microeconomics Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...

, the expenditure minimization problem is the dual of the

utility maximization problem Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my uti ...

: "how much money do I need to reach a certain level of happiness?". This question comes in two parts. Given a

consumer A consumer is a person or a group who intends to order, or uses purchased goods, products, or services primarily for personal, social, family, household and similar needs, who is not directly related to entrepreneurial or business activities. ...

utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophe ...

, prices, and a utility target, * how much money would the consumer need? This is answered by the

expenditure function 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 ...

. * what could the consumer buy to meet this utility target while minimizing expenditure? This is answered by the

Hicksian demand function In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essenti ...

Expenditure function

Formally, the

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 of at least

u^*

, :

e(p, u^*) = \min_ p \cdot x

where :

\geq = \

is the set of all packages that give utility at least as good as

u^*

Hicksian demand correspondence

Hicksian demand is defined by :

h : \mathbb^L_+ \times \mathbb_+ \to P(\mathbb^L_+)

h(p, u^*) = \underset\ p \cdot x

. 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^*)). \,

The relationship between the

and

Marshallian demand In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the s ...

in the utility maximization problem mirrors the relationship between the

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 In microeconomics, the property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is strictly preferred to it.''Microeconomic Theory'', by A. Mas-Col ...

References

*{{cite book , author1-link=Andreu Mas-Colell, author2-link=Michael Whinston, author3-link=Jerry Green (economist), last=Mas-Colell , first=Andreu , last2=Whinston , first2=Michael , name-list-style=amp , last3=Green , first3=Jerry , year=1995 , title=Microeconomic Theory , location=Oxford , publisher=Oxford University Press , isbn=0-19-507340-1

External links

Anatomy of Cobb-Douglas Type Utility Functions in 3D
Consumer theory Optimal decisions Expenditure

Expenditure function

Hicksian demand correspondence

See also

References

External links