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 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
where
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 function of and the prices; then the expenditure function is

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 and income level :