Consider the moral hazard problem in the context of an insurance contract. Suppose that all customers...

Consider the moral hazard problem in the context of an insurance contract. Suppose that all customers are identical with utility function U(w)=√w. Each customer has the choice to drive safely or dangerously, and this behavior is not observable to the insurance company. Consider a driver, call him Antonio, for whom driving safely costs 5 units of utility (in terms of increased travel times) but decreases the risk of an accident for Antonio from 10% to 2%. If there is no accident, Antonio’s wealth is $90,000. If there is an accident, his wealth decreases to $40,000.

Suppose the insurance company offers a Premium of $400 and a deductible of $30,000. Given the choice between this policy and not getting insurance, does Antonio buy this policy? Does he drive carefully? What are the insurance company’s expected profits from this policy?

Solutions

Expert Solution

a. The utility function of the customers is U(w)=√w. If there is no accident then the wealth is $90,000 and if there is an accident then the total wealth is $40,000. Driving safely costs 5 utils of utility and decreases the risk of accidents from 10 percentages to 2 percentages.

U(w) = √90,000 = 300.

U(w) = √40,000 = 200.

Driving safely decreases the utile by 5 units. Therefore, total utils for driving safely will be 300 - 5 = 295 utils.

If the person takes the policy of which a deductible is of $30,000 and a premium of $400. Then the total money he has to pay is $30,000+$4000 = $34,000. So deducting this value from $90,0000 = $90,0000 - $34,0000= $56,0000.

U(w) = √56,000 = 236.64

When the person has paid an insurance he won't be driving safely, do e don't be losing his $5 utils. So, if he pays an insurance he will get 235.64 units of utility.

Antonio should buy the insurance as he will be secured and even after paying the insurance the utility he is getting is more than that of without insurance. Even if he meets an accident the utility which he is getting by doing an insurance is more than not doing it.

Antonio will not be driving carefully as he knows he has already paid an insurance amount which will secure him. Hence, he will not drive carefully.

Driving carefully has decreased the risk of an accident from 10% to 2%. If Antonio pays an insurance then the company will get $30,000+400 = $34,000. After paying an insurance amount Antonio will not drive carefully as that reduces his travel time. Therefore, expected profit from the insurance is,

2%of 34,000 + 10% of 0.

= 2*34,0000/100 + 0

= $680.

Rahul Sunny answered 1 year ago

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