Lecture Notes. 1 What is Asymmetric Information and why it matters for Economics?

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1 Lecture Notes The most important development in economics in the last forty years has been the study of incentives to achieve potential mutual gain when the parties have different degrees of knowledge. (Ken Arrow, Nobel Laureate in Economic Sciences) 1 What is Asymmetric Information and why it matters for Economics? Information is asymmetric in situations where one agent knows something that another agent does not. Asymmetric information affects agents behavior in many economic contexts, for example, when a firm knows more about the potential rewards of a business project than a bank that finances this project; a credit card owner knows more about her ability and willingness to repay her debt than the bank that issues this card; a seller of a used car (or any other durable good) is better informed about its quality than a buyer of this car; a customer knows her taste for a good or a service better than the firm that supplies and prices it; an employee knows his ability and his work ethics better than a firm that hires him; a person knows more about her health than a company that provides her life insurance or sells her an annuity (the annuity is a financial contract opposite to a life insurance: the buyer makes a lump-sum payment and gets an income flow as long as she lives); a person knows more about his driving habits than the company that provides his auto insurance. Asymmetric information has two general effects on economic behavior: adverse selection and moral hazard. Adverse selection arises when better-informed agents use their knowledge before signing a contract with a less-informed party. For example, a seller of used car may accept a low price but only if he knows that his car has a bad quality (that is, a lemon ); a person may accept a credit card with a very high interest rate but only if she knows that she is likely to default on the card debt; a person may buy extensive health insurance but only if she knows that her health is likely to deteriorate soon;

2 a business traveller may travel in a cheaper coach class rather than in the more expensive and luxurious first-class. On the other hand, moral hazard arises when agents take some hidden actions after signing a contract with a less-informed party. For example, a person may change his driving habits after buying a full auto insurance; a person may use a firm s computers, printers etc. for her own private gain. Note that in order to promote his or her self-interest, the better-informed party may rely on lying and cheating. Economists usually assume that people are unscrupulous about such behavior, and would agree with Machiavelli 1 who writes A wise ruler, therefore, cannot and should not keep his word when such an observance of faith would be to his disadvantage and when the reasons which made him promise are removed. And if men were all good, this rule would not be good; but since men are a contemptible lot and will not keep their promises to you, you likewise need not keep yours to them. While lying and cheating may seem immoral to some, it is definitely a fact of life. 2 Adverse Selection in the Used-Car Market First we focus on the market for used cars where sellers are better informed about the quality of their cars than potential buyers are. Assume that (i) the quality of any used car that may be traded in the market can be measured by a number 0 q 1, (ii) a seller is willing to trade a car of quality q at any price p $10000 q, (iii) a buyer who knows the quality q of a used car would accept any price p $15000 q for this car, (iv) a buyer who knows only the average quality q of a used car, would accept any price p $15000 q for this car. (In economists jargon, the buyer is risk neutral, that is, she does not mind gambling and cares only about the average payoff.) Consider first the perfect information case, where the quality of each car is known both to all agents. Then every car will be traded at a price between $10000 q and $15000 q. We will usually assume that the sellers compete more aggressively than buyers, and hence, they drive the price down to $10000 q, which is the bottom end of the possible price range. Next imagine that buyers are unable to determine the quality q of any particular used car, but they do know the probability distribution of the quality q among all 1 Machiavelli is a famous medieval political advisor. The quotation is from his book The Prince, which makes a good reading for any student of political science and economics.

3 potential sellers. For example, assume that 1 of all used cars have high quality q = 1, 3 1 have medium quality q = 0.5, and 1 have low quality q = In this market, there must be a common price p for all used cars because buyers cannot distinguish between cars of different quality, and hence, such cars cannot be sold at a different price. In order to determine what the price p could be, consider several cases. Case 1. Suppose that p This price is high enough to induce all sellers to trade their cars (remember that owners of the high-quality cars with q = 1 will trade only if p ) Then the average quality of a car which is for sale at this price is q = = 0.5. So the maximal price that the buyers will want to pay is = It turns out that there is no market equilibrium for any price above Case p < This price will suffice only for sellers who own cars with inferior qualities q = 0 and q = 0.5. Owners of the top quality cars with q = 1 will want to keep them, rather than to sell. Therefore the average quality of a car which is for sale at this price is q = = Note that that we rescale the probabilities for q = 0.5 and q = 0 from 1 to 1. We do 3 2 this because the share of medium quality (or bad quality) increases when top quality is withdrawn from the market. This is exactly what the name adverse selection suggests: the average quality of a used car deteriorates when the price is reduced. It follows that the maximal price that the buyers will want to pay is = Again no equilibrium is possible at prices 5000 p < Case 3. 0 p < This price will suffice only for sellers of lemons with q = 0. Accordingly the average quality will be q = 0, and the buyers will be willing to pay only p = 0. So this is the only possible equilibrium in this market, where only the lowest quality cars are sold at a very low price. The adverse selection in this example reduces trade severely, and also reduces welfare because buyers and sellers can no longer benefit from trading cars of high and medium quality. 2.1 The role of the quality distribution More trade may survive in equilibrium when the quality q has a different probability distribution. For example, assume that 10% of used cars have high quality q = 1, 80% have medium quality q = 0.5, and 10% have low quality q = 0. Again, there is no equilibrium at any price p because q = 1 and the buyers are willing to 2 pay at most However, there is an equilibrium for 5000 p < In this case, q = = Note that when the top quality cars are withdrawn from the market, the share of the medium quality cars increases from 8 to 8, and the share of lemons increases 10 9 from 1 to 1. Accordingly, buyers are willing to pay up to = 6666, and

4 an equilibrium is possible. In this equilibrium, all cars that have medium and low quality are traded at a price between 5000 and (This price is p = 5000 if sellers compete aggressively.) Note that there is another equilibrium where the market price p = 0 and only lemons with q = 0 are traded. The model does not say exactly which of the two equilibria will prevail. Of course, the equilibrium with p = 5000 is better in terms of welfare that the equilibrium with p = 0 because the society benefits from the trade of the medium quality cars. Yet even if p = 5000, the outcome is still not completely efficient because the high-quality cars are not traded. Moreover, this outcome presents another problem because some buyers are unhappy when they buy a lemon at the price If these buyers can return and resell their newly acquired lemons, then they will certainly do so at any positive price p > 0. Such reselling is called arbitrage. Because of arbitrage, one can expect the share of the low quality cars that are for sale to increase until the trade of the medium quality cars becomes impossible as well. 2.2 Mechanisms that reduce adverse selection Fortunately, there is a number of mechanism that may reduce adverse selection, and increase the volume of trade and the total welfare. These mechanisms include (i) regulation; for instance, the government regulates the markets for food and drugs, so you can be sure that even at a very low price you will not buy a lemon that will poison you, (ii) reputation; well-established businesses, such as big car dealerships, cannot afford to sell lemons because it hurts their reputation and future profits; (iii) assurance; the less-informed party can pay for an expert opinion, for example, it is common to hire a mechanic to inspect a used car, and to check the history of the car; (iv) warranty. For example, the seller of a used car who claims it to have a top quality may commit to pay a compensation to the buyer if the car needs a repair within a certain time period. In order for warranty to work, it must incur different costs for owners of cars of different qualities. For example, suppose that the cost of a warranty is zero if q = 1, but this cost is 5500 if q = 0 or q = 0.5. Then it is possible to have an equilibrium where top quality cars with q = 1 are sold for with a warranty, and all other cars are sold for 5000 without a warranty. In this case, it does not make sense for sellers with q = 0 or q = 0.5 to sell their cars for with a warranty because they would end up with < Note that in some of the above mechanisms, such as reputation or warranty, it is the better-informed party (sellers) who attempt to convey their knowledge to the lessinformed agents (buyers). In general, this kind of activities is called signalling. In other mechanisms, such as assurance, it is the less-informed agents who try to collect relevant information. This kind of activity is called screening.

5 2.3 Exam Problem Exam Problem: Suppose that 20% of used cars have the high quality q = 1, 60% have the medium quality q = 0.5, and 20% have the low quality q = 0. (a) In the perfect information case, every car will be traded at a price close to $10000 q because seller compete agressively. So top quality cars will be sold at $10000, medium quality at $5000, and low quality at $0. (b) If the buyers cannot observe the quality of each car, then there are two possible equilibria. One equilibrium is with p = 5000 when only the low and medium quality cars are traded. The average quality q = 60% 20% + 60% % 20% + 60% 0 = 3 8 so the buyers are willing to pay up to $ = The price is 5000 because 8 sellers compete aggressively. The other equilibrium is with p = 0. In this case only lemons are traded. (c) The perfect information equilibrium is efficient. In the case of asymmetric information, the equilibrium with p = 5000 is the best in terms of efficiency but not fully efficient because top quality cars are not traded. The equilibrium with q = 0 is the worst. 3 Non-Linear Pricing Firms often sell their good or service in two or more different bundles. Each bundle specifies (i) the amount and/or quality of the good, and (ii) the fee that the consumer has to pay. For example, T-Mobile charges $40 for a 300-minute plan, and only $50 for a 600-minute plan. An airline company will sell coach and first-class fares. Microsoft offers Windows Home and Windows Professional editions. One possible explanation for this phenomenon is based on asymmetric information: firms knows less about the willingness to pay of any customer than the customer herself. For simplicity, assume that there is just one firm in the market (a monopolist), which can produce any quantity q of the good at the cost c q. In other words, the marginal cost is constant. Then Π(q) = t c q is the profit that the monopolist extracts from a consumer who buys q units of the good and makes a payment t. Assume that in this case, the consumer derives utility U(q) = k u(q) t. (1) Here the function u is taken to be concave so that the marginal utility k u (q) is decreasing. Let also u(0) = 0. It means that if the consumer does not trade with the monopolist, she derives the reservation utility 0. We adopt the utility form (1) mainly because of its simplicity rather than because it has some deep meaning. 2 2 One may possibly interpret k as the number of times that the consumer plans to use the good, and u(q) as the utility that she derives from a single usage (think about music downloads).

6 What is the maximal profit that the monopolist can receive from a consumer with a known coefficient k in the utility function? This maximization problem can be written as max[t c q] (2) q,t subject to the participation constraint k u(q) t 0. The participation constraint reflects the fact that the monopolist cannot enforce the customer to trade at a loss, and hence, must offer a contract (q,t) that will deliver at least the reservation utility 0. Profit maximization dictates that k u(q) t = 0. Otherwise, if k u(q) t > 0, then the profit can be raised by increasing the payment t. Thus, the maximization problem (2) reduces to max[k u(q) c q]. q The solution q to this problem is given by the first order condition The profit maximizing fee t satisfies and the maximal profit is k u (q ) c = 0. t = k u(q ), Π = t c q. Note that both q and t positively depend on the coefficient k and negatively on the cost c. (Recall that u (q) is decreasing). For example, if u(q) = q, then u (q) = 1 2 q, 1 k 2 q c = 0, that is, q = k2, t k = k 2 = k2, and 4c 2 4c 2 2c Π = k2 2c c k2 4c 2 = k2 2c k2 4c = k2 4c. The solution to the profit maximization problem can be analyzed on a graph, where q is put on the horizontal axis, and t on the vertical axis. The profit maximizing contract is then a point where the participation indifference curve is tangent to an isoprofit line. The first order condition k u (q ) = c then reflects the equality of the slopes of the indifference curve and the isoprofit line.

7 6 5 t Π = 2 Π = 0.5 Π = U = 0 U = 1 U = q Figure 1: Indifference curves and Isoprofit lines for k = 1, c = 0.5, and u(q) = q Note that the profit maximizing contract under achieves overall efficiency as it maximizes the total surplus of the monopolist and the consumer: Π(q) + U(q) = (t c q) + (k u(q) t). Yet, all of the gains from trade go to the monopolist: the consumer gets the reservation utility 0. The adverse selection problem arises when consumers have different k s in their utility functions, and the monopolist cannot tell high-demand consumers (with high k) from low-demand consumers (with low k). For simplicity, suppose that there are two consumers: one with utility k 1 u(q) t and the other with utility k 2 u(q) t. Assume that k 2 > k 1 so that the second consumer has a higher evaluation of the good than the first. Imagine that the monopolist decides to offer both contracts (q 1,t 1) and (q 2,t 2) to these consumers. Then both of them will choose (q 1,t 1)!! This is an adverse selection from the monopolist s point of view: the high-demand customers from the second group select a contract which generates low profits. This outcome is not efficient either because the high-demand customers have a marginal utility that is greater than the marginal cost. There are several possible approaches that the monopolist may use to raise his profits. Ignore the less profitable consumers. Offer (q 2,t 2) to all consumers. Then only the second group of customers will accept the contract, and the others will walk away.

8 Restore efficiency. Keep the efficient quantity q 2 but lower the fee for the highdemand consumers so that they have no incentive to switch to the contract (q 1,t 1). The highest fee t 2 that will restore efficiency is t e 2 = k 2 (u(q 2) u(q 1)) k 1. Maximize profits. subject to the participation constraints (t 2 c q 2 ) + (t 1 c q 1 ) k 1 u(q 1 ) t 1 0; k 2 u(q 2 ) t 2 0, and the incentive compatibility constraints k 1 u(q 1 ) t 1 k 1 u(q 2 ) t 2 ; k 2 u(q 2 ) t 2 k 2 u(q 1 ) t 1. The participation constraints guarantee that each customer prefers to accept her bundle rather than to walk away. The incentive compatibility constraints guarantee that each customer prefers her bundle rather than the other available one. The solution (q1,t 1 ), (q1,t 2 ) to this problem is given by k 2 u (q2 ) = c (2k 1 k 2 )u (q1 ) = c t 1 = k 1 u(q1 ) t 2 = k 2 u(q2 ) (k 2 k 1 )u(q1 ). See that the high-demand customer will obtain the efficient quantity and a positive information rent: Information Rent = (k 2 k 1 )u(q 1 ) = k 2 u(q 1 ) t 1. This is exactly how much the high-demand customer can obtain by switching to the contract (q1,t 1 ) designed for the low-demand group. The low-demand customer will obtain less than efficient quantity and will have a zero information rent. So there is a tradeoff between efficiency and profit maximization. Again we can analyze this solution on a graph. Example: u(e) = e, k 1 = 4, k 2 = 6, c = 1. Then q1 = 4, t 1 = 8, q2 = 9, t 2 = 18. In the full information case, the profit is Π = (t 1 cq 1) + (t 2 cq 2) = = 13. When both groups switch to the first contract, the profit is Π = (t 1 cq 1) + (t 1 cq 1) = = 8. When the first group is ignored, the profit is Π = 0 + (t 2 cq 2) = = 9.

9 20 16 t (q 2,t 2) (q 2,t 2 ) U 2 = 0 U 2 = 2 U 1 = (q 1,t 1) (q 1,t 1 ) q Figure 2: Finding the profit maximizing contract. When efficiency is restored, then t e 2 = 14 and the profit is Π = (t 1 cq 1) + (t e 2 cq 2) = = 9. Finally, the contract q 1 = 1, t 1 = 4, q 2 = 9, t 2 = 16 delivers the profit 3.1 Exam Problem Π = (t 1 cq 1 ) + (t 2 cq 2 ) = = 10. A monopoly can produce a good at a constant marginal cost 1. It faces two equally large groups of consumers: the low demand group with utility U 1 (q) = 10 q t, and the high demand group with utility U 2 (q) = 16 q t. Both types have reservation utility 0. Then q1 = 25, t 1 = 50, q2 = 64, t 2 = 128. q1 = 4, t 1 = 20, q2 = 64, t 2 = 116 information rent is 12 and the loss in profit is = Signalling in the Labor Market In many economic contexts, the assumptions of the non-linear pricing model are violated because of competition. For example, if there are many firms which can

10 produce the good with marginal cost c, then competition will make them sell it at a price close to c per unit instead of offering non-linear contracts. However, asymmetric information may still have interesting implications. Consider a labor market where half of all job applicants are able workers with the high productivity and half are less able workers with the low productivity The workers effort is fixed (say 40 hours a week) and plays no role in the signalling model. The employers are perfectly competitive (rather than being a monopoly). In the perfect information case when the productivity of each worker is known to the employers, the able workers will receive the wage and the unable the wage However, if the productivity is not observable, then for all workers, the wage will equal the average productivity = Prior to taking the job, each worker can choose to obtain y 0 years of education. The cost of such education is lower for the able than for the unable: the difference in cost can be explained by psychological reasons as well as by the fact that the more able students usually receive more financial support. Assume that the cost of each year of education is for the able, and for the unable. For simplicity, assume that the education does not affect productivity at all and serves only the signalling purpose. (Do not worry: the model can be extended to the case when education increases productivity). Next we describe signalling equilibria. Each equilibrium of this sort has three ingredients: (1) employer s beliefs about the ability of a person with education y; (2) the wage that the employer pays a recruit with education y; (3) the education that the able people choose, and the education that the unable people choose. In equilibrium, the beliefs determine the wages, the wages determine the education choices, and the education choices justify the beliefs. First, we focus on the separating equilibria. In each equilibrium of this sort, the employers believe that the education above a certain critical level y provides an accurate signal for the applicant s high ability. More precisely, the employers believe that any person who has education y y has a high ability, and anybody who has education y < y has a low ability. Because of competition, they offer a wage of to any applicant with y y and to any applicant with y < y. Accordingly, each worker has a choice: either get the education y and then receive the wage of 80000, or get no education and receive the wage Note that if y > 4 then neither the able nor the unable will want to get the education y because it is not worth it: the cost of such education is above even for the able, and hence is greater than the wage differential. Thus the employer s beliefs are not accurate (why?) On the other hand, if y < 2, then both able and unable will choose y years of education because the cost is below even for the unable and hence, is less than the wage differential. Again, the employer s beliefs are not accurate (why?) Finally, if 4 y 2, then the able will choose to get the education y rather than 0 because y ;

11 the unable will choose to get no education rather than y because y Therefore, the employer s beliefs are accurate in the equilibrium: anybody who has education y y is indeed an able person; anybody who has education less than y is indeed unable. The separating equilibria can be ranked in terms of welfare. The lower y, the higher is the total welfare because the education plays only a signalling purpose, and hence, the education costs decrease the total welfare. The best separating equilibrium in our example is for y = 2. Next, there is another kind of equilibrium where the education plays no signalling role whatsoever. For example, if the employers believe that regardless of education, a person is equally likely to be unable or able, then the employers will offer a wage = to every applicant. But then neither able nor unable will 2 2 want to receive any education because it cannot add to their salary. So the employers beliefs are again accurate. This kind of equilibrium is called pooling because it does not distinguish between different groups of workers. This equilibrium is the best in terms of welfare because no resources are spent on education. 4.1 The Intuitive Refinement of Equilibria The beliefs that the employers have in the separating and pooling equilibria can be justified by the workers actions in the equilibrium. However, these beliefs may be less reasonable if out-of-equilibrium actions may take place. Look at any separating equilibrium with y > 2. What if the employer meets an worker with education y between 2 and y? (This never happens in the equilibrium, but the equilibrium is not a law of nature, and should prevail only in the long run.) Is it intuitive to believe the low ability in this case? It seems very implausible that an unable worker chooses y because she can never gain by doing so. The cost of such education is above for her, but the greatest possible wage differential in any kind of equilibrium is only On the other hand, it does not seem that crazy if some able worker chooses y. For instance, such behavior can be explained if this worker is not fully adjusted to the equilibrium and thinks that education somewhat less than y is still sufficient to get the high salary Thus it is not intuitive to believe that the workers with education y have low ability. Accordingly, the separating equilibrium with y = 2 specifies the most intuitive beliefs. Similarly, one can show that the beliefs in the pooling equilibrium is not intuitive. The intuitive refinement of beliefs suggests that the most reasonable equilibrium is the separating one with y = 2. The experimental evidence provides some support. 5 Moral Hazard in Insurance Moral hazard occurs if, after signing a contract, an agent takes a hidden action that the other party cannot observe. For example, after buying a zero-deductible liability insurance, the agent may start driving recklessly because she bears no direct cost of an accident. One can view insurance fraud, like arson, as a severe case of moral

12 500 u u is concave u is linear x u from example 1500 Figure 3: Risk aversion and Risk neutrality hazard. Labor contracts are also subject to moral hazard: workers whose effort is not directly observable to the employer often choose to provide less effort than those who are closely supervised. 5.1 Risk Aversion and Expected Utility Before modelling moral hazard in insurance contracts, we need to explain why consumers buy any insurance at all. The reason is risk aversion: a typical person would rather have $x for sure than to take a gamble that pays $(x + y) or $(x y) with equal probabilities 1. More generally, a risk averse person prefers $(πy +(1 π)z) for 2 sure rather a lottery that pays $y with probability π and $z with probability 1 π. We also assume that the agent has expected utility. Roughly, this means that the agent assigns a utility index u(x) to all relevant monetary prizes x and then evaluates any gamble that pays $y with probability π and $z with probability 1 π via utility πu(y) + (1 π)u(z). (Expected utility applies also to more complex gambles that have more than two possible outcomes. There is a strong rationale that motivates using expected utility, but there are some problems as well. These issues are out of the scope of our class. If you are interested, you should check a course about risk and uncertainty.) There is a connection between risk aversion and the utility index u: risk aversion holds if and only if u is concave. On the other hand, the insurance company is assumed to be risk neutral, that is, indifferent between getting sure $(πy + (1 π)z)

13 utility grows here the participation constraint L isoprofit line B 45% line A profit grows here Figure 4: The optimal insurance contract without moral hazard and a lottery that pays $y with probability π and $z with probability 1 π. The company is risk neutral (or at least almost risk neutral) because (i) the loss L is very small relative to the company s cash flow (but L may be large relative to the driver s wealth); (ii) the managers have little personal stake in any individual risk and maximize the average profits. Risk neutrality is equivalent to expected utility with a linear u. (See Figure 3.) 5.2 Moral Hazard in Auto Insurance Consider a driver who gets into an accident (say, over a six-month period) with probability π if he drives cautiously, and with probability π > π if he drives recklessly. He gets an extra utility amount g from reckless driving. The monetary loss in the case of an accident is L (the assumption that all accidents are equally costly is not essential, but makes things simpler). Thus, if the driver has no insurance, then his expected utility is U = πu( L) + (1 π)u(0) if he drives safely, and U = π u( L) + (1 π )u(0) + g if he drives recklessly. Assume that U > U, which means that g is not worth the greater risk of an accident implied by reckless driving. Then the uninsured agent will choose to drive safely. Example: Let L = $ Let u(x) = x if x $1000, and u(x) = 2x if x < $1000. This index means that the disutility from every dollar of a loss beyond

14 $1000 is twice as much as the disutility from every dollar of a loss less than $1000. Let π = 0.05, π = 0.01, and g = 100. Then U = 0.01( ) = 190 U = 0.05( ) = 850. Suppose now that the driver buys insurance. The insurance contract consists of a premium R that the driver pays upfront, and a deductible D that the driver has to pay out-of-pocket in the case of an accident (so the insurance company provides the compensation C = L D.) Thus, the agent s wealth is A = R D if there is an accident, and W = R if not. If the agent drives safely, then his expected utility is and the company s profit is πu(a) + (1 π)u(b) π(r + D L) + (1 π)r = πa (1 π)b πl. If the agent drives recklessly, then his expected utility is and the company s profit is π u(a) + (1 π )u(b) + g, π (R + D L) + (1 π )R = π A (1 π )B π L. Next, suppose that the insurance company is a monopolist, and can observe whether the agent drives safely or not. Then it offers a contract that requires safe driving and maximizes the profit subject to the participation constraint πa (1 π)b πl πu(a) + (1 π)u(b) U. The constraint is necessary so that the agent is willing to buy the insurance at all. To solve this maximization problem, we use Figure 4. The slope of the agent s indifference curve equals πu (A) MRS = (1 π)u (B) = π 1 π π along the 45% degree line. The slope of the company s isoprofit lines is. So 1 π the optimal contract must lie at the intersection of the 45% degree line and the indifference curve that passes through the point ( L, 0) of no insurance. Example (continued): u(x) = x if x $1000, and u(x) = 2x if x < $1000. The utility at the ( L, 0) of no insurance for the safe driver is 190. On the other hand, if he has a zero-deductible insurance and pays a premium R, then his utility is u( R). The company s optimal contract solves u(r) = 190. The solution is R = $190. The profit of the insurance company is $ = $90.

15 In reality, the company may be able to observe some information about the agent s driving, such as the number of speeding tickets. But this information is too little, and the company cannot require that the agent drives safely in the insurance contract. In this case, the zero-deductible insurance will not work because of moral hazard: the agent who buys this insurance will choose to drive recklessly and obtain utility u( R)+100 rather than u( R). But because of the higher probability of an accident, the insurance company will lose money. Example (continued): π = 0.05, π = 0.01, and R = $190. Then the company s profit when the agent drives recklessly is negative 0.05 ( ) = 310. Note that the lowest premium that the company may charge to insure a reckless driver is $500. But if the premium is that high, the agent will choose to stay uninsured and drive safely. Then his utility is 190 rather than = 400 that he gets if he buys the insurance and drives recklessly. So moral hazard puts another constraint on the insurance contract that the company may offer without inducing reckless driving. This condition requires that if the agent buys the insurance, he still prefers to drive safely rather than to drive recklessly. It simplifies to π u(a) + (1 π) u(b) π u(a) + (1 π ) u(b) + g. Thus, the contract lies above the curve (π π )(u(a) u(b)) g, or u(b) u(a) g π π. u(b) u(a) = g π π. The red hatched area in Figure 5 illustrates the contracts that satisfy both the participation and the moral hazard constraints. This area has a kink at the intersection of the indifference curve passing through the no insurance point and the moral hazard curve. At the kink point the slope of the indifference curve is greater than the slope π of the company s isoprofit line, which is. So the only way how the company 1 π could increase its profit at this point without breaking the participation constraint is to decrease B and increase A. But this is impossible because of the moral hazard constraint. Thus, we have found that the kink point in Figure 5 is the optimal insurance contract for the company. Under moral hazard, the optimal contract can be found as a solution to two equations: πu(a) + (1 π)u(b) = U, u(b) u(a) = g π π.

16 B 45% line L the participation constraint A the moral hazard constraint Figure 5: The optimal insurance contract with moral hazard This solution is given by (1 π)g u(a) = U π π, u(b) = U + πg π π. Note that the moral hazard decreases the profit of the insurance company because it makes the original best contract unfeasible. Example (continued): u(x) = x if x $1000, and u(x) = 2x if x < $1000. π = 0.05, π = 0.01, and g = $100. The optimal contract under moral hazard satisfies (1 π)g u(a) = U = = 2665, π π u(b) = U + πg = = 165. π π It follows that 2A = 2665 and B = 165, Thus, R = B = 165 and D = B A = = The profit of the insurance company is $ $ = $ The moral hazard reduces the profit of the insurance company from $90 to $ Bargaining under Asymmetric Information Our models of adverse selection and moral hazard stay within the standard frameworks where the price is either determined by competition or set by a monopolist.

17 Yet there are other mechanisms for handling economic transactions. We discuss some aspects of mechanism design in a simple bargaining problem. Consider two agents a seller and a buyer who bargain over the price of a good that initially belongs to the seller. The agents value the good at V and V respectively. They have no control over their own values. When the object is sold at a price p, the seller and the buyer derive utilities (gains from trade) equal to p V and V p respectively. The values V and V are private: the buyer does not know V and the seller does not know V. Yet the probability distributions of V and V are known to both agents, who are assumed risk neutral For concreteness, take the following numbers. The seller puts a high value V = 80 or a low value V = 0 on the good with equal probabilities 1. The buyer puts a high 2 value V = 100 or a low value V = 20 on the good with equal probabilities 1. The 2 distributions of V and V are independent. The following table summarizes the four possible cases: V = 100 V = 20 V = 80 High V - High V High V - Low V V = 0 Low V - High V Low V - Low V Each of these cases HH, HL, LH, and LL arises with probability 1. 4 Imagine first that there is a benevolent mediator who knows both V and V and seeks the highest possible total welfare for the buyer and the seller. Then this mediator would arrange trade in cases HH, LH, and LL, but not in HL (why?) The mediator could also divide gains from trade equally by setting a price p = = 90 for HH, 2 p = = 50 for LH, and p = 20+0 = 10 for LL. 2 2 What happens if there is no such mediator? Then the outcome depends on how the trade is arranged. For example, the seller may have the bargaining power to set a price p and then let the buyer either accept or reject. It is easy to check that the seller will set a price p = 100 regardless of her true value V. Let V = 80. If p < 80 or p > 100, then the seller does not benefit from trade (why?). If 80 p 100, then the seller gets 1 (p 80) because the trade occurs 2 only if V = 100, that is, with probability 1. Thus, to derive the highest possible 2 utility equal to 10, the seller should set p = 100. Let V = 0. If p 20, then the seller gets p because she sells the object for sure; so she gets at most 20 in this case. If 20 < p 100, then she gets 1 2 p (why?). Thus, to derive the highest possible utility equal to 50, the seller should set p = 100 again. So the trade will occur only in cases HH and LH at the price p = 100. All the benefits from trade in LL are lost. Moreover, the division of the remaining gains is very unfair because the seller receives all of them. Note the presence of adverse selection: the seller cannot offer a low price p 20 for a buyer with V = 20 because a buyer with V = 100 can pretend he has a low value as well. Alternatively, the buyer may have the bargaining power to set a price p that the seller may either accept or reject. Then the buyer will set p = 0 (this is a homework

18 problem). The trade will occur only in cases LH and LL, and the benefits from trade in HH will be lost. In order to extract more gains from trade and to divide them more equally, the agents may rely on more symmetric bargaining mechanisms. For example, they may use the split-the-difference mechanism. It works in two stages. First, the seller and the buyer announce their values to be v and v respectively. Assume that the seller can announce v = 80 or v = 0, and the buyer only v = 100 or v = 20. They can lie: v need not equal V and v need not equal V. After the announcements are made, the trade is determined as follows: v = 100 v = 20 v = 80 p = 90 no trade v = 0 p = 50 p = 10 So the mechanism prescribes trade only if v v at a price p = v+v 2 that splits the difference between v and v. In order to understand the outcome of this mechanism, we identify the strategies that the two agents might use when making their announcements. The seller has only two reasonable strategies to choose from: tell the truth v = V ; always report the high value v = 80 (we call this strategy lie because it implies that the seller lies when V = 0). The buyer also has two reasonable strategies tell the truth v = V ; always report a low value v = 20 (again, we call this strategy lie ) So there are four possible outcomes: (i) the seller and the buyer tell the truth; then they receive benefits V = 100 V = 20 V = 80 U = 10 and U = 10 U = 0 and U = 0 V = 0 U = 50 and U = 50 U = 10 and U = 10 The total benefit for either agent is = 70. (ii) the seller lies, and the buyer always tells the truth; then their benefits are V = 100 V = 20 V = 80 U = 10 and U = 10 U = 0 and U = 0 V = 0 U = 90 and U = 10 U = 0 and U = 0 The seller s benefit from trade is = 100, and the buyer s benefit is = 20.

19 (iii) the seller tells the truth, and the buyer lies; then their benefits are V = 100 V = 20 V = 80 U = 0 and U = 0 u = 0 and U = 0 V = 0 U = 10 and U = 90 U = 10 and U = 10 The seller s benefit from trade is 20 and the buyer s benefit is 100. (iv) both the seller and the buyer lie. Then there is no trade (why?). So both agents get zero benefits from trade. So the seller and the buyer play a game with the following payoff matrix: truth lie truth U = 70,U = 70 U = 20,U = 100 lie U = 100,U = 20 U = 0,U = 0 Note that the situation when both agents always tell the truth is not a Nash equilibrium. For instance, if the seller lies and the buyer remains truthful, then the seller s payoff increases from 70 to 100. Similarly, the buyer can do better by lying if the seller remains truthful. This game has two Nash equilibria in pure strategies where one of the agents is always truthful, and the other is not. The seller lies and the buyer tells the truth. Then U = 100 and U = 20. Accordingly, it is not worth for the buyer to use the other strategy v = 20 because then he will get 0 rather than 20. Also the seller does not gain by telling the truth: then her payoff is reduced from 100 to 70. Trade occurs in HH and LH, but not in LL (why?) So this equilibrium is just as inefficient as the situation when the seller could set the price. The only difference is that the buyer now gets a small share of the gains from trade. The seller tells the truth but the buyer lies (why is this a Nash equilibrium?) Then U = 20 and U = 100. The trade happens in LH and LL, but not in HH (why?) So this equilibrium is as inefficient as the situation when the buyer could set the price, though it is somewhat fairer for the seller. To summarize, the split-the-difference mechanism may seem simple and fair, but in fact it still produces inefficient and unfair outcomes. 6.1 Revelation Principle Is it possible to design a better mechanism for bargaining? Before answering this question, we formulate a simple but powerful principle that simplifies mechanism design. The revelation principle states that any equilibrium outcome in any mechanism can be replicated in another mechanism in an equilibrium where all agents always tell the truth about their type. To understand why this principle is true, introduce a mediator who

20 asks both agents about their private information (their values) and also about their equilibrium strategies (such as telling the truth, or lying); executes these strategies based on the private information that he receives from the agents. Then the seller (or the buyer) has no incentive to lie to the mediator because by doing so, she can only mislead the mediator to execute a wrong strategy that will reduce her own utility. In some sense, lying to the mediator would be the same folly as lying to oneself in the equilibrium in the original mechanism. Analogously, it cannot in the interest of a defendant to lie to her lawyer if she thinks that the lawyer executes a strategy that is best for her. Thus, we have designed a new mechanism with the built-in mediator, where truthtelling is an equilibrium that produces the same outcome as the equilibrium in the original mechanism. For example, consider the equilibrium in the split-the-difference mechanism where the seller lies, and the buyer is always truthful. By adding a mediator who executes these equilibrium strategies, we obtain the following new mechanism: v = 100 v = 20 v = 80 p = 90 no trade v = 0 p = 90 no trade Telling the truth is an equilibrium in this mechanism (why?) Similarly, we can start from the equilibrium where the seller is always truthful, and the buyer lies, and obtain another mecahnism with a truth-telling equilibrium. v = 100 v = 20 v = 80 no trade no trade v = 0 p = 10 p = An optimal bargaining mechanism The revelation principle asserts that in order to find an optimal mechanism, we may focus only on truth-telling equilibria. Let us return to the bargaining problem and find an optimal mechanism of the following sort: v = 100 v = 20 v = 80 p = 80 with prob. q no trade v = 0 p = 50 with prob. 1 p = 20 with prob. q By revelation principle, we restrict q so that truth telling is an equilibrium. Then this equilibrium extracts all the gains from trade in the most lucrative case LH, but wastes some gains in cases HH and LL to deter the agents from lying about their types. Let us find the optimal q. There is a truth-telling constraint that requires than neither the seller nor the buyer can benefit from lying as long as their opponent always tells the truth. Suppose that the buyer always tells the truth. Then the seller s payoffs are

21 V = 100 V = 20 V = V = q if she is truthful, and V = 100 V = 20 V = V = 0 80 q 0 if she lies. So the seller s truth telling constraint is q 80 q, which reduces to q 5. (Another homework problem is to show that the buyer s 6 truth telling constraint is the same.) Note that q is the probability of trade in cases HH and LL, where trade is beneficial. Hence, the highest efficiency is achieved when q is maximal: q = 5. 6 The truth-telling equilibrium in the above mechanism is much better than either of the split-the-difference equilibria. First, it extracts 5 rather than 1 of the combined 6 2 gains from trade in cases HH and LL. Second, it is more equitable as it divides gains from trade equally between the seller and the buyer. Yet even the best mechanism does not achieve full efficiency! This is a general finding. It turns out that asymmetric information prevents the society from achieving full efficiency. 7 Auctions Auctions are wide-spread selling mechanisms where, in contrast with the competitive markets, the price is determined exclusively by bids made by potential buyers. Auctions are used because the values that bidders attach to the object being sold are not known to the seller. If the seller knew the values of all bidders, he could simply offer the object to the bidder with the highest value at the price equal or just below this value. By using an auction, the seller lets competition to determine the price and simultaneously, to collect some information about the bidders s values. There are several common auction forms. Perhaps, the most prevalent form is the English (or open ascending price) auction. In this auction, the price of the object is raised, typically by small increments, until only one bidder remains who is still interested at buying the object at the current price. Another auction form is the Dutch (or open descending price) auction. In the Dutch auction, starting from a prohibitively high level, the price of the object is gradually lowered until there is a bidder willing to buy the object. There are several auction forms where the bids are sealed (say, in envelopes). In this case, each buyer s bid is not known to her opponents. In all forms that we consider, the highest bid wins the object. However, the payments may differ. In the first-price auction, the winner pays her bid. In the second-price auction, the winner pays the second highest bid.

22 The analysis of auctions depends on the nature of values that are attached to the object being sold. In the case of private values, the value of each bidder is determined by her taste alone and is independent of the values that other bidders might have. This assumption is reasonable as long as each bidder needs the good for private consumption and does not plan to resell it. We use the following model for private-value auctions. Prior to the auction each bidder i learns her private value V i. The value V i is independent of the values of other bidders and is known only to bidder i. After learning V i, bidder i makes a bid B i that depends on V i. If she wins the object, the payoff is her value V i minus the payment she has to make (this payment will depend on the auction form). If the bidder does not win the object, her payoff is zero. For simplicity, assume that there are only two bidders. Then the payoffs in the first-price sealed-bid auction auction are V 1 B 1 if B 1 > B 2 V 2 B 2 if B 2 > B 1 V Π 1 (B 1,B 2 ) = 1 B 1 V if B 2 1 = B 2 and Π 2 (B 1,B 2 ) = 2 B 2 if B 2 1 = B 2 0 if B 1 < B 2 0 if B 2 < B 1. Here, we assume that if two identical bids are made, then each bidder gets the object with probability 1. Note that the same payoffs are obtained in the Dutch auction 2 when the first buyer decides to buy when the price is lowered to B 1, and the second buyer decides to buy when the price is lowered to B 2. Thus, from the two auction formats are the same in terms of possible strategies and payoffs. So game theory predicts that the outcomes should be the same, but it is not exactly true. There is empirical evidence that the bids in the Dutch auction are lower than in the firstprice sealed-bid auction presumably because participants derive extra utility from the process of seeing the price go down during the Dutch auction. In the second-price auction, the winner pays the opponent s bid rather than her own. Hence, V 1 B 2 if B 1 > B 2 V 2 B 1 if B 2 > B 1 V Π 1 (B 1,B 2 ) = 1 B 2 V if B 2 1 = B 2 and Π 2 (B 1,B 2 ) = 2 B 1 if B 2 1 = B 2 0 if B 1 < B 2 0 if B 2 < B 1. As usual, economists look for an equilibrium. In equilibrium, each buyer should make a bid that is optimal for her given her opponent s bid. In the second-price auction, B 1 = V 1 and B 2 = V 2. Indeed, if the value of bidder 1 is V 1, then by submitting B 1 V 1 she cannot get a payoff which would be higher than when she simply submits V 1. To see this consider two possible cases. Case 1. V 1 > B 2. If B 1 > B 2, then the first buyer wins the auction, pays B 2, and gains Π 1 (B 1,B 2 ) = V 1 B 2 = Π 1 (V 1,B 2 ). If B 1 B 2, then the first buyer does not win the auction (or wins it with probability 1), and gains Π 2 1(B 1,B 2 ) < V 1 B 2 = Π 1 (V 1,B 2 ). Thus, she cannot do better by submitting B 1 instead of V 1. Case 2. V 1 B 2. If B 1 B 2, then the first buyer wins the auction with a positive probability, pays B 2, and gains Π 1 (B 1,B 2 ) V 1 B 2 0 = Π 2 1 (V 1,B 2 ). If B 1 < B 2, then the first buyer does not win the auction and gains Π 1 (B 1,B 2 ) = 0 = Π 1 (V 1,B 2 ). Thus, she cannot gain by submitting the bid B 1 instead of V 1.

23 Similarly, it is optimal for the second bidder to submit V 2. In other words, in the second-price auction, no matter what opponents do, it is optimal to submit a bid equal to the true private value (honesty is the best policy). Thus, there is an equilibrium in dominating strategies. In this equilibrium, the buyer with the highest value wins and pays the second highest value. The same outcome occurs in the equilibrium in the English auction (why?) The analysis of equilibrium in the Dutch and the first-price sealed-bid auction is more complex. First, there is no longer a dominating strategy which remains optimal regardless of the opponent s bids (why?) However, there is a Nash equilibrium, where each bidder s strategy is optimal given the opponent s play. Let us find the Nash equilibrium in a simple setting where both V 1 and V 2 are uniformly and independently distributed on the interval [0, 1]. By symmetry, each buyer should use the same function b(v) to determine how much to bid when the private value equals v. Then the marginal benefit for the first player of pretending that her value is V 1 + v rather than V 1 is The marginal cost is (V 1 b(v 1 )) } {{ } how much she gains when she changes a loss to a win whenever she wins the auction v }{{} the probability that she changes a loss to a win. This happens only when V 1 < V 2 < V 1 + v (b(v 1 + v) b(v 1 )) } {{ } V }{{} 1 how much extra she pays the probability that she wins. when V 2 < V 1 This happens only As b(v 1 ) is the optimal bid, the marginal cost must equal the marginal benefit. Thus, V 1 b(v 1 ) = b(v 1 + v) b(v 1 ) V 1 = b (V 1 )V 1. v It is easy to check that b(v 1 ) = V 1 2 is the solution. Similarly, b(v 2 ) = V 2 More generally, if there are N bidders with independently and uniformly distributed private values, then the equilibrium strategies are b(v i ) = N 1V N i. The revelation principle asserts that any equilibrium outcome in any auction can be replicated in another auction in a truth-telling equilibrium. For example, To replicate the outcome of the first-price auction, modify its rules so that the highest bidder wins, but pays only half of his bid. Then truth-telling becomes an equilibrium (in the case when there are two bidders with independently and uniformly distributed private values). Note that the principle of revenue equivalence does not mean that any two auctions generate the same revenue for any realizations of private values. For example, if V 1 = 0.7 and V 2 = 0.6, then the revenue is 0.6 in the second-price auction, but only 0.35 in the first-price. Yet, if V 1 = 0.7 and V 2 = 0.1, then the revenue is 0.1 in the second-price auction, but still equals 0.35 in the first-price. 2.