Problem 6 - Health Insurance Consider an insurance market with two kinds of individuals x {,} where x is the individual's expected health cost.

Problem 6 - Health Insurance

Consider an insurance market with two kinds of individuals x ∈ {¼,¾}

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. Insurance companies can offer health insurance, meaning they sell a contract where they promise to pay the health costs of the policy holder. The willingness

to pay for insurance of a type x individual is W (x) = 1.5x. There is a total of 2,400 individuals in the market.

1. Suppose an individual's health type x can also be observed by the insurance company and the insurance premium can be made conditional on x. What is the aggregate surplus in the health insurance market in this case?

2. Now, suppose there is asymmetric information and the insurer cannot tell the health type of an individual. Suppose insurers price insurance actuarially fair.

(a) 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(¼) is the price the insurance company will charge if it expects to insure types {¼,¾}

in equal proportion, and p(¾) is the price it charges

if it expect to have only type x = ¾ in its insurance pool. Determine p(¼) and p(¾)

(b) 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. 

(c) What is the deadweight loss due to asymmetric information relative to the market where x is observable to everyone?