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

$e(p,u^{*}):{\mathbf{\text{<mi fromhbox="1">R</mi>}}}_{+}^L\times \mathbf{\text{<mi fromhbox="1">R</mi>}}\to \mathbf{\text{<mi fromhbox="1">R</mi>}}$

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

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

where

$\ge (u^{*})=\{x\in {\mathbf{\text{<mi fromhbox="1">R</mi>}}}_{+}^L:u(x)\ge u^{*}\}$

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