Expenditure function

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 ${\displaystyle u}$ that describes preferences over n goods, the expenditure function ${\displaystyle e(p,u^{*})}$ is defined as:

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

where ${\displaystyle p}$ is the price vector ${\displaystyle u^{*}}$ is the desired utility level, ${\displaystyle \geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}}$ is the set of providing at least utility ${\displaystyle u^{*}}$.

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

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