5. Adverse Selection, numerical example. In the Shire, every hobbit has an incom

Question

5. Adverse Selection, numerical example. In the Shire, every hobbit has an income of 10 each year and a utility function represented by U(I) I-0.0512. Hobbits occasionally get sick, and when they do, they go to the hospital and are made immediately better A hospital visit costs 8. There are three types of hobbits and each has a different probability of getting sick (types cannot be changed): (14 points) Annual Probability of Illness 0.12 0.30 0.60 Type 6 meals per day 7 meals per day Smoker (a) In the absence of health insurance, find the expected utility of each type of hobbit (b) A private health plan, Green Cross, is set up. Membership is optional, but since everyone is risk averse, the managers assume that everybody will join. Green Cross sets their premiums according to the chance that an average hobbit gets sick (Green Cross cannot distinguish between types). Assume there 10% of the population of Shire are six meal a day hobbits 10% of the population of the Shire are 7 meal a day hobbits, and the remaining 80% are smoking hobbits, what is the actuarially fair premium? (c) If Green Cross charges the actuarially fair rate, will every type of hobbit join the plan? What is the expected utility for each type? (d) After the 1st year of operation, Green Cross finds they have lost money. Why? They commission a study to discover the probability that a member of Green Cross gets sick. What is the probability? They fix a new actuarial fair premium based on their study. What is the new rate? (e) Who joins the plan in the 2nd year? Why? Is the plan sustainable? (f) What will the eventual equilibrium be? Who w buy insurance and how much will it cost under the sustainable plan? (g) Would an individual mandate to buy insurance make the hobbits better or worse off relative to a world with no individual mandate?

Explanation / Answer

a) If the hobbit does not get sick, his income is 10

U = 10 - 0.05 (10)2 = 10 - (0.05*100) = 10 - 5 = 5

If he hobit get sick, his net income is = 10 - 8 = 2

Usick = 2 - 0.05(4) = 2 - 0.2 = 1.8

Therefore, the expected utility of each type of hobbit is :

EU(6 meals) = 0.88* 5 + 0.12*1.8 = 4.4 + 0.216 = 4.616

EU(7 months) = 0.7*5 + 0.3*1.8 = 3.5 + 0.54 = 4.04

EU(smoker) = 0.4*5 + 0.6*1.8 = 2 + 1.08 = 3.08

b) The actuarially fair premium is equal to the expected value of the compensation recieved in the event of the loss. Since the premiums are set accorging to the probability of each hobit getting sick, the expected value of compensation is given by the sum of the expected loss for each type multiplied by their proportion in the population as follows:

E(compensation) = The loss incurred in case of sickness* (0.12*0.1 + 0.3*0.1 + 0.6*0.8) = 8* (0.012+0.03 + 0.48) = 4.176

Therefore, the premium charged is 4.176

c) the expected loss for the 6 meals hobit = 0.12*8 = 0.96

the expected loss for the 7 meals hobit = 2.4

expected loss for smoker = 4.8

Since the expected loss for only the smoker is greater than the actuarially fair premium, only the smoker will join the programme.

Utility after insurance = (10- 4.176) - 0.05*(10-4.176)2 = 5.824 - 0.05*(5.824)2 = 4.128

Since each hobit gets the same amount of 5.824 whether or not he gets sick, the expected utility of all types of hobits from purcahsing the insurance at the actuarially fair premium is 4.128.

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