Editing Maximal and minimal elements (section)

===Consumer theory===
In economics, one may relax the axiom of antisymmetry, using preorders (generally [[total preorder]]s) instead of partial orders; the notion analogous to maximal element is very similar, but different terminology is used, as detailed below.

In [[consumer theory]] the consumption space is some set <math>X</math>, usually the positive orthant of some vector space so that each <math>x\in X</math> represents a quantity of consumption specified for each existing commodity in the
economy. [[Preferences]] of a consumer are usually represented by a [[total preorder]] <math>\preceq </math> so that <math>x, y \in X</math> and <math>x \preceq y</math> reads: <math>x</math> is at most as preferred as <math>y</math>. When <math>x\preceq y</math> and <math>y \preceq x</math> it is interpreted that the consumer is indifferent between <math>x</math> and <math>y</math> but is no reason to conclude that <math>x = y.</math> preference relations are never assumed to be antisymmetric. In this context, for any <math>B \subseteq X,</math> an element <math>x \in B</math> is said to be a '''maximal element''' if
<math display="block">y \in B</math> implies <math>y \preceq x</math>
where it is interpreted as a consumption bundle that is not dominated by any other bundle in the sense that <math>x \prec y,</math> that is <math>x \preceq y</math> and not <math>y \preceq x.</math>

It should be remarked that the formal definition looks very much like that of a greatest element for an ordered set. However, when <math>\preceq </math> is only a preorder, an element <math>x</math> with the property above behaves very much like a maximal element in an ordering. For instance, a maximal element <math>x \in B</math> is not unique for <math>y \preceq x</math> does not preclude the possibility that <math>x \preceq y</math> (while <math>y \preceq x</math> and <math>x \preceq y</math> do not imply <math>x = y</math> but simply indifference <math>x \sim y</math>). The notion of greatest element for a preference preorder would be that of '''most preferred''' choice. That is, some <math>x\in B</math> with
<math display="block">y \in B</math> implies <math>y \prec x.</math>

An obvious application is to the definition of demand correspondence. Let <math>P</math> be the class of functionals on <math>X</math>. An element <math>p\in P</math> is called a '''price functional''' or '''price system''' and maps every consumption bundle <math>x\in X</math> into its market value <math>p(x)\in \R_+</math>. The '''budget correspondence''' is a correspondence <math>\Gamma \colon P \times \R_{+} \rightarrow X</math> mapping any price system and any level of income into a subset
<math display="block">\Gamma (p,m) = \{ x \in X ~:~ p(x) \leq m \}.</math>

The '''demand correspondence''' maps any price <math>p</math> and any level of income <math>m</math> into the set of <math>\preceq </math>-maximal elements of <math>\Gamma (p, m)</math>.
<math display="block">D(p,m) = \left\{ x \in X ~:~ x \text{ is a maximal element of } \Gamma(p,m) \right\}.</math>

It is called demand correspondence because the theory predicts that for <math>p</math> and <math>m</math> given, the [[rational choice]] of a consumer <math>x^*</math> will be some element <math>x^* \in D(p,m).</math>