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:

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

Suppose ${\displaystyle u}$ is a continuous utility function representing a locally non-satiated preference relation on ${\displaystyle {\textbf {R}}_{+}^{n}}$. Then ${\displaystyle e(p,u^{*})}$ is

(1) As in the above proposition, note that

${\displaystyle e(\lambda p,u)=\min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}$ ${\displaystyle \lambda p\cdot x=\lambda \min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}$ ${\displaystyle p\cdot x=\lambda e(p,u)}$

(2) Continue on the domain ${\displaystyle e}$: ${\displaystyle {\textbf {R}}_{++}^{N}*{\textbf {R}}\rightarrow {\textbf {R}}}$