# Finding the minimum and maxium price given utility index [closed]

Say you have the expected utility index $U(Y ) = \sqrt{Y}$ and an initial wealth $Y = 10$. Consider the lottery with a payoff of $10$ with probability $\pi\in (0,1)$ and a payoff of $5$ with probability $1-\pi$ Note: Derive general expressions in $\pi$ and then use $\pi = 0.5$ to get specific answers.

If you own this lottery, what is the minimum price and maximum price you would be willing to sell it for?

I am not sure how to proceed any suggestions are greatly appreciated

Suppose you are selling the lottery for price $P$. Then your wealth would be $Y'=10+P$. You are looking for the minimum price $P$ such that $$U(10+P) \geq \pi U(10+10)+(1-\pi)U(10+5).$$
• I see so we are essentially solving for $P$ in the equation $$\sqrt{10+P} = \pi \sqrt{20} + (1-\pi)\sqrt{15}$$? – Wolfy Jan 16 '17 at 17:27