Insurance market - full coverage insurance, Microeconomics

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There are two individuals in town, one is high risk and the other is low risk. 1 The probabilities of having an accident for the low risk individual and high risk individual are pL = 025 0. and pH = 050 0 respectively. Assume for now that no individual will have more than one accident during the year. Each accident costs 2000 TL.

a. Suppose that the insurance company is only offering full-coverage insurance. And also assume that high risk individual is risk neutral; and hence is willing to pay a premium equal to his expected costs, but no more. But the low risk individual is risk averse and hence willing to pay more than his expected costs to avoid the risk; and say that his risk premium is 25 TL [he is willing to pay an additional 25 TL over his expected costs just to avoid the risk]. Since the insurance company cannot observe the type, it offers a single insurance contract for all. Insurance company is trying to decide on the premium to charge. Determine, which individual(s) buys the insurance, and the profits of the insurance company if the premium charged is 50; if the premium charged is 75 and if it is 100.

Can the insurance company provide insurance to both parties without making a loss?

b. What if the risk premium of the low risk individual was lower than 25 TL? Can the insurance company serve all ensuring that it is not making any losses? What is the name of this problem? Why does it arises?

c. Suppose that now the insurance company charges a low premium of 50 TL for an individual who passes a "safe driving test". And charges a premium of 100 TL otherwise, To take a test a payment of 20 TL to the certification agency, For the low risk individual the driving test is easy, since he is slow and cautious he pass the test at his first trial. But for the high risk individual passing the test requires at least 3 trials. Check if passing the test can act as a credible signal or not: Would risky individual take the test? What about the low risk individual? Is this a separating or a pooling equilibrium then?

d. Suppose that during the year both individual were insured. At the end of the year the insurance company checks its records and finds out that the number of accidents were quite higher than expected for both of the individuals. They are surprised. What might be going wrong, and how can this be corrected?

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