Relationship To Other Functions
Hicksian demand functions are often convenient for mathematical manipulation because they do not require income or wealth to be represented. Additionally, the function to be minimized is linear in the, which gives a simpler optimization problem. However, Marshallian demand functions of the form that describe demand given prices p and income are easier to observe directly. The two are trivially related by
where is the expenditure function (the function that gives the minimum wealth required to get to a given utility level), and by
where is the indirect utility function (which gives the utility level of having a given wealth under a fixed price regime). Their derivatives are more fundamentally related by the Slutsky equation.
Whereas Marshallian demand comes from the Utility Maximization Problem, Hicksian Demand comes from the Expenditure Minimization Problem. The two problems are mathematical duals, and hence the Duality Theorem provides a method of proving the relationships described above.
The Hicksian demand function is intimately related to the expenditure function. If the consumer's utility function is locally nonsatiated and strictly convex, then
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