Axiom: More is Better; But when is more better?

Question

I'm taking an introductory microeconomics course and have been introduced to the 3 axioms of economic preferences. These include

• Completeness
• Transitivity
• Non-satiation

My understanding of non-satiation is incomplete. This is because of how the lecturer has described it. I am told that non-satiation means

1. More is better
2. Indifference curves further from the origin are better

From some research on the internet I find that non-satiation is also known as the Axiom of Dominance, described as:

If any combination A has more of one or of both the goods than B, then it is said A dominates B. This axiom states that if A dominates B, then the consumer will prefer A to B. This axiom is also known as the axiom of non-satiation or of monotonicity.

Given this, my understanding is that given 2 bundles $A(x,y)$ and $B(a,b)$, $A > B$ if $x > a$ and $y > b$.

My questions:

• Q1) Is my understanding correct?
• Q2) What if $x<a$ but $y>b$? Is the agent indifferent? He will have to be, otherwise completeness will be violated?

Thanks for your help.

Answer 1

You seem to be talking about (strong) monotonicity rather than non-satiation (which is closely related to but different from monotonicity).

(Strong) monotonicity says that we prefer one bundle to another if the first bundle has strictly more of at least one good and no less of any good. A bit more precisely and in a two-good setting,

Let $B_1=(x_1,y_1)$ and $B_2=(x_2,y_2)$ be two bundles. Suppose the preference $\succsim$ satisfies (strong) monotonicity. Then $(x_1,y_1)\succ(x_2,y_2)$ if either of the following is true:

• $x_1>x_2$ and $y_1\geq y_2$
• $x_1\geq x_2$ and $y_1>y_2$

So, to your Q2: If $x_1>x_2$ but $y_1 < y_2$, then (strong) monotonicity says nothing about which bundle is preferred. (This is not to say that the preference does not have any ordering over the two bundles, but merely that strong monotonicity alone won't tell us anything about this ordering.)