Lexicographic preferences

Lexicographic preferences (lexicographical order based on the order of amount of each good) describe comparative preferences where an economic agent infinitely prefers one good (X) to another (Y). Thus if offered several bundles of goods, the agent will choose the bundle that offers the most X, no matter how much Y there is. Only when there is a tie of Xs between bundles will the agent start comparing Ys.

For example, if for a given bundle (X;Y;Z) an agent orders his preferences according to the rule X>Y>Z, then the bundles {(5;3;3), (5;1;6), (3,5,3)} would be ordered, from most to least preferred:

1. 5;3;3
2. 5;1;6
3. 3;5;3

• Even though the first option contains fewer total goods than the second option, it is preferred because it has more Y. Note that the number of X's is the same, and so the agent is comparing Y's.
• Even though the third option has the same total goods as the first option, the first option is still preferred.
• Even though the third option has far more Y than the second option, the second option is still preferred because it has slightly more X.

Implications

If all agents have the same lexicographic preferences, then general equilibrium cannot exist because agents won't sell to each other (as long as price of the less preferred is more than zero). But if the price of the less wanted is zero, then all agents want an infinite amount of the good. Equilibrium cannot be attained.

Lexicographic preferences can still exist with general equilibrium. For example,

• Different people have different bundles of lexicographic preferences such that different individuals value items in different orders.
• Some, but not all people have lexicographic preferences.
• Lexicographic preferences extend only to a certain quantity of the good.

Lexicographic preferences are the classical example of rational preferences that are not representable by a utility function, if amounts can be any non-negative real value. If there were such a function U then, e.g. for 2 goods, the intervals [U(x,0),U(x,1)] would have a non-zero width and be disjoint for all x, which is not possible for an uncountable set of x-values. If there are a finite number of goods and amounts can only be rational numbers, utility functions do exist.

The relation is not continuous because for a decreasing convergent sequence $x_n \rightarrow 0$ we have (x_n,0) > (0,1), while the limit (0,0) is smaller than (0,1).

Origin of term

"Lexicography" refers to the compilation of dictionaries, and is meant to invoke the fact that a dictionary is organized alphabetically: with infinite attention to the first letter of each word, and only in the event of ties with attention to the second letter of each word, etc.