# Preference relations based on Varian

I understand that there is no universally agreed terminology for preference relations. However I need to pin down a definitive way to think about them (both for my exam, and my own sanity). Please can we help me decipher the following.

• Note: I apologise for the extended nature of this question. Hours of internet research, book reading, and past papers has still not given ultimate clarity. Please just answer individual components as and when you have time. Thanks!

Using Varian Microeconomic Analysis 3rd edition as follows:

p.96 states:

• Weak Monotonicity: If $$x \ge y$$ then $$x \succeq y$$
• Strong Monotonicity: if $$x \ge y$$ and $$x ≠ y$$ then $$x > y$$
• Questions
• Why does weak monotonicity give me an inequality that implies a preference relation $$x \succeq y$$, but Strong Monotonicity gives me an inequality that just implies another inequality $$x > y$$.

My course pack supposedly based of Varian states that:

"The non-satiation assumption on preferences is that for any $$x,y \in \mathbb{R^{n+}}$$, if $$x > y$$ that is $$x_i \ge y_i \; \forall i$$ and $$x_i > y_i$$ for some $$i$$ then $$x \succ y$$"

• Questions
• I understand how this definition is equivalent to Varian's Strong Monotonicity. However Varian makes no reference of the $$\succ$$ sign used here. Does Varian mean to use this? Why is one using strict preference and the other using weak preference?

My course pack also gives another definition of Non-satiation at the very start that Varian makes no mention of:

Preferences satisfy non-satiation if for any $$x,y \in \mathbb{R^{n+}} \;\; x >>y$$ that is $$x_i > y_i \; \forall i\in [1,n]$$ implies $$x \succ y$$

• Questions
1. How do these two definitions of non-satiation match up. Is one implicitly referring to strong monotonicity and the second one weak monotonicity? How should i interpret this? Both course pack definitions claim to imply $$\succ$$ so how/why should I differentiate the two?
2. Both non-satiation definitions from my course pack end up with $$x \succ y$$ is there a condition which specifically references $$x \succeq y$$ or do i rely on something like $$x \succ y \implies x \succeq y$$. In general what's a guide for using $$\succ$$ vs $$\succeq$$ in the context of Varian and what i've shown here from my course pack.

Monotonicity, preference, and non-satiation

My course pack says "some books refer to non-satiation as 'more is better,' others use the mathematical term 'monotonicity'"

I understand monotonicity to just be a bit like not changing direction/order i.e.

• Monotonically increasing - is a non decreasing function.

And a Monotonic transformation is a transformation that preserves the order of the function it's transforming.

So i have 5 terms:

• Non-Satiation
• Weak Monotonicity
• Strong Monotonicity
• Weak preference
• Strict preference.

Almost like a game of snap, Can you help me match them up, with their appropriate symbols? E.g. Strong Monotonicity, implies weak preference, implies satiation??

Thanks!

There must be something wrong with your copy of Varian's book, here strong monotonicity is correctly written with a strict preference symbol:

The term non-satiation is usually used differently than in your coursepack; the usual interpretation is that there is no bundle $$x^*$$ such that $$x^*\succeq x$$ for all $$x$$; there is no "bliss-point."

The way your coursepack uses the term, it is weaker than Varian's strong monotonicity (because the assumption $$x>y$$ is weaker than the assumption $$x\gg y$$).

Strong monotonicity always implies weak monotonicity. If preferences are continuous and satisfy the coursepack version of non-satiation, they also satisfy weak monotonicity. To see this, let $$e=(1,1,\ldots,1)\in\mathbb{R}^n$$. If $$x\geq y$$, then $$x+(1/n) e\gg y$$, so $$x+(1/n) e\succeq y$$. Taking limits and using the continuity of $$\succeq$$, we get $$x\geq y$$.

It is possible to construct preferences that satisfy the coursepack version of non-satiation but not weak monotonicity. They can, of course, not be continuous. Here is an example with $$n=2$$, a slight twist on lexicographic preferences. We let $$\succeq$$ be defined by $$(x_1,x_2)\succeq (y_1,y_2)$$ if either $$x_1>y_1$$ or $$x_1=y_1$$ and $$x_2\leq y_2$$. Intuitively, more of good 1 is better, and more of good 2 is worse, but the amount of good 1 matters infinitely much more than good 2. These preferences satisfy the coursepack version of non-satiation but do not satisfy weak monotonicity.

• Hi Michael, again another superb answer and so helpful: 1) Did you mean this the other way round? "because the assumption 𝑥>𝑦 is weaker than the assumption 𝑥≫𝑦". I would have thought the double arrow was a healer condition because it's more demanding? Also the context of what you say seems to imply you think my course-pack is weaker? And my course pack is the double arrow >>? Commented May 11, 2023 at 14:28
• 2) Also given that Varian doesn't actually state x > y anywhere. Are you taking the condition. x > y to mean 𝑥 ≥ 𝑦 and 𝑥≠𝑦 then 𝑥>𝑦 Commented May 11, 2023 at 14:29
• ^^As stated by my course pack. As in, I understand that out of the two definitions for non-satiation my course pack gives. This one is identical in meaning to to Varian's definition of strong montonoicty. It's just that Varian doesn't explicitly state that strong Monotonocity can be described as x > y. Where as my course pack basically says x > y means [varian's definition of strong montonocoity] and this implies strict preference $x \succ y$. I'm just checking we are on the same page of using x > y for strong monotnocity. Which both sources agree implies strict preference $x \succ y$ Commented May 11, 2023 at 14:40
• The condition $x\geq y$ and $x\neq y$ (which I indeed wrote as $x>y$) means that every coordinate of $x$ is at least as large as the corresponding coordinate of $y$ and at least one coordinate must be strictly larger. The condition $x\gg y$ means that every coordinate of $x$ is strictly larger than the corresponding coordinate of $y$, a stronger condition. Since it requires a stronger assumption, the coursepack condition is weaker. And yes, $\gg$ is a typographically nicer way to write $>>$. Commented May 11, 2023 at 16:26
• And, yes, the first coursepack definition you cite in the comment is the same as Varian's strong monotonicity. And the second one is weaker. Commented May 11, 2023 at 16:28