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Finding x(p, w) is the '''Utility Maximization Problem'''.
The solution x(p, w) need not be unique. If u is continuous and no commodities are free of charge, then x(p, w) is nonempty. Proof: B(p, w) is a [[compact space]]. So if u is [[continuous]], then the [[Karl Weierstraß|Weierstrass]] theorem implies that u(B(p, w)) is a compact subset of <math>\textbf R</math>, and hence contains an upper bound.
If a consumer always picks an optimal package as defined above, then x(p, w) is called the [[Marshallian demand correspondence]]. If there is always a unique maximizer, then it is called the [[Marshallian demand function]]. The relationship between utility and Marshallian demand in the Utility Maximization Problem mirrors the relationship between indirect utility and [[Hicksian demand]] in the [[Expenditure Minimization Problem]].
==References==
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