ps6sol - 14.03 Fall 2000 Problem Set 6 Solutions 1....

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Unformatted text preview: 14.03 Fall 2000 Problem Set 6 Solutions 1. Expected utility for consumer with health p is (1 - p ) U (1) + pU (1 - 1) = 1 - p . Expected utility for this consumer if she purchases full insurance at premium r is (1 - p ) U (1 - r ) + pU (1 - r - 1 + 1) = U (1 - r ) = 1 - r . So a consumer purchases insurance iff 1 - r ‡ 1 - p p ‡ 1 - 1 - r . Define p c to be the probability of sickness for the most healthy person to buy insurance, i.e. p c = 1 - 1 - r . Then the average health of those who enroll when the premium is r is (1 + p c )/ 2 = (2 - 1 - r ) / 2. The average profit for the insurance plan is given by average revenue (equal to the premium) minus average cost (equal to average health), or r - (1 + p c )/ 2 = (2 r - 2 + 1 - r )/ 2. A. premium = ½ p c = (2 - 2)/ 2 most healthy enrollee: p = (2 - 2) / 2 least healthy enrollee: p = 1 average health of enrollees = (4 - 2) / 4 average profit = ( 2 - 2) / 4 < 0 , plan loses money. B. premium = (4 - 2) / 4 p c = (2 - 4 2)/ 2 most healthy enrollee: p = (2 - 4 2) / 2 least healthy enrollee: p = 1 average health of enrollees = (4 - 4 2) / 4 average profit = ( 4 2 - 2) / 4 < 0 , plan loses money. C. premium = (4 - 4 2) / 4 p c = (2 - 8 2)/ 2 most healthy enrollee: p = (2 - 8 2) / 2 least healthy enrollee: p = 1 average health of enrollees = (4 - 8 2) / 4 average profit = ( 8 2 - 4 2) / 4 < 0 , plan loses money. D.,E. The premium such that the pool of citizens who enroll at that premium cost on average exactly that premium is r such that r = (2 - 1 - r ) / 2. 1 - r = 2 - 2 r 1 - r = 4 - 8 r + 4 r 2 0 = 3 - 7 r + 4 r 2 0 = (4 r - 3)( r - 1) The relevant root gives us a premium of ¾. Most healthy enrollee: p = ½ Least healthy enrollee: p = 1 Average health of enrollees = ¾ F. 1) If there is no health plan, expected utility is 1- p , so average expected utility is ½. 2) Under the current (voluntary) break-even plan, r = ¾ and p c = 1/2. The expected utility for enrollees is U (1-3/4) = ½. The expected utility for non-enrollees is 1- p , and since p<1/2 for non-enrollees, the average expected utility for non-enrollees is ¾. The average expected utility for all consumers (which is equal to the average expected utility for enrollees times the probability of being an enrollee, plus the average expected utility for non-enrollees times the probability of being an non-enrollee) is (1/2)(1/2)+(3/4)(1/2) = 5/8....
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ps6sol - 14.03 Fall 2000 Problem Set 6 Solutions 1....

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