$2.$ Jax is a motorcycle enthusiast, whose preference over lotteries is described by Bernoulli

utility function u$($w$)=$w$.$ Riding a motorcycle can result in a serious accident.

Assume that Jax$$s wealth is $100$ if no accident occurs, but if an accident occurs, Jax$$s

wealth is $36($a drop of $64$ due to the accident$).$ The probability of accident depends

on whether Jax decides to ride cautiously or aggressively. In particular, if Jax rides

cautiously, then the probability of an accident $1/10.$ But riding cautiously is less fun$$

the decision to ride cautiously imposes a cost, effectively reducing Jax$$s wealth by $36.$

If Jax rides aggressively, then he enjoys himself $($there is no cost$),$ but the probability

of an accident is $3/4.$

Thus, aggressive and cautious riding can be represented as lotteries over wealth levels:

$$ Aggressive: w $=100$ with probability $1/4,$ and w $=36$ with probability $3/4.$

$$ Cautious: w $=64$ with probability $9/10,$ and w $=0$ with probability $1/10.$

$($a$)$ What is Jax$$s expected utility associated with the $$aggressive$$ lottery?

$($b$)$ What is Jax$$s expected utility associated with the$$cautious$$ lottery?

$($c$)$ Will Jax choose to be cautious or aggressive? What is his expected utility?

$($d$)$ What is the probability that Jax has an accident?

Suppose an insurer enters the market, and offers Jax full insurance at price p $=48.$ In

other words, if Jax buys the insurance, then his wealth will be $52,$ whether he has an

accident or not. If he buys the insurance, he must also decide whether to be aggressive

or cautious. Let$$s consider the choice of whether to be cautious or aggressive if Jax

buys insurance.

$$ Insurance, Aggressive: w $=52$ with probability $1.$

$$ Insurance, Cautious: w $=5236=16$ with probability $1.$

$($e$)$ What is Jax$$s expected utility if he buys insurance and is aggressive?

$($f$)$ What is Jax$$s expected utility if he buys insurance and is cautious?

$($g$)$ Will Jax choose to be cautious or aggressive if he buys insurance? What is his

expected utility? Explain your answer in a sentence or two.

$($h$)$ Compare your answers to $($c$)$ and $($g$).$ Does Jax prefer not to buy insurance and

ride as in part $($c$),$ or does he prefer to buy insurance and ride as in part $($g$)?$

$($i$)$ Given the choices that Jax makes about buying insurance and riding aggressively

or cautiously, how does the existence of the insurance company affect the proba$-$

bility that he has an accident? Explain in a sentence or two.

$($j$)$ What is the insurance company$$s expected profit, given the choice that Jax makes?

Now suppose that a new regulation is introduced, which restricts the amount of insur$-$

ance coverage that a company can offer. In particular, suppose that in exchange for an

initial payment of $1/2,$ the insurance company will pay Jax $5$ if an accident occurs, and

will pay $0$ if no accident occurs. Let$$s consider the choice of whether to be cautious or

aggressive with such a partial insurance policy.

$$ Partial Insurance, Aggressive: w $=99\mathrm{.}5$ with probability $1/4,$ and w $=40\mathrm{.}5$ with

probability $3/4.$

$$ Partial Insurance, Cautious: w $=63\mathrm{.}5$ with probability $9/10,$ and w $=4\mathrm{.}5$ with

probability $1/10.$

$($k$)$ What is Jax$$s expected utility if he buys partial insurance and is aggressive?

$($l$)$ What is Jax$$s expected utility if he buys partial insurance and is cautious?

$($m$)$ Will Jax choose to be cautious or aggressive if he buys partial insurance? What

is his expected utility? Explain your answer in a sentence or two.

$($n$)$ Compare your answers to $($g$)$ and $($m$).$ Which version of insurance is better for

Jax?

$($o$)$ Given the choices that Jax makes about buying insurance and riding aggressively

or cautiously, how does the regulation affect the probability that he has an acci$-$

dent? Explain in a sentence or two.

$($p$)($Bonus$)$ What is the insurance company$$s expected profit, given the choices that

Jax makes? $($Hint: if Jax buys, the insurance company collects $1/2,$ but has to

pay $5$ whenever Jax has an accident$).$