Asymmetric Information in Competitive Markets

Chapter 22 Asymmetric Information in Competitive Markets In our treatment of externalities in Chapter 21, we introduced into our model for the first time an economic force (other than government-induced price distortions) that causes a competitive market to allocate scarce resources inefficiently in the absence of some other market or non-market institution. 1 We furthermore illustrated that the problems raised by externalities are problems related to the non-existence of some market, necessitating either the establishment of a new market or the fine-tuning of market forces by some non-market institution. In this chapter, we will see another example of an economic force that can result in the non-existence of certain markets and in an inefficient allocation of scarce resources in existing markets. This economic force arises from certain types of information being distributed asymmetrically across potential market participants and, as we will see, it relates closely to a particular type of externality that is generated in the process. Information is, of course, always different for buyers and sellers with buyers knowing about the tastes and economic circumstances that underlie their demand for a good and sellers knowing the costs of production that underly their supply decisions. One of the great advantages of markets is that, through the formation of market prices, such information is utilized in an efficient manner as the price sends just the right signal to buyers and sellers about how scarce goods should be allocated in the market. Information asymmetries that cause externality problems in markets, however, are different than simply different sets of knowledge about our own individual tastes and costs. They involve hidden information that impacts others adversely because the information can be used to take advantage of the person on the other side of the market. We will then say that information asymmetries occur whenever buyers and sellers have different information regarding the nature of the product (or service) that is being traded or the true costs of providing that product (or service). A common example of this occurs in insurance markets. Suppose, for instance, I approach a health insurance company about my interest in purchasing health insurance. I have inherently more information than the insurance company. In particular, I know more about my own health status and thus the likelihood that I will need health care, 1 This chapter presumes a good understanding of the partial equilibrium model from Chapters 14 and 15 and makes conceptual references to material on externalities from Chapter 21. Section B of the chapter also builds on the non-general equilibrium parts of Chapter 17.

808 Chapter 22. Asymmetric Information in Competitive Markets than the insurance company, and I know more about how my lifestyle might change if I know that I am insured. This is information the insurance company would very much like to have in order to ascertain the likely cost of providing insurance to me. The worse my health is and the more likely I am to engage in risky behavior if I am insured, the more costly it is likely to be for the insurance company to provide health insurance to me. And I have every incentive to hide bad health or a tendency toward risky behavior as I approach the insurance company to get a good deal on health insurance. If the insurance company cannot distinguish between people who are hiding information about their health and those who simply want insurance but have nothing to hide, it may end up finding it impossible to provide insurance packages that healthy individuals would be willing to buy. Thus, the problem of asymmetric information and the associated problem of those with hidden information adversely selecting into insurance markets, can lead to missing markets. Similar problems arise in other markets. In the used car market, for instance, the owner of a used car may have significantly more information about the quality of the car than do potential buyers. In labor markets, workers know more about their real qualifications than employers may be able to ascertain. In mortgage markets, potential homeowners may know more about their real ability to make mortgage payments in the future than do the banks that lend money. In pharmaceutical markets, drug companies may know much more about the real effectiveness of particular drugs than do patients or even doctors. And in financial markets, corporate officers know more about the true financial health of a corporation than does the average shareholder. Each of these cases shares some of the characteristics of insurance markets in that one side of the market has inherently more information that is relevant for the market transaction than does the other side, which then may make the other side hesitate about entering a transaction. And in each case there may exist other market mechanisms, civil society institutions or government policies that can alleviate the problems markets face in dealing with such information asymmetries. This chapter is organized somewhat differently from other chapters in that Section A is written without requiring that you have covered the topic of risk in Chapter 17. You can gain an appreciation for the problems markets encounter under asymmetric information without understanding fully how we model risk, and Section A attempts to provide such an understanding. However, since information asymmetries represent particular problems for insurance markets that deal with risk (as described in Chapter 17), Section B of the chapter builds on the framework for insurance under risk and we introduced in Chapter 17. If you have covered only the intuitive first part of Chapter 17, you can still read the subsections (of Section B below) that focus on a graphical exposition of the impact of asymmetric information in insurance markets. For this reason, the mathematical exposition in Section B is confined to separate subsections. 22A Asymmetric Information and Efficiency We will discover in this Section that the presence of hidden information on one side of the market can generate inefficiencies by resulting in externality problems. In some cases, this will lead to the non-existence of markets that, if information were more generally available, would make everyone better off. In other cases, it will lead to market distortions in which we can see in principle how more information will lead to greater efficiency. We will develop these ideas initially through a treatment of one hypothetical insurance market before illustrating the deadweight losses in a set of more familiar graphs. Then, in the final two sections of Part A of this chapter, we will return to real world examples that exhibit the phenomena introduced earlier in more abstract settings.

22A. Asymmetric Information and Efficiency 809 22A.1 Grade Insurance Markets Let s begin with a somewhat silly example. Suppose I approached your professor the day before the beginning of the semester and told him I wanted to sell grade insurance in your class. Here is how it would work: If a student wants to insure that he gets at least a grade x in the class, he can purchase insurance that guarantees him grade x as a minimum grade for a price p x. Higher grade guarantees will carry with it a higher price. At the end of the semester, the professor and I will sit down and look at the legitimate grade distribution and particularly at the grades earned by those who bought insurance from me. If an earned grade falls below x for which a student bought insurance at the beginning of the semester, I have to pay the professor to overcome his scruples and raise the grade, with the size of the payment depending on how much the grade needs to be raised in order to get to the grade for which the student had bought insurance. If, on the other hand, a student who bought insurance for grade x actually earned a grade at or above x, no grade adjustment is necessary and no cost is incurred by my grade insurance company I just get to keep what the student paid me without dishing out anything to the professor. To make this example more concrete, let s suppose that the grade insurance business is perfectly competitive (which implies that each grade insurance company will end up making zero economic profit in equilibrium), and let s suppose that grades in your course are curved (prior to me paying off the instructor to raise some grades) around a C, with 10% of all students earning an A, 25% earning a B, 30% earning a C, 25% earning a D and 10% earning an F. 2 Finally, let s suppose that your professor s scruples are such that it costs a minimum of c for him to raise your grade by one letter grade (and 2c to raise it by two letter grades, 3c to raise it three letter grades, etc.). 22A.1.1 A-Insurance and The Adverse Selection Problem To focus on one particular problem that the grade insurance market faces, suppose first that only A-insurance can be offered and that student behavior will be exactly the same whether or not a student has insurance. Students who buy insurance at the beginning of the semester thus study and work just as hard in the class as they would have in the absence of having insurance. Students themselves have a pretty good idea whether they are likely to do well or poorly in the class, but as an outsider coming in, I don t know anything about any individual student and only know the distribution of grades that will emerge at the end. If everyone were forced to buy the A-insurance, it would not be difficult to determine the equilibrium insurance premium p A if we know that everyone in the grade insurance business makes zero profit in equilibrium. We would know that I would have to pay 4c for everyone in the 10% of the class that earns an F, 3c for everyone in the 25% of the class that earns a D, 2c for everyone in the 30% of the class that earns a C and c for everyone in the 25% of the class that earns a B. The insurance premium would then be p A = 0.1(4c) + 0.25(3c) + 0.3(2c) + 0.25c = 2c. (22.1) The price of A-insurance would thus simply be determined by how much it takes to pay off your professor to raise a grade by 1 level. If that price is $100, the premium would be equal to $200 per student. 2 Note to my students at Duke: I understand that we have grade inflation at Duke so please don t write me e-mails telling me that this is not a Duke curve.

810 Chapter 22. Asymmetric Information in Competitive Markets Exercise 22A.1 What would be the equilibrium insurance premium if, in a system that forced all students to buy insurance, the only insurance policy offered were one that guarantees a B? What if the only policy that were offered was one that guaranteed a C? Suppose, however, that we do not force everyone to buy a particular policy but simply left it up to individual students to determine whether or not to buy insurance. If it were reasonable to expect the set of students who choose to buy insurance to be a random sample of the class, the exact same logic that we used above would result in exactly the same premium. 3 It seems likely, however, that those students choosing to buy insurance will not represent a random sample, with students who are expecting an A in the class anyhow uninterested in purchasing insurance. Thus, if I charged the insurance premium in equation (22.1), I would lose money. Now suppose that all students are willing to pay as much as 2c to raise their grade by one level and 0.5c for any additional increase in the grade by another level. Put differently, an F student is willing to pay 2c to raise his grade to a D, 2.5c to raise his grade to a C, 3c to raise his grade to a B and 3.5c to raise his grade to an A. Exercise 22A.2 In an efficient allocation of grade insurance (when only A-insurance is offered), who would have A-insurance? (Hint: Compare the total cost of raising each student type s grade to the total benefit that this would yield for each student type.) Exercise 22A.3 If all types of insurance policies were available i.e. A-insurance, B-insurance, etc. who would have what type of insurance under efficiency? (Hint: Compare the marginal cost of raising each student type s grade by each level to the marginal benefit of doing so.) This would imply that 90% of the class would be willing to buy the A insurance if it were offered at a premium of 2c. But my insurance company would now incur higher costs. If the class has 100 students in it, I would incur a cost of c for the 25 B students, a cost of 2c for the 30 C students, a cost of 3c for the 25 D students and a cost of 4c for the 10 D students for an overall cost of 200c or an average cost of 2.22c for each of the 90 students that buy the insurance. In order for me to make zero profit, I therefore have to now charge a premium of 2.22c for the A-insurance. But at that price, the B-students would no longer be willing to pay for the A-insurance because the price is above what they are willing to pay for a 1 letter grade increase in their grade. This means that I would have to charge a premium of approximately 2.69c for the same insurance policy in order to break even if only C, D and F students bought my insurance. Exercise 22A.4 Verify that my break-even insurance premium for A-insurance would have to be approximately 2.69c if only the 65 C, D and F students bought the insurance. But now the C students are no longer willing to pay for the insurance since they are willing to pay only 2.5c to raise their grade by two levels 2c for the first level and 0.5c for the second. Thus, only D and F students are willing to pay 2.69c for my A-insurance. But if they are the only ones buying, you can verify that my premium has to go up to approximately 3.29c sufficient to get only F students to be interested in the A insurance, which would then necessitate a premium of 4c which not even F students are willing to pay. Thus, if students are allowed to choose whether or not to buy A-insurance, I will not be able to sell any insurance in equilibrium if the students know what kind of students they are and I do not. This is an example of a more general problem known as the adverse selection problem that can arise in markets with asymmetric (or hidden) information. 3 It is true that this would involve some risk for the insurance company since a random sample will sometimes contain relatively more good students and other times relative bad students, but if the insurance company sells many of these types of contracts in different classrooms, that risk would disappear.

22A. Asymmetric Information and Efficiency 811 The adverse selection problem arises in our example because each student has more information than my insurance company about how much of a cost I will incur if I sell him grade insurance. As a result, students will adversely select into buying insurance from me with high cost students more likely to demand insurance than low cost students. It would be efficient (as you should have concluded in exercise 22A.2) for B and C students to hold A insurance in our example, but neither does. 4 As in the case of the externalities in Chapter 21, the competitive equilibrium is inefficient. Even if students cannot perfectly predict what grade they will earn in the absence of insurance, they will have more information than I do about the probability that they will earn a good grade. Thus, even if students that end up earning an A in the absence of insurance are willing to buy insurance at the beginning of the term, they will still be willing on average to pay less than those who end up with a worse grade. Because of the adverse selection problem, students who line up to buy insurance from me therefore impose a negative externality in the market by raising the average cost of insurance (and thus the premium I have to charge). Their decision to enter the market adversely impacts the other students. It is this negative externality that arises from asymmetric information, and it is because of the presence of this externality that a market equilibrium does not exist in our example. Exercise 22A.5 Would I be able to sell A insurance if students were always willing to pay 2c for every increase in their letter grade? Would the resulting equilibrium be efficient? 22A.1.2 Information, Adverse Selection and Statistical Discrimination We have seen above how the asymmetry of information in the A-insurance market can lead to a non-existence of the insurance market due to the negative externality generated through adverse selection. To focus a little further on how asymmetric information causes this, we can consider how the equilibrium (or lack thereof) will change if I am able to obtain the information that we have so far assumed only students possess. Suppose first that I can observe student transcripts at the beginning of the semester and, from them, I can perfectly infer what grade each student will make at the end of the term in the absence of insurance. I could then offer each student a menu of insurance policies and price them with that information in mind. For a B student, for instance, I could offer the A-insurance at a price of c which the student would be more than willing to pay (with me making zero profit). For C, D and F students, I could similarly price A-insurance at 2c, 3c and 4c respectively, with C and D students willing to pay the price but F students unwilling (since such insurance is worth only 3.5c to them). We have thus restored the market for A-insurance by eliminating the informational asymmetry. We have furthermore done so in an efficient way, with insurance sold only to students whose willingness to pay is above the cost of the insurance product. The real world, of course, is never that certain, and neither students nor I can perfectly predict what grade they will end up earning at the end of the term in the absence of insurance. Suppose, then, that I observe from transcripts what grades a student has made on average and am therefore able to classify students into A students, B students, C students and D students. Suppose I also know by looking at the past performance of students in your course that A students earn an A 75% of the time and a B 25% of the time, and all other students earn a grade one level above 4 It is efficient for B and C students to hold A insurance (when only A insurance is an option) because the cost of raising their grades is c and 2c respectively while their benefit from getting an A is 2c and 2.5c respectively. The benefit is equal to the cost for C students and it is therefore efficient for them to have or not have insurance. But F students benefit by 3.5c and cost 4c.

812 Chapter 22. Asymmetric Information in Competitive Markets their usual grade 25% of the time, their usual grade 50% of the time and a grade below their usual grade 25% of the time. Assuming that students have no more information than I do, I could then again offer the different insurance policies to each type of students at a premium that will result in an expected zero profit for me. For instance, since I know that I will incur a cost of c with 25% probability for an A student, I can price an A-insurance policy for an A student at 0.25c. Similarly, since I know a B student who purchases an A-insurance will cost me nothing with 25% probability, c with 50% probability and 2c with 25% probability, I can price an A-insurance for a B student at c. You can verify on your own that the equilibrium price for an A-insurance would again be 2c for a C student and 3c for a D student. Exercise 22A.6 What would be the equilibrium price p F A for an F student if that student will earn an F with 75% probability and a D with 25% probability? Notice that nothing has fundamentally changed if the grade outcome is uncertain so long as it is equally uncertain from the student s perspective as it is from mine. As long as the student has no more information than I do, whether that information involves uncertainty or not, no adverse selection problem will arise and an equilibrium price will emerge for A-insurance but will differ depending on what type of student is purchasing the insurance. When I have perfect information about each student and can perfectly predict the type of grade he will earn in the absence of insurance, I will discriminate based on the individual characteristics of the student. In the case where both I and the students are somewhat uncertain about what the semester will hold, however, I end up discriminating based on the statistical evidence I have regarding the probabilities that a particular student will earn particular grades. Such price discrimination that is based on the underlying characteristics of the group to which an individual belongs is called statistical discrimination. 22A.1.3 The Moral Hazard Problem Throughout our discussion of the problems in our silly A-insurance market we have made the heroic assumption that students will study just as hard and diligently if they have grade insurance as if they did not. But would they? Or would the knowledge of the guarantee of a certain grade offered by my insurance company cause some students to blow off the material, stop coming to class, stop studying perhaps even skip exams? If you have stuck with this course all the way through Chapter 22, chances are you are the kind of student that gets at least some satisfaction from actually learning rather than just getting a grade on a transcript. Perhaps you are even that rare student who would work just as hard if there were no exams and no grades given. But students will vary in terms of how much value they place on the grade relative to the actual learning in a course which implies that the degree to which students will change behavior under my grade insurance will differ across students. The problem of individuals changing behavior in this way after entering a contract is known as the moral hazard problem and it makes executing the contract more expensive for the other party to the contract. If all students react the same to being insured, then I can at least predict how much more they will cost me than they would if they continued to behave as if they were not insured. If, for instance, a random selection of half the class buys A-insurance from me, we calculated earlier that a premium of 2c would make my expected profit zero in the absence of moral hazard. But, if each of the students who bought insurance then changes behavior sufficiently to end up with one letter grade below where he would have ended up otherwise, I would have to charge a premium of 3c to have an expected profit of zero. The anticipation of moral hazard behavior by those I insure

22A. Asymmetric Information and Efficiency 813 therefore implies I must charge more than I otherwise would, and it arises in insurance markets whenever individuals engage in riskier behavior when insured. If students differ in their change in behavior once they have insurance, however, we have a bigger problem than simply higher insurance premiums assuming students know themselves better than I know them. Once again, I would possess less information about the student than the student himself possesses, and this will reinforce the adverse selection problem that we discussed in the absence of moral hazard. Even if I could identify the A, B, C, D and F students from their transcripts and knew precisely what grade each will earn in the absence of insurance, I would now have to worry about the fact that some of each type of student will exhibit greater moral hazard once they are insured than others. The B student that knows he can earn a B in the course and knows that he will work just as hard if he is insured will not, for instance, be willing to pay as much for A-insurance as the B student who knows he can enjoy the beach a whole lot more if he has A-insurance. Thus, students will adversely select into my insurance pool based on the level of moral hazard they will exhibit once insured. As long as they know this information and I do not, we can get the same kind of unraveling of the insurance market we saw in our initial example of adverse selection. Adverse selection, then, causes problems for insurance companies because of the adverse externality that high cost customers impose on low cost customers as they drive up the price of insurance and may cause insurance markets to no longer function in equilibrium. Moral hazard by itself, on the other hand, is a problem that insurance companies can, in our example, deal with through pricing of premiums. However, if moral hazard creates informational asymmetries because insurance companies cannot identify how different individuals will engage in different levels of risky behavior once insured, this creates another adverse selection problem that can once again undermine the existence of markets. Much has been written by economists about the optimal ways in which insurance companies (and others facing moral hazard problems on the other side of the market) can arrange contracts so as to minimize moral hazard behavior. Although we will not develop this formally in this chapter, you can think of some possible conditions my insurance company might place on those who buy grade insurance. For instance, I might require as part of the contract that your professor certifies at the end of the term that students who will benefit from owning grade insurance have in fact attended class, handed in assignments and taken exams. (Issues like this are often covered in courses on the economics of contracting.) For now we can simply note that, to the extent to which insurance companies can find ways of minimizing moral hazard through contractural arrangements as they sell insurance, they limit the adverse selection problem that accompanies the existence of moral hazard. 22A.1.4 Less Extreme Equilibria with Adverse Selection So far, we have demonstrated that the adverse selection problem may cause certain markets not to exist. This is an extreme manifestation of the problem of adverse selection, and not all markets that are subject to adverse selection will cease to exist entirely. Suppose, for instance, that your professor will not permit me to sell A-insurance but only agrees to let me sell B-insurance i.e. insurance that guarantees a student will earn at least a B in the course. To make the example as simple as possible, let s assume that there is no moral hazard problem, that students know exactly what grade they will earn, that I have no information about any individual student and that it is prohibitively costly for me to gather any useful information on individual students. We know right away, of course, that no A or B student would then be interested in buying insurance from me. In a class of 100 students, only the 65 C, D and F students are therefore

814 Chapter 22. Asymmetric Information in Competitive Markets potential customers. If they all end up buying the insurance from me, I know that I will incur a cost of c for the 30 C students, 2c for the 25 D students and 3c for the 10 F students. My average cost per customer is then 110c/65 or approximately 1.69c. Since students are willing to pay 2c for a one level increase in their grade and 0.5c for each additional level increase, we know that C, D and F students would be willing to pay 2c, 2.5c and 3c for B-insurance and thus are all willing to pay my break-even premium of 1.69c. In this case, the adverse selection problem is therefore not sufficiently large to eliminate the equilibrium in the B-insurance market. Exercise 22A.7 Conditional on only B insurance being allowed, is this equilibrium efficient? Now suppose that student demand for grade insurance was slightly different: suppose a student is willing to pay 1.5c for a one level increase in his grade and c for each additional increase. This implies that C students would only be willing to pay 1.5c for B-insurance, less than the premium of 1.69c I have to charge to break even when all C, D and F students buy insurance. If I therefore end up providing B-insurance to only the 35 D and F students, you can verify that I would have to charge a break-even premium of approximately 2.29c. Since this is less than the value D and F students place on B-insurance, the equilibrium would involve 35 B-insurance policies sold to just those students. Now, the externality of adverse selection causes fewer policies to be sold, but an equilibrium still exists. Exercise 22A.8 Conditional on only B insurance being allowed, is this equilibrium efficient? The example can, of course, get a lot more complex if the professor allows me to sell all forms of insurance i.e. A, B, C, D insurance. In end-of-chapter exercise 22.1, we will investigate this more closely under the assumption that individuals are uncertain about exactly the grade they will get and are willing to pay 1.5c to get their typical grade but only 0.5c more for each grade above their usual. In this case, it is inefficient for anyone to buy insurance other than insurance to guarantee his usual grade. This is because the cost of insuring your usual grade is c while the benefit is 1.5c but raising your grade each level above the usual is valued at only 0.5c but costs c. As we will demonstrate in the exercise, adverse selection will result in inefficiency once again. 22A.1.5 Signals and Screens to Uncover Information At this point, we have shown how asymmetric information can cause problems in our grade insurance market. It should be clear from our example, however, that good or low cost students have an incentive to find ways of credibly revealing information to my insurance company so that I can give them a better deal. Similarly, my insurance company has an incentive to invest in ways of uncovering information by getting access to transcripts, interviewing students, etc. Put differently, students have an incentive to signal information to me, and I have an incentive to screen the applicant pool. You can explore in end-of-chapter exercises 22.2 through 22.4 how such signals and screens can be efficiency enhancing and how they can be wasteful under different assumptions about the grade insurance market. We will furthermore revisit the issue in the next section after exploring a more graphical model that frames the ideas we have explored thus far in a different (and more realistic) setting. 22A.2 Revealing Information through Signals and Screens Let s now move away from the artificial grade insurance market and consider the case for insurance more generally. While our treatment in this section can be applied to all types of insurance, we ll

22A. Asymmetric Information and Efficiency 815 frame our discussion in terms of car insurance. Suppose that there are two types of potential consumers: high cost consumers that are likely to get into accidents, and low cost consumers that drive safely and are less likely to call upon insurance companies to pay for damages. We can then think of car insurance for type 1 consumers carrying an expected marginal cost of MC 1 and car insurance for type 2 consumers carrying an expected marginal cost of MC 2, with MC 1 > MC 2. To make the example as simple as possible, let s suppose further that demand curves are equal to marginal willingness to pay curves and that the aggregate demand curve D 1 for type 1 consumers is the same as the aggregate demand curve D 2 for type 2 consumers. Panel (a) of Graph 22.1 then illustrates what the car insurance market would be like if there were only type 1 consumers, and panel (b) illustrates what it would be like if there were only type 2 consumers. In each case, it is straightforward to predict how the competitive market would allocate resources (assuming there are no substantial recurring fixed costs to running insurance companies): In panel (a), the equilibrium price p 1 would cause consumers of type 1 to purchase x 1, the efficient quantity that maximizes social surplus. In panel (b), the equilibrium price p 2 would similarly cause type 2 consumers to buy x 2 insurance policies once again allocating resources efficiently. And if a competitive insurance industry can tell type 1 consumers apart from type 2 consumers, this is exactly the outcome that will emerge with all insurance policies priced at the marginal cost relevant for the type of consumer who is purchasing insurance. Graph 22.1: Adverse Selection in Car Insurance Market Panel (c) of Graph 22.1 then merges panels (a) and (b) into a single picture. If insurance companies can tell safe drivers apart from unsafe drivers, type 1 consumers will get consumer surplus equal to area (a) while consumers of type 2 will get consumer surplus equal to area (a+b+c+d+e+f). Since insurance firms are making zero profit, the overall social surplus would then be equal to (2a + b + c + d + e + f). 22A.2.1 Deadweight Loss from Asymmetric Information Now suppose that firms cannot distinguish between type 1 and type 2 drivers and thus cannot price car insurance based on the expected marginal cost of each consumer that walks through the door. Rather, the only information that firms have is that half of all drivers are of type 1 and half

816 Chapter 22. Asymmetric Information in Competitive Markets are of type 2. Each insurance company then gets a random selection of drivers to insure and thus knows that half their customers are high cost and half are low cost. Under perfect competition that drives profits for insurance companies to zero, this implies that the single price charged for car insurance will lie halfway between MC 1 and MC 2 indicated by p in panel (c) of Graph 22.1. Exercise 22A.9 Suppose the current market price for car insurance were less than p. What would happen under perfect competition with free entry and exit? What if instead the market price for car insurance were greater than p? Is is easy to see immediately that high cost consumers will benefit from the information asymmetry we have introduced their price for car insurance drops from p 1 under full information to p. Consumers of type 2 will analogously be hurt by the informational asymmetry seeing their price increase from p 2 to p. The fact that some consumers are better off and some are worse off does not, however, itself raise an efficiency problem. Rather, the efficiency problem emerges from the fact that overall consumer surplus falls as a result of the informational asymmetry. To be more precise, we can see in panel (c) of Graph 22.1 that consumer surplus for type 1 consumers increases to (a + b + c) while consumer surplus for type 2 consumers falls to (a + b + c) giving us an overall surplus of (2a + 2b + 2c). Note that area (b) is equal in size to area (d) which means we can re-write this overall surplus as (2a+b+2c+d). Note further that the triangle (c) is equal in size to triangle (f) which means we can further re-write the overall surplus as (2a + b + c + d + f). Comparing this to the full information surplus of (2a + b + c + d + e + f), we have lost area (e) which is therefore the size of the deadweight loss from introducing asymmetric information that keeps firms from pricing insurance policies differently for consumers of type 1 and 2. 5 To provide some intuition as to where this deadweight loss comes from, we can note two further geometric facts in Graph 22.1: Area (g) is equal to half of area (e), and area (f) is equal to area (g) (and thus also equal to half of area (e).) Thus, the deadweight loss can equivalently be stated as area (f + g). Panel (a) of the graph places area (g) into the graph for just consumers of type 1 where we originally said that consumers would buy x 1 insurance policies when they are priced at marginal cost. All the way up to x 1, the marginal benefit (as indicated by the demand curve) exceeds the marginal cost and it is therefore efficient to provide policies up to x 1. For policies after x 1, however, the marginal cost of providing additional insurance policies exceeds the marginal benefit making it inefficient to provide policies beyond x 1. When x policies are bought by type 1 consumers, the deadweight loss from this over-consumption of insurance is then area (g). The reverse holds in panel (b) for low cost consumers whose marginal benefit exceeds marginal cost until x 2 but who reduce their consumption to x under the uniform price p. Thus, consumers of type 2 are now under-consuming insurance with the deadweight loss (f) emerging directly from this under-consumption. Exercise 22A.10 True or False: The greater the difference between MC 1 and MC 2, the greater the deadweight loss from the introduction of asymmetric information. Exercise 22A.11 Suppose that type 1 consumers valued car insurance more highly implying D 1 lies above D 2. Can you illustrate a case where the introduction of asymmetric information causes type 2 consumers to no longer purchase any car insurance? What price would type 1 consumers then pay? 5 It may seem that our analysis relies too heavily on symmetries that emerge from the assumption that type 1 and 2 consumers do not differ in overall number or demand. End-of-chapter exercise 22.5 illustrates that the analysis, while notationally more complex, is similar when these assumptions are relaxed.

22A. Asymmetric Information and Efficiency 817 Notice that the adverse selection problem in our car insurance market is very much like the problem we first encountered in the grade insurance market of the last section: consumers that cost less to insure safer drivers or better students are driven out of the insurance market by rising premiums due to the adverse selection of consumers who cost more to insure. The result in Graph 22.1 is less extreme in the sense that not all low cost consumers are driven out of the market and not all high cost consumers come into the market. But the basic economic forces are the same. 22A.2.2 Screening Consumers The asymmetric information equilibrium in Graph 22.1 (which is replicated in panel (a) of Graph 22.2) is called a pooling equilibrium because all consumer types end up in the same insurance pool with the same insurance contract while the full information equilibrium in which the different types are charged based on their marginal cost is called a separating equilibrium (because the types end up in separate insurance contracts). When asymmetric information leads to pooling of different types, however, it would be to the advantage of an insurance company to find a way of screening out high cost customers and providing insurance to only low cost types. Given that there is a demand for screening services that identify who the safe drivers are, we might then imagine that a screening industry will form a competitive industry that screens consumers and sells information to insurance companies. Suppose first that this screening industry becomes very good at gathering information on consumers so good, in fact, that the marginal cost of gathering information on any particular driver is virtually zero. In that case, competition in the screening industry will drive the price of screening services (paid by insurance companies) to zero. Put differently, if the screening industry becomes very good at gathering information on drivers, information will be revealed to insurance companies at roughly zero cost. This then leads us back to the full information separating equilibrium in which high cost drivers are charged a price p 1 and low cost drivers are charged p 2. The emergence of a screening industry that screens consumers at low cost therefore restores the efficient equilibrium and recovers the dead weight loss from the pooling equilibrium. Exercise 22A.12 How much do type 1 consumers lose? How much do type 2 consumers gain? What is the net effect on overall consumer surplus? But now suppose that information is not all that easy to gather. In particular, suppose it costs q per driver to gather sufficient information to allow the screening firms to tell type 1 drivers apart from type 2 drivers. If insurance companies buy this information for all drivers that apply for policies, insurance companies will have to pass this screening cost onto consumers in order to maintain zero profits. But they can t pass it onto type 1 consumers because if the price for high cost insurance policies rose above p 1, a new insurance company could emerge and simply sell insurance at p 1. So, in order for insurance companies to make zero profit, they will have to price the policies of low cost customers above MC 2 to pay for the screening price charged by the screening firms for both type 1 and type 2 consumers. Thus, the new separating equilibrium will have p 1 = MC 1 and p 2 = MC 2 + β where β > q and sufficient to cover all the screening costs for both types of consumers. Suppose, then, that the screening cost q per driver is such that β = (p MC 2 ) is required in order for insurance companies to make zero profit in the separating equilibrium where they charge p 1 = MC 1 to type 1 consumers. This implies that p 2 = p i.e. the insurance premiums for low cost drivers remain unchanged from the pooling equilibrium because of the screening cost, But the premiums for high cost drivers rise to MC 1 because insurance companies can now tell who the

818 Chapter 22. Asymmetric Information in Competitive Markets Graph 22.2: Insurance Companies Screening Drivers unsafe drivers are and thus will no longer insure them below marginal cost. In panel (a) of Graph 22.2, consumer surplus for type 1 drivers then falls by (b + c) (from (a + b + c) to just (a)) while consumer surplus for type 2 drivers remains unchanged. Overall consumer surplus therefore falls by (b + c) raising the deadweight loss that already existed in the initial pooling equilibrium. But wait it gets worse! The cost of screening customers is paid to screening firms who make zero profit and thus is not a benefit to anyone. In panel (a) of Graph 22.2, this cost is equal to area (d+ e), which means that the increase in deadweight loss from moving to the separating equilibrium is (b + c + d + e). Exercise 22A.13 Why is the screening cost equal to area (d + e)? Exercise 22A.14 * Why do firms in this case pay a screening cost that does not allow them to lower any premiums? (Hint: Think about whether given that everyone else pays for the screening costs and discovers who are the safe and unsafe drivers an individual firm can do better by not discovering which of its potential customers are type 1 and which are type 2.) Thus, as screening costs rise, the move from a pooling equilibrium with asymmetric information to a separating equilibrium (where the asymmetric information is eliminated through screening) becomes inefficient. This is because gathering information is itself costly to society, and someone will have to bear that cost. While the pooling equilibrium without screening gives rise to deadweight losses, these deadweight losses can then be reduced through screening only if the cost of gathering information is relatively low. Panel (b) of Graph 22.2 illustrates a less extreme case where the separating equilibrium price p 2 lies below the pooling equilibrium price p because screening costs are lower than previously assumed. Type 1 consumers still lose (b + c) in consumer surplus as their premium rises to MC 1, but type 2 consumers now gain (h + i) in consumer surplus. Thus, overall consumer surplus changes by (h + i b c). Screening costs are furthermore equal to (j + k) implying an overall change in social surplus of (h + i b c j k) as we move to the screening equilibrium. Note that as screening costs fall toward zero, (j + k) approaches zero while (h + i) approaches (d + e + f). Since

22A. Asymmetric Information and Efficiency 819 (d + e + f) is unambiguously greater than (b + c), overall surplus therefore increases for sufficiently low screening costs. Exercise 22A.15 Could there be a screening-induced separating equilibrium in which p 2 is higher than p? Exercise 22A.16 Would your analysis be any different if the insurance companies did the screening themselves rather than hiring firms in a separate industry to do it for them? 22A.2.3 Consumer Signals Suppose next that insurance companies find it too costly to screen consumers and we are therefore in our pooling equilibrium where p is charged to all drivers. As we have already shown, this implies that low cost drivers are paying too much and high cost drivers are paying too little. It is therefore in the interest of low cost drivers to find a way to signal insurance companies that they are a safe bet and, if they succeed in signaling their type, it becomes in the interest of high cost types to falsely signal that they, too, are safe drivers. Whether a separating equilibrium can emerge in the insurance market through consumer signals then depends on the cost of signaling your true type as well as the cost of falsely signaling that you are a different type than you actually are. Consider first the extreme case where it is costless for type 2 drivers to signal that they are safe but it is very costly for type 1 drivers to falsely signal that they too are safe drivers. Because it is easy for type 2 drivers to reveal information that can then not easily be obscured by type 1 drivers, a full information separating equilibrium with insurance premiums p 1 = MC 1 and p 2 = MC 2 will emerge and the deadweight loss from pooling will be eliminated through consumer signaling. If, on the other hand, it is equally costless for type 1 drivers to pretend to be type 2 drivers, this cannot happen and we simply remain in the pooling equilibrium where no useful information is conveyed to the insurance companies. Exercise 22A.17 True or False: When it is costless to tell the truth and very costly to lie, consumer signaling will unambiguously eliminate the inefficiency from adverse selection. Now suppose that things get a little murkier in that it costs δ for type 2 consumers to signal that they are safe drivers and it costs γ for type 1 consumers to pretend to be safe drivers. If the industry is currently pooling all drivers into a single insurance contract with price p, type 2 drivers would be able to reduce their premiums to MC 2 if they can credibly signal that they are safe drivers, thus each getting a benefit of (p MC 2 ). So long as δ < (p MC 2 ), it therefore makes sense for a type 2 consumer who is currently paying p to absorb the cost of signaling his type and get his premium lowered to MC 2. Suppose, then, that the type 2 consumers successfully signal their type and induce a separating equilibrium where the industry charges MC 2 to type 2 consumers and MC 1 to type 1 consumers. The only way this can truly be an equilibrium is if it is too costly for the type 1 consumers to falsely signal that they, too, are safe drivers and a type 1 consumer in a separating equilibrium would be willing to pay as much as (MC 1 MC 2 ) the difference between the low and high insurance premiums to pretend to be a safe type! Thus, we can get a separating equilibrium if δ < (p MC 2 ) and γ > (MC 2 MC 1 ) i.e. if the signaling cost plus the low cost insurance premium is less than the pooling insurance premium for safe drivers, and if the cost of lying is greater than the difference between the low and high cost insurance rates. Is this outcome necessarily efficient? Just as in the case of screening, the answer again depends on how high δ the cost of revealing information is.

820 Chapter 22. Asymmetric Information in Competitive Markets Exercise 22A.18 Suppose δ = (p MC 2 ) and γ > (MC 1 MC 2 ). What is the increase in dead weight loss in going from the initial pooling equilibrium to the separating equilibrium? Exercise 22A.19 True or False: If δ and γ are such that a separating equilibrium emerges from consumer signaling, the question of whether the resulting resolution of asymmetric information enhances efficiency rests only on the size of δ, not the size of γ. But there is another possibility: Suppose δ < (p MC 2 ) and γ < (MC 1 MC 2 ); i.e. suppose the cost of truthfully signaling that you are a safe driver is less than the amount that safe drivers are overpaying in our initial pooling equilibrium and the cost of lying is less than the difference between the marginal costs imposed on insurance companies by the two types. It is then possible to get a pooling equilibrium with signaling where both types send signals that they are safe drivers but because both types send these signals, no actual information is conveyed to the insurance companies who therefore continue to price all policies at p. Given that everyone is sending an I am safe signal, not sending such a signal might be interpreted as you being unsafe and thus everyone will send them because everyone else is sending them. 6 This is of course unambiguously inefficient consumers are sending costly signals without revealing any actual information and thus without changing anything in the insurance industry. Exercise 22A.20 * Is it possible under these conditions for there to also be a pooling equilibrium in which no one sends any signals? (Hint: What would insurance companies have to believe in such an equilibrium if they did see someone holding up the I am safe sign?) Exercise 22A.21 * Suppose (p MC 2 ) < δ = γ < (MC 1 MC 2 ). Will there be a separating equilibrium? (Hint: The answer is no.) Exercise 22A.22 Why is it possible for a signaling equilibrium to result in a pooling equilibrium in which no information is revealed but it is not possible to have such a pooling equilibrium emerge when firms screen? 22A.2.4 Information Costs and Deadweight Losses under Asymmetric Information Our example of car insurance has illustrated two fundamental points: First, as already shown in our grade insurance examples, the presence of asymmetric information may cause pooling equilibria in which behavior is based on average characteristics rather than individual characteristics. This will lead to the emergence of deadweight losses as some will over-consume while others will underconsume (relative to the efficient level) or, if the problem is sufficiently severe, entire markets will cease to exist. Second, it may be possible for information asymmetries to be remedied through the revelation of information either because the informed side of the market signals or because the uninformed side of the market screens. But this only leads to greater efficiency if the cost of transmitting information is relatively low and if the information that is exchanged is actually informative (and thus leads to a separating equilibrium). We will explore these ideas further in end-of-chapter exercises, including some where we will investigate the possible outcomes of signals and screens within our grade insurance markets. But now we turn to a discussion of some of the most prevalent real world situations in which asymmetric information plays an important role. As you will see, many of these have nothing to do with insurance even though they can be understood with the tools we have developed within the insurance context. 6 It is not clear what insurance companies should believe in this case about someone who deviates from the behavior of everyone else and does not send an I am safe signal, but it is certainly possible that insurance companies would believe such individuals to be of type 1. We will discuss how economists might think about such out-of-equilibrium beliefs in Section B of Chapter 24.

22A. Asymmetric Information and Efficiency 821 22A.3 Real World Adverse Selection Problems In our development of the basic demand and supply model of markets earlier in the book, we distinguished between three different types of markets: output markets in which consumers demand goods supplied by producers, labor markets in which producers demand labor supplied by workers, and financial markets in which producers demand capital from investors (or savers). Asymmetric information can appear in any of these markets, and we will therefore treat each of these separately below. As before, we will point to three types of institutions that can then ameliorate the externality problem created by adverse selection. New markets like the screening firms in our car insurance example might appear and facilitate the exchange of hidden information; non-market civil society institutions might play a similar role, or government policy might be crafted to address the problem. And in many instances a combination of these approaches is utilized in the real world. 22A.3.1 Adverse Selection in Output Markets We have already discussed extensively the problems of adverse selection in one particular output market where the output is insurance. In some insurance markets, there is much that insurance companies can observe about individuals (thus giving rise to a relatively small adverse selection problem), while in other insurance markets much remains hidden information. In the case of life insurance, for instance, the chances of a consumer using the insurance can be predicted reasonably well so long as the insurance company knows a few basics such as the consumer s age, gender, health condition and whether or not she smokes. (For life insurance policies with high benefits, they might also require a basic health exam.) While some consumers might behave more recklessly if their life is insured (thus giving rise to a moral hazard problem that can strengthen adverse selection), most consumers probably will not change behavior significantly just because their heirs will receive a payment if they die. 7 Life insurance companies can therefore use relatively costless screens to categorize consumers into different risk types and then price life insurance policies accordingly. As a result, we rarely hear of calls for government intervention in life insurance markets, with insurance providers employing an army of actuaries who predict the probability of premature death for different types of consumers. Exercise 22A.23 Another factor that lessens the adverse selection problem in life insurance markets is that the bulk of demand for life insurance comes from people who are young to middle aged and not from the elderly. How does this matter? In the case of unemployment insurance, on the other hand, markets may face considerably more difficulty in overcoming the adverse selection problem. As someone approaches an insurance company to inquire about unemployment insurance policies, it is difficult for the insurance company to tell whether the consumer is asking for this insurance because she knows that she is about to get laid off. Age or health exams do not provide a useful screen (as they do in the case of life insurance) the hidden knowledge is much more difficult to unearth. Consumers themselves may also not find easy ways to signal their type. It may therefore be the case that signaling and screening are too costly for widespread unemployment insurance markets to form without some nonmarket institution to spur such a market. Before governments became involved in insuring everyone, certain civil society institutions, for instance, utilized local knowledge of individual reputations 7 An exception to this involves individuals contemplating suicide, and suicide is therefore typically excluded as a cause of death that would trigger an insurance payment.

822 Chapter 22. Asymmetric Information in Competitive Markets to provide insurance within small communities where individual reputations were relatively wellknown. In most developed countries, such institutions disappeared when governments instituted mandatory unemployment insurance for everyone using compulsory unemployment insurance taxes to fund the system. Tenured professors with lifetime job security (who would not voluntarily purchase unemployment insurance) as well as workers in industries whose fortunes fluctuate greatly with the business cycle then all pay into the system in hopes that overall consumer surplus is increased even as some are paying for a service they do not require all because the adverse selection problem may be sufficiently severe for private markets and civil society institutions to offer too little insurance. Exercise 22A.24 In our car insurance example, asymmetric information caused the market to create a pooling equilibrium in which some over-consumed and others under-consumed. Why might this not be the case in the unemployment insurance market where those with high demand are much more likely to be those with high probability of being laid off? (Hint: Can you imagine an unraveling of the market for reasons similar to what we explored in the grade insurance case?) Exercise 22A.25 Is mandatory participation in government unemployment insurance efficient or do you think it might just be more efficient than market provision? In yet other insurance markets, a combination of approaches has emerged. For instance, in the US, health insurance for the non-elderly is provided largely by private insurance companies. However, the government covers some segments of the population (the elderly and the poor) directly through Medicare and Medicaid, and it subsidizes employers to provide health insurance to their employees. Large employers then enjoy an additional advantage in that they have a large pool of workers that is less risky to insure than individuals. And an ethical civil society standard (often also codified into laws) in the medical profession requires doctors in emergency rooms to treat uninsured patients thus effectively providing at least some form of implicit insurance to the formally uninsured. Debates over whether this is the right balance of markets, civil society and government in the health insurance market continue in the US, while in other countries governments have approached health insurance much as the US has approached unemployment insurance. My goal is not to offer an answer as to what the best approach to a fairly complicated set of issues is but merely to point out that adverse selection (and moral hazard) has something to do with the policy debates surrounding this issue. You can learn more about this in public finance and health economics courses and in end-of-chapter exercises 22.7, 22.8 and 22.9. Exercise 22A.26 What is the adverse selection problem in health insurance markets? What is the moral hazard problem for such markets? Exercise 22A.27 It is often proposed that health insurance companies not be allowed to discriminate based on pre-existing health conditions. Does this ameliorate or aggravate the adverse selection problem? Can you see why such proposals are often accompanied by proposals that everyone be required to carry health insurance? Insurance markets, however, are not the only output markets that might suffer from adverse selection problems. The used car market, for instance, is plagued by adverse selection but this time the hidden information resides with the supplier rather than the consumer. You may have heard that, when you buy a new car, its value drops by several thousand dollars the moment you drive it off the lot. Why? Because if you were to try to sell this car to someone else the week after you bought it, potential buyers would (rightfully) wonder whether you have discovered something about the car that is not observable to them and whether you might not be adversely selecting

22A. Asymmetric Information and Efficiency 823 (as a seller) into the used car market. Consumers in the used car market can then employ various screens to try to get to the potentially hidden information screens such as taking the used car to a trusted mechanic who can give an independent third party certification of quality. Or used car dealerships might offer warranties that signal to consumers the quality of the used car. Some brands of cars are known to have fewer problems and so brand names can signal quality. Brand names, warranties and third party certifications therefore all represent ways that hidden information can be unearthed and at least partially overcome the adverse selection problem. Exercise 22A.28 Consider used car dealerships in small towns. How might reputation play a role similar to brand names in addressing the asymmetric information problem? In a world with increasingly complex products, the issue of product quality that is potentially hidden from consumers of course extends far beyond the used car market. The quality of much of what I see in stores from computers to televisions to kitchen appliances to over-the-counter medications is difficult for me to evaluate. Again, warranties can signal quality, as can the brand names that have good reputations. Third party certification groups (such as the magazine Consumer Reports) have emerged. They routinely test products and sell the information to me in a separate market (through, for instance, the Consumer Reports magazine or web-site), and consumer advocacy groups outside the market provide similar services. The American Heart Association puts its seal of approval on certain foods. And industry groups have often established industry standards, sometimes requiring third party certification to insure quality. Even my underwear has stickers that try to signal quality informing me that Inspector 10 had done his job. While all these signals are costly and thus use some of society s resources, they nevertheless can be (and often are) socially beneficial if they are not too costly and lead to more widespread information that can overcome adverse selection externalities in markets. At the same time, some producers might be able, at least in the short run, to signal that their products are of higher quality than they actually are, expending wasteful effort to hide their true type in order to end up in a pooling equilibrium with high quality producers. Thus, just as in the example of car insurance, signals may in some instances represent a socially wasteful use of resources aimed at deceiving rather than informing, or they may be too costly even when they result in a resolution of the information asymmetry. Exercise 22A.29 What is Consumer Reports analogous to in our discussion of car insurance? Finally, as in insurance markets, the government often steps in as well. Cigarette packages contain dire warnings required by law, and my barber has a sign on his mirror telling me that he is licensed to cut hair. We will see in later chapters that there may be other, less benign reasons why my barber had to get a license to operate and we therefore might be careful in interpreting such government involvement as solely serving the purpose of reducing adverse selection. Our goal here, however, is not to sort out which of the various signals and screens aimed at adverse selection problems are good and which are bad which truly raise social surplus and which are socially wasteful. Rather, I simply want to persuade you that a variety of market, civil society and government supported signals and screens in fact operate at least in part because markets by themselves might not perform optimally in the presence of adverse selection. 22A.3.2 Adverse Selection in Labor and Capital Markets There is only so much that an employer can ascertain about a potential employee before hiring him. The adverse selection problem in labor markets therefore occurs when workers have hidden

824 Chapter 22. Asymmetric Information in Competitive Markets information about their own productivity. Education, work experience and letters of reference offer ways for us to signal information to our employers, but workers with identical resumes may still be quite different on the job. Additional information might be signaled less formally in job interviews aimed at screening applicants. Depending on the cost of the signal relative to the benefit, such efforts may once again be socially productive in the sense that they convey true information or socially wasteful if they signal false information or are simply too costly. We are often led to believe, for instance, that more education is always better. This may be true if the only reason for someone to get more education is to truly increase productivity on the job (and if the marginal cost of additional education is greater than the marginal benefit for the student). But in some instances, education may simply serve as a signal masking the underlying productivity of a worker. If the cost of getting the signal of having attained a certain level of education is sufficiently low, then low-productivity workers might get an education simply to end up in a pooling equilibrium with truly high-productivity workers. While this may make the unproductive worker better off, it dilutes the information of the signal and does not serve to convey the information that employers seek. 8 If you take a course on the economics of education or in labor economics, you will probably find yourself debating the issue of whether your college increases your real productivity or simply serves as a screening institutions that signals something about you that was already there when you started as a freshman. (This is explored in more detail in exercise 24.14.) Exercise 22A.30 Which of the following possibilities makes it more likely that widespread college attendance is efficient: (1) Colleges primarily provide skills that raise marginal product, or (2) colleges primarily certify who has high marginal product. The same issues arise in financial markets. Banks and mortgage companies have less information than those who apply for loans. Applicants therefore seek ways of signaling their creditworthiness and banks seek ways of screening applicants. In the past when individuals moved less often and resided more within small communities, one s informal reputation was an important signal if everyone knows Joe is a liar and a cheat, there is not much point to lending him money. In today s world, such informal mechanisms are less effective, but other institutions have taken their place. Credit companies keep detailed records on anyone who has ever had a credit card or a loan or a bank account. We are often told to be sure to build a credit history precisely because this signals something about us that may come in handy when the time comes to apply for a mortgage. Thus, as informal reputations became less effective, new markets formed markets that gather and sell information about our creditworthiness. In many ways, our credit report has become our reputation in credit markets. We face similar information problems when we try to decide where to invest our money. Companies try to get us to buy their stocks, and banks try to sell us various types of savings instruments with different risks and returns. Often, the places we consider investing have much more information about their true value than we do, and we therefore have to expend effort, or hire someone to expend effort in our place, to gather information that might be hidden. Again, there exist many different financial advising firms that now specialize in gathering such information and selling it to us for a price (or a commission), and non-profit ( civil society ) institutions provide information on firms (often on web sites accessible to potential investors). In addition, the government has 8 Note that the adverse selection problem is less severe if it is easy for firms to fire workers who prove less productive than they initially appeared, but many laws and regulations as well as union protections for workers often make firing workers costly for firms.

22A. Asymmetric Information and Efficiency 825 created its own oversight mechanism, requiring financial disclosure statements by publicly traded companies and offering their seal of approval in terms of deposit insurance to banks. 22A.4 Racial and Gender Discrimination Many societies, including the US, continue to struggle with overcoming social problems arising from the legacy of racial and gender discrimination. Such discrimination has deep historical roots, dating back to some of the darker periods in history when prejudice was endemic and often explicitly supported by government policy. Despite legislation that now outlaws such discrimination, studies continue to suggest instances when applicants for employment (in labor markets) or credit (in financial markets) are offered different wages or interest rates despite identical observable qualifications, with less favorable deals offered to women and minorities. We will see in this section that such discrimination may persist in markets even when old prejudices have died out if markets are characterized by asymmetric information of the type discussed throughout this chapter. 22A.4.1 Statistical Discrimination and Gender Consider first a case where gender discrimination characterizes market transactions in the life insurance market. We have already discussed how life insurance companies calculate the expected probability of premature death for individuals. Smokers, for instance, are required to pay higher life insurance premiums than non-smokers because, on average, smokers die earlier than non-smokers. At the same time, many of us know of people who smoked all their life and ended up living to a ripe old age. Smoking appears to be more damaging to some than to others, with some individuals being fortunate to have genes that protect them from the adverse consequences of smoking. Even if I know that my family tends to be able to smoke like chimneys and still survive to an old age, insurance companies will discriminate against me in their pricing policies if they know that I smoke. Because they lack information on my individual probability of being affected by smoking, they discriminate based on the statistical evidence on smokers as a group they engage in statistical discrimination because of the informational asymmetry that keeps them from knowing fully my individual characteristics. The same reason that causes statistical discrimination against smokers in life insurance markets then also causes statistical discrimination against men in these markets. Women on average live longer than men and so my wife, despite the fact that her family seems more predisposed to cancer and heart disease than mine, ends up getting a better deal on life insurance than I do. The same is true of young people in car insurance markets you might be a much better driver than I am, but because I am older and on average people my age get into fewer accidents, you end up having to pay a higher car insurance premium than I do. Statistical discrimination discrimination based on the average statistics of the demographic groups to which individuals belong is therefore economically rational in insurance markets that are characterized by asymmetric information. Exercise 22A.31 What are we implicitly assuming about the costs of screening applicants in these markets? While we may not see a big moral issue arising from such statistical discrimination in insurance markets, we might be considerably more disturbed when the same type of discrimination emerges in other markets. On average, for instance, women are more likely to exit the labor force for some period in order to raise children. This is not at all true for some women, and an increasing number of men are also taking larger responsibility for child rearing. Employers, however, have a difficult time identifying which women and men are individually more likely to exit the labor force for child

826 Chapter 22. Asymmetric Information in Competitive Markets rearing, but it is easy for them to identify whether employees or potential employees are men or women. As a result of this asymmetric information, employers may therefore use the underlying statistics of average behavior by men and women to infer the likelihood that a particular employee will be with the company for a long period. As a result, they may statistically discriminate against female employees, offering them lower wages or less job training in anticipation of the greater likelihood that they will leave the company. Notice that, from a purely economic perspective, this is no different than the insurance company statistically discriminating against me when my wife and I apply for life insurance because the company does not have full information, it uses the available statistical evidence to infer information that is true on average but may be false for any given individual. And, just as in the case of life insurance, the discrimination that results in equilibrium may have nothing to do with companies inherently preferring one gender over another. Exercise 22A.32 True or False: Statistical Discrimination leads to equilibria that have both separating and pooling features. 22A.4.2 Gender Discrimination based on Prejudice versus Statistical Discrimination When we observe incidences of gender discrimination, it is therefore difficult to know whether the discrimination arises from inherent prejudices or from economic considerations due to asymmetric information. Discrimination based on prejudice is defined as discrimination that arises from tastes that inherently prefer one group over another while statistical discrimination arises from asymmetric information. Life insurance companies that charge lower premiums to women do not do so because they like women more than men they do so because women on average live longer than men. Similarly, employers who discriminate against women in labor markets may be motivated solely by economic considerations rooted in asymmetric information. Let me be clear: I am not arguing that such discrimination may not be due to more pernicious causes related to good-old-boys on corporate boards feeling uncomfortable about allowing women more economic opportunities. I am simply pointing out that the same logic that causes life insurance companies to discriminate in favor of women (and against smokers) may also lie behind some of the discrimination against women we might observe in labor markets. Nor am I saying that only taste discrimination based on prejudice should disturb us but understanding the root causes of discrimination may help us better formulate solutions that eliminate all forms of gender discrimination. Exercise 22A.33 Suppose public schools invested more resources into gender sensitivity training in hopes of lessening gender discrimination in the future. Would you recommend this if you knew that gender discrimination was purely a form of statistical discrimination? Markets, for instance, tend to punish employers for discriminating based on prejudice. Suppose that companies A and B in a competitive market are identical in every way except for the fact that company A is governed by a corporate board that is prejudiced against working with women while company B is not. This implies that company B has a larger pool of talent to draw from and will be able to gain a competitive advantage over company A by employing qualified women. Both companies may operate in equilibrium, but the prejudiced company will earn lower dollar profits because part of its profit comes in the form of prejudiced corporate leaders getting utility from excluding women. Shareholders should prefer to invest in company B that makes more dollar profits, which implies that the stock of company B will have higher market value than the stock

22A. Asymmetric Information and Efficiency 827 of company A. 9 Now consider a third company C that is just like company B but suppose that C is willing to engage in statistical discrimination while B is not. If the labor market is characterized by asymmetric information and if women on average are more likely to leave the labor force to rear children, then company C will engage in statistical discrimination that will likely make it more profitable. While the market thus tends to punish companies that engage in taste discrimination based on prejudice, it will reward companies that engage in statistical discrimination. Finally, suppose there exists yet a fourth company D that has developed an effective screening tool which can differentiate individually among applicants (of both genders) that can differentiate between those that are likely to leave the labor force and those that are not. This company can, of course, do even better than company C by using its information and eliminate all forms of discrimination. Exercise 22A.34 In the past, gender discrimination was often enshrined in statutory laws making it illegal for firms to hire women into certain roles or schools to admit women as students. If you are one of the corporate board members in company A, why might you favor such laws even if all you care about is not having women in your own company? If you are one of the corporate board members in company C, would you similarly favor such a law? As societies consider ways of eliminating all forms of gender discrimination in labor markets, the appropriate strategies then differ depending on what form the discrimination takes. Both taste discrimination (due to prejudice) and statistical discrimination (due to asymmetric information) can persist in markets, but markets tend to punish the former while rewarding the latter. Taste discrimination disappears as old prejudices disappear from people s tastes, but statistical discrimination persists so long as companies are economically rewarded by discriminating in the presence of asymmetric information. Statistical discrimination will therefore tend to persist so long as underlying statistical differences between the genders persist unless other institutions are put in place to make statistical discrimination less profitable. If, for instance, men on average demand equal amounts of time away from the labor force in order to rear children, the root cause of statistical gender discrimination in labor markets disappears. Alternatively, some governments have instituted mandatory parental leave for both genders when children enter a household, some have focused on subsidizing child care to make it easier for women to return to the labor force, and some have instituted rigorous anti-discrimination laws that offset the rewards from statistical discrimination with government sanctions. Finally, there exists an incentive for companies (such as company D in our example) to figure out more effective ways of differentiating between potential employees of both genders and for potential employees to signal whether they are likely to leave the labor force or not. Again, the goal here is not to advocate one form of institutional solution over another but simply to suggest that there are a variety of government and non-government institutions that might emerge to address the asymmetric information problem which results in statistical gender discrimination in labor markets. 22A.4.3 Racial Discrimination Just as gender discrimination in labor markets can result from either inherent prejudice or from asymmetric information, persistent racial discrimination can have the same two root causes. We 9 This presumes, of course, that not all shareholders are similarly prejudiced. But even if some shareholders are prejudiced and get utility from owning stock in companies that discriminate against women, the stock market will reward the non-discriminating company with higher stock values so long as not all shareholders are prejudiced.

828 Chapter 22. Asymmetric Information in Competitive Markets began our discussion of gender discrimination in the context of life insurance markets where insurance companies price discriminate against men because of the higher average life expectancy of women. For a variety of complex reasons, it turns out that African Americans have shorter average life expectancy in the US than whites. Gender discrimination in insurance markets, however, is legal, while racial discrimination is not. Thus, the statistical discrimination that would tend to make life insurance premiums higher for African Americans is not permitted, causing insurance companies not to explicitly price-discriminate against African Americans as they do against men. Even in the absence of legal barriers, the bad publicity from explicit racial discrimination in the pricing of life insurance premiums might be sufficient to keep this from happening so long as large numbers of potential customers would be offended by seeing insurance premium tables that have separate columns for different races. At the same time, it may well be the case that insurance companies discriminate below the radar screen by being less aggressive in advertising their life insurance products to African Americans. Despite the legal barriers to racial discrimination and despite much progress over the past decades, however, it appears that such racial discrimination continues to persist in other markets. But it again becomes difficult to ascertain what fraction of the observed discrimination in those markets is due to taste discrimination based on prejudice as opposed to statistical discrimination based on asymmetric information. In the case of racial discrimination, such statistical discrimination may well be due to average differences between groups that emerge from the historical legacy of past (and present) racial discrimination elsewhere. It is well-documented, for instance, that African American children on average attend worse public schools than non-minority children. In the past, this resulted from explicit public policy which, at least in the American South, set up different school systems for African Americans, systems that were funded at vastly different levels and, as the Supreme Court stated explicitly (in 1954) in Brown vs Board of Education, resulted in separate and unequal education for African American children. But even today, entry into public schools is determined by where a child s parents live, with schools that serve disproportionate numbers of minority children (on average) systematically worse than schools that serve primarily non-minority children. A variety of economic factors therefore continue to cause minority children to on average attend worse public schools than non-minority children even as the public school system overall has become officially more integrated. Now suppose that an employer is faced with identical high school transcripts from two applicants, one non-minority and one African American. For all the employer knows, the African American applicant has many unobservable characteristics that will make him a much better employee than the non-minority applicant. But the employer also knows that on average, African American children attend worse public schools and thus have not had the same opportunity to gain skills as non-minority children. The employer then faces the same asymmetric information problem we have discussed throughout this chapter and will be tempted to statistically discriminate against the African American applicant even if she has no prejudice (derived from pernicious tastes) in her heart. Recognizing that it may thus be economically rational for her to discriminate does not imply moral approval for such discrimination whether racial discrimination in labor markets results from inherent prejudice or from asymmetric information, it is deeply disturbing to many of us. Rather, recognizing that such discrimination can persist even in the absence of explicit taste discrimination simply suggests that market forces by themselves may be insufficient to stamp out racial discrimination when underlying average group difference arise from discrimination elsewhere. It furthermore suggests that, even if all forms of racial discrimination are illegal, it is likely that subtle and difficult-to-detect racial discrimination may persist in markets so long as these markets

22B. Insurance Contracts with Two Risk Types 829 are characterized by such asymmetric information. Exercise 22A.35 True or False: In the above example, the asymmetric information that leads to statistical discrimination against African Americans is still rooted in discrimination based on prejudice but it may be rooted primarily in prejudice-based discrimination from the past. In the short run, societies can combat such discrimination through a variety of civil society and government institutions. For instance, if a decline in inherent prejudice due to pernicious tastes leads to an increasing number of individuals placing explicit value on diversity, employers might overcome their temptation to statistically discriminate because their non-minority employees gain utility from knowing that they are working in a diverse environment and because their customers are offended if civil society advocacy groups advertise that a particular company has a non-diverse labor force. Alternatively, governments have instituted a variety of different forms of affirmative action policies to explicitly encourage more diverse work environments. In the long run, however, the temptation to engage in statistical discrimination of the kind we have raised here subsides only when more equal access to educational opportunities is offered to all irrespective of race and ethnicity. A society that successfully equalizes such opportunities will therefore eliminate the very statistical group differences that lead to informational asymmetries that in turn lead to statistical discrimination. The tendency of racial discrimination to persist in markets is therefore not fully eliminated until attitudes in people s tastes are non-discriminatory and opportunities for different groups are truly equal. 22B Insurance Contracts with Two Risk Types As noted at the beginning, we deviate in this Chapter somewhat from our usual practice of formalizing mathematically in Section B what we did intuitively in Section A. Section A was written without the presumption that you have covered the sometimes optional topic of risk (from Chapter 17), and this constrained us to thinking only about whether or not a consumer will buy insurance not how much insurance coverage each consumer might buy. But now we will build a model of adverse selection directly on the topics related to insurance markets that we introduced in Chapter 17, models in which we considered a whole menu of actuarily fair insurance contracts ranging from no insurance to full insurance. If you have previously covered only Section A of Chapter 17, you can focus solely on the non-mathematical parts of this Section to build adverse selection into the graphical insurance models you have previously seen. For this reason, the mathematical sections 22B.1.2, 22B.2.2 and 22B.3.3 are put in separate sub-sections allowing you simply to skip them if you d prefer to focus on just the graphical exposition. While we will develop some new intuitions and insights with this model, we should note however that the car insurance model in the previous section could in fact be re-interpreted to yield similar insights. We leave you to do this in the context of health insurance in end-of-chapter exercises 22.7 and 22.8. Suppose, then, that consumers (like my wife in Chapter 17) face the possibility of a bad outcome in which their consumption is x 1 and the possibility of a good outcome in which their consumption is x 2. Suppose further that there are two consumer types, with consumers of type δ facing outcome x 1 with probability δ (and outcome x 2 with probability (1 δ)), and consumers of type θ facing outcome x 1 with probability θ (and outcome x 2 with probability (1 θ).) We will adopt the convention that δ<θ, implying that the δ types face less risk than the θ types. Otherwise, the two consumer types are identical in every way, with x 1 and x 2 the same for both types and with each type having the same underlying tastes which we will assume throughout are independent

830 Chapter 22. Asymmetric Information in Competitive Markets of which state of the world occurs. As in Chapter 17, we will furthermore assume that each individual s tastes over risky gambles can be expressed as an expected utility. And we will assume that each type knows the risk she faces but that insurance companies do not necessarily know which type any given individual represents. In most of what follows, the insurance companies only know that a fraction γ of the population is of type δ and the remaining fraction (1 γ) is of type θ. Insurance companies offer insurance contracts that are defined (as in Chapter 17) by an insurance premium p and an insurance benefit b. If a consumer purchases an insurance contract (p, b), her consumption in the good state falls to (x 2 p) while her consumption in the bad state rises to (x 1 + b p). As we showed in Chapter 17, since we assume that tastes over consumption are stateindependent, each consumer type would then choose to fully insure so long as she faced complete and actuarily fair insurance markets. 22B.1 Equilibrium without Adverse Selection In Chapter 17, we graphed indifference curves in graphs with x 2 on the horizontal and x 1 on the vertical axis, and we graphed the menu of actuarily fair insurance contracts in the same graphs. We will return to this way of modeling insurance in end-of-chapter exercises 22.4, 22.5 and 22.8. In endof-chapter exercise 17.3, however, we showed that we can alternatively graph indifference curves on a graph with the insurance benefit b on the horizontal and the insurance premium p on the vertical. And, if insurance companies were able to offer actuarily fair (and thus zero-profit) contracts to each type separately, such contracts could similarly be graphed in such a graph. From our work in Chapter 17 we know that such contracts would have the feature that p = δb for consumer type δ and p = θb for consumer type θ. Exercise 22B.1 Explain why such contracts are actuarily fair. 22B.1.1 A Graphical Depiction of Equilibrium without Adverse Selection Panel (a) of Graph 22.3 does this for a consumer of type δ where x 1 = 10, x 2 = 250 and δ = 0.25 as it was for the example of my wife deciding on life insurance in Chapter 17. Notice that this consumer becomes better off as she moves southeast on the graph because moving southeast implies greater insurance benefits and lower insurance premiums. The graph also contains the line p = δb that represents the menu of actuarily fair insurance contracts for this consumer type. Since tastes are state-independent in this example, our work in Chapter 17 implies that our risk averse consumer will fully insure, purchasing a policy (b, p) = (240, 60) at which her indifference curve must be tangent to the line representing her insurance options. Exercise 22B.2 Why is (b, p) = (240, 60) an insurance contract that provides full insurance to a δ type consumer? Exercise 22B.3 What would indifference curves look like for risk neutral consumers? What about risk loving consumers? Panel (b) of the graph then illustrates exactly the same for consumer type θ assuming that θ = 0.5 i.e. assuming that this consumer type is twice as likely to encounter the bad state. Risk aversion again implies that the consumer will choose to fully insure when faced with a menu of actuarily fair insurance contracts, but such contracts are twice as expensive for type θ since the insurance company is twice as likely to have to pay out benefits.

22B. Insurance Contracts with Two Risk Types 831 Graph 22.3: Equilibrium Insurance Policies in the Absence of Asymmetric Information Exercise 22B.4 Demonstrate that full insurance for type θ implies the same benefit level as for type δ. If insurance companies can tell which consumer type they are facing when they enter an insurance contract, then panel (c) depicts the competitive equilibrium in which the full insurance contract (b δ, p δ ) = (240, 60) is sold to type δ and the full insurance contract (b θ, p θ ) = (240, 120) is sold to type θ, with insurance companies earning zero profit. This equilibrium is efficient there is no way to make anyone consumers or firms better off without making someone else worse off. 22B.1.2 Calculating the Equilibrium without Adverse Selection Graph 22.3 (and the remaining graphs in this chapter) assume that the (state-independent) utility of consumption can be described by the function u(x) = α lnx (again as in Chapter 17). This results in an expected utility from the insurance contract (b, p) for type δ of and for type θ U δ (b, p) = δα ln(x 1 + b p) + (1 δ)α ln(x 2 p), (22.2) U θ (b, p) = θα ln(x 1 + b p) + (1 θ)α ln(x 2 p). 10 (22.3) Exercise 22B.5 Are these consumer types risk averse? If consumer type δ faces an actuarily fair menu of insurance contracts described by p = δb, she will choose (b, p) to maximize equation (22.2) subject to p = δb. Solving this problem results in an optimal choice of which fully insures the consumer. b = x 2 x 1 and p = δ(x 2 x 1 ), (22.4) 10 If you have trouble seeing how we arrive at this as the expected utility, you should review the concepts in Chapter 17.

832 Chapter 22. Asymmetric Information in Competitive Markets Exercise 22B.6 Set up the expected utility maximization problem for θ types and derive the optimal choice assuming they face an actuarily fair insurance menu? Exercise 22B.7 How do these results relate to the values in Graph 22.3? 22B.2 Self-Selecting Separating Equilibria Now suppose that insurance companies cannot tell the low risk type δ consumers apart from high risk type θ consumers unless some information is revealed through signaling or screening. In part A of the chapter, we investigated how consumers can send explicit signals to try to reveal their type and how firms can invest in screens that reveal information and we implicitly assumed that such signals and screens could be bought at some cost. But there is another way that consumers of insurance can identify themselves when multiple different insurance contracts are offered to all customers: they could simply choose different contracts depending on which risk type they are and thus self-select into different insurance pools. Firms may therefore want to design the set of contracts that are offered in such a way that consumers reveal their type through their actions. Note that we could not investigate this possibility in our car insurance example of part A because we assumed there that the decision to insure was a discrete decision either you bought insurance or you did not and not one that involved choices over how much insurance to buy. The full information equilibrium depicted in Graph 22.3c can then no longer be an equilibrium when firms do not know who is what type. Under full information, there was no problem having insurance companies offer all actuarily fair insurance contracts p = δb to δ types because they knew who the θ types were and could simply prevent them from buying insurance contracts intended for low cost δ types. But if insurance companies cannot tell who the high cost types are, they can no longer offer all the p = δb contracts because type θ consumers would end up buying one of those contracts rather than those intended for them. Insurance companies would then make negative profits as they incur higher costs on type θ consumers while selling them low cost insurance. In the absence of knowing who is what type, the insurance industry will therefore have to restrict what types of contracts it offers. 22B.2.1 A Graphical Exposition of Self-Selecting Separating Equilibrium We can then ask which insurance contracts will not be offered in an equilibrium in which insurance companies achieve the outcome that individuals self-select into different insurance pools based on their risk types. First, note that it must be the case that high risk types still get fully insured at actuarily fair rates in such an equilibrium. If this were not the case, there would be room for new insurance companies to enter and offer such actuarily fair full insurance to high risk types. This implies that the insurance contracts that will be restricted are those for low-risk types. Since those types face less risk, it is less costly for them to forego some insurance in order to be able to get a better deal on their insurance contract than they could if they chose from contracts intended for high risk types. This then opens the door for low risk types to signal that they are in fact low risk types by choosing an insurance contract that is actuarily fair for them but does not fully insure with insurance companies simply not making actuarily fair full insurance available for low risk types. This is illustrated in panel (a) of Graph 22.4 where we again have two (green) actuarily fair contract lines, one for high risk θ types and another for low risk δ types. The high risk θ types once again optimize along the actuarily fair set of insurance contracts aimed at them settling at

22B. Insurance Contracts with Two Risk Types 833 the full insurance contract A. All the contracts that lie in the shaded area below the magenta U θ, however, are preferred by high risk types to their actuarily fair full insurance contract A. They would therefore much prefer to choose an insurance contract from the portion of the p = δb line that lies within the shaded region, with any contract on that line to the right of B strictly preferred by them to A. Thus, if insurance companies want to induce high and low risk types to self-select into separate actuarily fair insurance contracts, they cannot offer any of the p = δb contracts to the right of B. Graph 22.4: Self-Selecting Separating Equilibrium with Asymmetric Information In a separating equilibrium in which risk types identify themselves through the insurance contracts that they purchase, the only actuarily fair insurance contracts that can then be offered are those that are located on the bold portion of the p = δb line in Graph 22.4a. And of these, risk averse consumers of type δ will demand only the contract represented by point B since all other contracts that are offered involve greater risk (without a change in the expected value of the outcome). Exercise 22B.8 Suppose insurance companies offer all actuarily fair insurance contracts to type θ. Can you identify in panel (b) of Graph 22.4 the area representing all insurance contracts that consumers of type δ would purchase rather than choosing from the menu of contracts aimed at type θ? Exercise 22B.9 From the area of contracts you identified in exercise 22B.8, can you identify the subset which insurance companies would be interested to offer assuming they are aware that high risk types might try to get low cost insurance? Exercise 22B.10 From the contracts identified in exercise 22B.9, can you identify which of these contracts could not be offered in equilibrium when the insurance industry is perfectly competitive? You should be able to see straight away that the competitive separating equilibrium in this example is inefficient. In particular, the competitive equilibrium in the absence of asymmetric information (depicted in Graph 22.3c) has low risk types δ with higher utility without anyone else doing worse (since high risk θ types do equally well and firms make zero profits in either case.) The inefficiency arises from the fact that there are missing markets not all the actuarily fair insurance contracts for δ types are offered under asymmetric information. And the missing markets arise from the adverse selection problem; i.e. the problem that high risk types would adversely

834 Chapter 22. Asymmetric Information in Competitive Markets select into the low risk insurance market if the missing market for fuller insurance targeted at low risk customers emerged. 22B.2.2 Calculating the Separating Equilibrium The mathematics behind Graph 22.4a is in principle relatively straightforward: the insurance contract B is identified as the intersection of the indifference curve of high risk θ types who fully insure under actuarily fair insurance with the line representing all actuarily fair insurance contracts for the low risk δ types. Full insurance for a θ type implies a consumption level of ((1 θ)x 2 + θx 1 ) in each state with certainty, which implies that the full insurance utility for type θ is U θ f = u((1 θ)x 2 + θx 1 ) = α ln ((1 θ)x 2 + θx 1 ). (22.5) Exercise 22B.11 Can you verify that full insurance implies consumption of ((1 θ)x 2 + θx 1)? The indifference curve that gives all combinations of b and p such that a θ type is indifferent to the full insurance outcome is then given by all (b, p) under which her expected utility U θ (b, p) is equal to Uf θ from equation (22.5); i.e. U θ (b, p) = αθ ln(x 1 + b p) + α(1 θ)ln(x 2 p) = α ln ((1 θ)x 2 + θx 1 ) = U θ f. (22.6) We can then cancel the α terms and use the rules of logarithms to rewrite the middle part of this equation as which we can solve for b to get (x 1 + b p) θ (x 2 p) (1 θ) = (1 θ)x 2 + θx 1 (22.7) b = ( ) 1/θ (1 θ)x2 + θx 1 + p x (x 2 p) (1 θ) 1. (22.8) Although Graph 22.4 is not drawn using this precise function, this is the (inverse of the) equation for the magenta indifference curve in Graph 22.4a when we substitute in θ = 0.5, x 2 = 250 and x 1 = 10; i.e. the equivalent to the magenta indifference curve in our graph is described by the equation ( ) 1/(0.5) (1 0.5)250 + 0.5(10) b = + p 10 = 1302 + p 10. (22.9) (250 p) (1 0.5) 250 p Our logic above told us that the highest actuarily fair insurance policy for the low risk δ types that can exist in a separating equilibrium is given by the intersection of this indifference curve with the line p = δb that represents the menu of all actuarily fair insurance contracts for low risk types. Written in terms of b, this line is b = p/δ or b = 4p when δ = 0.25 (as we assumed in our graph). Thus the premium at point B in the graph is given by the intersection of equation (22.9) and the actuarily fair insurance menu b = 4p (represented by the lower green line in Graph 22.4a). This means we need to solve the equation 4p = 1302 + p 10 (22.10) 250 p

22B. Insurance Contracts with Two Risk Types 835 Equilibrium Insurance for Low Risk δ Types θ p b x 1 + b p x 2 p 0.25 60.00 240.00 190.00 190.00 0.33 31.05 124.20 102.10 219.30 0.50 21.30 85.20 73.90 228.70 0.75 12.19 48.76 46.57 237.81 0.90 5.77 23.08 27.31 244.23 0.99 0.68 2.70 12.03 249.32 1.00 0.00 0.00 10.00 250.00 Table 22.1: δ = 0.25, x 1 = 10, x 2 = 250 which can be rewritten as 3p 2 740p + 14400 = 0. (22.11) Applying the quadratic formula, we get p = 225.37 and p = 21.30 which represent the two premiums at which the magenta indifference curve crosses the lower green line in Graph 22.4a. Point B in our graph lies at the lower of these premiums, with p = 21.30 and corresponding b = 4p = 85.20. In a competitive separating equilibrium, we therefore have two insurance contracts that are sold (b θ, p θ ) = (240, 120) and (b δ, p δ ) = (85.2, 21.3), with high risk θ types fully insuring under the former and low risk δ types revealing their type by purchasing less than full insurance under the latter contract. Exercise 22B.12 Can you show mathematically (by evaluating utilities) that this equilibrium is inefficient relative to the equilibrium identified in Graph 22.3c? Exercise 22B.13 True or False: Under perfect competition (and assuming that insurance companies incur no costs other than the benefits they pay out), risk averse individuals with state-independent tastes will fully insure in the absence of asymmetric information but may insure less than fully in its presence. Exercise 22B.14 Can you verify the intercepts for point C in Graph 22.4b? Table 22.1 then presents the equilibrium insurance contracts for low risk δ types as the high risk type becomes riskier i.e. as θ increases. For our particular example, low risk types continue to find some insurance regardless of how risky the θ types are (unless θ reaches 1), but low risk types clearly purchase less insurance in separating equilibria as high risk types become riskier. Put differently, the externality from adverse selection increases in severity as high risk types become riskier. In cases where insurance can only be sold in discrete units such as cases like those in Section A where grade insurance was not continuous low risk types might be frozen out of the insurance market altogether. Exercise 22B.15 Draw a graph with b on the horizontal and p on the vertical axis illustrating the separating equilibrium in row 4 of Table 22.1. 22B.3 Pooling Contracts with Asymmetric Information In our treatment of self-selecting separating equilibria, we have implicitly assumed that insurance companies cannot earn positive profit by offering an insurance contract that attracts both high and

836 Chapter 22. Asymmetric Information in Competitive Markets low risk types into the same insurance pool. We will now explore how such a possibility might emerge and how it might make the self-selecting equilibrium we have analyzed so far impossible to achieve. And we will see shortly that this possibility depends crucially on the number of high risk types relative to the number of low risk types in the economy. Suppose an insurance company were to offer a contract that was more attractive for both risk types than the separating equilibrium contracts we previously identified. If a fraction γ of the population is of type δ (and the remaining fraction (1 γ) is of type θ), then such an insurance company would expect on average to pay δb for the fraction γ of its customers that are low risk types and θb to the fraction (1 γ) of its customers who are high risk types. Thus, the insurance company would expect to make zero profits when p = γδb + (1 γ)θb = [γδ + (1 γ)θ] b. (22.12) 22B.3.1 Pooling Contracts that Eliminate Self-Selecting Separating Equilibria Note that, when γ = 0, this simply reduces to the equation p = θb that defines the zero profit line for high risk types, and when γ = 1 it reduces to the zero profit line for low risk types. As γ increases from zero to 1, the zero profit line from having both types buy the same policy therefore rotates from the high-risk zero profit line to the low-risk zero profit line. In Graph 22.5a, for instance, the zero profit pooling line is depicted for the case where γ = 0.5 with this (green) line lying exactly midway between the zero profit lines for the individual risk types. Graph 22.5: A Pooling Equilibrium Does Not Exist We can now think about the possibility of a pooling insurance contract that breaks the selfselection separating equilibrium in this example. Panel (b) of Graph 22.5 replicates panel (a) from Graph 22.4 and illustrates the contracts A and B that would be bought by types θ and δ in a separating equilibrium. In addition, panel (b) of Graph 22.5 includes the set of zero-profit pooling contracts that emerges when half the consumers are of type δ and half the consumers are of type θ (i.e. when γ = 0.5). Note that in this case, the (blue) indifference curve for δ types that goes through contract B lies to the southeast of the (grey) zero profit pooling line which implies that

22B. Insurance Contracts with Two Risk Types 837 the low-risk δ types prefer to identify themselves as low risk types by choosing the contract B over any possible zero-profit pooling contract. Thus, there is no pooling contract that would attract both risk types and result in non-negative profit for insurance companies when B is currently offered. The self-selecting separating equilibrium stands. But now suppose that γ is equal to 2/3 instead of 1/2 i.e. suppose that two thirds of the population was low risk and one third of the population was high risk. What changes as a result in Graph 22.5b? The zero-profit lines aimed at the two types individually are given by p = δb and p = θb and thus are unaffected by changes in γ. Similarly, the tastes of the two types are unchanged (since individual tastes have nothing to do with how many others of each type there are in the economy) which implies the blue and magenta indifference curves remain unchanged. The only thing that changes is the (grey) line representing the possible pooling contracts that give insurance companies zero profit! In particular, as γ increases, this line becomes shallower (without a change in the intercept), and as it becomes shallower, it will eventually cross the blue indifference curve for δ types. Exercise 22B.16 Can you show, using equation (22.12), that the last sentence is correct? Panel (a) of Graph 22.6 then illustrates the zero-profit pooling line for γ = 2/3, and it illustrates the (blue) indifference curve for δ types that is tangent to this line at point D. Point B the best possible contract that would allow δ types to identify themselves without θ types wanting to imitate them, now lies slightly to the northwest of this indifference curve implying that low risk δ types would (slightly) prefer D even though this contract is not actuarily fair from their perspective. Similarly, θ types prefer D to the actuarily fair full insurance contract A; i.e. while the contract D does not fully insure them, it represents terms that are better (from their perspective) than actuarial fairness. Thus, we have identified a contract D that is strictly preferred by both risk types to the contracts B and A in the previous separating equilibrium, and the same is true for contracts slightly to the northwest of D which would result in positive profits for insurance companies. This then makes it impossible to sustain the separating equilibrium we were able to sustain when γ was 0.5 by raising γ to 2/3, we have made it sufficiently easy to find pooling contracts that everyone prefers. And this of course becomes even easier as γ increases further. Exercise 22B.17 What is the expected value of consumption for θ types at point D? Is it higher or lower than under full insurance? Explain. Exercise 22B.18 What is the expected value of consumption for δ types at point D? Is it higher or lower than the expected value of consumption without insurance? Explain. 22B.3.2 Almost A Pooling Equilibrium We have so far shown that the separating equilibrium breaks down when there are sufficiently many low risk types relative to high risk types in the economy because this allows firms to offer pooling contracts that are both preferred to the separating equilibrium contracts by all types and result in positive profit. To check whether there exists a pooling equilibrium, however, is trickier. Not only would we have to identify a zero-profit contract (such as D in Graph 22.5) that breaks the separating equilibrium, but we would further need to demonstrate that no other contract could result in positive profits for a firm that offers such a contract when all other firms offer D. Exercise 22B.19 Why must any potential pooling equilibrium contract D lie on the zero-profit pooling line?

838 Chapter 22. Asymmetric Information in Competitive Markets Panel (b) of Graph 22.6 illustrates once again point D on the zero-profit pooling line but this time shows both the (magenta) indifference curve for high risk types and the (blue) indifference curve for low risk types that contain point D. We can then ask whether there exist insurance contracts in each of the areas (labeled by lower case letters) that would earn an individual insurance company positive profits given that all other companies offer the contract D. Graph 22.6: A Pooling Equilibrium? First, note that all insurance contracts that fall in the regions (a), (b), (c) or (d) lie to the northeast of both the blue and the magenta indifference curves and thus any company that offers a contract in those regions would attract no customers. Second, contracts that lie in the regions (e) and (f) lie to the northeast of the blue indifference curve and to the southeast of the magenta indifference curve which implies that such contracts would attract only high risk θ types and thus yield negative profit (given that all these contracts lie below the zero-profit line for high risk types). Third, contracts that fall in the regions (g) and (h) lie to the southeast of both the blue and the magenta indifference curves which implies they will attract both high and low risk types. But all such contracts lie below the zero-profit pooling line, which implies that an insurance company would earn negative profits when offering such contracts. Finally, this leaves regions (i) and (j) that lie to the southeast of the blue indifference curve and the northeast of the magenta indifference curve implying that such contracts would attract only low risk δ types. Those contracts falling in region (i), however, lie below the zero-profit line for δ types and would thus earn negative profit. We are then left with only contracts in the shaded region (j) that could potentially earn positive profit for a firm that offers insurance contracts in this region while other companies all offer the policy D. Without some friction in the market, everyone offering policy D is therefore not a competitive equilibrium. However, there are several ways in which we might still have D emerge as a pooling equilibrium: First, it might be that there are some start-up costs to offering an insurance policy different than what the market offers costs of advertising and alerting consumers about the new policy. If those costs are sufficiently high, it may well be that contracts in region (j) will not result in positive profits for individual insurance companies (when all others are offering D). Second, it might be that there is some search cost that consumers incur when looking for something other than the prevalent market policy and if this cost is sufficiently high, the policies

22B. Insurance Contracts with Two Risk Types 839 in region (j) might not lie to the southeast of the blue indifference curve once the search cost is taken into account. Finally, if firms in the market adjust quickly to changing circumstances, it might be that firms who currently offer D know that, as soon as they make a positive profit in region (j), other firms will offer policies closer to the zero-profit line p = δb and will thus drive profits to zero. If the firms anticipate this, they may not offer policies in regions (j). This, however, begins to get us into the area of strategic thinking on the part of firms, a topic for later chapters. Exercise 22B.20 * Can you think of what would have to be true about how the blue and magenta indifference curves relate to one another at D in order for the problematic area (j) to disappear? Explain why this would then imply that D is a competitive equilibrium pooling contract. Exercise 22B.21 For the case where γ = 1/2 and where a pooling equilibrium therefore does not exist (as shown in Graph 22.5b), can you divide the set of possible insurance contracts into different regions and illustrate that no firm would have an incentive to offer any contracts other than those that are provided in the separating equilibrium? 22B.3.3 Calculating the Almost Pooling Equilibrium From our graphical exposition, it is clear that a competitive pooling equilibrium can arise only if the optimal insurance contract for low risk δ types from the set of zero-profit pooling contracts (given in equation (22.12)) yields greater utility for δ types than the insurance contract that allows δ types to separate from high risk θ types. Thus, we can begin by calculating the optimal contract from the set of contracts (b, p) satisfying p = [γδ + (1 γ)θ]b; i.e. we can solve the optimization problem max b,p Uδ (b, p) = αδ ln(x 1 + b p) + α(1 δ)ln(x 2 p) subject to p = [γδ + (1 γ)θ] b. (22.13) Solving this in the usual way, we get and b = (1 δ)x 1 γδ + (1 γ)θ 1 + δx 2 γδ + (1 γ)θ (22.14) p = (γδ + (1 γ)θ)(1 δ)x 1 γδ + (1 γ)θ 1 + δx 2. (22.15) In Graph 22.6a, we assumed γ = 2/3 (with δ = 0.25, θ = 0.5, x 1 = 10 and x 2 = 250). Plugging these into equations (22.14) and (22.15), we get (b, p)=(176.25, 58.75) which is point D in the graph. Substituting these back into the utility function for δ types, we get utility of 5.1522α. Low risk δ types could alternatively purchase the contract (b, p)=(85.2, 21.3) (represented by point B) that allows them to separate from high risk types but plugging this contract into the expected utility function for δ types gives utility of 5.1500α which is just below what the same types can attain by pooling with high risk types. Thus, δ individuals prefer D to B when γ = 2/3 and by implication for all γ > 2/3. Exercise 22B.22 Can you demonstrate mathematically that θ types also prefer D to their separating contract A (which has (b,p)=(240,120)?

840 Chapter 22. Asymmetric Information in Competitive Markets Pooling Contracts γ p b x 1 + b p x 2 p 2/3 58.75 176.25 127.50 191.25 0.80 59.29 197.62 148.33 190.71 0.85 59.47 206.86 157.39 190.52 0.90 59.66 216.93 167.27 190.34 0.95 59.83 227.93 178.10 190.17 1.00 60.00 240.00 190.00 190.00 Table 22.2: δ = 0.25, θ = 0.5, x 1 = 10, x 2 = 250 Exercise 22B.23 When γ = 0.5 (as in Graph 22.5), equations (22.14) and (22.15) give (b, p)=(154.67,58). Can you demonstrate that the indifference curve containing this point lies below the indifference curve that δ types can attain by purchasing the contract B that allows them to separate? Table 22.2 then reports results for higher values of γ, with the insurance contract approaching that of actuarily fair full insurance for the low risk δ types as the fraction of δ types in the population approaches 1. Exercise 22B.24 Can you explain intuitively the change in pooling contracts as you move down Table 22.2? What happens to the problematic (j) region from our graph as we go down the table? 22B.4 Non-Existence of a Competitive Equilibrium In Graph 22.6b, we gave an example of how competitive markets may have difficulty sustaining a pooling equilibrium when γ is sufficiently high such that a separating equilibrium does not exist. In particular, we illustrated for a particular set of indifference curves that, unless there are some frictions that make it difficult for individual insurance companies in competitive markets to deviate from the commonly offered pooled insurance contract, there exists an incentive for firms to find contracts in the region denoted (j) that is preferred by low-risk types to the pooled contract D and that would earn the deviating firm a positive profit. But none of the policies in the (j) region of the graph represent policies that can be sustained as an equilibrium either. Thus, if γ is sufficiently high to make the potential pooling preferable to separating for low risk types, a competitive equilibrium may in fact not exist in this set-up. (For other sets of indifference curves, such an equilibrium does exist as you might have already worked out in within-chapter-exercise 22B.20.) How should we interpret such a non-existence of an equilibrium? It may lead us to conclude that insurance markets like this will simply shift back and forth with firms moving policies around to attract customers, earning profits briefly before shifting policies again to adjust to changing market conditions. It may imply that markets will search for other ways more explicit signals and screens to separate different risk types into different insurance pools. As we have argued in Section A, there may be instances when firms can gain only noisy information that can lead to statistical discrimination. The insurance industry may also develop particular norms or industry standards that constrain the set of insurance contracts that can be offered. Alternatively, you can see how the government could, in principle, solve the non-existence (or instability) problem by simply offering a single insurance contract (like D) and not permitting an insurance industry to operate in this

22B. Insurance Contracts with Two Risk Types 841 market or it could regulate the insurance market and mandate that only D is offered within that market. None of these solutions, however, will implement efficiency unless they find ways of costlessly revealing the asymmetric information to all parties and thus allowing the industry to reach the full information competitive equilibrium. Conclusion The primary problem raised by asymmetric information is what we have called the adverse selection problem. High cost consumers, for instance, adversely select into markets with low cost consumers and thus impose a negative externality on low cost consumers by driving up price; or low quality producers adversely select into markets with high quality producers thus lowering price and making it difficult to sustain high quality. We have shown that such adverse selection sometimes aggravated by moral hazard will cause over-consumption by some and under-consumption by others, with deadweight losses for society overall. In some instances, we have even seen that asymmetric information can cause entire markets to disappear. Our primary application has been the insurance market where the concept of adverse selection can be presented in a variety of different ways, as can the pooling equilibira that arise in the absence of a resolution to the asymmetric information problem and separating equilibria that may emerge through signals and screens (or, as discussed in Section B, through self-selection when firms restrict the set of contracts they offer). But we have also seen how understanding adverse selection and information asymmetries can help us understand some fundamental struggles that societies experience struggles like overcoming the legacy of discrimination. In some of the end-of-chapter exercises, we will further illustrate some tensions between efficiency goals (which have been the focus of the chapter) and other societal priorities (such as those advocated by proponents of universal health insurance based on the premise that everyone is in some moral sense entitled to such insurance.) This chapter concludes our treatment of inefficiencies that may arise in competitive markets. In Chapters 18 through 20, such inefficiencies resulted from policy-induced distortions of market prices; in Chapter 21, they arose from market prices not fully capturing all marginal social benefits or costs due to the presence of externalities; and in this chapter, inefficiencies emerged from the presence of asymmetric information, with one side of the market able to potentially take advantage of the other side because of more knowledge that is directly relevant to the market transaction. In the case of policy-induced price distortions, we suggested that an understanding of how these distortions arise may allow governments to find less distortionary ways to accomplish their goals. In the case of externalities or asymmetric information, on the other hand, we discussed ways in which additional markets, non-market civil society institutions and governments may find ways of improving (in terms of efficiency) on market outcomes. We will now move to Part V where we will begin to think about how to model behavior in economic settings where individuals are not small and where strategic thinking becomes important. To some extent, we have already begun to head down this road: In our treatment of adverse selection, for instance, we thought about whether individual firms might be able to benefit by deviating from the equilibrium behavior of other firms, and in our treatment of the Coase Theorem in the previous chapter, we thought about individuals negotiating after courts assign property rights. But from now on, we will let go of any notion of perfectly competitive behavior and focus more squarely on the strategic element of economic life. In the settings we will investigate, individuals can no longer take their economic environment as given because their actions help shape the economic environment in discernable ways. This will introduce the concept of market power into

842 Chapter 22. Asymmetric Information in Competitive Markets our thinking and will lead us away from thinking of price-taking behavior. It will also open another way in which markets fail to achieve efficient outcomes when markets are no longer perfectly competitive and thus some agents employ market power to advance their own interests. End of Chapter Exercises 22.1 Consider again the example of grade insurance. Suppose students know whether they are typically A, B, C, D or F students, with A students having a 75% chance of getting an A and a 25% chance of getting a B; with B, C and D students having a 25% chance of getting a grade above their usual, a 50% chance of getting their usual grade and a 25% chance of getting a grade below their usual; and with F students having a 25% chance of getting a D and a 75% chance of getting an F. Assume the same bell-shaped grade distribution as in the text i.e. in the absence of grade insurance, 10% of grades are A s, 25% are B s, 30% are C s, 25% are D s and 10% are F s. A: Suppose, as in the text, that grade insurance companies operate in a competitive market and incur a cost c for every level of grade that is changed for those holding an insurance policy. And suppose that A through D students are willing to pay 1.5c to ensure they get their usual grade and 0.5c for each grade level above the usual; F students are willing to pay 2c to get a D and 0.5c for each grade level above that. (a) Suppose next that your professor only allowed me to sell B insurance. Would I be able to sell any? (b) What if I were only allowed to sell C or only D insurance? (c) * If they were the only policies offered, could policies A and D attract customers in a competitive equilibrium at the same time? In equilibrium, who would buy which policy?(hint: Only C, D and F students buy insurance in equilibrium.) (d) * If they were the only policies offered, could policies A and C attract customers in a competitive equilibrium at the same time? (Hint: The answer is no.) (e) * If they were the only policies offered, could policies B and D attract customers in a competitive equilibrium at the same time? (Hint: The answer is again no.) (f) Without doing any further analysis, do you think it is possible to have an equilibrium in which more than 2 insurance policies could attract customers? (g) Are any of the equilibria you identified efficient? (Hint: Consider the marginal cost and marginal benefit of each level of insurance above insuring that each student gets his/her typical grade.) B: In A(c), you identified a particular equilibrium in which A and D insurance are sold when it was not possible to sell just A insurance. (a) How is this conceptually similar to the self-selecting separating equilibrium we introduced in Section B of the text? (b) How is it different? 22.2 Suppose that everything in the grade insurance market is as described in exercise 22.1. But instead of taking the asymmetric information as fixed, we will now ask what can happen if students can transmit information. Assume throughout that no insurance company will sell A insurance to students other than A students, B insurance to students other than B students, etc. whenever they know what type students are. A: Suppose that a student can send an accurate signal to me about the type of student he is by expending effort that costs c. Furthermore, suppose that each student can signal that he is a better student than he actually is by expending additional effort c for each level above his true level. For instance, a C student can signal his true type by expending effort c but can falsely signal that he is a B student by expending effort 2c and that he is an A student by expending 3c. (a) Suppose everyone sends truthful signals to insurance companies and that insurance companies know the signals to be truthful. What will be the prices of A-insurance, B-insurance, C-insurance and D-insurance? (b) How much surplus does each student type get (taking into account the cost c of sending the truthful signal). (c) Now investigate whether this truth-telling can be part of a real equilibrium. Could B students get more surplus by sending a costlier false signal? Could C, D or F students? (d) Would the equilibrium be any different if it was costless to tell the truth but it costs c to exaggerate the truth by each level? (Assume F-students would be willing to pay 1.5c for getting an F just as other students are willing to pay 1.5c to get their usual grade.)

22B. Insurance Contracts with Two Risk Types 843 (e) Is the equilibrium in part (d) efficient? What about the equilibrium in part (c)? (Hint: Think about the marginal cost and marginal benefit of providing more insurance to any type.) (f) Can you explain intuitively why signaling in this case addresses the problem faced by the insurance market? B: In Section B of the text, we considered the case of insurance policies (b, p) in an environment where the bad outcome in the absence of insurance is x 1 and the good outcome in the absence of insurance is x 2. We further assumed two risk types: δ types that face the bad outcome with probability δ and θ types that face the bad outcome with probability θ where θ > δ. (a) Suppose that both types are risk averse and have state-independent tastes. Show that, under actuarily fair insurance contracts, they will choose the same benefit level b but will pay different insurance premiums. (b) Suppose throughout the rest of the problem that insurance companies never sell more than full insurance; i.e. they never sell policies with b higher than what you determined in (a). In Section B we focused on self-selecting equilibria where insurance companies restrict the contracts they offer in order to get different types of consumers to self-select into different insurance policies. In Section A, as in part A of this question, we focused on explicit signals that consumers might be able to send to let insurance companies know what type they are. How much would a θ type be willing to pay to send a credible signal that he is a δ type if this will permit him access to the actuarily fair full insurance contract for δ types? (c) Suppose for the rest of the problem that u(x) = ln x is a function that permits us to represent everyone s tastes over gambles in the expected utility form. Let x 1 = 10, x 2 = 250, δ = 0.25 and θ = 0.5 as in the text. Suppose further that we are currently in a self-selecting equilibrium of the type that was discussed in the text (where not all actuarily fair policies are offered to δ types). 11 How much would a δ type be willing to pay to send a credible signal to an insurance company to let them know he is in fact a δ type? (d) Suppose we are currently in the separating equilibrium but a new way of signaling your type has just been discovered. Let c t be the cost of a signal that reveals your true type and let c f be the cost of sending a false signal that you are a different type. For what ranges of c t and c f will the efficient allocation of insurance in this market be restored through consumer signaling? (e) Suppose c t and c f are within the ranges you specified in (d). Has efficiency been restored? 22.3 In exercise 22.2, we showed how an efficient equilibrium with a complete set of insurance markets can be reestablished with truthful signaling of information by consumers. We now illustrate that signaling might not always accomplish this. A: Begin by once again assuming the same set-up as in exercise 22.1. Suppose that it costs c to truthfully reveal who you are and 0.25c more for each level of exaggeration; i.e. for a C student, it costs c to reveal that he is a C student, 1.25c to falsely signal that he is a B student and 1.5c to falsely signal that he is an A student. (a) Begin by assuming that insurance companies are pricing A, B, C and D insurance competitively under the assumption that the signals they receive are truthful. Would any student wish to send false signals in this case? (b) * Could A insurance be sold in equilibrium (where premiums have to end up at zero-profit rates given who is buying insurance)? (Hint: Illustrate what happens to surplus for students as premiums adjust to reach the zero profit level.) (c) * Could B insurance be sold in equilibrium? What about C and D insurance? (d) * Based on your answers to (b) and (c), can you explain why the equilibrium in this case is to have only D-insurance sold and bought by both D and F students? Is it efficient? (e) Now suppose that the value students attach to grades is different: They would be willing to pay as much as 4c to guarantee their usual grade and 0.9c more for each level of grade above that. Suppose further that the cost of telling the truth about yourself is still c but the cost of exaggerating is 0.1c for each level of exaggeration about the truth. How much surplus does each student type get from signaling that he is an A student if A-insurance is priced at 2c? (f) Suppose that insurance companies believe that any applicant for B insurance is a random student from the population of B, C, D and F students; that any applicant for C insurance is a random student from the population of C, D and F students; and any applicant for D insurance is a random student from the population of D and F students. How would they competitively price B, C and D insurance? 11 Recall from the text that, in this separating equilibrium, δ types bought the insurance policy (b, p) = (85.2, 21.3). While the u function in the text is multiplied by α, we showed that the indifference curves are immune to the value α takes and so we lose nothing in this problem by setting it to 1.

844 Chapter 22. Asymmetric Information in Competitive Markets (g) Suppose that, in addition, insurance companies do not sell insurance to students who did not send a signal as to what type they are. Under these assumptions, is it an equilibrium for everyone to signal that they are A students? (h) There are two sources of inefficiency in this equilibrium. Can you distinguish between them? B: In exercise 22.2B, we introduced a new signaling technology that restored the efficient allocation of insurance from an initially inefficient allocation in a self-selecting separating equilibrium. Suppose that insurance companies believe anyone who does not send a signal that he is a δ type must be a θ type. (a) Suppose that c f is below the range you calculated in B(d) of exercise 22.2. Can you describe a pooling equilibrium in which both types fully insure and both types send a signal that they are δ types? (b) In order for this to be an equilibrium, why are the beliefs about what a non-signal would mean important? What would happen if companies believed that both types are equally likely not to signal? (c) True or False: For an equilibrium like the one you described in part (a) to be an equilibrium, it matters what firms believe about events that never happen in equilibrium. 22.4 Assume again the basic set-up from exercise 22.1. A: We will now investigate the role of firm screens as opposed to consumer signals. (a) Suppose that insurance companies were interested in offering A, B, C and D insurance. In the absence of insurers knowing something about student types, will any of these insurance policies be provided in equilibrium (assuming the same bell-shaped grade distribution as the one used in the text)? (b) Suppose that an insurance company can screen students. More precisely, suppose an insurance company can, for a fee of c, obtain a student s transcript and thus know what type a student is. If insurance companies will only sell insurance of type i to students who have been screened as type i, what would be the equilibrium insurance premium for each insurance assuming perfect competition (and no recurring fixed costs)? (c) Would each insurance type be offered and bought in equilibrium? (d) How high would the cost of obtaining transcripts have to be in order for the insurance market to collapse? (e) In the case of signaling, we had to consider the possibility of pooling equilibria in which the same insurance is sold to different types of students who care sufficiently for the higher grade to each be willing to pay the zero-profit premium as well as, for some, to pay the cost of falsely signaling their type. If insurance companies can screen for the relevant information, could it ever be the case assuming that individuals care sufficiently much about higher grades that several types will get the same insurance? (Hint: Suppose an insurance company attempted to price a policy such that several types would get positive surplus by buying this policy. Does another insurance company have an incentive to compete some of the potential customers for that policy away?) (f) Does the separating equilibrium that results from screening of customers depend on how many of each different type are in the class and what exactly the curve is that is imposed in the class? (g) Suppose we currently have a market in which a large number of insurers sell the different insurance types at the zero-profit price after screening customers to make sure insurance of type i is only sold to type i. Now suppose a new insurance company enters the market and devises B insurance for C students. Will the new company succeed in finding customers? (h) Would your answer to (f) change if students are willing to pay 1.5c to insure their usual grade and c (rather than 0.5c) for each grade above the usual? (i) True or False: When insurance companies screen, the same insurance policy will never be sold to different student types at the same price, but it may be the case that students of different types will insure for the same grade. B: * Now consider the introduction of screening into the self-selction separating equilibrium of Section B of the text. As in the text, suppose that consumption in the absence of insurance is 10 in the bad state and 250 in the good state and that δ types have a probability of 0.25 of reaching the bad state while θ types have a probability of 0.5 of reaching that state. Suppose further that individuals are risk averse and their tastes are state-independent. (a) Instead of graphing b on the horizontal and p on the vertical axis, begin by graphing x 2 i.e. consumption in the good state on the horizontal and x 1 i.e. consumption in the bad state on the vertical. Indicate with an endowment point E where consumption would be in the absence of insurance.

22B. Insurance Contracts with Two Risk Types 845 (b) Illustrate the actuarily fair insurance contracts for the two types of consumers and indicate the two insurance policies that are offered in a self-selection separating equilibrium. (c) Suppose a screening industry i.e. an industry of firms that can identify what type an insurance applicant is for a cost of k per applicant emerges. If an insurance firm gives applicants the option of paying k (as an application fee) to enable the company to pay a screening firm for this information, would θ types pay it? (d) What is the highest that k can be in order for δ types to agree to pay the fee. Illustrate this in your graph. (e) The applicant s decision of whether or not to pay the fee is really a decision of whether to send a signal. How is this different from the type of signal we analyzes in exercise 22.3? In particular, why does θ s signaling behavior matter in exercise 22.3 but not here? (f) Suppose that instead of asking applicants to pay the screening fee, the insurance company paid to get the information from the screening firms for all applicants before determining the terms of the insurance contract they offered. Will the highest that k can be to change the self-selection separating equilibrium differ from what you concluded in part (d)? (g) Will the insurance allocation be efficient if the screening industry ends up selling information to insurance firms? 22.5 * We developed our first graphical model of adverse selection in the context of car insurance in section 22A.2 where we assumed that the marginal cost MC 1 of providing car insurance to unsafe drivers of type 1 is greater than the MC 2 of providing insurance to safe drivers of type 2. A: Continue with the assumption that MC 1 > MC 2. In this exercise, we will investigate how our conclusions in the text are affected by altering our assumption that D 1 = D 2 i.e. our assumption that the demand (and marginal willingness to pay) curves for our two driver types are the same. (a) Suppose demand curves continue to be linear with slope α but the vertical intercept for type 1 drivers is A 1 while the intercept for type 2 drivers is A 2. Suppose first that A 1 > A 2 > MC 1 > MC 2. Illustrate the equilibrium. Would p still be halfway between MC 1 and MC 2 as was the case in the text? (b) Identify the deadweight loss from asymmetric information in your graph. (c) What is the equilibrium if instead A 2 > A 1 > MC 1 > MC 2? How does p compare to what you depicted in (a)? (d) Identify again the deadweight loss from asymmetric information. (e) What would have to be true about the relationship of A 1, A 2, MC 1 and MC 2 for safe drivers not to buy insurance in equilibrium? (f) What would have to be true about the relationship of A 1, A 2, MC 1 and MC 2 for unsafe drivers not to buy insurance in equilibrium? B: In our model of Section B, we assumed that the same consumption/utiltiy relationship u(x) can be used for high cost θ and low cost δ types to represent their tastes over risky gambles with an expected utility function. (a) Did this assumption imply that tastes over risky gambles were the same for the two types? (b) Illustrate the actuarily fair insurance contracts in a graph with x 2 the consumption in the good state on the horizontal and x 1 the consumption in the bad state on the vertical. Then illustrate the choice set created by a set of insurance contracts that all satisfy the same terms i.e. insurance contracts of the form p = βb (where b is the benefit level and p is the premium). (c) Can you tell whether θ or δ types will demand more insurance along this choice set? (d) True or False: Our θ types would be analogous to the car insurance consumers of type 1 in part A of the exercise while our δ types would be analogous to consumers of type 2. (e) Suppose there are an equal number of δ and θ types and suppose that the insurance industry for some reason offered a single full set of insurance contracts p = βb and that this allowed them to earn zero profits. Would the p = βb line lie halfway between the actuarily fair contract lines for the two risk types? (f) Suppose instead that the insurance industry offered a single insurance policy that provides full insurance and that firms again make zero profits. Would the contract line that contains this policy lie halfway between the two actuarily fair contract lines in your graph? What is different from the previous part?

846 Chapter 22. Asymmetric Information in Competitive Markets 22.6 Everyday Application: Non-Random Selection is Everywhere: The problem in our initial discussion of A- grade insurance markets was that adverse selection led to non-randomness in the insurance pool: Although almost everyone was willing to pay the insurance premium that would have made zero expected profit for insurance companies with a randomly selected insurance pool, no one was willing to pay as higher cost students adversely selected into the pool. This kind of non-random selection is, however, not confined to insurance markets but lies at the heart of much that we see around us. 12 (Both part A and part B of this exercise can be done without having done section B in the chapter.) A: Consider the following examples and describe the non-random selection that can cause observers to reach the wrong conclusion just as insurance companies would charge the wrong premiums if they did not take into account the effect of non-random selection. (a) Suppose I want to know the average weight of fish in a lake. So you take out a boat and fish with a net that has 1-inch holes. You fish all day, weight the fish, take the average and report back to me. (b) A TV report tells us the following: A recent study revealed that people who eat broccoli twice a week live an average of 6 years longer than people who do not. The reporter concludes that eating broccoli increases live expectancy. (c) A cigarette company commissions a study on the impact of smoking on fitness. To compare the average fitness of smokers to that of non-smokers, they recruit smokers and non-smokers at a fitness center. In particular, they recruit smokers from the aerobics program and they recruit non-smokers from a weight-loss class. They find the surprising result that that smokers are more fit than non-smokers? (d) Four out of five dentists recommend a particular toothbrush from a sample of dentists that are provided free dental products by the company that makes the toothbrushes. (e) When surveyed after 1 year of buying and using a facial cream, 95% of women attest to its effectiveness at making their skin look younger. (f) Children in private schools perform better than children in public schools. Thus, concludes an observer, private schools are better than public schools. (Careful: The selection bias may go in either direction!) (g) A study compares the test scores of children from high income and low income households and demonstrates that children from high income households score significantly higher than children from low income households. An observer concludes that we can narrow this test score gap by re-distributing income from high income families to low income families. B: It is often said that the gold standard of social science research is to have a randomized experiment where some subjects are assigned to the treatment group while others are randomly assigned to the control group. Here is an example: A school voucher program is limited to 1,000 voucher participants but 2,000 families apply, with each having their child tested on a standard exam. The administrators of the program then randomly select 1,000 families that get the voucher or the treatment and treat the remaining 1,000 families as the control group. One year later, they test the children again and compare the change in average test scores of children from the two groups. They find that those who were randomly assigned to the treatment group have, on average, significantly higher test scores. (a) Suppose that all 1,000 children in each group participated in the testing that led to the computation of average score changes for each group. Would you be comfortable concluding that it was likely that access to the voucher program caused an increase in student performance? (b) Suppose that only 800 students in each group participated in the testing at the end of the first year of the program, but they were randomly selected within each group. Would your answer to (a) change if only the average change in test scores for these students were used? (c) Suppose that families had a choice in terms of whether to participate in the testing at the end of the year. But families in the treatment group were told that the only way they can continue using the voucher for another year is to have their child tested; and families in the control group were told that some new slots in the voucher program would open up (because some of the voucher families have dropped out of the program) but the only way the families in the control group get another chance to be picked to receive a voucher is to have their child tested. In the treatment group, who do you think is more likely to self-select to have their child tested: Families that had a good experience with their voucher, or families that had a bad experience? 12 Research studies often refer to the erroneous conclusions one might draw as a result of such non-random selection as selection bias. If you take an econometrics course, you will learn much about how to statistically adjust for such biases. Many of the these techniques emanate from work by Nobel Laureate James Heckman (1944-).

22B. Insurance Contracts with Two Risk Types 847 (d) In the control group, who do you think is more likely to self-select to have their child tested: Families that had a good experience the previous year outside the voucher program, or families that had a bad experience? (e) Suppose again that 800 students from each group participated in the testing but now you know about the incentives that families have for showing up to have their child tested. How does this affect your answer to (b)? (f) From a researcher s perspective, how can the non-random selection into testing be described as adverse selection that clouds what you can conclude from looking at average test score differences between the two groups? How is this example similar to part A(c)? 22.7 Business Application: Competitive Provision of Health Insurance: Consider the challenge of providing health insurance to a population with different probabilities of getting sick. A: Suppose that, as in our car insurance example, there are two consumer types consumers of type 1 that are likely to get sick, and consumers of type 2 that are relatively healthy. Let x represent the level of health insurance, with x = 0 implying no insurance and higher levels of x indicating increasingly generous health insurance benefits. Assume that each consumer type has linear demand curves (equal to marginal willingness to pay), with d 1 representing the demand curve for a single consumer of type 1 and d 2 representing the demand curve for a single consumer of type 2. Suppose further that the marginal cost of providing additional health coverage to an individual is constant, with MC 1 > MC 2. (a) For simplicity, suppose throughout that d 1 and d 2 have the same slope. Suppose further, unless otherwise stated, that d 1 has higher intercept than d 2. Do you think it is reasonable to assume that type 1 has higher demand for insurance? (b) Begin by drawing a graph with d 1, d 2, MC 1 and MC 2 assuming that the vertical intercepts of both demand curves lie above MC 1. Indicate the efficient level of insurance x 1 and x 2 for the two types. (c) Suppose the industry offers any level of x at price p = MC 1. Illustrate on your graph the consumer surplus that type 1 individuals will get if this were the only way to buy insurance and they buy there optimal policy A. How much consumer surplus will type 2 individuals get? (d) Next, suppose you want to offer an additional insurance contract B that earns zero profit if bought only by type 2 consumers, that is preferred by type 2 individuals to A and that makes type 1 consumers just as well off as they are under the options from part (c). Identify B in your graph. (e) Suppose for a moment that it is an equilibrium for the industry to offer only contracts A and B (and suppose that the actual B is just slightly to the left of the B you identified in part (d)). True or False: While insurance companies do not know what type consumers are when they walk into the insurance office to buy a policy, the companies will know what type of consumer they made a contract with after the consumer leaves. (f) In order for this to be an equilibrium, it must be the case that it is not possible for an insurance company to offer a pooling price that makes at least zero profit while attracting both type 1 and 2 consumers. (Such a policy has a single price p that lies between MC 1 and MC 2.) Note that the demand curves graphed thus far were for only one individual of each type. What additional information would you have to know in order to know whether the zero-profit price p would attract both types? (g) True or False: The greater the fraction of consumers that are of type 1, the less likely it is that such a pooling price exists. (h) Suppose that no such pooling price exists. Assuming that health insurance firms cannot observe the health conditions of their customers, would it be a competitive equilibrium for the industry to offer contracts A and B? Would this be a pooling or a separating equilibrium? (i) Would you still be able to identify a contract B that satisfies the conditions in (d) if d 1 = d 2? What if d 1 < d 2? B: Part A of this exercise attempts to formalize a key intuition we covered in section B of the text with a different type of model for insurance. (a) Rather than starting our analysis by distinguishing between marginal costs of different types, our model from section B starts by specifying the probabilities θ and δ that type 1 and type 2 individuals will find themselves in the bad state that they are insuring against. Mapping this to our model from part A of this exercise, with type 1 and 2 defined as in part A, what is the relationship between δ and θ?

848 Chapter 22. Asymmetric Information in Competitive Markets (b) To fit the story with the model from section B, we can assume that what matters about bad health shocks is only the impact they have on consumption and that tastes are state independent. (We will relax this assumption in exercise 22.8). Suppose we can, for both types, write tastes over risky gambles as von-neumann Morgenstern expected utility functions that employ the same function u(y) as utility of consumption (with consumption denoted y). Write out the expected utility functions for the two types. (c) Does the fact that we can use the same u(y) to express expected utilities for both types imply that the two types have the same tastes over risky gambles and thus the same demand for insurance? (d) If insurance companies could tell who is what type, they would (in a competitive equilibrium) simply charge a price equal to each type s marginal cost. How is this captured in the model developed in section B of the text? (e) In the separating equilibrium we identified in part A, we had insurance companies providing the contract A that is efficient for type 1 individuals but providing an inefficient contract B to type 2. Draw the model from section B of the text and illustrate the same A and B contracts. How are they exactly analogous to what we derived in part A? (f) In part A we also investigated the possibility of a potential pooling price or pooling contract breaking the separating equilibrium in which A and B are offered. Illustrate in the different model here how the same factors are at play in determining whether such a pooling price or contract exists. (g) Evaluate again the True/False statement in part A(g). 22.8 * Policy Application: Expanding Health Insurance Coverage: Some countries are struggling with the problem of expanding the fraction of the population that has good health insurance. A: Continue with the set-up first introduced in exercise 22.7 including the definition of x as the amount of insurance coverage bought by an inidividual. Assume throughout that demand for health insurance by the relatively healthy (type 2) is lower than demand for health insurance by the relatively sick (type 1) i.e. d 1 > d 2. (a) Illustrate d 1, d 2, MC 1 and MC 2 and identify the contracts A and B from exercise 22.7. (b) Suppose that the fraction of relatively sick (type 2) consumers is sufficiently high such that no pooling contract can keep this from being an equilibrium. On the MC 1 line, indicate all the contracts that can be offered in this equilibrium (even though only A is chosen). Similarly, indicate on the MC 2 line all the contracts that can be offered in this equilibrium (even though only B is chosen). (c) True or False: Insurance companies in this equilibrium restrict the amount of insurance that can be bought at the price p = MC 2 in order to keep type 1 consumers from buying at that price. (d) Why is the resulting separating equilibrium inefficient? How big is the deadweight loss? (e) Suppose that the government regulates this health insurance market in the following way: It identifies the zero-profit pooling price p and requires insurance companies to charge p for each unit of x but does not mandate how much x every consumer consumes. Illustrate in your graph how much insurance type 1 and type 2 consumers will consume under this policy? Does overall insurance coverage increase or decrease? (f) How much does consumer surplus for each type change as a result of this regulation? Does overall surplus increase? (g) True or False: This policy is efficiency enhancing but does not lead to efficiency. (h) It may be difficult for the government to implement the above price regulation p because it does not have enough information to do so. Some have suggested that the government instead set the insurance level to some x and then let insurance companies compete on pricing this insurance level. Could you suggest, in a new graph, a level of x that will result in greater efficiency than regulating price? (You need to do this on a new graph for the following reason: If the government sets x between the amounts consumed by type 1 and 2 under the zero-profit price regulation p, the resulting competitive price p should be lower than p )? B: Now consider again whether we can find analogous conclusions in the model from Section B as modified in exercise 22.7. (a) Interpreting the model as in exercise 22.7, illustrate the separating equilibrium in a graph with the insurance benefit b on the horizontal axis and the insurance premium p on the vertical. Include in your graph a zero-profit pooling contract line that makes the separation of types an equilibrium outcome. (b) How would you interpret the price regulation proposed in A(e) in the context of this model? (c) Illustrate in your graph how insurance coverage will increase if the government implements this policy.

22B. Insurance Contracts with Two Risk Types 849 (d) Now consider the same problem in a graph with y 2 the consumption level when healthy on the horizontal axis and y 1 the consumption level when sick on the vertical. Illustrate the endowment point E = (y 1, y 2 ) that both types face in the absence of insurance. (e) Illustrate the actuarily fair insurance contracts for type 1 and 2 consumers. Then indicate where the separating equilibrium contracts A and B lie in the graph assuming state-independent tastes. (f) Introduce into your graph a zero-profit pooling contract line such that the separating equilibrium is indeed an equilibrium. Then illustrate how the proposed government regulation affects the choices of both types of consumers. (g) Suppose that, instead of regulating price, the government set an insurance benefit level b (as in part A(h)) and then allowed the competitive price to emerge. Where in your graph would the resulting contract lie if it fully insures both types? (h) Suppose next that tastes were state-dependent with u 1 (y) and u 2 (y) the functions (for evaluating consumption when sick and when healthy) that we need to use in order to arrive at our expected utility function. If u 1 and u 2 are the same for both consumer types, does our main conclusion that the price regulation will cause an increase in insurance coverage change? 22.9 Policy Application: Moral Hazard versus Adverse Selection in Health Care Reform: We mentioned moral hazard only briefly and primarily in the context of how this might aggravate the adverse selection problem. In this exercise, we explore moral hazard a bit more in the context of health insurance. (Both part A and part B of this exercise can be done without having done section B in the chapter.) A: Suppose throughout that individuals do not engage in riskier life-styles as a result of obtaining health insurance. (a) How does this assumption eliminate one form of moral hazard that we might worry about? (b) Suppose that a unit of health care x is such that it can be provided at constant marginal cost that is the same for all patients. Illustrate a patient s demand curve for x as well as the MC curve for providing x. (c) Suppose demand for health care services is equal to marginal willingness to pay. If the patient pays out-ofpocket for health care, how much would she consume assuming that health care services are competitively priced (with health care providers facing negligible recurring fixed costs)? (d) Suppose next that the patient has insurance coverage that pays for all health related expenses. How much x does she consume now? (e) Moral hazard refers to the change in behavior that arises once a person enters a contract. Have you just uncovered a source of moral hazard in the health insurance market? Explain how this results in inefficiency. (f) Now replicate your picture two times: Once for a patient where the moral hazard problem is small, and once for a patient where it is large. If insurance companies cannot tell the difference between these two individuals, how does this asymmetric information potentially give rise to adverse selection? B: Consider two alternative proposals for health care reform: Under proposal A, the government mandates that everyone buy health insurance, restricts insurance companies to provide a single type of policy with generous benefits and then lets the companies compete for customers to sell that policy. Under proposal B, the government sets up health care savings accounts for everyone and allows insurance companies to offer only policies with high deductibles. Under this latter policy, consumers would then pay for most health related expenditures using funds in their health care savings accounts and could convert any balance to retirement accounts when they reach the age of 65 (and thus become eligible for government health care for the elderly called Medicare in the U.S.) Insurance under policy B is therefore aimed only at catastrophic events that cost more than the deductible of the policy. (a) Suppose you were concerned about excessive health care costs. How would the two different proposals aim at addressing this? (b) If you thought the primary problem arose from the moral hazard analyzed in part A of this exercise, which policy would you favor? (c) Suppose instead that you thought the primary problem arose from the rising cost of health insurance linked to increasingly severe adverse selection (unrelated to the moral hazard problem analyzed in part A) and a growing pool of uninsured people. Which policy might you more likely favor? 22.10 Policy Application: Statistical Profiling in Random Car Searches: Local law enforcement officials sometimes engage in random searches of cars to look for illegal substances. When one looks at the data of who is actually searched, however, the pattern of searches oftentimes does not look random.

850 Chapter 22. Asymmetric Information in Competitive Markets A: In what follows, assume that random searches have a deterrent effect i.e. the more likely someone believes he is going to be searched, the less likely he is to engage in transporting illegal substances. (a) Suppose first that it has been documented that, all else being equal, illegal substances are more likely to be transported in pick-up trucks than in passenger cars. Put differently, if pick-up truck owners are searched with the same probability as passenger car owners, law enforcement officials will be more likely to find illegal substances when they randomly search a pick-up truck than when they randomly search a passenger vehicle. If the objective by police is to find the most illegal substances given that they have limited resources (and thus cannot search everyone), is it optimal for them to search randomly? (b) Suppose the police decides to allocate its limited resources by searching pick-up trucks with probability δ and passenger cars with probability γ (where δ > γ). After a few months of this policy, the police discovers that they find on average 2.9 grams of illegal substances per pick-up-truck search and 1.5 grams of illegal substances per passenger vehicle. How would you advise the police given their limited resources to change their search policy in order to increase the amount of drugs found? (c) Given your answer to (b), what has to be true about the probability of finding illegal substances in pick-up trucks and passenger cars if the search probabilities for the two types of vehicles are set optimally (relative to the police s objective to find the most illegal substances)? (d) If you simply observe that δ > γ, can you conclude that the police are inherently biased against pick-up trucks owners? Why or why not? (e) What would have to be true about the average yield of illegal substances per search for the different types of vehicles for you to argue that the police was inherently biased against pick-up trucks? (f) Could it be the case that δ > γ and the police show behavior inherently biased against passenger cars? (g) We have used the emotionally neutral categories of pick-up trucks and passenger vehicles. Now consider he more empirically relevant case of minority neighborhoods and non-minority neighborhoods with law enforcement often searching cars in the former with significantly higher probability than in the latter. Can you argue that such behavior by law enforcement officials is not inherently racist in the sense of being motivated by animosity against one group, but that instead it could be explained simply as a matter of statistical discrimination that maximizes the effectiveness of car searches in deterring the trafficking in illegal substances? What evidence might you look for to make your case? B: Suppose that the police has sufficient resources to conduct 100 car searches per day and that half of all vehicles are pick-up trucks and half are passenger cars. The probability of finding an illegal substance in a pickup truck is p t(n t) = 9/(90+n t) where n t is the number of pick-up truck searches conducted. The probability of finding an illegal substance in a passenger car is p c(n c) = 1/(10 + n c) (where n c is the number of car searches conducted). (a) Suppose that the objective of the police is to maximize the number of interdictions of illegal substances. Write down the optimization problem with n t and n c as choice variables and the constraint that n t+n c = 100. (b) According to the police s objective function, how many trucks should be searched per day? How many passenger vehicles? (c) If law enforcement conducts searches as calculated in (b), what is the probability of interdicting illegal substances in pick-up trucks? What is the probability of interdicting such substances in passenger cars? (d) If law enforcement officials search trucks and cars at the rates you derived in (b), how many illegal substance interdictions would on average occur every day? (e) How many of each type of car would on average be searched each day if the police instead searched vehicles randomly? (f) If the police conducted random searches, what would be the probability of finding illegal substances in each of the two vehicle types? How does this compare to your answer to (c)? (g) How many illegal substance interdictions per day would on average occur if the police conducted random searches instead of what you derived in (b)? (h) Why is your answer to (c) different than your answer to (g)? (i) Insurance companies charge higher insurance rates to young drivers than to middle aged drivers. How is their behavior similar to the behavior by law enforcement that searches pick-up trucks more than passenger cars in (b)?