Grain growth

Lexicographic preferences or lexicographic orderings (lexicographical order based on the order of amount of each good) describe comparative preferences where an economic agent prefers an amount of one good (X) to any amount of another (Y). More generally, 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. It represents the same generalization of utility theory as nonstandard infinitesimals extend the real numbers. With lexicographic preference, the utility of certain goods is infinitesimal in comparison to others.

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.

A distinctive feature of such lexicographic preferences is that a domain of real-valued functions of agents does not map into a real-valued range. That is, there is no real-valued representation of a social welfare function.

In terms of real valued utility, one would say that the utility of Y and Z is infinitesimal compared with X, and the utility of Z is infinitesimal compared to Y. The model of real numbers is always logically ambiguous; one is allowed to adjoin infinitesimal quantities to make a nonstandard model. Standard models of the real numbers exclude infinitesimals, so lexicographic preferences are not precisely described by standard reals. But by assigning a utility to X which is much much larger than the utility of Y, which in turn is much much larger than the utility of Z, the infinitesimal order relation can be approximated arbitrarily closely, which means this is a problem of idealized limits only.

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 with standard prices. The utilities are infinitesimal, but the prices are not. Allowing infinitesimal prices resolves this.

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.

The nonstandard equilibrium prices for exchange can be determined for lexicographic order using standard equilibrium methods, except using nonstandard reals as the range of both utilities and prices. All the theorems regarding existence of prices and equilibria extend to the case of nonstandard utilities, since the nonstandard reals form a conservative extension, meaning that any theorem which is true for reals can be extended to the nonstandard reals and remains true.

Lexicographic preferences are the classical example of rational preferences that are not representable by a utility function over the standard reals. 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, simply by taking 1/N to be the size of the infinitesimal, where N is sufficiently large, to approximate nonstandard numbers.

The relation is not continuous because for a decreasing convergent sequence $x_n\to 0$ 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. See lexicographic preference.

Notes

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