Lexicographic preferences
Encyclopedia
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:
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,
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
we have 
, while the limit (0,0) is smaller than (0,1).
Lexicographical order
In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...
based on the order of amount of each good) describe comparative preferences where an economic agent
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....
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:
- 5;3;3
- 5;1;6
- 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 equilibriumGeneral equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
cannot exist because agents won't sell to each other (as long as price
Price
-Definition:In ordinary usage, price is the quantity of payment or compensation given by one party to another in return for goods or services.In modern economies, prices are generally expressed in units of some form of currency...
of the less preferred is more than zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
). 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
we have , while the limit (0,0) is smaller than (0,1).