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

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2022-04-17T22:05:27.697Z
2022-04-17T22:45:46.457Z
2022-04-17T22:45:46.457Z
Herr K.
<p>Bob is an expected utility maximizer with utility function <span class="math-container">$u(x) = −e^{−ax}$</span>, where <span class="math-container">$a &gt; 0$</span> is a parameter. Bob has wealth <span class="math-container">$w$</span>. There are two states of the world, a good state and a bad state. The probability that the state is bad is <span class="math-container">$π$</span>. If the state is bad, Bob gets sick,
and would need to spend an amount <span class="math-container">$L$</span> 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 <span class="math-container">$P$</span>.</p>
<ol>
<li>Find Bob’s coefficient of absolute risk aversion</li>
<li>Find the maximum price <span class="math-container">$ \overline{P}$</span> Bob would be willing to pay for the insurance</li>
<li>How does <span class="math-container">$\overline{P}$</span> changes with the parameters w, L, and π</li>
</ol>
<p>So my attempt was for the first one was that as the coefficient of absolute risk aversion</p>
<ol>
<li><span class="math-container">$u'(w)= ae^{-aw}$</span></li>
</ol>
<p><span class="math-container">$u''(w)= -a^2e^{-aw}$</span></p>
<p>Hence <span class="math-container">$A(w) = \frac{-a^2e^{-aw}}{ae^{-aw}} = a$</span></p>
<ol start="2">
<li><span class="math-container">$\max_P \pi u(w-L-P+L)+(1-\pi)u(w-P)$</span></li>
</ol>
<p><span class="math-container">$=\max_P \pi u(w-P)+(1-\pi)u(w-P)$</span></p>
<p><span class="math-container">$=\max_P u(w-P)$</span></p>
<p>From here I felt this was weird I know this is full insurance, but then when the optimizing equation comes out like this how do you proceed??</p>
<p>Also for question 3, how would you find the change of P according to L and <span class="math-container">$\pi$</span>, if it just get removed like that. Please take a hand to help Economic Geniuses.</p>

<microeconomics><consumer-theory><uncertainty>
Expected utility function and Full insurance
user40996
# Expected utility function and Full insurance

(Asked by: username_0 on 2022-04-17)

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

2. $\max\_P \pi u(w-L-P+L)+(1-\pi)u(w-P)$

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

$=\max\_P u(w-P)$

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.

**Tags:** microeconomics, consumer-theory, uncertainty


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## 1 answers

### Answer by username_1 on 2022-04-17. Score: 1 | Accepted answer

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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