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 expenditure? This is answered by the Hicksian demand correspondence.

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

[edit] 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 $u *$ ,

$e(p, u^*) = \min_{x \in \geq{u^*}} p \cdot x$

where

$\geq{u^*} = \{x \in \textbf R^L_+ : u(x) \geq u^*\}$

is the set of all packages that give utility at least as good as $u *$ .

[edit] 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.