Suppose that there are two types of individuals: a share d of them carries the T gene, which will make them become sick with probability k_T ; a share 1 d of them carries the G gene, which will make them become sick with probability k_G < k_T . All individuals have income W, but if they become sick the cost of treatment is m such that their disposable income in that case is only W m. In the questions below, assume that there is no moral hazard (unless mentioned otherwise) and that individuals maximise their expected utility, which in absence of health insurance is: EU = k U(W m) + (1 k) U(W) Assume first that neither insurance companies nor individuals know who has a T gene and who has a G gene. However, everybody knows the proportion of individuals with T and G genes in the population, and everybody knows the probability to become sick with each gene. In this situation, insurance companies offer a "pooling contract" in which any individual receive a payment b if they becomes sick, depending on the amount of premium p paid by the individual (irrespective of whether they become sick or not).

5a) How would this insurance contract be priced (i.e., what would be the relationship between b and p given d, k_T , k_G) if it was actuarially fair? Would the insurance contract allow individuals to buy full insurance? If your answer to this second question is "No", explain how the insurance companies could calculate the maximum amount of insurance coverage - i.e., the maximum premium p - that they will allow individuals to buy given d, k_T, k_G, and m.

5b) What would be different in the contract offered by insurance companies if there was moral hazard?

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Now, suppose that insurance companies offer the actuarially fair contract in 5a. [Note: we continue to assume that there is no moral hazard in the questions below.]

5c) Would both types buy the insurance contract (i.e., would both types think that they are better off with the insurance contract than without insurance), given that they don't know their type? Explain what the answer to such a question depends on.

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Now, suppose that a firm developed a simple genetic test that allows individuals to learn their type, and that all individuals do it because it is free. However, for ethical reasons, insurance companies are not allowed to learn the result of such tests.

5d) Would both types buy the insurance contract in 5a (i.e., would both types think that they are beer o with the insurance contract than without insurance) given that they know their type? Explain what the answer to such a question depends on.

5e) If individuals know their type, then insurance companies could offer two different insurance contracts called "separating contracts" instead of the pooling contract in 5a. How would these contracts be priced (i.e., what would be the relationship between b and p given d, k_T, k_G) if they were actuarially fair? Would these contracts allow individuals to buy full insurance? If your answer to this second question is "No", explain how the insurance companies could calculate the maximum amount of insurance coverage - i.e., the maximum premium p - that they will allow individuals to buy given d, k_T, k_G, and m. Which type would buy which contract?

5f) Which type(s) will be better off with the separating contracts in 5e than with the pooling contract in 5a? Explain what the answer to such a question depends on.

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Now, suppose that insurance companies are not allowed to ask the result of the genetic tests, but that individuals are allowed to volunteer that information to insurance companies.

5g) Which individuals will have an incentive to reveal the result of their genetic tests? What will likely happen to the insurance market if all such individuals reveal the result of their genetic tests? Which individuals will better off in the resulting insurance market compared to an insurance market in which only the "pooling contract" in 5a would be offered? Which individuals will be better off in the resulting insurance market compared to an insurance market in which only the "separating contracts" in 5e would be offered? [Note: we continue to assume that all contracts will be actuarially fair.]