Expenditure function

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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 that describes preferences over n commodities, the expenditure function

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


is the set of all bundles that give utility at least as good as .

Expressed equivalently, the individual minimizes expenditure subject to the minimal utility constraint that giving optimal quantities to consume of the various goods as as functions of and the prices; then the expenditure function is

Expenditure and indirect utility[edit]

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

See also[edit]


  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.