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 ${\displaystyle u}$ that describes preferences over n commodities, the expenditure function

${\displaystyle e(p,u^{*}):{\textbf {R}}_{+}^{n}\times {\textbf {R}}\rightarrow {\textbf {R}}}$

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

${\displaystyle e(p,u^{*})=\min _{x\in \geq (u^{*})}p\cdot x}$

where

${\displaystyle \geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}}$

is the set of all bundles that give utility at least as good as ${\displaystyle u^{*}}$.

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

${\displaystyle e(p_{1},\dots ,p_{n};u^{*})=p_{1}x_{1}^{*}+\dots +p_{n}x_{n}^{*}.}$

## 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 ${\displaystyle p}$ and income level ${\displaystyle I}$:[1]:106

${\displaystyle e(p,v(p,I))\equiv I}$