I'm struggling with understanding a problem in finance:
We have 2 states:
$S_1$: bad state $Y-K = 2000$; probability $\pi$ = 10%
$S_2$: normal state $Y = 5000$; probability $1-\pi$ = 90 %
If the insurance premium is $p = 0.2$, what would be the optimal amount of insurance $q$ for the following utility functions?
a) $u(x) = x$ (risk-neutral)
b) $u(x) = x^2$ [This one I find particularly tricky?] (risk-loving)
I know that we have to max $\pi \cdot u(Y-K+q-q \cdot p)+(1- \pi) \cdot u(Y-qp)$ and then take the derivative w.r.t. $q$ and set it equal to $0$ for risk-averse people, but how about these cases .
I also have the answers, but cannot figure out how exactly we reach them, so they are not so useful to me. ( a) $q=0$ or $q= -500$ if short sales are allowed; b) $q=22000$
I suppose that in both cases we would have a corner solution but at what value if at all this is true. I would be really grateful if somebody could explain this exercise to me :)
Best regards!