# 3) Let 2) denote the expected healthcare costs of a given individual. Suppose that v is distributed uniformly over the interval [0,10] in the young…

Let v denote the expected healthcare costs of a given individual.

3) Let 2) denote the expected healthcare costs of a given individual. Suppose that v isdistributed uniformly over the interval [0,10] in the young population, and uniformly overthe interval [10, 30] in the old population. Suppose further that because of risk aversion, aperson With expected healthcare cost 1) is Willing to pay up to v + 4 to purchase (complete)health insurance. Assume that % of the total population are young and that % are old.Each person knows his or her expected healthcare costs, but insurance companies can observeonly Whether a person is young or old. Assume that the insurance industry is perfectlycompetitive. a) Suppose all insurance companies set premium p for the young. Identify the set of youngpeople Who purchase health insurance at that premium, and ﬁnd expected insurance-companyproﬁts as a function of p. Do the same for the old, When all companies set premium q forthem. b) Find equilibrium premiums 19* for the young and q* for the old. Show that at thesepremiums, the proportion of the old people Who purchase health insurance is smaller thanthat of the young. Hint: Note that the proﬁts of the insurance company from young con-sumers are 7r(p) = p — E [v|insured young given p] and the proﬁts from old consumers are7r(q) = q — E [v|insured old given q]. Use the fact that there is perfect competition amonginsurance companies (so that they earn zero proﬁts from both types of consumers) to derive the equilibrium prices.

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