# Consider an insurance market with three kinds of individuals

April 14, 2018

Consider an insurance market with three kinds of individuals x belongs to (1/4, 1/2, 3/4) where x is the individual’s expected health cost. We will also refer to x as an individual’s health type. There are equally many of each health type in the population. So each type’s representation in the population is 1/3. Insurance companies can offer health insurance, meaning they sell a contract where they promise to pay the health costs of the policy holder. The wiliness to pay for insurance of a type x individual is W(x).5x. There is a total of 2400 individuals in the market.Now suppose there is asymmetric information and the insurer cannot tell the health type of an individual. Suppose insurers price insurance actuarially fair.As pricing is actuarially fair, the insurance price is set equal to the expected cost of insuring an individual. Denote by p(x) the price the insurer will charge if it is insuring all types above and including x. So, p(1/2) is the price the insurer will charge if it is insuring all types above and including x. So, p(1/2) is the price the insurance company will charge if it expects to insure types(1/2, 3/4) in equal proportion. Determine p(1/4), p(1/2) and p(3/4).Only individuals with willingness to pay W(x) greater than or equal to the price of insurance will purchase. Determine the equilibrium price in the market and the types of individuals that are insured.Relative to the case where health types are observed by all, how much lower is the aggregate surplus in the asymmetric information case where health type is only known by the Indi dual.

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