Editing Ordinal utility (section)

== Monotonicity ==
Suppose, from now on, that the set <math>X</math> is the set of all non-negative real two-dimensional vectors. So an element of <math>X</math> is a pair <math>(x,y)</math> that represents the amounts consumed from two products, e.g., apples and bananas.

Then under certain circumstances a preference relation <math>\preceq</math> is represented by a utility function <math>v(x,y)</math>.

Suppose the preference relation is ''monotonically increasing'', which means that "more is always better":
:<math>x<x' \implies (x,y)\prec(x',y)</math>
:<math>y<y' \implies (x,y)\prec(x,y')</math>

Then, both partial derivatives, if they exist, of ''v'' are positive. In short: 
::''If a utility function represents a monotonically increasing preference relation, then the utility function is monotonically increasing.''