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If the utility function of an individual is $u(w) = 10 \sqrt{w}$ and the individual starts with $w = 100$ (where $w$ denotes the wealth available to him). If he buys a lottery that costs him $51$ and gets $351$ with probability $p$ and $0$ with probability $(1-p)$, when does the individual buy the lottery if he is an expected utility maximizer?

I attempted it this way. If he buys the lottery, then his expected utility will be $$p \cdot u(100 + 351 - 51) + (1-p) \cdot u(100 - 51) = 200p + 70(1-p) = 130p + 70.$$

If he does not buy the lottery, his utility is $u(100) = 100$.

He buys the lottery if and only if $130p + 70 \geq 100 \iff {\displaystyle p \geq \frac{3}{13}}$.

Did I do it correctly?

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  • $\begingroup$ Yes, you did it correctly. $\endgroup$
    – Amit
    Feb 2, 2023 at 5:05
  • $\begingroup$ @Amit Thank you. $\endgroup$
    – user43302
    Feb 2, 2023 at 5:21

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