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I'm a first year Finance grad student and we're learning utility maximization problem now. We have been assuming that a solution to these problems exists. Are there any general theorem about the existence of maximization solution on $\mathbb{R}^n$? I have encountered once that "A utility maximization problem with a continuous utility function on a compact set has solution". Thanks in advance!

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Yes. There are general theorems about existence and uniqueness. The one you cited is Weierstrass' extreme value theorem (notice budget sets are compact only if no price is zero). See Mas Colell's Microeconomic Theory or Varian's Microeconomic Analysis or any graduate level book on microeconomics for references.

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