Expenditure Minimization Probl

Expenditure Minimization Problem

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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 my expenditure? This is answered by the Hicksian demand correspondence.

Contents

1 Expenditure function 2 Hicksian demand correspondence 3 See also 4 References

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.