Question

Consider the moral hazard problem in the context of an insurance contract. Suppose that all customers are identical with utility function . 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.

If Antonio does not buy insurance, will he drive safely or dangerously? What is his EU from each of these choices?

In designing a partial insurance policy to sell to Antonio that would incentivize careful driving, what constraints would the insurance company have to meet with the premium (P) and deductible (D) so that Antonio buys the insurance and drives safely? HINT: Think of the deductible as a negative bonus Antonio must pay in the event of an accident.

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?

Answer 1

1. IF THERE IS NO ACCIDENT ANTONIO'S WEALTH = $90000
   IF THERE IS AN ACCIDENT, THEN ANTONIO'S WEALTH= $40000
   THE EXPECTED WEALTH FROM THE INSURANCE = (.5*90000+.5*40000)= (45000+20000)= 65000
   THE UTILITY FOR WHOM, WHO DRIVE SAFELY= 5 UTILES (UNIT OF UTILITY)
   THEN THE EXPECTED UTILITY FOR THE SAFELY DRIVING= (5*.10)= .5
   THE EXPECTED UTILITY FOR THE DRIVING LESS SAFELY= (5*.2)= 1
   IF ANTONIO DID'T BUY THE INSURANCE,HE WILL DRIVE SAFELY
2. ANTONIO DID'T BUY THIS POLICY BECAUSE HE HAS MORE EXPECTED WEALTH FROM THE INSURANCE
   HE DID DRIVE CAREFULLY
   THEY CAN EARN MORE THAN INSURANCE BY THIS POLICY