Editing Expenditure function (section)

In [[microeconomics]], the '''expenditure function''' represents the minimum amount of expenditure needed to achieve a given level of [[utility]], given a [[utility function]] and the prices of goods.

Formally, if there is a utility function <math>u</math> that describes [[Preference (economics)|preferences]] over ''n ''goods, the expenditure function <math>e(p, u^*)</math> is defined as:

:<math>e(p, u^*) = \min_{x \in \geq(u^*)} p \cdot x</math>

where <math>p</math> is the price vector <math>u^*</math> is the desired utility level, <math>\geq(u^*) = \{x \in \textbf R^n_+ : u(x) \geq u^*\}</math> is the set of providing at least utility <math>u^*</math>.

Expressed equivalently, the individual minimizes expenditure <math> x_1p_1+\dots +x_n p_n</math> subject to the minimal utility constraint that <math>u(x_1, \dots , x_n) \ge u^*,</math> giving optimal quantities to consume of the various goods as <math> x_1^*, \dots x_n^*</math> as function of <math>u^*</math>  and the prices; then the expenditure function is

:<math>e(p_1, \dots , p_n ; u^*)=p_1 x_1^*+\dots + p_n x_n^*.</math>