Topics in Information Economics: Adverse Selection

Transcription

1 Chapter 3 Topics in Information Economics: Adverse Selection 1. The adverse selection problem.. Akerlof s application to the second-hand car market. 3. Screening: A price discrimination model. 3.1 Introduction For a wide range of products (for example, things you buy from a grocery store or a drug store) the markets are large and decentralized, and the exchanges are anonymous. Usually you do not know who owns the grocery store when you go in to buy a tube of toothpaste or a zucchini, nor does the owner bother to keep track of every customer. There are, however, important exceptions. Consider the market for used cars. It is difficult to tell how good or bad a used car is unless you use it for some time. The seller, on the other hand, knows how good the car is. In such situations, you would care about the identity of the seller. In fact, in such cases anonymous exchange is often impossible, as that would lead to a breakdown of the market. The customers are willing to pay a price based on the average quality. However, this price is not acceptable to the better than average quality car owner. This leads the owners of better than average cars to leave the market. But the same phenomenon then applies to the rest of the market, and so on. 1

2 The problem is called lemons problem or adverse selection. The same problem arises in selling insurance. The insurance company must know the customer and her past record of illnesses. Anonymous exchange would, in general, lead to a market price (in this case insurance premium) that is too high - and only the people who feel that they are very likely to claim the insurance would buy insurance. This would lead, as in the case of used cars above, the market to fail. A different sort of problem arises when you, as the manager of a company, are trying to hire some salespeople. As the job requires a door to door sales campaign, you cannot supervise them directly. And if the workers choose not to work very hard, they can always blame it on the mood of the customers. If you pay them a market clearing flat wage, they would (assuming away saintliness) not work hard. This problem is known as moral hazard. Note the difference between adverse selection and moral hazard. In the first case, the asymmetry in information exists before you enter into the exchange (buy used car, sell insurance). In the latter, however, the asymmetry in information arises after the wage contract is signed. This is why another name for adverse selection is hidden information and another name for moral hazard is hidden action. In the following sections, we will consider certain remedies to adverse selection and moral hazard. In what follows, attitudes towards risk will play a role. First, a fuller description of the lemons problem. This was first noted by Akerlof (1970). At the time, hardly anyone understood the importance of his ideas. Five top journals rejected his paper. Today, of course, it is recognized as a classic. It has spawned a huge literature on information economics that has significantly advanced our knowledge of economic institutions. 3. Akerlof s Model of the Automobile Market Suppose there are four kinds of cars there are new cars and old cars, and in each of these two categories there may be good cars and bad cars. Buyers of new cars purchase them without knowing whether they are good or page of 19

3 bad, but believe that with probability x they will get a good car, and with the residual probability (1 x), a bad car. After using a car for some time the owners can find out whether the car is good. This leads to an asymmetry: sellers of used cars have more information about the quality of the specific car they offer for sale. But good cars and bad cars must sell at the same price since it is impossible for the buyer to differentiate. This leads to market failure. A Generalization to the case with continuous grades of quality Suppose we can index quality by some number q which is uniformly distributed on the interval [0,1]. Hence the average quality of the cars is 1/. Suppose that there are a large number of risk neutral buyers who are prepared to pay 3q/ for a car of quality q, and sellers are prepared to sell a car of quality q at price q. If quality was observable, any price in this range would be admissible. If quality of an individual car cannot be observed by the buyers, they base their decisions on the expected quality of the cars in the market. If the expected quality is given by Q, then the buyers would be willing to pay up to 3Q/. What is the likely outcome in this market? If the price were p, all sellers with cars of quality less than p would offer their cars for sale. Hence, average quality of the cars that appear in the market would be Q = p/. But this implies that the reservation price of the buyers would be 3Q/ = 3p/4. In other words, there is no price such that the buyers are prepared to pay the asking price, and the market fails. Implications This is an instance of market failure owing to adverse selection, and if the welfare gains from trade in the market are sufficiently great, there is scope for private institutions to evolve to serve as guarantors of quality. Of course, it is costly to set up these institutions, but these costs may be more than made up by the gains from trade in the relevant market. Standard applications of this model include the medical insurance markets (old people cannot buy insurance at any price), credit markets etc. It is also interesting in this context to think of the specific institutions set up to counteract these problems. page 3 of 19

4 3.3 Price Discrimination under Asymmetric information This section is adapted from Salanie, The Economics of Contracts, MIT Press, Chapter The Model There is a seller and a buyer. The seller sells a unit of quality q at price t. Cost of producing quality q is c(q), where c (q) > 0, and c (q) > 0. The profit of the seller is given by π(t, q) = t c(q) and the utility of the buyer is given by u(t, q, θ) = θq t. θ is a parameter that reflects how much the buyer cares about quality. θ is usually referred to as the buyer s type. This is the buyer s private information. Suppose θ can take two values, θ 1 and θ, where θ > θ 1. A contract is a pair (q, t) offered by the seller. The buyer gets 0 utility if he does not buy. Thus any contract must give the buyer at least 0. Assume, for simplicity, that the seller has all the bargaining power. The indifference curve of a buyer of type θ i, i {1, } is given by θ i q t = constant. Thus the slope of the indifference curve is given by dq dt = θ i. utility constant Figure 3.1 shows the indifference curves of type θ. Next, an iso-profit curve for the seller is given by t c(q) = constant. The slope of the iso-profit curve is given by dq dt = c (q). profit constant page 4 of 19

5 t θ q - t < 0 θ q - t = 0 t θ q - t > 0 θ q - t < 0 1 θ q - t = 0 1 θ q - t > 0 1 q q Figure 3.1: Indifference maps of types θ and θ 1. The arrows show the direction of improvement. Note that all iso-profit curves have the same slope at points along any vertical line. t t A B q q~ q Figure 3.: Iso-profit map of the seller. The arrow shows the direction of improvement. Iso-profit curves have the same slope at points along any vertical line. At points A and B the slope of the iso profit curves is c ( q). page 5 of 19

6 3.3. The Full Information Benchmark Under full information, the seller offers a contract (q 1, t 1 ) to type θ 1 and another contract (q, t ) to type θ. To determine the optimal contract for each type θ i, i {1, }, the seller solves max q i,t i t i c(q i ) subject to θ i q i t i 0. However, since the seller has all the bargaining power, there is no reason to give the buyer any more than 0, and thus the constraint holds with equality. Thus the seller solves: which can be rewritten as max q i,t i t i c(q i ) subject to θ i q i t i = 0, max q i θ i q i c(q i ). Thus qi is such that c (qi ) = θ i, and ti = θ i qi. Figure 3.3 shows the optimal contracts under full information. page 6 of 19

7 !h t θ q - t = 0 t* iso-profit θ q - t = 0 1 t* 1 q* q* 1 q Figure 3.3: The full information solution. The optimal contract for type θ i is obtained at the point of tangency between the iso-profit curve and the indifference curve of type θ i at the reservation utility level (here the reservation utility is 0). page 7 of 19

8 3.3.3 Contracts Under Asymmetric Information Under asymmetric information, the full information contracts are not incentive compatible. To see this, note that θ q t = 0, but θ q 1 t 1 > θ 1 q 1 t 1 = 0. Thus type θ prefer contract (q 1, t 1 ) to contract (q, t ). To preserve incentive compatibility, the seller must offer contracts ( ˆq 1, ˆt 1 ) and ( ˆq, ˆt ) such that θ ˆq ˆt θ ˆq 1 ˆt 1. Given any (q 1, t 1 ), the best that the seller can do is to offer a contract (q, t ) such that the above is satisfied with equality. In other words, given any contract (q 1, t 1 ), the incentive compatible contract (q, t ) lies somewhere on the indifference curve of type θ passing through the contract (q 1, t 1 ). The best contract for the seller on this indifference curve is of course the point at which an iso-profit curve is tangent to the indifference curve. Since the slope of all indifference curves of type θ is θ, the tangency occurs exactly at the quantity q. Thus under asymmetric information, the optimal contract for type θ involves ˆq = q. What about ˆt? Given any ( ˆq 1, ˆt 1 ), this is determined by the incentive compatibility condition θ q ˆt = θ ˆq 1 ˆt 1. The remaining problem is to determine the optimal ( ˆq 1, ˆt 1 ). We will not derive this here (for a full derivation, see Salanie, ch )- but simply note the following: First, there is no reason for the seller to give any surplus to type θ 1. Thus ( ˆq 1, ˆt 1 ) is such that ˆt 1 = θ 1 ˆq 1. Second, it must be that 0 ˆq 1 < q1. To see this, note that under full information, q1 is derived by equating marginal revenue (given by θ 1) to marginal cost (given by c (q 1 )). Under asymmetric information, the marginal revenue is the same, but marginal cost is greater than c (q 1 ). This is because, for any additional q 1, in addition to the direct production cost there is an indirect cost arising through incentive compatibility. To maintain incentive compatibility, t must be lowered. This is shown in figure 3.5. Thus new marginal cost of q 1 is equal to c (q 1 ) + where > 0 and denotes the information cost (revenue lost through lower t ). Thus the optimal choice of q 1 must be lower than q 1. page 8 of 19

9 t θ q - t = 0 θ q - t > 0 θ q - t = 0 1 t 1 q 1 q Figure 3.4: Given (q 1, t 1 ), incentive compatibility requires that (q, t ) must be on the dotted indifference curve of type θ. t a t b t c t iso-profit θ q-t=0 θ q-t = K >0 θ q-t = K >0 θ q-t= q 1 q 1 q q Figure 3.5: If no other contract is offered, (q, t a ) is incentive compatible (note that q = q, and ta = t ). If a low quality-low price contract is offered so that q 1 = q 1 1 (with corresponding change in t 1 to keep the contract on the 0 utility indifference curve of type θ 1 ), to preserve incentive compatibility, (q, t ) must lie on θ q t = K 1. The best such contract for the seller is (q, t b). Thus t is reduced. As q 1 increases from q 1 1 to q 1, to preserve incentive compatibility, t must be further reduced to t c. Thus under asymmetric information, there is an additional indirect marginal cost (or marginal information cost) of increasing q 1. page 9 of 19

10 Figure 3.6 shows the contracts under asymmetric information. t θ q-t=0 θ q-t>0 Information Rent of type θ { t * ^ t iso-profit θ 1q-t=0 ^ t 1 ^ q q* 1 q*=q ^ 1 q Figure 3.6: Contracts under asymmetric information. Note that type θ 1 gets a 0 utility under both full and asymmetric information. Type θ gets a 0 utility under full information, but positive utility under asymmetric information whenever ˆq 1 > 0. The information rent of type θ is given by (θ ˆq ˆt ) (θ q t ). Since ˆq = q, the information rent is given simply by ˆt t. page 10 of 19

11 Chapter 4 Topics in Information Economics: Moral Hazard 1. The principal-agent problem, moral hazard.. Risk neutral agents. 3. Risk averse agents and agency costs. 4.1 Introduction Suppose a risk-averse individual has purchased an insurance policy that promises to compensate her fully in case his bike is stolen. Once insured, the individual has no incentive to be careful about securing the bike because if it is stolen, the insurance company, and not he, will bear the loss. (Assume, for the sake of argument, that he does not mind the bother of reporting the loss to the police, filling claim-forms, etc.). Indeed, if being security-minded causes disutility, he might choose to be downright careless, and so expose the insurance company to an especially high level of risk. Of course, the insurance contract may require that he take due care to prevent theft, but then it may be hard to observe carelessness, or to prove it in a court of law. This phenomenon that the very act of insurance blunts the incentives of the insured party to be careful, and so increases the overall risk to the insurer is described as moral hazard in the insurance literature. We know how insurance companies react to this hazard. To preserve the right incentives, they may provide only partial insurance. This exposes the 11

12 insured party to at least some residual risk, and thus prompts more careful behavior. This feature of insurance contracts, namely that the risk is effectively shared between the insurance company and the insured party is known as co-insurance. Note that the consequence of the moral hazard problem is to reduce the amount of insurance that is available to individuals. The moral hazard problem is a direct consequence of an informational asymmetry: in the example, the true level of care (or effort) is hidden from the firm. The asymmetry in information here is described as hidden action, as distinct from that of hidden type. The issue can be studied more generally as a principal-agent problem. Many economic transactions have the feature that unobservable actions of one individual have direct consequences for another, and the affected party may seek to influence behavior through a contract with the right incentives. In the above example, the insurance company (the principal) is affected by the unobservable carelessness of the insured (the agent): it then chooses a contract with only partial insurance to preserve the right incentives. Other economic relationships of this type include, shareholders and managers, manager and salespersons, landlord and tenants, patient and doctors, etc. We consider the principal-agent problem a bit more generally. 4. A formalization A principal (or just P) hires an agent (A) to carry out a particular project. Once hired, A chooses an effort level, and his choice affects the outcome of the project in a probabilistic sense: higher effort leads to a probability of success, and that translates into higher expected profit for P. If the choice of effort was observable P could stipulate, as part of the contract, the level of effort that is optimal for her (that is, for P). When effort cannot be monitored, P may yet be able to induce a desired effort level, by using a wage contract with the right incentive structure. What should these contracts look like? The formal structure can be set up as follows: 1. Let e denote the effort exerted by the agent on the project. Assume, for the moment, that A can either work hard (choose e = e H ) or take it easy (choose e = e L ); the story can later be generalized for more page 1 of 19

13 than two, but finite number of, effort choices. The choice of e cannot be monitored by P: this feature hidden action characterizes the informational asymmetry in the problem.. Let the random variable π denote the (observable) profit of the project. Profit is affected by the effort level chosen, but is not fully determined by it. (If it was fully determined, the principal can infer the true effort choices by observing the realization of π: we would no longer be in a situation of hidden actions.) Higher effort leads to higher profit in a probabilistic sense. Assume that π can take a value in some finite set, {π i }. Let f(π e) be the conditional probability of π under effort e, and F(π e) be the associated cumulative distribution. We assume that the distribution of π conditional on e H dominates (in the first order stochastic sense) the distribution conditional on e L : we have F(π e H ) F(π e L ), with strict inequality for some π. 3. The principal is risk neutral: she maximizes expected profit net of any wage payments to the agent. 4. The agent is (weakly) risk-averse in wage-income, and dislikes effort. His utility function takes the form u(w, e) = v(w) g(e), where w is wage received from the principal, and g(e) is the disutility of effort. Assume v > 0, v 0, and g(e H ) > g(e L ). Note the central conflict of interest here. The principal would like the agent to choose higher effort, since this leads to higher expected profits. But, other things being the same, the agent prefers low effort. However, (to anticipate our conclusion), if the compensation w package is carefully designed, their interests could be more closely aligned. For instance, if the agent s compensation increases with profitability (say, by means of a profit-related bonus), the agent would be tempted to work hard (because high effort will increase the relative likelihood of more profitable outcomes). However, this will also make the agent s compensation variable (risky), and that is not so efficient from a risk-sharing perspective. The inefficiency in risk-sharing will be an unavoidable consequence of asymmetric information. 5. The principal chooses a wage schedule w(π) that depends on π. Note this is a (variable) payment schedule rather than a constant page 13 of 19

14 payment, precisely because the principal may wish to influence the effort choices of the agent. (More generally, the wage schedule could condition payments jointly on π and any other observable signal m that is correlated with the effort choice. With the right wage schedule, it may be possible to induce a particular effort choice. To persuade the agent to accept the contract and to induce e H, for instance, the wage contract must satisfy the following constraints: The contract should be acceptable to the agent. Let u 0 be the reservation utility (the expected utility the agent gets from alternative occupations). The compensation package should be such that its expected utility under e H is at least as large as u 0. This is the participation constraint (PC), or the individual rationality constraint. The compensation package should be such that the agent prefers to exert e H rather than any other effort level. In other words, w(π) must create the right incentives for choosing e H over e L. This is called the incentive compatibility(ic), or relative incentive constraint. (NB: If there are K possible effort choices, there would be K 1 (IC) constraints to satisfy.) 4.3 The principal s problem Given the wage schedule w(π), the principal gets the surplus π w(π) for outcome π. Define w(π i ) = w i. So the choice of a wage schedule boils down to the choice of a set of numbers {w i }, one w i for each π i, in order to maximize (π i w i ) f(π i e). i This expected surplus depends on {w i }, and on the agent s choice of e. If effort was observable, it could be stipulated in the contract: the principal then needs to worry about satisfying only the participation constraint. If effort is not observable, the agent s choice of e will depend on the {w i } offered, and to induce any level of effort, the principal must try to satisfy both the participation and incentive compatibility constraints. The principal s problem is solved in two steps. page 14 of 19

15 Step 1: For each effort level, e H or e L, determine the wage schedule that would implement that effort level most cheaply. That is, for given e j, minimize i w i f(π i e j ), subject to the relevant constraints. (If effort is observable, only (PC) is relevant; if not observable both (PC) and (IC) are relevant). Let C(e j ) be the minimized cost of implementing e j. If e j can never be implemented, define C(e j ) to be infinite. Step : Choose to implement the effort level that yields the highest expected surplus. That is, choose e j, to maximize π i f(π i e j ) C(e j ). i 4.4 Observable effort Suppose effort is observable. The principal can then implement any level of effort subject only to the participation constraint. To implement e j, she must choose {w i } to minimize w i f(π i e j ), i subject to i f(π i e j )v(w i ) g(e j ) u 0. (PC) Proposition 1. With observable effort, a constant wage level would implement e j most cheaply. Having solved the above problem for both e H or e L, she must choose to implement that level of effort which yields a higher net surplus. 4.5 Unobservable effort If effort is unobservable, to implement any e j the principal must choose a wage that satisfies both the participation constrain and incentive compatibility constraint. To implement e j, she must choose w(.) to minimize w i f(π i e j ), i page 15 of 19

16 subject to i i f(π i e j )v(w i ) g(e j ) i f(π i e j )v(w i ) g(e j ) u 0, (PC) f(π i e k )v(w i ) g(e k ), k = j (IC) Note: as stated earlier, if there were K possible effort levels, we would have K 1 (IC) constraints. We now consider two cases Case 1: Risk neutral agent Proposition. If the agent is risk neutral, and effort is unobservable, the optimal contract leads to the same outcome as when effort is observable. Why? Choose a wage schedule of the form w i = π i α, where α is fixed, and so chosen that the participation constraint is just satisfied. This amounts to selling the project to the agent for a price α: then the agent then has all the incentive to maximize profits. Case : Risk averse agent When effort is not observable, and the agent is risk averse, to implement e L is easy: offer a constant wage that just satisfies the participation constraint. Given a constant wage, the agent will have no incentive to work hard and will choose e L. To implement e H is harder: we demonstrate this with an example. Example: Consider the following principal-agent model. A principal hires an agent to work on a project in return for wage payment w > 0. The agent s utility function is separable in the effort and wage received: we have u(w, e i ) = v(w) g(e i ), where v( ) is his von-neumann Morgenstern utility function for money, and g(e i ) is the disutility associated with effort level e i exerted on the project. Assume that the agent can choose one of two possible effort levels, e 1 or e, with associated disutility levels g(e 1 ) = 5 3, and g(e ) = 4 3. The value of the project s output depends on the agent s chosen effort level in a probabilistic fashion: If the agent chooses effort level e 1, the project yields output π H = 10 with probability p(h e 1 ) = 3, and π L = 0 with the residual probability. If the agent chooses effort level e, the project yields π H = 10 with probability p(h e ) = 1 3, and π L = 0 with the residual probability. page 16 of 19

17 The principal is risk neutral: she aims to maximize the expected value of the output, net of any wage payments to the agent. The agent is riskaverse, with v(w) = w 1, and his reservation utility equals 0. (a) Suppose, first, that the effort level chosen by the agent is observable by the principal. A wage contract then specifies an effort level e, and an output-contingent wage schedule {w H, w L }. Here w H is the wage paid if π = π H, and w L the wage if π = π L. Show that if effort is observable, it is optimal for the principal to choose a fixed wage contract (that is, w H = w L = w). Provide brief intuition for this result. (b) If effort is observable, which wage w should the principal offer if she wants to implement e 1? Which wage implements e? Which induced effort level provides a higher expected return to the principal, net of wage costs? (c) Suppose, next, that the agent s choice of effort level is not observable. In this circumstance, a contract consists of an output-contingent wage schedule {w H, w L }. Which wage schedule will implement e 1 in this case? Which expected net return does the principal get in this case? How does this compare with the value in part (b), where effort was observable? (d) If effort is not observable, which contract is best for the principal? Should she implement e 1 or e? Answer: (a) Since the agent is risk-averse and the principal is risk-neutral, an optimal risk-sharing result is for the principal to take all risk. If the agent were to carry unwanted risk, he would have to be given a higher expected wage than in the optimal case. Since this is not necessary for incentive reasons because effort is observable, the first best solution is for the principal to take all risk and completely insure the agent. The principal s problem is to minimize the agent s expected wages. Suppose the principal offers state dependent wages {w H, w L }. Let p i denote the probability of π H under effort e i, i {1, }. page 17 of 19

18 Formally, the principal s problem when offering state dependent wages {w H, w L } is, for a particular e i, to min w H,w L p i w H + (1 p i )w L s.t.(ir i ) p i v(w H ) + (1 p i )v(w L ) g(e i ) = 0. We solve this using the Lagrangian L = p i w H + (1 p i )w L λ [p i v(w H ) + (1 p i )v(w L ) g(e i )] which has the foc and Therefore L w L = 0 1 λ = v (w L ) L w H = 0 1 λ = v (w H ). v (w L ) = 1 λ = v (w H ) Since v > 0, this implies that w L = w H = w as required. Further, given that v is concave, the second order condition for a maximum is satisfied. (b) Because of (a), we know that in the first best contracts the wage depends on the observable effort e i {1, } and not on the state of the world j {L, H}. To implement e i at minimum cost, the principal simply needs to pay a wage w i to satisfy the participation constraint for that effort level, given by v(w i ) g(e i ) = u 0 = 0. (4.5.1) page 18 of 19

19 We get 1. (high effort) e 1 :. (low effort) e : agent s participation: w1 = 5 3 w 1 = 5 9, 0 principal s profit: π(e 1 ) = agent s participation: principal s profit: w = 4 3 w = 16 9, π(e ) = Since π(e 1 ) > π(e ), the principal will implement e 1. (c) State dependent contracts are {w H, w L }. Implementing e 1 at minimum expected wage cost requires minimizing 3 w H w L subject to the participation constraint as well as incentive constraint of the agent. Solving the two constraints completely determine w H and w L, so that there is no further minimization to be done. (IR 1 ) 3 wh wl 5 3 0, and (IC 1 ) 3 wh wl wh + 3 wl 4 3. Since there are two variables to be determined, and two inequalities, it is possible to find w H and w L so that both bind. (IC 1 ) implies wh = 1 + w L. We can use this in (IR 1 ) and obtain (1 + w L ) + w L = 5 (w L = 1, w H = 4) with an associated profit of π(e 1 ) = 3 (10 4) (0 1) π(e 1) = (d) Implementing e : Any flat wage that satisfies the agent s participation constraint implements e. The cheapest way for the principal to do this is to offer the first best flat wage w 1 = 16/9. As before, π(e ) = Since π(e 1) > π(e ), the principal s optimal choice under asymmetric information is e 1. page 19 of 19