Expenditure function

In microeconomics, the expenditure function gives the minimum amount of money an individual needs to spend to achieve some level of utility, given a utility function and the prices of the available goods.

Formally, if there is a utility function u that describes preferences over n commodities, the expenditure function

e(p, u^*) : \textbf R^n_+ \times \textbf R \rightarrow \textbf R

says what amount of money is needed to achieve a utility u^* if the n prices are given by the price vector p. This function is defined by

e(p, u^*) = \min_{x \in \geq(u^*)} p \cdot x

where

\geq(u^*) = \{x \in \textbf R^n_+ : u(x) \geq u^*\}

is the set of all bundles that give utility at least as good as u^*.

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

e(p_1, \dots , p_n ; u^*)=p_1 x_1^*+\dots + p_n x_n^*.

See also

References