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(Preference Relation/Set) Continuous $\succsim$ imply closedness of upper and lower contour sets

Looking more closely at your question, I think things should not be overly complicated. From Mas-Colell et.al. Definition 3.C.1: The preference relation $\succsim$ on X is continuous if it is ...
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Can the Certainty Equivalent be negative?

If you start out with €0, then the certainty equivalent of losing €2.5 with probability 1 is -€2.5. Your exercise basically asks you to calculate what difference winning the lottery with a small ...
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Proof of the tangency condition in UMP

The proof below is not very rigorous - I do not discuss discontinuities - but I think it captures the logic well. From Wikipedia: Suppose $f$ is a function of one real variable defined on an ...
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State dependent preferences vs state independent preferences in utility theory

Starting from the textbook, I would highly recommend any textbook for stochastic dynamic optimization. Then I would recommend you to get acquainted with markov chains, because it is relatively good ...
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I take it that $u: \mathbb{R} \to \mathbb{R}$ and not $u: \mathbb{R}^N \to \mathbb{R}$ (as in the question). Otherwise $u(c)$ for $c \in \mathbb{R}$ does not make sense. tldr: if $u$ is continuous, a ...