# Reference for monotonicity: $x\geq y\implies x\succsim y$ and $x>y\implies x\succ y$

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?

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