Expenditure function

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

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

$e(p, u^*) : \textbf R^n_+ \times \textbf R \rightarrow \textbf R$

says what amount of money is needed to achieve a utility $u^*$ if the n prices are given by the price vector $p$. This function is defined by

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

where

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

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

Expressed equivalently, the individual minimizes expenditure $x_1p_1+\dots +x_n p_n$ subject to the minimal utility constraint that $u(x_1, \dots , x_n) \ge u^*,$ giving optimal quantities to consume of the various goods as $x_1^*, \dots x_n^*$ as functions of $u^*$ and the prices; then the expenditure function is

$e(p_1, \dots , p_n ; u^*)=p_1 x_1^*+\dots + p_n x_n^*.$

Expenditure and indirect utility

The expenditure function is the inverse of the indirect utility function when the prices are kept constant. I.e, for every price vector $p$ and income level $I$:^[1]:106

$e(p, v(p,I)) \equiv I$

See also

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

References

1. Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0393957357.

• Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (2007). Microeconomic Theory. pp. 59–60. ISBN 0-19-510268-1.
• Mathis, Stephen A.; Koscianski, Janet (2002). Microeconomic Theory: An Integrated Approach. Upper Saddle River: Prentice Hall. pp. 132–133. ISBN 0-13-011418-9.
• Varian, Hal R. (1984). Microeconomic Analysis (Second ed.). New York: W. W. Norton. pp. 121–123. ISBN 0-393-95282-7.