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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.

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1 Answer 1

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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.

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