# Expected utility function and Full insurance

Bob is an expected utility maximizer with utility function $$u(x) = −e^{−ax}$$, where $$a > 0$$ is a parameter. Bob has wealth $$w$$. There are two states of the world, a good state and a bad state. The probability that the state is bad is $$π$$. If the state is bad, Bob gets sick, and would need to spend an amount $$L$$ on his health. There is a health insurance policy available that fully covers health care expenses in case of sickness. The price of the policy is $$P$$.

1. Find Bob’s coefficient of absolute risk aversion
2. Find the maximum price $$\overline{P}$$ Bob would be willing to pay for the insurance
3. How does $$\overline{P}$$ changes with the parameters w, L, and π

So my attempt was for the first one was that as the coefficient of absolute risk aversion

1. $$u'(w)= ae^{-aw}$$

$$u''(w)= -a^2e^{-aw}$$

Hence $$A(w) = \frac{-a^2e^{-aw}}{ae^{-aw}} = a$$

1. $$\max_P \pi u(w-L-P+L)+(1-\pi)u(w-P)$$

$$=\max_P \pi u(w-P)+(1-\pi)u(w-P)$$

$$=\max_P u(w-P)$$

From here I felt this was weird I know this is full insurance, but then when the optimizing equation comes out like this how do you proceed??

Also for question 3, how would you find the change of P according to L and $$\pi$$, if it just get removed like that. Please take a hand to help Economic Geniuses.

Hint for part 2: Compare Bob's expected utility when he has no insurance (think of this as $$P=0$$) and when he has full insurance at price $$P$$. For the insurance to be preferred, $$P$$ must be such that he is no worse off with insurance than without. $$\overline P$$ is the price that makes him indifferent between having and not having insurance.
Hint for part 3: In part 2 you should get $$\overline P$$ as a function of $$L, w, \pi$$. Simply take partial derivatives with respect to these variables to examine their effects.