View source for Expenditure minimization problem

{{otheruses|Minimisation (disambiguation){{!}}Minimisation}}
In [[microeconomics]], the '''expenditure minimization problem''' is the [[Duality (optimization)|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]].

==Expenditure function==
Formally, the [[expenditure function]] is defined as follows.  Suppose  the consumer has a utility function <math>u</math> defined on <math>L</math> commodities.  Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices <math>p</math> that give utility of at least <math>u^*</math>,

:<math>e(p, u^*) = \min_{x \in \geq{u^*}} p \cdot x</math>

where

:<math>\geq{u^*} = \{x \in \mathbb{R}^L_+ : u(x) \geq u^*\}</math>

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

==Hicksian demand correspondence==
'''Hicksian demand''' is defined by
: <math>h : \mathbb{R}^L_+ \times \mathbb{R}_+ \to P(\mathbb{R}^L_+)</math> 
: <math>h(p, u^*) = \underset{x \in \geq u^* }{\operatorname{argmin}}\ p \cdot x</math>.<ref>{{Cite web| title=Consumer Theory |url=https://web.stanford.edu/~jdlevin/Econ%20202/Consumer%20Theory.pdf |author=Jonathan Levin |author2=Paul Milgrom}}</ref>
Hicksian demand function gives the cheapest package that gives the desired utility.  It is related to Marshallian demand function by and expenditure function by
:<math>h(p, u^*) = x(p, e(p, u^*)). \,</math>
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 [[1:1 correspondence|correspondence]], and not a function. This does not happen, and the demands are functions, under the assumption of [[local nonsatiation]].

==See also==
* [[Utility maximization problem]]

==References==
{{Reflist}}
*{{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==
*[http://www2.hawaii.edu/~fuleky/anatomy/anatomy.html Anatomy of Cobb-Douglas Type Utility Functions in 3D]

[[Category:Consumer theory]]
[[Category:Optimal decisions]]
[[Category:Expenditure]]