In microeconomics, the expenditure minimization problem is the dual problem to the utility maximization problem: "how much money do I need to be happy?". This question comes in two parts. Given a consumer's utility function, prices, and a utility target,

• how much money would the consumer need? This is answered by the expenditure function.
• what could the consumer buy to meet this utility target while minimizing expenditure? This is answered by the Hicksian demand correspondence .

Expenditure function

Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function u defined on L commodities. Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices p that give utility greater than u ^* ,

where

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

Hicksian demand correspondence

Secondly, the Hicksian demand correspondence h(p,u ^* ) is defined as the cheapest package that gives the desired utility. It can be defined in terms of the expenditure function with the Marshallian demand correspondence

h(p,u ^* ) = x(p,e(p,u ^* )).

If the Marshallian demand correspondence x(p,w) is a function (i.e. always gives a unique answer), then h(p,u ^* ) is also called the Hicksian demand function.

See also

• Utility maximization problem

References

• Mas-Colell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 0195073401