To receive credit for this homework, it must be handed in class

M. QuinziiFall 2015Econ. 103 : Economics of UncertaintyHomework 5To receive credit for this homework, it must be handed in class on Monday November 16. The day whenthe homework is due is unusual because there is no class on Wednesday November 11 (Veteran’s day).The next midterm is on Wednesday November 18. The material covered by the midterm is (i) theoptimal choice of insurance of a risk adverse agent and (ii) the equilibria in the insurance market depending onthe assumptions on information. This corresponds to Chapters 2 and 3 of the class notes and homework 3 to 5.(i) was already in midterm 1 so you should not have too much work to review it. However for those who did notdo too well on the first midterm, please make sure that you know this material well.1. Consider the following scenario: the population is composed of two types of agents, the “Healthies” and the “Sicklies”. The Healthies have a good immune system and rarely catch diseases: inany given year only 1/10 of them get sick. The Sicklies have a deficient immune system and easilycatch diseases: in a given year half of them get sick. The population is composed of three quartersof Healthies and a quarter of Sicklies. All agents have the same income of $60,000, incur the samecost of $20,000 if they are sick, and are expected-utility maximizers with a logarithmic (naturallogarithm) utility function for wealth. Suppose that there is a large number of insurance companiesand free entry in the insurance business so that competition drives profits to zero.(a) Suppose that there exists a reliable screening test whose result indicates with certainty if anagent is a Healthy or a Sickly. If the test is not performed on an agent, neither the agentnor the insurance company can know if the agent is a Healthy or a Sickly. The governmentis afraid that if agents are screened, those who turn out to be Sicklies will be discriminatedagainst in the work place. In consequence it issues a regulation forbidding the hospitals toperform the test. What will be the equilibrium of the insurance market?(b) The insurance companies are complaining that the Sicklies cost too much and need to paymore for their insurance. They are lobbying Congress to change of the legislation and enable1them to test their customers. Will consumers join the insurance companies’ lobby for theremoval of the regulation or will they lobby to keep the regulation in place? Explain carefullyyou answer.(c) Now suppose the regulation preventing insurance companies to determine the type of theircustomers is maintained, but that agents, by observing their own medical history, can knowtheir own type. Suppose that the only insurance contracts which have been invented untilnow specify the price of a dollar of insurance and consumers can choose how much insurancethey want to purchase (provided the amount that they buy is non negative and not largerthan the loss of $20,000). Show that the zero-profit equilibrium is such that only the Sickliesget insured. What is the price of insurance? How much insurance each agent buy? Arguecarefully why this is the only equilibrium of this type. [Here an equilibrium is such that insurancecompanies make zero profit by offering a per-dollar price for insurance and all other per-dollar pricesmake losses. The possibility of entry with a different contract is not yet taken into account.](d) Assume that the situation is the same as in question (c), except that an insurance company,InsurMagic, introduces a contract where the price per dollar of insurance is only 12.5 centsbut the quantity of insurance that an agent can take is limited to $4,000. Moreover thecompany requires exclusivity (i.e. if another insurance company reimburses part or all of themedical costs of the insured agent, InsurMagic does not reimburse anything). Explain whyInsurMagic, which seemed to be such a successful company with a wide pool of customers,filed for bankruptcy soon after introducing this contract.(e) Another insurance company, SmartInsurance, witnessed the downfall of InsurMagic, butthought that the idea of selling insurance with a limited coverage and exclusivity was interesting. After careful study it proposed a contract at the price of 12.5 cents for each dollarof insurance, with a limit of $ 2,000 on the amount of insurance that can be bought, and withexclusivity. Explain why SmartInsurance made a lot of profit out of this contract.(f) Other insurance companies, witnessing the success of SmartInsurance, began to imitate itand to offer contracts with limited coverage. Show that the equilibrium of the market is suchthat all insurance companies offer two types of contracts: one with unlimited coverage at the2price of 50 cents for a dollar of insurance, and another with a coverage of $2007.25 at theprice of 10 cents by dollar of insurance. Argue carefully why this pair of contracts constitutesan equilibrium.2. Consider a population composed of two types of agents, the Explorers and the Sedentaries. TheExplorers like to move around and explore the world, while the Sedentaries like staying home andenjoying the company of family and friends. From the point of view of a car insurance company,the Explorers have a higher probability of accident than the Sedentaries since they spend more timeon the road. More precisely, the probability of accident of an Explorer is 0.5 while the probabilityof accident of a Sedentary is 0.2. Both types have the same average income of $70,000 and thesame average cost of accident: in case of an accident it costs $30,000 to fix the cars involvedin the accident and pay the medical bills. Not surprisingly the Explorers and the Sedentarieshave different preferences. The Sedentaries maximize expected utility with a (ex post) utility forconsumption u(x) = −1/x, while the Explorers maximize expected utility with a (ex post) utilityfor consumption v(x) = ln(x).(a) What is the coefficient of relative risk aversion of the Sedentaries? of the Explorers? Whichtype of agents is more risk averse? Give an example of a situation—which is not necessarilyrelated to driving an automobile— where the two types of agents will behave differentlybecause of the difference in their risk aversion.(b) Suppose that, by inspecting the odometers of their customers’ cars, the insurance companiescan distinguish between the Explorers and the Sedentaries. Suppose in addition that theinsurance market is perfectly competitive with free entry. Explain what will be the equilibrium on the car insurance market i.e which price the insurance companies will charge theircustomers and which amount of insurance these customers will choose.(c) To motivate better your answer to (b) represent in a graph with co-ordinates (Xg , Xb ) whereXg is the consumption in case of no-accident and Xb the consumption in case of an accident(i) the no-insurance point for both types of agents(ii) the indifference curve of the Explorers through this no-insurance point: give the equationof this indifference curve and its slope at the no-insurance point3(iii) the indifference curve of the Sedentaries through the no-insurance point: give the equation of this indifference curve and its slope at the no-insurance point(iv) the indifference curve of the Explorers through their equilibrium consumption (labelthis point: E): give the coordinates of this point, the slope and the equation of theindifference curve through this point.(v) the indifference curve of the Sedentaries through their equilibrium consumption (labelthis point: S): give the coordinates of this point, the slope and the equation of theindifference curve through this point.Note: It is recommended that you refer to this graph when answering the following questionsUnfortunately the Explorers learn how to take back miles from their odometers so that theinsurance companies can no longer rely on the examination of these odometers to find out the typeof a customer. Now they have to sell the same insurance contract(s) to all customers, and let thecustomers choose what they want to buy.(d) Suppose that it is legally impossible for an insurance company to restrict the amount ofinsurance that an agent wants to buy (up to the amount of the loss). Thus the insurancecompanies can only offer a contract of the following type: they fix a price per dollar ofinsurance and the customer chooses how much insurance he/she wants to buy. Explain whatwill be the equilibrium in the car insurance market. ( Be precise: indicate the price ofinsurance, how much insurance an agent of each type will choose, what will be his/her finalconsumption in each circumstance. Explain why it is an equilibrium).(e) Explain why there is an inducement for the insurance companies to introduce other contracts,with a lower price per dollar of insurance but a limited amount of insurance, and an exclusivityclause (i.e. agents choosing this type of contract cannot buy more insurance from otherinsurance companies).(f) Suppose now that the insurance companies have lobbied effectively and the regulation ischanged: that can can sell contracts with or without limitation of the amount of insuranceand with or without exclusivity clause. Explain what will be the equilibrium in these circumstances, and indicate in the graph drawn in question (c) the equilibrium consumption of each4type of agent (label E the equilibrium consumption of the Explorers and S the equilibriumconsumption of the Sedentaries.(g) Calculate the coordinates of E and S . Indicate exactly which insurance contracts the insurance companies are offering to their consumers and which contract each type of agentschooses.5

 

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