Field of sets

In microeconomics, the expenditure function describes the minimum amount of money an individual needs to achieve some level of utility, given a utility function and prices.

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 bundles that give utility at least as good as $u^{*}$.

See also

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

References

• Andreu Mas-Colell, Michael D. Whinston and Jerry R. Green Microeconomic Theory, 2007, ISBN 0-19-510268-1, pp. 59-60, The Expenditure Function