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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?

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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.
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  • $\begingroup$ $y>x$ means $y>>x$. I am interested in the reference for the first definition in the question. $\endgroup$
    – High GPA
    Aug 16, 2023 at 15:05
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    $\begingroup$ @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. $\endgroup$
    – tdm
    Aug 17, 2023 at 7:13
  • $\begingroup$ This is great!! $\endgroup$
    – High GPA
    Aug 17, 2023 at 7:46

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