Puneet Varma (Editor)

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 u that describes preferences over n commodities, the expenditure function

e ( p , u ∗ ) : R + n × R → 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 ∈≥ ( u ∗ ) p ⋅ x

where

≥ ( u ∗ ) = { x ∈ R + n : u ( x ) ≥ u ∗ }

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

Expressed equivalently, the individual minimizes expenditure x 1 p 1 + ⋯ + x n p n subject to the minimal utility constraint that u ( x 1 , … , x n ) ≥ u ∗ , giving optimal quantities to consume of the various goods as x 1 ∗ , … x n ∗ as functions of u ∗ and the prices; then the expenditure function is

e ( p 1 , … , p n ; u ∗ ) = p 1 x 1 ∗ + ⋯ + 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 :

e ( p , v ( p , I ) ) ≡ I

References

Expenditure function Wikipedia
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