A health insurance company knows that there are two types of customers (smokers and non-smokers), each facing different health risks. The probabilities of getting sick and the healthcare costs in the case of illness for the two customer types is given in the table below.
Group
Healthcare costs
Probability of getting sick
Smokers
$1200
50%
Non-smokers
$1200
20%
Assume that each customer has a monthly income of $1600 and has a utility function given by U(x)=sqrt(x), where x is the remaining income after medical/health insurance expenses have been paid.
a. Explain the problem of “adverse selection” and the problem of “moral hazard.” Give one example of a market in which adverse selection occurs and one example of a market in which moral hazard can be observed. What is the difference between the two information problems? Explain the consequences of adverse selection and moral hazard for the principal – agent relationship.
b. Discuss in detail the two ways (screening and signalling) in which the principal or the agent can try to resolve the problem of adverse selection. Give an example of screening and signalling and explain how the parties involved benefit from these strategies.
c. For the numerical example above construct the lotteries associated with the income that remains after healthcare expenses have been paid for smokers and for non-smokers (assuming agents do not have health insurance). Calculate the expected utility associated with the lotteries for smokers and non-smokers. Calculate the certainty equivalent of the two lotteries. What is maximum amount that smokers and non-smokers are willing to pay for full insurance? Calculate the actuarially fair health insurance premium for the two groups of customers (i.e. the expected healthcare costs).
d. Assume the company cannot distinguish between smokers and non-smokers and offers a full insurance contract to all customers. How much should the company charge for the full insurance contract? Which customer group will purchase full insurance?