Wikipedia

# 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 $x_{1}p_{1}+\dots +x_{n}p_{n}$ subject to the minimal utility constraint that $u(x_{1},\dots ,x_{n})\geq u^{*},$ giving optimal quantities to consume of the various goods as $x_{1}^{*},\dots x_{n}^{*}$ as function of $u^{*}$ and the prices; then the expenditure function is

$e(p_{1},\dots ,p_{n};u^{*})=p_{1}x_{1}^{*}+\dots +p_{n}x_{n}^{*}.$ ## Expenditure and indirect utility

The expenditure function is the inverse of the indirect utility function when the prices are kept constant. I.e, for every price vector $p$ and income level $I$ ::106

$e(p,v(p,I))\equiv I$ 