(Solution) - Sometimes economists speak of the certainty equivalent of a risk -(2025 Original AI-Free Solution)

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Sometimes economists speak of the certainty equivalent of a risky stream of income. This problem asks you to compute the certainty equivalent of a risky bet that promises a 50-50 chance of winning or losing $5,000 for someone with a starting income of $50,000. We know that a certain income of somewhat less than $50,000 will provide the same expected utility as will taking this bet. You are asked to calculate precisely the certain income (that is, the certainty equivalent income) that provides the same utility as does this bet for three simple utility functions:
a. U (I) = ?I.
b. U (I) = ln (I) (where ln means ??natural logarithm??)
c. U (I) = -1/ I
What do you conclude about these utility functions by comparing these three cases?