Expenditure minimization problem

Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices

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

Hicksian demand is defined by Hicksian demand function gives the cheapest package that gives the desired utility.

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.