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If u is continuous and no commodities are free of charge, then x(p, w) is nonempty. |
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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 [[Weierstrass theorem]] implies that u(B(p, w)) is a compact subset of <math>\textbf R</math>, and hence contains an upper bound.
==References==
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