Editing Utility maximization problem (section)

== A technical representation ==
Suppose the consumer's [[consumption set]], or the enumeration of all possible consumption bundles that could be selected if there were a budget constraint.

The consumption set = <math>   \mathbb{R}^n_+ \ .</math> (a set of positive real numbers, the consumer cannot preference negative amount of commodities).

<math>x \in \mathbb{R}^n_+ \ .</math>

Suppose also that the price vector (''p'') of the n commodities is positive,
[[File:Utility_maximisation_with_a_budget_line.png|thumb|284x284px|Figure 2: This shows the optimal amounts of goods x and y that maximise utility given a budget constraint.]]
<math>p \in \mathbb{R}^n_+ \ ,</math>

and that the consumer's income is <math>I</math>; then the set of all affordable packages, the [[budget set]] is,

<math>B(p, I) = \{x \in \mathbb{R}^n_+ | \mathbb{\Sigma}^n_{i=1} p_i x_i  \leq I\} \ ,</math>

The consumer would like to buy the best affordable package of commodities.

It is assumed that the consumer has an [[ordinal utility]] function, called ''u''. It is a real-valued function with domain being the set of all commodity bundles, or

:<math>u : \mathbb{R}^n_+ \rightarrow \mathbb{R}_+ \ .</math>

Then the consumer's optimal choice <math>x(p,I)</math> is the utility maximizing bundle of all bundles in the budget set if <math>x\in B(p,I)</math> then the consumers optimal demand function is:

<math>x(p, I) = \{x \in B(p,I)| U(x) \geq U(y) \forall y \in B(p,I)\}</math>

Finding <math>x(p,I)</math> is the '''utility maximization problem'''.

If ''u'' is continuous and no commodities are free of charge, then <math>x(p,I)</math> exists,<ref>{{Cite book|title=Choice, preference and Utility|publisher=Princeton university press|year=n.d.|pages=14}}</ref> but it is not necessarily unique. If the preferences of the consumer are complete, transitive and strictly convex then the demand of the consumer contains a unique maximiser for all values of the price and wealth parameters. If this is satisfied then <math>x(p,I)</math> is called the [[Marshallian demand function]]. Otherwise, <math>x(p,I)</math> is set-valued and it is called the [[Marshallian demand correspondence]].