Editing Utility maximization problem (section)

=== 1) Walras's Law ===
[[Walras's law]] states that if a consumers preferences are complete, monotone and transitive then the optimal demand will lie on the [[Budget constraint|budget line]].<ref>{{Cite book|last=Levin|first=Jonothan|title=Consumer theory|publisher=Stanford university|year=2004|pages=4–6}}</ref>

==== Preferences of the consumer ====
For a utility representation to exist the preferences of the consumer must be complete and transitive (necessary conditions).<ref>{{Cite book|last=Salcedo|first=Bruno|title=Utility representations|publisher=Cornell university|year=2017|pages=18–19}}</ref>

===== Complete =====
Completeness of preferences indicates that all bundles in the consumption set can be compared by the consumer. For example, if the consumer has 3 bundles A,B and C then;

A <math>\succcurlyeq</math> B, A <math>\succcurlyeq</math> C, B <math>\succcurlyeq</math> A, B <math>\succcurlyeq</math>C, C <math>\succcurlyeq</math>B, C <math>\succcurlyeq</math>A, A <math>\succcurlyeq</math>A, B <math>\succcurlyeq</math>B, C <math>\succcurlyeq</math>C.  Therefore, the consumer has complete preferences as they can compare every bundle.

===== Transitive =====
Transitivity states that individuals preferences are consistent across the bundles.

therefore, if the consumer weakly prefers A over B (A <math>\succcurlyeq</math> B) and B <math>\succcurlyeq</math>C this means that A <math>\succcurlyeq</math> C (A is weakly preferred to C)

===== Monotone =====
For a preference relation to be [[Monotone preferences|monotone]]  increasing the quantity of both goods should make the consumer strictly better off (increase their utility), and increasing the quantity of one good holding the other quantity constant should not make the consumer worse off (same utility).

The preference <math>\succcurlyeq</math> is monotone if and only if;

1)<math>(x+\epsilon, y)\succcurlyeq(x,y)</math>

2) <math>(x,y+\epsilon)\succcurlyeq(x,y)</math>

3) <math>(x+\epsilon, y+\epsilon)\succ(x,y)</math>

where  <math>\epsilon</math> > 0