Discontinuity
A lexicographic preference relation is not a continuous relation. This is because, for a decreasing convergent sequence we have , while the limit (0,0) is smaller than (0,1).
In economics, lexicographic preferences or lexicographic orderings describe comparative preferences where an agent prefers any amount of one good (X) to any amount of another (Y). Specifically, 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 between bundles with regard to the number of units of X will the agent start comparing the number of units of Y across bundles. Lexicographic preferences extend utility theory analogously to the way that nonstandard infinitesimals extend the real numbers. With lexicographic preferences, the utility of certain goods is infinitesimal in comparison to others.
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.
As an 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:
A lexicographic preference relation is not a continuous relation. This is because, for a decreasing convergent sequence we have , while the limit (0,0) is smaller than (0,1).
A distinctive feature of such lexicographic preferences is that a multivariate real domain of an agent's preferences does not map into a real-valued range. That is, there is no real-valued representation of a preference relation by a utility function, whether continuous or not. [1] Lexicographic preferences are the classical example of rational preferences that are not representable by a utility function.
Proof: suppose by contradiction that there exists a utility function U representing lexicographic preferences, e.g. over two goods. Then U(x,1)>U(x,0) must hold, so the intervals [U(x,0),U(x,1)] must have a non-zero width. Moreover, since U(x,1)<U(z,1) whenever x<z, these intervals must be disjoint for all x. This 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.
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. Thus, lexicographic preferences can be represented by utility functions returning nonstandard real numbers.
If all agents have the same lexicographic preferences, then general equilibrium cannot exist because agents will not sell to each other[ clarification needed ] (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,
The nonstandard (infinitesimal) 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.
In economics, utility is a measure of the satisfaction that a certain person has from a certain state of the world. Over time, the term has been used in at least two different meanings.
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In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number is said to be finite if, and only if, for some integer . is said to be infinitesimal if, and only if, for all positive integers . The term "hyper-real" was introduced by Edwin Hewitt in 1948.
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