Expenditure function

In microeconomics, a consumer's expenditure function describes how much money a consumer needs achieve a given level of happiness. That is, given a utility function and prices, it says how much wealth the consumer would need to reach a desired utility level.

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

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

says what amount of money is needed to achieve a utility $u^{*}$ if prices are set by $p$. This function is defined by

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

where

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

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

See also

• Expenditure minimization problem
• Hicksian demand function
• Utility maximization problem