Hicksian Demand Correspondence
Secondly, the Hicksian demand function 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 function
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 demand 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.
Read more about this topic: Expenditure Minimization Problem
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