Assignemnt
1. In Reidland, ever citizen has a wealth of 10 and utility represented by U(I) = I - 0.3 *I2. Reidlanders 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 Reidlanders and each has a different probability of getting sick (types are genetic and cannot be changed):
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Type
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Probability of Illness
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Vegetarian
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0.12
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Carnivore
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0.28
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Smoker
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0.667
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(Parts a-f are on adverse selection.)
a. In the absence of health insurance, find the expected utility each type of Reidlander.
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 Reidlander gets sick (Green Cross cannot distinguish between types).
Assume there are equal proportions of each type in the population. What is the actuarial fair premium?
c. If Green Cross charges the actuarial fair rate, what is the expected utility for each type?
d. Will every type of Reidlander join the plan?
e. (Adverse Selection) 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? Who joins the plan in the 2nd year?
f. Is the plan in e. sustainable? What will the eventual equilibrium be? Who will buy insurance and how much will it cost under the sustainable plan?
g. (Separating Equilibrium) Suppose now that Green Cross gets smart and offers two plans. The first plan is based on the plan in (0. Call it Plan I. The second plan offers a small insurance plan with a co-payment system (Plan II). For a cost of .1 (paid if a person is sick or ell), Plan II pays .5 if an individual gets sick. The individual then pays the rest. Who buys which plan? Why?
h. Are these plans sustainable? (ie, the firm is not losing money). How does this strategy solve the asymmetric information problem?
2. Owners of a firm know that their profits depend on two things: how hard the manager works and the state of the economy. Managers exert either maximum or minimum effort. The economy can be either good or bad. Outcomes are given below.
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Good Economy
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Bad Economy
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Maximum Effort
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700,000
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400,000
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Minimum Effort
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400,000
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200,000
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The economy is good or bad with equal probability. The manager faces a cost of effort of C = 55,000 for maximum effort and C=0 for minimum effort (she can always spend her time selling craft goods over the Internet and make 55,000; if she exerts maximum effort, she doesn't have time to peddle anything). The firm considers paying the manager according to one of the schemes below Evaluate each scheme for its incentive effects on the manager's performance and the firm's expected profits.
Assume the manager is risk-neutral.
a. A flat salary of 30,000 that is not tied to firm performance.
b. A bonus of 0 if profits equal 200,000 or 400,000 and a bonus of 120,000 if profits equal 700,000. In addition, the employee gets a 10,000 salary
c. A bonus given by the formula Bonus = .20*(Profit - 300,000) if Profit > 300,000. Also, the contract gives a 10,000 salary.
d. A bonus given by the formula: Bonus = .30*(Profit - 300,000) if Profit > 300,000. Also, the contract gives a 10,000 salary.