:</p>
<p><span class="math-container">$$If \ \  x\geq y \ and \ x\neq y, \ then \ x \succ y.$$</span></p>
<p>Meaning that if the bundle x has at least as much for every good than y but they are not equal (meaning that x has more than y for at least one good), then x is stricly preferred to y.</p>
<p>But then Jehle (Advanced Microeconomic Theory) says that strict monotonicity is:</p>
<p><span class="math-container">$$If \  x\geq y, then \ x\gtrsim y, while \ if \ x\gg y, then \ x \succ y.  $$</span></p>
<p>Given that <span class="math-container">$x \gg y$</span> means that x contains stricly more than y for every good in the bundle.</p>
<p>That said, don't these definitions conflict? I know that conflicting definitions are not necessarily wrong, especially among different authors, but I am still confused wether I am just misinterpreting the definitions and they are the same or they are actually different.
<br />
Jehles' definition requires x to have more goods than y for every good so that you can say that x is preferred to y, while Varian's implies that x is preferred to y if it has more goods than y for at least one good, given that for all other goods x has at least as much as y.<br />
So, to Varian, if x = (1,2) and y = (1,1) and <span class="math-container">$\succeq$</span> is stricly monotonic, you can already say that x is preferred to y, while in Jehles' definition you can't.</p>
<p>Am I right about the different definitions? Is this actually a thing? Varian's definition seems to make more sense to me.</p>
<p>Obs: I'am an undergrad in economics introducing myself to intermediate microeconomics, so please be kind lol. Thanks</p>

CC BY-SA 4.0
2024-10-13T21:21:35.187Z
They are, indeed, not equivalent. The utility function given by $u(x,y)=x$ represents preferences that are strictly monotone in the sense of Jehle and Reny but not strongly monotone in Varian's sense.
Michael Greinecker
<microeconomics>
Different Concepts of Strict Monotonicity of Preferences?
zzaeg
# Different Concepts of Strict Monotonicity of Preferences?

(Asked by: username_0 on 2024-10-13)

While studying microeconomics using many books at the same time (Mas-Colell, Kreps, Varian and Jehle) I have encountered a problem that I don't seem to understand.

Varian defines strong(or strict) monotonicity of a preference relation as:

$$If \ \ x\geq y \ and \ x\neq y, \ then \ x \succ y.$$

Meaning that if the bundle x has at least as much for every good than y but they are not equal (meaning that x has more than y for at least one good), then x is stricly preferred to y.

But then Jehle (Advanced Microeconomic Theory) says that strict monotonicity is:

$$If \ x\geq y, then \ x\gtrsim y, while \ if \ x\gg y, then \ x \succ y. $$

Given that $x \gg y$ means that x contains stricly more than y for every good in the bundle.

That said, don't these definitions conflict? I know that conflicting definitions are not necessarily wrong, especially among different authors, but I am still confused wether I am just misinterpreting the definitions and they are the same or they are actually different.
  
Jehles' definition requires x to have more goods than y for every good so that you can say that x is preferred to y, while Varian's implies that x is preferred to y if it has more goods than y for at least one good, given that for all other goods x has at least as much as y.  
So, to Varian, if x = (1,2) and y = (1,1) and $\succeq$ is stricly monotonic, you can already say that x is preferred to y, while in Jehles' definition you can't.

Am I right about the different definitions? Is this actually a thing? Varian's definition seems to make more sense to me.

Obs: I'am an undergrad in economics introducing myself to intermediate microeconomics, so please be kind lol. Thanks

**Tags:** microeconomics

> **Comments:**
> **username_1**: They are, indeed, not equivalent. The utility function given by $u(x,y)=x$ represents preferences that are strictly monotone in the sense of Jehle and Reny but not strongly monotone in Varian's sense.  

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## 1 answers

### Answer by username_2 on 2024-10-16. Score: 2 | Accepted answer

It's not necessarily a contradiction, but more like Varian's definition is more permissive, while Jehle's is stricter. (These 2 are indeed different definitions)

For Varian:

the preference relation ≻ is strictly monotonic for bundles and , if ≥ and and ≠ , *then* ≻ .

For Jehle:

if ≥ then, ≳ (weak preference). However, for strict preference ≻ , Jehle requires >> , which means that has strictly more of every good than .

Random example:

If = (2,1) and = (1,1), Varian says ≻ because has more of the first group, even though its a tie for the second. However, Jehle wouldn't agree with ≻ since doesn't dominate in the second good. I.e. ≻ would only hold if was (2,2) instead.

So yes your intuition is correct.


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