Suppose a person has a Bernoulli utility function $u(\cdot)$ and an initial wealth $w_0$. A lottery $L$ offers a payoff $A$ with probability $p$ and payoff $B$ with probability $q$, where $q = 1-p$. If the person owns the lottery, what is the minimum price they would sell it for? On the other hand, if the person does not own the lottery, what is the maximum price they would be willing to pay for it? Are these prices equal (and why)? If $A = 100$, $B = 50$, $w_0 = 100$, and $u(k) = \sqrt{k}$, find the buying and selling prices for this lottery under this utility function.
I have done similar problems involving insurance but those were often about maximizing expected utility and this one is different. I tried to tackle this question and was stumped. I figured that for the first part, if the person owns the lottery, they would only sell if the expected utility of selling at some price $x$ is greater than or equal to their current utility:
$$E[u(w_0+x)] \geq u(L)$$
This implies that $u(w_0+x) \geq p\cdot u(w_0+A)+q\cdot u(w_0+B)$.
Likewise, the buying price would be the price $y$ such that their expected utility of buying the lottery is greater than or equal to their current utility without said lottery:
$$E[u(L-y)] \geq u(w_0)$$
This implies $p\cdot u(w_0-y+A) + q\cdot u(w_0-y+B) \geq u(w_0)$.
Assuming the above are correct then these prices are not necessarily equal; intuitively it makes sense since selling the lottery and buying the lottery have different functions. Economically I can understand it but can't really put it into words.
If I substitute the values given I get:
$$\sqrt{100+x} \geq p\cdot \sqrt{100+100} + q\cdot \sqrt{100+50}$$
$$\sqrt{100+x} \geq p \sqrt{200} + (1-p)\sqrt{150}$$
How do I solve for $x$ without values of $p$? The problem doesn't give any value for $p$.
If I substitute the values given for the buying I get:
$$p\cdot \sqrt{100-y+100} + q\cdot \sqrt{100-y+50} \geq \sqrt{100}$$
$$p\sqrt{200-y} + (1-p)\sqrt{150-y} \geq \sqrt{100}$$
I run into the same problem again. Not sure if I set up the wrong equation or if there's another way to think about this.
I tried searched for alternate solutions elsewhere and keep getting different answers from different people that could also be right. One person mentioned using the certainty equivalent (not sure how to do this) whereas another didn't even use the utility function to determine the buying price. If anyone can shed some light on this problem it would be greatly appreciated!