Econ 152 Fall 2023: Problem Set 2 Due Date: Day: December 5, Time: by noon. Please upload your completed assignment to Canvas in the Assignments tab. Part A: Exercises Question 1 [Moral Hazard versus Adverse Selection] Explain whether each of the following is an example of moral hazard or adverse selection. a) Unemployed workers search less actively for a job when they are covered by unemployment insurance. b) A world traveler who intends to engage in extreme sports across the world purchases extended international travelers’ insurance. c) Someone whose parents lived to 110 is anxious to buy a life annuity (i.e., a policy that costs a fixed amount upfront and then pays a certain amount annually until death). d) An employee in a firm with a generous workers’ compensation package, which extended its old coverage to include minor back pain issues, decides to stop using an ergonomic chair because it is too bulky for the small office space. e) People who are eligible for Social Security retire earlier than they would otherwise. 2 Question 2 [Social Security and Private Retirement Savings] Consider an economy that is composed of individuals who live for two periods, and who have the following preferences over consumption in period 1 (C1) and period 2 (C2): U = ln(C1) + ln(C2) These individuals all receive an income Y in period 1, and no income in period 2. They can save as much of their period 1 income as they like in bank accounts, earning a real interest rate equal to r. a) How much will a typical individual choose to save under this system? [Hint: First find the budget constraint relating consumption in the second period to consumption in the first period. Solve for optimal consumption in period 1, and then back out optimal savings.] b) Now, the government decides to set up a social security system. This system will take a (small) amount, T, from each individual in the first period, put it in the bank, and transfer it to them (with interest) in the second period. What does this do to the amount of savings done by individuals? What does it do to total savings in society? What would we call this type of a social security system? c) Now assume that instead of putting the money from the tax (T) in the bank, the government simply transfers it from the current young to the current old in any period (keep assuming that T is small). Assume that the population grows at n% per period, and that there is no wage growth (g=0). (What do we call this type of social security system?) Solve for consumption in period 1 for individuals who enter period 1 after this social security system is already in place. How much will individuals choose to save in period 1 under this system? [Hint: there are different returns on private investment and on taxes paid to the social security system. Your answer should be formulas for individuals’ optimal choices in terms of the problem parameters: Y, T, r, n]. d) How do savings by individuals under the system in c) compare to savings by individuals under the systems in a) and b) under the following three cases: i) n = r ii) n > r iii) n < r Explain your findings in each case. How do total societal savings in c) compare to total societal savings in a) and b)? 3 Question 3 [Insurance: Workers’ Compensation] All residents in Smallville work at the only local factory. They all earn the same annual wage W. There is some risk that a worker may suffer a minor injury on the job because the factory has a lot of dangerous equipment. If a worker is hurt on the job during the year, the worker suffers a financial loss of L (this includes doctor bills and other costs). There are three different types of residents. Type A residents are quite accident-prone, type B residents are moderately careful, and type C residents are extremely careful. The chance that a worker of type A has an accident during the year is 2/5, the chance that a worker of type B has an accident is 1/5, and the chance that a worker of type C has an accident is zero. a) Insurance companies offer to sell accident insurance to residents in this town. Assume that companies can tell what type each resident is. Describe the insurance contracts (the relationship between the insurance premium, a, paid by the worker and the net benefit, b, received by the worker in the case of an accident) that competitive firms will offer to workers of each type. [Hint: You are simply finding the zero-profit condition for insurance firms when dealing with each type of worker.] b) Each individual has an expected utility function that looks like the following: EU = p ln(YA) + (1–p) ln(YNA), where p is the probability of an accident, YA is an individual’s net income in case of an accident, and YNA is an individual’s net income if no accident occurs. Assume that W is equal to 100 and L is equal to 50. If a person of type B is offered an actuarially fair insurance contract, derive the amount of insurance this person would choose to buy. Interpret your result. [Hint: Find the optimal amount of insurance by choosing either a or b to maximize a type B person’s expected utility subject to the appropriate no-profit insurance constraint.] c) Would a type A person prefer full insurance at actuarial fair rates appropriate to their own risk or the insurance contract chosen by type B individuals in part b)? [Do not work this out algebraically. Derive your answer using a graph with net income in the no-accident state on the xaxis and net income in the accident state on the y-axis.] Would an insurance company be willing to offer full insurance at rates appropriate to type B’s risk if it could not tell type A and type B individuals apart? d) Explain how the government can help to improve problems of adverse selection like the one encountered in part c). 4 Question 4 [Actuarially Unfair Insurance] Assume there is a single private flood insurance company in the city. Let’s consider the decision to purchase flood insurance from the perspective of the owner of a bookstore—whom we will call Susan. Both Susan and the insurance company know that the risk of flood in the bookstore is equal to p, which is fairly high because of the local climate. Susan receives an income of W from the business if there is no flood and would have no income (a financial loss of the entire W) in the event of a flood. Susan’s expected utility (EU) depends on income (Y) in each of the two states as follows: EU = p × ln(Yflood) + (1–p) × ln(Yno flood) Because of the lack of competition, private flood insurance is not offered at actuarially fair rates and the insurer makes a profit. Assume that the insurance firm sets its policy such that expected profits (that is, expected revenue minus expected costs) are equal to k times the premium collected. So, if a is the premium paid if there is no flood and b is the net payout if there is a flood, expected profits for the firm are equal to ka. a) Find the equation describing the set of flood insurance contracts the firm offers Susan. Show graphically how the implied budget constraint compares to the budget constraint associated with actuarially fair insurance. Set up your graph so that income in the event of no flood is shown on the x-axis and income in the case of a flood is shown on the y-axis. b) Given the set of contracts offered by the flood insurance company, how much insurance would Susan choose to buy? Does she choose to buy less or more than full insurance? Why? Derive your answer algebraically and explain using a graph. 5 Part B: Short-Answer and Multiple-Choice Questions Use the following to answer questions 1-3: You enjoy riding bicycles down steep mountains. You estimate that in the next year, there is a 4% probability you will crash and incur medical bills of $20,000, which is equal to your salary, and a 96% chance that you will not be injured at all. Assume your utility level depends only on your consumption/income in either possible states of nature (and that your utility function itself is the same in each state of nature). 1. If you buy full insurance, which of the following combinations could represent your utility if you are injured or not injured, respectively? A) uninjured: 34; injured: 1 B) uninjured: 500; injured: 20 C) uninjured: 180; injured: 190 D) Both a and b are possible. E) None of the above is possible. 2. Suppose that you instead buy only partial insurance. Which of the following combinations could represent your utility if you are injured or not injured, respectively? A) uninjured: 34; injured: 1 B) uninjured: 500; injured: 20 C) uninjured: 180; injured: 190 D) Both a and b are possible. E) None of the above is possible. 3. If you buy full insurance at an actuarially fair premium, which of the following represents the dollar value of goods consumed if you are injured or not injured, respectively? A) uninjured: $20,000; injured: $20,000 B) uninjured: $19,400; injured: $19,400 C) uninjured: $19,200; injured: $19,200 D) uninjured: $20,000; injured: $0 E) Both b and c are possible. 4. Consumers aim for ____________ because of ______________. A) self-insurance; diminishing marginal utility B) consumption smoothing; diminishing marginal utility C) self-insurance; means-tested program benefits D) consumption smoothing; means-tested program benefits E) moral hazard; diminishing marginal utility 6 5. Insurance allows individuals to do which of the following? A) consume more in every period in the future B) have more steady consumption across periods in the future which could involve either bad or good possible outcomes C) shift consumption from bad possible outcomes to good possible outcomes D) all of the above E) both b and c 6. Which of the following are examples of self-insurance? A) borrowing B) drawing on savings C) receiving unemployment insurance D) both a and b E) both b and c 7. Is self-insurance likely to be greater for rich or poor people? Explain. 8. Suppose you and your immediate relatives agree to support each other in the event one of you loses your job. This is an example of which of the following? A) disability insurance B) Social Security C) worker’s compensation D) self-insurance E) unemployment insurance 9. Suppose you go from being uninsured to buying a health insurance policy. You respond by jumping out of a perfectly good airplane for the first time—attached to a parachute, of course. This behavior is best characterized as which of the following? A) adverse selection B) means testing C) moral hazard D) both a and b 10. Suppose you go to buy a life insurance policy and the underwriter asks you to allow the company the right to access all your medical records. What is the underwriter concerned about? A) adverse selection B) rent-seeking C) moral hazard D) self-insurance 7 11. Suppose that you are selling life insurance, and a 40 year-old male nonsmoker wants to buy life insurance and asks you for a price on a $1 million policy for the next ten years. Your co-worker suggests that the rate should be based on the death rates of male nonsmokers in their 40s in the entire general population. What do you say? 12. Suppose all your health care expenses are covered under your health insurance policy, so that you face no out-of-pocket expenditures. You will use medical care up to the point where: A) your total benefits equal the costs of providing the medical care. B) your marginal benefit is zero. C) your marginal benefit is equal to the marginal cost of the medical care. D) your marginal benefit is equal to the total costs of providing the medical care. E) your total benefits are equal to the cost of your health care insurance. 13. People buying insurance aim to shift consumption from periods where consumption is ___________ to periods where consumption is ___________ when marginal utility is relatively ___________. A) high; low; low B) high; low; high C) low; high; high D) low; high; low E) None of the above is correct. 14. If the government pays part of the health care costs of those citizens determined by the government to be economically deprived, which of the following justifications for government intervention is being used? A) high administrative costs B) redistribution C) paternalism D) externalities E) adverse selection 15. You lower the deductible on your car insurance and start driving less carefully. This is an example of which of the following? A) adverse selection B) rent-seeking C) moral hazard D) self-insurance