In
economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
, 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
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings.
* In a Normative economics, normative context, utility refers to a goal or ob ...
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.
Etymology
''
Lexicography
Lexicography is the study of lexicons and the art of compiling dictionaries. It is divided into two separate academic disciplines:
* Practical lexicography is the art or craft of compiling, writing and editing dictionaries.
* Theoretical le ...
'' 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.
Example
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 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 because it has more X.
*Even though the third option has far more Y than the second option, the second option is still preferred because it has more X.
Discontinuity
A lexicographic preference relation is not a
continuous relation. This is because, for a decreasing
convergent sequence
As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n \times \sin\left(\tfrac1\right) equals 1."
In mathematics, the li ...
we have
, while the limit (0,0) is smaller than (0,1).
Utility function representation
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
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
. That is, there is no real-valued representation of a preference relation by a
utility function
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings.
* In a Normative economics, normative context, utility refers to a goal or ob ...
, whether continuous or not.
[ Amartya K. Sen, 1970 984 ''Collective Choice and Social Welfare'', ch. 3, "Collective Rationality," pp. 34-35]
Description.
Lexicographic preferences are the classical example of rational preferences that are not representable by a
utility function
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings.
* In a Normative economics, normative context, utility refers to a goal or ob ...
.
''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)
infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
compared to Y. Thus, lexicographic preferences can be represented by utility functions returning
). 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
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 (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
, meaning that any theorem which is true for reals can be extended to the nonstandard reals and remains true.
In Appl. Math. and Computation