Expenditure minimization problem

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.

Expenditure function

Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function Failed to parse (Missing texvc executable; please see math/README to configure.): u

defined on Failed to parse (Missing texvc executable; please see math/README to configure.): L
commodities.  Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices Failed to parse (Missing texvc executable; please see math/README to configure.): p
that give utility greater than Failed to parse (Missing texvc executable; please see math/README to configure.): u^*

,

Failed to parse (Missing texvc executable; please see math/README to configure.): e(p, u^*) = \min_{x \in \geq{u^*}} p \cdot x

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): u^* .

Hicksian demand correspondence

Secondly, the Hicksian demand correspondence Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): h(p, u^*) = x(p, e(p, u^*)).

If the Marshallian demand correspondence Failed to parse (Missing texvc executable; please see math/README to configure.): x(p, w)

is a function (i.e. always gives a unique answer), then Failed to parse (Missing texvc executable; please see math/README to configure.): h(p, u^*)
is also called the Hicksian demand function.

See also

• Utility maximization problem

References

• Mas-Colell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 0-19-507340-1

External links

• Anatomy of Cobb-Douglas Type Utility Functions in 3Dlo:ບັນຫາການຈຳກັດລາຍຈ່າຍໃຫ້ຕໍ່າສຸດ