In microeconomics, the expenditure minimization problem is another perspective on the utility maximization problem: "how much money do I need to reach a certain level of happiness?". This question comes in two parts. Given a consumer's utility function, prices, and a utility target,
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Expenditure function
Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function
where
is the set of all packages that give utility at least as good as
Hicksian demand correspondence
Secondly, the Hicksian demand function
The relationship between the utility function and Marshallian demand in the utility maximization problem mirrors the relationship between the expenditure function and Hicksian demand in the expenditure minimization problem. It is also possible that the Hicksian and Marshallian demands are not unique (i.e. there is more than one commodity bundle that satisfies the expenditure minimization problem); then the demand is a correspondence, and not a function. This does not happen, and the demands are functions, under the assumption of local nonsatiation.