# Are lexicographic preferences strictly monotonic?

I'm a bit confused about lexicographic preferences and whether they abide by the strict monotonicity axiom.

The definition we were given for strict monotonicity is:

For any two bundles $x$ and $y$, if $x_i \succsim y_i$ for each good $i$, then $x$ is strictly preferred to $y$

The preferences are summarized below:

(1) The bundle that has more of good 1 is better, regardless of the amount of good 2.

(2) If the amount of good 1 is the same, the bundle that has more of good 2 is better.

I'm confused about part (1), because a bundle can have much more of good 2 but still be preferred if it has a bit more of good 1, e.g. $(2,5)$ is preferred to $(1,100)$, even though it has so much more in it. Surely that can't follow the provided definition of strict monotonicity (or am I really thick)?

Thanks!

• Your definition of strict monotonicity looks incomplete as it would suggest that if $x=y$ then $x$ is strictly preferred to $y$ and $y$ is strictly preferred to $x$. Do you need to add "and $x_j \succ y_j$ for some good $j$"? – Henry Mar 20 '17 at 8:37

It seems you are not confused about part (1) because that's exactly what it means.

Lexicographic preferences are monotone. Monotonicity means more is better. If I have more of every good in the bundle, then I like that bundle more. This is still true for lexicographic preferences, even though parts of the bundle may not matter. A condition for monotonicity is e.g. that (5,4) is preferred to (3,1). In other words if 5>3 and 4>1 then the first bundle is preferred. This is true for lexicographic preferences.

Your definitions of strict monotonicity and lexicographic preferences seem to deviate a bit from conventional definitions though. Here is a proof using your definitions.

Proof:

Let us compare two bundles with $n$ elements each: $x=(x_1, x_2, ..., x_n)$ and $y=(y_1, y_2, ...,y_n)$

It is best to start with the if statement when going about proving such claims. The condition and therefore starting point is $x_1>y_1$ and $x_2>y_2$, .... , and $x_n>y_n$.

In that case since $x_1>y_1$ we have according to your preference definition that bundle $x$ is preferred to bundle $y$. Furthermore we have that bundle $y$ is not preferred to bundle $x$. So bundle $x$ is strictly preferred.

Hence, we have shown that if $x_i$>$y_i$ for each $i$, then $x$ is preferred (because in that case $x_1 > y_1$), which is your definition of monotonicity. The preferences are therefore monotonic.

• To complete this (correct) answer, my impression is that the OP was confused about the direction of the implication. Strict monotonicity means that $(x_1>y_1,x_2>y_2)$ implies that $x$ is stricly preferred to $y$. But it does not means that the reverse implication is true. In other words, $x$ being strictly preferred to $y$ does not imply that $(x_1>y_1,x_2>y_2)$, which is why the example with $(2,5)$ and $(1,100)$ does not violate strict monotonicity. – Oliv Mar 19 '17 at 11:58