Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function defined on commodities. Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices that give utility of at least ,
where
is the set of all packages that give utility at least as good as .
Hicksian demand function gives the cheapest package that gives the desired utility. It is related to Marshallian demand function by and expenditure function by
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.
Anatomy of Cobb-Douglas Type Utility Functions in 3D
January 01, 1970
expenditure, minimization, problem, other, uses, minimisation, microeconomics, expenditure, minimization, problem, dual, utility, maximization, problem, much, money, need, reach, certain, level, happiness, this, question, comes, parts, given, consumer, utility. For other uses see Minimisation In microeconomics the expenditure minimization problem is the dual of 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 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 function Contents 1 Expenditure function 2 Hicksian demand correspondence 3 See also 4 References 5 External linksExpenditure function editFormally the expenditure function is defined as follows Suppose the consumer has a utility function u displaystyle u nbsp defined on L displaystyle L nbsp commodities Then the consumer s expenditure function gives the amount of money required to buy a package of commodities at given prices p displaystyle p nbsp that give utility of at least u displaystyle u nbsp e p u min x u p x displaystyle e p u min x in geq u p cdot x nbsp where u x R L u x u displaystyle geq u x in mathbb R L u x geq u nbsp is the set of all packages that give utility at least as good as u displaystyle u nbsp Hicksian demand correspondence editHicksian demand is defined by h R L R P R L displaystyle h mathbb R L times mathbb R to P mathbb R L nbsp h p u argmin x u p x displaystyle h p u underset x in geq u operatorname argmin p cdot x nbsp 1 Hicksian demand function gives the cheapest package that gives the desired utility It is related to Marshallian demand function by and expenditure function by h p u x p e p u displaystyle h p u x p e p u nbsp 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 See also editUtility maximization problemReferences edit Jonathan Levin Paul Milgrom Consumer Theory PDF Mas Colell Andreu Whinston Michael amp Green Jerry 1995 Microeconomic Theory Oxford Oxford University Press ISBN 0 19 507340 1 External links editAnatomy of Cobb Douglas Type Utility Functions in 3D Retrieved from https en wikipedia org w index php title Expenditure minimization problem amp oldid 1174752537, wikipedia, wiki, book, books, library,