Implementation of First-Best Allocations in Insurance Economies with Asymmetric information

Transcription

1 Implementation of First-Best Allocations in Insurance Economies with Asymmetric information Aristotelis Boukouras Kostas Koufopoulos This Version: 07 Octoer 2009 First Version: 24 July 2007 Astract In a Rothschild-Stiglitz insurance economy with a continuum of agents information aggregation can e used in order to achieve first-est allocation of resources, despite the presence of an adverse selection prolem. First, we show this result for a direct mechanism, which asks agents to sumit messages and provides allocations contingent on the aggregate messages of all agents. Then, we present an alternative indirect mechanism, which allocates consumption rights and allows agents to trade these rights amongst themselves. In this case, the allocation of resources is contingent on the final price and quantities of consumption rights each agent holds. For the second case, we show that the first-est contracts can e supported as an incentive compatile Perfect Bayesian- Nash Equilirium of a game in which multiple firms compete in offering contracts and attracting insurees. Keywords: adverse selection, first-est allocations, information aggregation, insurance, mechanism design JEL Classification: D41, D82, D86, We would like to thank Herakles Polemarchakis, Eric Maskin, Bhaskar Dutta, Peter Hammond, Thodoris Diasakos, Christopher Hennessy and Claudio Mezzetti for their very useful comments and remarks, for which we feel grateful. University of Warwick, Department of Economics, a.oukouras@warwick.ac.uk University of Warwick, Warwick Business School, Kostas.Koufopoulos@ws.ac.uk 1

2 1 Introduction We consider the insurance economy with asymmetric information (hidden types) introduced y Rothschild and Stiglitz (1976). Each insuree knows his own risk type (ut not the type of any other insuree) and the distriution of types and insurers know only the distriution of types. Rothschild and Stiglitz argued that insurance companies can distinguish high risks from low risks y offering contracts specifying oth the price and the quantity (non-linear pricing). Because the utility cost of under insurance is lower for the low risks, the low risks choose less than full insurance to prevent the high risks from mimicking them. The utility cost of under insurance for low risks is more than offset y the lower per unit premium they are charged for the coverage they purchase. Although the suggested solution work towards mitigating the adverse selection prolem, in the screening model of Rothschild and Stiglitz two key issues arise. First, the possiility of non-existence of a Nash equilirium. Second, the inefficiency of the equilirium (in the second-est sense) even if it exists. Both issues have attracted a lot of interest and have een analyzed in the context of the Walrasian mechanism and from a game-theoretic point of view. In the Walrasian tradition, Prescott and Townsend (1984) focus on constrained efficient equiliria and show that, in general, no equilirium exists 1. In Gale (1992, 1996) and Duey and Geanakoplos (2002) a competitive equilirium always exists ut, in general, it is inefficient (in the second-est sense). More recently, Bisin and Gottardi (2006) introduce consumption rights and show that (second-est) allocations can e implemented as competitive equiliria. Wilson (1977) and Riley (1979) propose new equilirium concepts (called anticipatory and reactive equilirium respectively) and show that an equilirium always exists 2. However, in oth cases the equilirium is not, in general, (second-est) efficient. Maskin and Tirole (1992) consider a general principal-agent framework (which has the Rothschild and Stiglitz economy as a special case) and show that any allocation which weakly Pareto-dominates the Rothschild-Stiglitz allocation can e supported as a Nash equilirium 3. Thus, in general, the equilirium is not second-est efficient. Koufopoulos (2008) proposes a mechanism under which a separating Nash equilirium always exists (even when a Rothschild-Stiglitz equilirium does not), it is unique and always (second-est) efficient. He also shows that oth the non-existence prolem and the inefficiency of the equilirium in Rothschild-Stiglitz are due to aritrary restrictions imposed on insurers (they are not due to the informational frictions of the economy). In this paper, we consider the Rothschild-Stiglitz insurance economy with a continuum of insurees. Our aim is to revisit the efficiency issue. The key question is whether first-est allocations can e implemented as a Nash equilirium, in the sense that all the equilirium 1 See also Rustichini and Siconolfi (2008). 2 In the original papers, the equilirium concepts of Wilson and Riley did not have game-theoretic foundations. Hellwig (1987) and Engers and Fernadez (1987) provide game-theoretic foundations to Wilson and Riley equilirium respectively. 3 This result otains when eliefs are completely unrestricted. If the intuitive criterion (Cho and Creps (1987)) is used to refine eliefs, then the Rothschild-Stiglitz allocation is the unique equilirium and so, in general, it is inefficient. 2

3 allocations, which are the outcomes of the mechanisms presented in this paper, reside on the first-est Pareto frontier of the economy under study 4. Our result indicates that the existing papers do not fully exploit the information availale. The channel through which we achieve a etter use of information is information aggregation: since the ex-ante distriution, which generates types, is known and due to the law of large numers, one can find out whether some insurees are lying aout their types or not. An appropriately designed mechanism can use this information in order to provide allocations conditional not only on the contracts (or messages) of agents ut also on the aggregate choices of the insurees and, thus, make them internalize the (negative) externalities they generate to other insurees. In order to demonstrate the general idea, in Section 3 we apply our framework in a Rothschild-Stiglitz economy with finite numer of insurees and two types 5. The main difference from the usual mechanism design framework is that we assume that the interim cumulative distriution of types is common knowledge, i.e. the exact numer of low-risk and high-risk insurees in the economy is pulic information ut the individual identity of each one of them (what type is each individual) remains private information. We then present a simple direct mechanism which implements the first-est allocation of resources as its unique perfect Bayesian-Nash equilirium. Agents report their types and receive contracts (or allocations) conditional on the aggregate numer of reports made for each type. If the numer of reports matches the known distriution, then each agent receives the first-est contract of the reported-type. Otherwise, agents receive incentive compatile allocations which provide lower expected utility for each type than the first-est contracts ut strictly higher utility than zero insurance. We show that, under this mechanism, it is a strictly dominant strategy for low-risk types to truthfully report their type, irrespectively of the reports of any other agent. Thus, there are no coordination failures and the equilirium strategy profile is unique. One might argue that this result otains ecause, in the case that a lie is detected, we impose very severe collective penalties to agents, forcing them to comply with the intended first-est allocations. However, this is not correct. In Section 3, we consider a mechanism imposing more severe collective punishments (no insurance) if a lie is detected and we show that the relationship etween truth-telling and the severity of collective penalties is not monotonic. Precisely ecause of the severity of the penalties, low-risk insurees may strictly prefer to lie aout their types than remain uninsured. This generates coordination failures and multiple equiliria, in which incentive compatiility is typically violated. Instead, our result is due to the design of the off-the-equilirium path allocations so as to make truth-telling strictly dominant strategy for low-risk insurees. It should also e pointed out that this result can e generalized to economies with multiple types, multiple signals, multiple states of nature, and heterogeneous preferences across types (even when the single-crossing is violated) 6. 4 In fact, the direct mechanism we provide in section 3, has a unique perfect Bayesian equilirium which implements the first-est allocation of the full information economy. 5 We ignore issues of aggregate uncertainty until section 4, y assuming that there is enough reserve wealth in the economy to cover even for the small proaility that all agents suffer the accident at the same time. 6 We are currently working on a more general case, where we identify the necessary and sufficient conditions of the prolem. 3

4 Section 4 is concerned with the case of a competitive insurance market and how the allocation of consumption rights can e used as an indirect mechanism of information aggregation for insurance firms. The game is played in several stages. In stage one, consumption rights are issued and distriuted to agents, who can trade them in stage two in a competitive market. Information aggregation takes place indirectly, in stage three, through the price, at which consumption rights are traded, and the quantity of rights each agent holds. While we assume that the quantity of rights of an insuree is private information, we allow any insuree to pulicly declare and verify the numer of rights he holds. For each type of insuree there is a specific quantity of rights required for the information aggregation process to work. If any agent pulicly declares that he possesses any other quantity of consumption rights, the information aggregation fails and consumption rights are invalidated. They are also invalidated if their price goes out of a specific price range. Stage four depends on the outcome of the previous stage. If consumption rights are valid, then insurance companies decide whether to enter the market or not and provide insurance contracts in exchange for a price and a specific quantity of consumption rights per contract. If consumption rights are not valid, then firms offer the Rothschild-Stiglitz equilirium set of contracts. In stage five, uncertainty is resolved and consumption takes place. We prove that the equilirium allocations of the game descried aove reside on the unconstrained (first-est) Pareto frontier of the economy and so we replicate the results of the previous section. However, some comments are due for this result. First, for deriving this result we use the Schmeidler (1973) definition of equilirium, which requires that deviations from the equilirium path are made y an infinitesimally small ut strictly positive mass of agents. This is done in order to avoid the conceptual complications and paradoxes associated with atomless agents. Second, the assumption that all firms offer the Rothschild-Stiglitz contracts off-the- equilirium path is ad hoc ut it is not crucial for the validity of the main result. On one hand, it can e justified as the outcome of a two-stage signaling game where agents propose contracts to firms and firms decide to accept or reject the offers. On the other hand, it can e replaced y Koufopoulos (2008) three-stage game, in which firms are allowed to offer menu of contracts and commit to them ex-ante, since this games has a unique equilirium outcome which provides agents with the second-est allocations. The two cases imply different price ranges for information aggregation to e successful and susequently different final allocations, ut in oth cases the equilirium is unique and the final allocations reside on the first-est frontier. Finally, the information aggregation process is taken as given and insurance firms do not decide on how it works. However, this can also e endogenized y a roader game in which insurees propose mechanisms to firms (which can e interpreted as market rules or structure) and firms either accept or reject the mechanism offered or y a game in which insurees propose and vote mechanisms. Since information aggregation helps agents to overcome the prolem of asymmetric information in this economy and achieve first-est allocations, it generates a Pareto improvement over standard mechanisms and thus it is preferred. In other words, our mechanism elongs to the equilirium set of mechanisms of the roader game and this is a possile way to rationalize our assumptions aout market structure. The issue of efficiency in mechanism design has also concerned other authors. Rustichini, 4

5 Satterthwaite and Williams (1994) show that the inefficiency of trade etween uyers and sellers of a good who are privately informed aout their preferences rapidly decreases with the numer of agents involved in the two sides of the market and in the limit it reaches zero. However, their model is concerned with a private values prolem while our papers features a common values prolem. In the area of prolems with common values, McLean and Postlewaite (2002), (2004) implement efficient outcomes when types across agents are correlated. The main difference with our framework is that we examine the case where types are independently distriuted and, as the papers y Dasgupta and Maskin (2000) and Jehiel and Moldovanu (2001) show, efficiency and incentive compatiility can not e simultaneously satisfied if the single crossing condition is violated or if signals are multidimensional. While the single crossing condition holds in ours setting, our mechanism can also e used for cases where it is violated 7. More recently, the papers y Mezzetti (2004) and Ausuel (2004),(2006) examine the issues of efficient implementation under interdependent valuations and independently distriuted types, ut they also assume that agents preferences are quasi-linear with respect to the transfers they receive, which is not the case in our model. Moreover, the mechanisms proposed in these papers may generate multiple equiliria (in most of which truth-telling is violated), while we are interested in a mechanism which has a unique truth-telling equilirium. Finally, from the Walrasian perspective, Bisin and Gottardi (2006) consider an insurance economy identical to ours and examine how markets for consumption rights can support an efficient outcome (second-est). Apart from the differences in the methodological approach, our paper shows how consumption rights can e used to implement first-est rather than second-est allocations as in Bisin and Gottardi. Section 2, elow, presents the main assumptions aout the economy and Section 3 introduces a direct mechanism for the case of an economy with finite numer of agents and common knowledge of the interim cumulative distriution. Section 4 analyzes the case of a continuum of agents when the ex-ante proaility distriution is common knowledge, which is the usual case considered in mechanism design, and Section 5 concludes. 2 The Economic Environment The economic environment in this paper follows closely the one adopted y Bisin and Gottardi (2006) and Rothschild and Stiglitz (1976). There is a continuum of agents in the economy. Each one of them is represented as a point in the [0,1] interval of real numers. The agents are separated into two types, and g, with respective proportions in the population φ and φ g, φ > 0, φ g > 0 and φ + φ g = 1. Agents receive an endowment of a single good y nature, which they consume. However, the initial endowment is uncertain. There are two possile states, H and L, for every individual, and his endowment when H (respectively L) is realized is ω H (ω L ). Let π s i e the proaility that individual state s S {H, L} is realized for an agent of type i {, g}. 7 We intend to prove this claim in the paper examining the more general formulation of the prolem. 5

6 These random variales are independently distriuted across all agents and identically distriuted across agents of the same type. Each agent is privately informed aout his type, ut the realization of individual states is commonly oserved. Let ω H > ω L 0 and 1 > πg H > π L > 0. Therefore, g is the low-risk type in this economy and type is the high-risk. The preferences of the agents are represented y a von NeumannMorgenstern utility function u : R + R + defined over consumption in each idiosyncratic state s S. Therefore, if c s i denotes the realized consumption of the good for type i in state s, U i (c i ) = π s i u(cs i ), for i {, g}, s {H, L}. We assume that preferences over s S consumption within a given state are independent of types(i.e. u is the same for all agents) and that they are strictly monotonic, strictly concave, and twice continuously differentiale, and lim x 0 u (c) =. Given the assumptions aove, each agent in the economy faces idiosyncratic risk, ut there is no aggregate uncertainty. Due to the law of large numers, the aggregate endowment is equal to Ω = [ ( )] φ i π s i ω s. Insurance companies can improve the welfare of agents i s y offering them transfers for each state of nature z i f = {z H i f, zl i f }, where i denotes an agent s type and f denotes the insurance company. These transfers are assumed to take the form of inding and enforceale contracts. The final consumption of a type i agent, who has signed a contract z j, is: c i = ω + z j. Let F 2 e the numer of insurance companies operating in the economy. Insurance companies are risk neutral and profit maximizing firms, which offer contracts of statecontingent transfers to agents. The aggregate profit for firm f is equal to Π f = [ ( λ i f π s i z j) ], i s where λ i f is the measure of type i agents that sign a z j contract with firm f. We are also allowing for free entry and exit of insurance companies from the market at the eginning of the economy (efore contracts are offered and signed), which implies that firms make zero profits in equilirium. 3 Presentation of the Mechanism under Finite Numer of Agents and a Benevolent Social Planner The purpose of this section is to present the main idea ehind the mechanism and to show why we achieve this result. In order to do so, we consider a slightly altered economic environment and we astract from economic competition. Suppose that instead of a continuum, there is a finite numer k of agents. Suppose also that m of these agents are of type and n are of type g, with m + n = k. The aggregate numer of agents in the economy and the numer of agents of each type are common knowledge, ut the type of each agent is private information. This means that all agents know k, m and n, ut they do not know who are of type or g, apart from their own type. Notice that this is a different information environment from the standard mechanism design prolem, where only the ex-ante proailities of each type are common knowledge and where the interim realization of types from the original distriution 6

7 is unknown. However, we consider this environment ecause, as the economy grows large (n ), the ex-ante and the interim distriutions coincide. We will also discuss this point later. The rest of the assumptions aout the economy remain the same as in the previous section, with the exception that there are no firms offering insurance contracts. We rather consider a centralized mechanism of risk-sharing that could potentially come aout y the intervention of a enevolent social planner. Since in our environment we include pulic information which is not availale in most cases, one could claim that there exists a very simple mechanism to achieve the main result. Suppose that each agent was asked to sumit a message aout her type. Once all messages are collected, agents are allocated insurance contracts conditional on the type they report, provided that exactly m agents report that they are of type and exactly n agents report that they are of type g. Otherwise, no contracts are provided at all. One can easily show that one of the equiliria that result from this mechanism is all agents to report their type truthfully if they elieve that the other agents will do so. But this type of mechanisms are prone to coordination failures, i.e. there are multiple equiliria where a suset of agents misreport their types ecause they elieve that other agents will also misreport their types. Specifically, there are two classes of equiliria with untruthful revelation of types. The first is the case where some low-risk agents report that they are highrisk types ecause they elieve that the same numer of high-risk types report that they are low-risk and vice versa. They do so ecause they prefer to receive the contract of high-risk types than to remain completely uninsured. The second type of equiliria present even more severe consequences in terms of economic efficiency. Agents misreport their types ecause they are indifferent etween a truthful and an untruthful report, elieving that many other agents will also misreport and hence their choice is inconsequential for the final allocation (this is akin to the pathogenic cases of voting, where any voting ehavior can e sustained as an equilirium if agents elieve that their votes do not affect the winner.). Thus, all agents remain uninsured. However, as we show elow, the undesirale equiliria are due to the severe punishment of the no-contract condition when the information aggregation procedure fails and can e eliminated with the appropriate selection of the outside option. Let c RS i denote the final allocation vector which an agent of type i receives under the Rothschild-Stiglitz equilirium of the economy and let ci FB denote the final allocation vector for type i under the first-est allocation of resources. Note that c RS = c FB. Let zrs i and zi FB denote the corresponding insurance contracts that the social planner provides in order to induce allocations c RS i and ci FB respectively. Figure 1 provides a graphical representation of these points on the space of state-contingent consumption. Consider now the following mechanism. The mechanism is similar to the one descried aove, the main difference etween the two eing the outside option when information aggregation fails. Agents are asked to send messages aout their types in stage one. In stage two, once all messages are collected, agents who reported that they are of type receive z FB and agents who reported that they are of type g receive zg FB. This allocation of contracts is conditional on exactly m messages eing -type messages and exactly n messages eing g- type. If this condition is violated then agents receive an outside option which is conditional 7

8 on the message they sent. Those who sent message g (i.e. claimed that are low-risk types) are left to choose etween the the Rothschild-Stiglitz set of contracts ( z RS, ) zrs g with some proaility 1 ɛ p, while with some proaility ɛ p they are provided with the contract z A, which induces the final consumption that corresponds to point A in Figure 1. Those who sent message also choose etween the set of contracts ( z RS, ) zrs g with some proaility 1 ɛp, and with the remainder proaility they are provided with the contract z B which induces the final allocation in point B. In stage three, agents make their choices, if their given the option, the state of nature is realized and consumption takes place. ɛ p is an aritrarily small proaility and the reason why it is used will ecome ovious elow. Point A is designed as a point on the indifference curve of the low-risk type which passes through point C and is aritrarily close ut strictly elow the indifference curve of the high-risk type which passes through c RS (see Figure 1). Point C, in turn, is designed as a point on the indifference curve of the high-risk type that passes through c RS and is aritrarily close to it on the right-hand side of the forty-five degree line (the point in Figure 1 is chosen so as to make the graph clear). Point B is designed as a point point on the forty-five degree line which resides strictly aove c RS ut strictly elow point D, which is the intersection of the indifference curve of the low-risk type that passes through point A with the forty-five degree line. In other words, a low-risk type of agent strictly prefers point A to point B and point c RS in terms of consumption, ut a high-risk type strictly prefers B to A, B to c RS and c RS to A. Such a construction is always possile as long as the degree of risk-aversion of the utility function is not infinite and the difference etween φ g and φ is strictly positive, as we have assumed. The effect of this intricate outside option is that low-risk type of agents have a strictly dominant strategy, namely reporting their type truthfully. In order to see this, consider the est-response of some agent g for every possile set of eliefs that she may have for the actions of the other agents. If g expects that all the other agents will report their type truthfully in stage one, then clearly her est-response is to report her type truthfully as well. By doing so, she achieves the final allocation cg FB, which she strictly prefers to any point of her outside option. Suppose now that she elieves that all agents will report truthfully apart from one highrisk agent, whom she expects to send the untruthful message g. If she chooses to send the message then she will receive c RS. But she can do strictly etter y sending the message g and making the process of information aggregation collapse. In such a case she receives A with proaility ɛ p and c RS g with proaility 1 ɛ p. Since she strictly prefers either of these allocations to c RS, she also prefers the convex comination of the two. Therefore, she will report her type truthfully. The same reasoning applies if she elieves that some numer l of high-risk agents will report g and l 1 low-risks will report, with 1 l min{m, n}. This deals with the first type of coordination failure, when two or more agents switch places in terms of messages. The second type of coordination failure is the one where agents ehave haphazardly ecause they elieve that others will do so and so they confirm their eliefs. Suppose that g elieves that information aggregation will fail regardless of the message she sends. Namely, she elieves that the distriution of messages of the rest of the agents is different from either 8

9 m 1 and n or m and n 1 (the two cases where her message can have an impact on information aggregation). Even in this case she prefers to report her type truthfully. Figure 1 c L ZPLg ο 45 u (RS) PZPL u g (A) (RS) u g FB c g ZPL c = c RS FB B D C A RS c g w c H If she sends the message she will choose c RS g whenever she is offered the choice ut she will receive allocation B with proaility ɛ p. She can do strictly etter y sending message g and receiving c RS g with proaility 1 ɛ p and allocation A with proaility ɛ p. Therefore, from the three oservations aove we conclude that it is a dominant strategy for any agent of type g to report truthfully, irrespectively of the eliefs she has for the actions of the other agents. This result significantly restricts the set of eliefs which are compatile with the perfect Bayesian equiliria of the game, ecause the only admissile equiliria are those where all 9

10 agents correctly elieve that low-risk types will truthfully report their type. This means that we need to consider cases where all agents elieve that the numer of g messages is at least equal to m. Consider now the est-response function of a high-risk type, conditional on these eliefs for the actions of low-risk types. There are two cases to consider. The first case is when elieves that the other high-risk types will report truthfully. Then, it is a est-response for her to report her type truthfully as well, in which case she receives c RS with certainty and avoids the small proaility of receiving allocation A (recall that c RS A). The second case is when elieves that some suset of high-risk types will report untruthfully their type and the information aggregation process will fail. But in this case, she also prefers to send message to message g, since she strictly prefers the lottery {c RS, B} with proailities {1 ɛ p, ɛ p } to the lottery {c RS, A} with the same proailities. Hence, conditional on all low-risk types reporting truthfully, it is the unique est-response for a high-risk type to also reveal her type truthfully. In other words, the mechanism provided aove has a unique equilirium, where all agents report their type truthfully. We summarize this result in the following proposition. Proposition 1: Under the assumptions made aout the economy in sections 2 and 3, the mechanism provided aove has a unique Perfect Bayesian-Nash equilirium, where all agents report their type truthfully. Proof: The proof follows from the results made aove: i) All low-risk type of agents have a strictly dominant strategy of reporting their type truthfully. ii) Conditional on that, the est-response for all high-risk types is also to report their type truthfully, irrespectively of their eliefs aout the actions of the other high-risk agents. Several oservations are due at this point. First, the construction of the outside option, which is required for the result of Proposition 1, is not unique. In fact, there are infinite many sets of points {A < B}, which can e used for this purpose. As long as A is chosen so as to e preferred to points c RS and B y a low-risk type and point B is chosen so as to e preferred to c RS and A y a high-risk type, the outside option implements the first-est allocation of resources as a unique equilirium. However, an issue of udget alance off the equilirium path may arise. If points A and B are located too closely to the zero-profit-line of the low-risk type (ZPL g in Figure 1) then the mechanism may not e alanced on the off-the-equiliriumpath su-games, where information aggregation fails. In fact, one can characterize the set of all udget-alanced points A : (cg H, cg) L and B : (c H, cl ) as the set that satisfies the previous conditions in addition to the equation m[φ g (ω H cg H ) + (1 φ g )(ω L cg)] L + n[φ (ω H c H ) + (1 φ )(ω L c L )] = 0. By choosing points A and B aritrarily close to crs we ensure that the alanced-udget condition is satisfied (in fact, the mechanism generates strictly positive expected profits off the equilirium path). Of course, udget alance may e violated due to the finite numer of agent, which implies that the economy faces aggregate uncertainty. In the analysis so far we have completely ignored this aspect of the prolem, ecause our main interest is in the case where the population is sufficiently large and aggregate uncertainty disappears. We simply use the finite 10

11 case to exemplify the workings of the mechanism and show how coordination failures can e avoided with the appropriate design of the outside option. The results of this section are susequently used in the original environment of section 2 in order to simplify the analysis and exposition, in which case the issue of aggregate uncertainty does not arise due to the continuum of agents. At this point it is noteworthy to mention that there are two ways through which one can move from the analysis of the case with aggregate uncertainty to the case without it. One way is to take the limit of the economy as the numer of agents grows to infinity and the other way is to take the case where there is a continuum of agents. In the section elow we adopt the latter method for comparison reasons with the related literature (Duey and Geanakoplos (2002), Bisin and Gottardi (2006), etc.). The prolem with the continuum of agents is that it is difficult to define what a zero-measure individual means and how information aggregation takes place if the individual mass can not e detected. In order to avoid these complications we use Aumann (1966) and Scmeidler (1973), and we assume that each action is taken y an infinitesimally small ɛ measure of agents of the same type. One could then derive the same results as the limit of ɛ tending to zero. We will return to this point in the discussion of the following section. Another point is that the proaility ɛ p is not required to e infinitesimally small in the design of the outside option. In fact, any proaility less or equal to one would yield the same results. However, y taking the limit as ɛ p goes to zero, we can maintain the result of Proposition 1 while having greatly simplified the outside option to the one where agents choose etween the Rothschild-Stiglitz allocations of the economy. This also simplifies the analysis of section 4, while giving microeconomic foundations on how to avoid the coordination failures of mechanisms like the one descried in the eginning of this section. Moreover, the mechanism presented aove is fairly general and can e easily extended to encompass multiple types of agents and multiple states of nature. For example, suppose that there are k different types of agents ranked from 1 to k according to the proaility of the high endowment state (ω H ) arising, with type 1 eing the highest-risk type (lowest proaility of ω H ) and type k eing the lowest-risk type (highest proaility of ω H ). The design of the mechanism remains the same, the only difference eing that the outside option consists of now of the k points corresponding to the Rothschild-Stiglitz equilirium plus k points, each one of which is strictly preferred y one type compared to the rest and to the allocations c RS k of all other types. Such a construction is always feasile as long as agents do not have infinite risk-aversion and the inframarginal proailities of the high-state are strictly positive for all types. One possile way of achieving this is the following. Find the Rothschild-Stiglitz allocation for the economy with the k types. For type 1, design a point A 1 on the forty-five degree line such that it is aritrarily close to point c RS 1. Take the indifference curve of the second highest-risk type (type 2) that passes through point A 1 and find its intersection with the in difference curve of type 1 which passes from the point c RS 1. Call this intersection point B. Design point A 2 aritrarily close to point B and inside the right gap formed y the indifference curves of type 1 and 2 which pass through B. For any other type j, 3 j k, design point A j aritrarily close to c RS j 1 and inside the right gap formed y the indifference curves of type 11

12 j 1 and j which pass through c RS j 1. Figure 2 elow, shows this design for the case of three types of agents. Similarly, one can define the mechanism to include multiple states of nature. Figure 2 c L ZPL 3 ο 45 ZPL 2 U ( B ) U ( RS ) U ( RS 2 3 ) 2 U 1 ( RS) ZPL 1 A 1 RS c 1 B A 2 RS c 2 A 3 RS c 3 w c H Regardless, the main point of this section is that, if the realized numer from each type is known, there exists a mechanism which can implement the first-est allocation of resources in a common values framework. The mechanism presented aove is a direct mechanism for achieving this. In the following section, we present an indirect mechanism for achieving the same result in an economy with a continuum of agents. Of course, due to the Direct Revelation Theorem, the two mechanisms are equivalent. 4 Competitive Insurance Markets and Consumption Rights In this section we return to the original environment of section 2, where there is a continuum of agents in the interval [0,1]. This guarantees that there is no aggregate uncertainty. A fraction φ g of the agents are low-risk and the remainder fraction φ are high-risk. In addition, we 12

13 consider a game where agents reveal their types indirectly, through the trading of consumption rights in a pulic market and insurance companies compete for attracting customers for their contracts. Information aggregation in this context corresponds to the case where lowrisk types purchase the consumption rights from high-risk types so that each one of them has a specific numer of rights at a price sufficiently low, so that an agent g strictly prefers to uy the rights at this price than to sell them in the market. More specifically, companies issue collectively a volume of papers with total mass equal to the mass of the population. The papers have no intrinsic value, ut can not e forfeited y the agents and we loosely interpret them as consumption rights to the first-est consumption level of the low risk type. These rights are perfectly divisile and are distriuted randomly and uniformly to the agents of the economy so that each agent receives one piece of paper 8 Agents are allowed to trade these rights among themselves in exchange for state contingent transfers p = {p H, p L } from the other party. Essentially, p is the price that an agent pays for uying one paper. Once again, it is assumed that these agreements are inding and enforceale, under the condition that the consumption rights will not e invalidated. Invalidation of consumption rights is equivalent to the information aggregation process failing according to the mechanism of the previous section. We also assume that the market for consumption rights is competitive. This means that there is only one price for one unit of consumption rights and this price is pulic information. In effect, the quantity of papers which each agent holds is the equivalent to sending a message aout ones type. In order to aggregate information, insurance companies demand that all agents hold specific quantities of consumption rights, so that sending a message aout eing a low-risk is equivalent to eing a uyer of consumption rights and sending a message aout eing high-risk is equivalent to eing a seller of consumption rights. Since low-risk types are of total mass φ g in a population of mass one, information aggregation implies that each one of them should hold 1 φ g consumption rights while each one of remainder population should hold zero rights. Define the state independent prices p = {p : U g (c RS + p) = U g (c RS g )}, p = {p : ( U g c FB g 1 φ g φ g p ) ( = U g (c RS g )} and p = {p : U g c FB g 1 φ g φ g p ) = U g (c RS + p)}. { p, p, p} are prices for the papers such that a low-risk type is indifferent etween eing a seller or a nonparticipant in the market, eing a uyer or a ( non-participant and eing a uyer or a seller respectively. Let p = min{ p, p, p}. Then, {U g c FB g 1 φ g φ g p ) U g (c RS g U g (c RS + p). This means that for any price strictly elow p, a low-risk agent strictly prefers to ecome a uyer of consumption rights than to ecome a seller or not to participate in the market. Insurance companies can ensure that low-risk agents will e the only uyers of the papers y requiring the price of the rights to e elow the threshold value defined aove and y defining an appropriately designed threat in case some agent is found not to hold the pre-specified numers of papers. Formally, the game is as follows. In stage one, nature determines the type of each agent 8 We make the assumption of perfect divisiility for the sake of simplicity. One can also assume that these papers are indivisile. In this case, a total mass κ of papers is issued, where φ g = κ ν, κ, ν. This, however, requires the proportion of low risk agents in the economy to e a rational numer, a limitation we wish to avoid. 13

14 and consumption rights are issued and distriuted to agents. In stage two, agents are left to trade these rights in a competitive market. The price for consumption rights is not paid immediately. It is paid in stage five, conditional on the rights eing valid at that stage. If the papers are invalidated, then the oligations of payment for their exchange are also nullified. In stage three, agents are given the option to pulicly declare (and verify) the numer of papers they hold. Consumption rights are invalidated under two conditions: i) if the price that prevails in the market for papers is greater or equal to p and ii) if at least one agent declares a numer of consumption rights different to 1 φ g or zero. Stage four unfolds depending on what happened in stage three. If consumption rights are valid, then insurance companies decide whether to enter or not, they offer menus of contracts z i f and agents choose etween z f and exchanging 1 φ g rights for a contract z g f. An agent who deposits a smaller numer of papers than 1 φ g receives the high-risk type contract z f. If consumption rights are not valid, then firms offer the Rothschild-Stiglitz set of contracts {z RS, zrs g } and agents are left to choose etween the two. In stage five, uncertainty is resolved, payments for trading of rights are made, if they are valid, and consumption takes place. For Proposition 2 we use the Schmeidler (1973) definition of Nash equilirium with continuum of agents, namely deviations from hypothetical equiliria are made y a continuum of agents of the same type with ɛ measure. In light of the analysis of the game in section three, if a low-risk type elieves that consumption rights will e invalidated due to some sort of coordination-failure, we interpret her indifference on what action to choose in favor of ecoming a uyer of 1 φ g φ g units of papers. Thus, she ehaves as if there was some small proaility of a paper-dependent outside option, similar to the one of the previous section. We will return to the interpretation of this in the comments that follow the proposition. Proposition 2: The game of section four, descried aove, has multiple Perfect Bayesian- Nash equiliria, however, in all of them all agents reveal their type truthfully. Each one of these equiliria is characterized y a price for consumption-rights p [0, p). In all of them, at least two firms enter the insurance market, low-risk agents uy papers from high-risk agents, consumption-rights are valid and all insurance companies offer the same set of contracts {z FB g, z FB }. Proof: We first show that in stage four all firms offer the same set of contracts and that this is the same as the set of contracts offered y the mechanism in section three, conditional of consumption rights not eing invalidated in stage three. Assume otherwise. Assume that there is one firm in the economy and it offers contracts {z g f, z f } which are different from set of contracts {zg FB, z FB }, which the enevolent social planner offers in section three (recall that zg FB and z FB correspond to the full insurance contracts that the two types of agents receive in the case of perfect information). We distinguish etween two different cases. Case 1: z g f is such that the contract does not provide full insurance (z H g f > z L g f and zh f > z L f ). Due to risk aversion, another firm can choose another set of contracts which makes profit for the new firm, ut give higher expected utility to oth types of agents, so that all agents in the economy move to the new firm. For example, 14

15 the set of contracts {z g j, z j } = {zg FB ɛ, z FB }, where ɛ is an aritrarily small numer, is a profitale deviation for firm j. Therefore, no firm will offer contracts which do not provide full insurance to oth types in equilirium. Case 2: Let z g f e such that it provides full insurance to low-risk types, ut z g f < zg FB. Let also z f e any full insurance contract for high-risk types compatile with the firm making non-negative profits. Because z g f is not the first-est full insurance contract, another firm j still has a profitale deviation y offering z g f + ɛ. This contract strictly increases the utility of low-risk types and therefore, they strictly prefer it. Also, as long as z g j < zg FB, the firm makes strictly positive profits. Therefore no contract z g j < zg FB can e part of an equilirium. A similar argument can e used to estalish that no firm which makes strictly positive profits y its -type contracts will receive any high-risk customer. These previous cases show that no other comination of contracts, apart from {z FB, zfb g } can e a potential equilirium set of contracts. It remains to see if {z FB, zfb g } is an equilirium. If firm f offers zg FB (which implies that the firm also offers z FB, otherwise it makes negative profits), no other firm can profitaly deviate y offering any other set of contract, ecause any other set will e either loss-making for the firm or low-risk agents will not prefer it. At least one more firm is required to enter in order to support this equilirium, however, otherwise the est-response of firm f is to set another set of contracts so that it increases its profits (from the previous example we know that such offers exist). Therefore, if at least two firms enter the insurance market and offer the set {z, z g }, there is no profitale deviation for any of them. Second, we show that all agents will truthfully reveal their types for any price p [0, p). Recall that, in stage three, property rights will e invalidated if at least one agent announces pulicly that she has a different numer of rights than 1 φ g or zero or if the price of the papers is not elow p. A low-risk agent who has acquired some numer of rights x < 1 φ g always prefers to announce this pulicly in stage 3, since, y invalidating consumption rights, she will e ale to choose z RS g, which she strictly prefers to z FB. Notice that, due to risk aversion, if agents elieve, in stage two, that consumption rights are valuale (i.e. they will not e invalidated in stage three), then their state-contingent price will e independent of state: p H = p L. Otherwise, there is loss of efficiency through the risk premium uyers of the rights pay to sellers. For the rest of the proof we consider only state-independent payments. Since oth types of agents start with one paper, each agent must ecome either a seller of one unit or a uyer of 1 φ g φ g units of the paper in order to e receive the first-est contracts y firms. Following a similar reasoning as in Proposition 1, it is a strictly dominant strategy for a low-risk agent to uy 1 φ g φ g units of rights than taking any other action. Suppose that g elieves that the market will function smoothly and the price of rights p is strictly elow p. Since she strictly prefers allocation cg FB to allocations c FB and c RS g, then her optimal action is to uy the ( correct amount of papers, which is implied y the fact that {U g c FB g 1 φ g φ g p ) U g (c RS g U g (c RS + p). If g elieves that the market for rights will not function smoothly (i.e. either the price will not e elow the threshold or some agents will not e allocated the quantity of papers they demand) she still prefers to demand the same quantity of papers, since she prefers allocation c RS g to c RS. 15

16 Given the dominant strategy for low-risk agents, the aggregate demand for papers y low-risk agents is 1 φ g. A high-risk agent chooses etween ecoming a seller of the paper or mimicking low-risk types and ecoming a uyer. If (or a small mass of type- agents) ecomes a seller of her paper, conditional on the other high-risks eing sellers, then the market clears and all agents have either 1 φ g or zero papers. She will receive the final allocation c RS + p. If ecomes a uyer of papers, however, then the market for papers does not clear, as aggregate demand exceeds supply, and she can not receive anything more than c RS for any possile case. This is ecause, if she is completely or partially rationed-out y the market (she receives some quantity of papers strictly less than 1 φ g ) she will still receive the contract z RS, irrespectively of whether consumption-rights are validated or not. If she receives the quantity demanded, then some mass of low-risk agents e will partially or fully rationed out y the market, and their est-response for them is to declare it in stage three, so that consumption-rights are invalidated. The again expects to receive contract z RS. Since allocation c RS + p is weakly preferred to allocation c RS y, her est-response is to sell all her consumption rights in the market, given the dominant strategy y agents of type g. This means that the unique Perfect-Bayesian Nash equilirium strategy for all agents is to truthfully reveal their type as long as p [0, p). We have also previously estalished that on the equilirium path property rights are valid and all firms that enter the market offer the set of contracts {zg FB, z FB }. This concludes the proof. Two remarks are useful for clarifying this result. First, the choice of the outside option is imposed exogenously in the game aove. However, it is possile to derive it endogenously. Consider the modified game in which a two-stage signaling game is played off-theequilirium path, if consumption rights are invalidated in stage three. In this su-game, agents offer move first and propose allocations (or contracts, it is equivalent) to insurance firms and firms decide whether to accept them or not. One can show (see for example Cho and Kreps (198?) and Maskin and Tirole (1994)) and the unique Perfect Bayesian-Nash equilirium of the game which passes the intuitive criterion is the Rothschild-Stiglitz pair of allocations and this su-game can give microeconomic foundations to the outside option of our game. Second, as we mentioned earlier, we assume in the proof that if a low-risk agent elieves that the market for consumption rights will fail and the papers will e invalidated, she will still demand 1 φ g φ g papers, despite the fact that this does not affect the final allocation she will consume. The justification for this is similar to the argument of the previous section. Suppose that agents elieve that if the market for rights fails there is a small proaility of government intervention. The government takes control of the economy and provides allocations according to demand or supply of papers y agents. By taking the limit of this small proaility as it goes to zero, one can justify the way we interpret indifference in favor of the first-est equilirium. Besides these technical remarks, the intuition for the result is simple. Asymmetric information aout characteristics acts like an externality, as any type can lie aout his type and reap the social enefits that accrue from the alleged characteristics. Incentive compatiility forces some of the cost to low-risk types, as they must ear some uncertainty in their con- 16

17 sumption in order to reveal their type. The allocation and trading of the papers allows the extraction of useful information aout types, in the same way that sending messages aout ones type does. Furthermore, lying aout ones types causes distortions in the allocation of the papers which is used in order to make the agent internalize the cost of his actions and satisfy incentive compatiility. Of course, this requires some information aggregation. This aggregation is done through the information aout the price of the consumption rights and the distriution of aggregate demand and supply. Given that agents anticipate this process of information aggregation to work, competition among firms for clients ensures that the first-est contracts will e the only ones offered in equilirium. Therefore, the game of this section can achieve in a rather decentralized setting the same outcome as the more centralized mechanism of section three. 5 Conclusion This paper demonstrates how information aggregation can improve the outcomes achieved in insurance markets with adverse selection either in the case where the interim cumulative distriution is common knowledge or in the case where the ex-ante proaility distriution is known and there is a continuum of agents. The main result is that information aggregation can e used to discover if there exists some agent who does not truthfully report his type and implement first-est allocations. This is done y making agents internalize the effects of mis-representing their types through the design of allocations off-the-equilirium path, which make low-risk agents strictly prefer truthful revelation irrespectively of the actions of other players. Finally, we show how the result can e applied to a competitive insurance market with free entry of insurance companies. Of course, there are certain limitations in our analysis. We have formally proven our results for the simple case, where there are only two types, two states of nature and agents have identical preferences (i.e. they differ only in terms of the proaility of the accident). However, we elieve that none of these assumptions is crucial for our result that that we can prove it even for cases where the single crossing condition is violated. We are currently working on relaxing these assumptions and on providing necessary and sufficient conditions for characterizing the more general prolem. 17

18 References Aumann J. Roert (1966): Existence of Competitive Equiliria in Markets with a Continuum of Traders, Econometrica, 34, Ausuel, L., (2004): An Efficient Ascending-Bid Auction for Multiple Ojects, American Economic Review, 94, Ausuel, L., (2006): An Efficient Dynamic Auction for Heterogeneous Commodities, American Economic Review, 96, Bisin, A., and P. Gottardi (2006): Efficient Competitive Equiliria with Adverse Selection, Journal of Political Economy, 114, Cho, I. K., and D. Kreps, (1987): Signalling Games and Stale Equiliria, Quarterly Journal of Economics, 102, Dasgupta, P., and E. Maskin (2000): Efficient Auctions, Quarterly Journal of Economics, 115, Engers, Maxim and Luis Fernandez (1987): Market Equilirium with Hidden Knowledge and Self-Selection, Econometrica, 55, Gale, D. (1992): A Walrasian Theory of Markets with Adverse Selection, Review of Economic Studies, 59, Gale, D. (1996): Equilira and Pareto Optima of Markets with Adverse Selection, Economic Theory, 7, Hellwig, M. (1987): Some Recent Developments in the Theory of Competition in Markets with Adverse Selection, European Economic Review, 31, Jehiel, P., and B. Moldovanu (2001): Efficient Design with Interdependent Valuations, Econometrica, 69, Koufopoulos, Kostas (2008): Existence and Efficiency of Nash Equiliria in Insurance Markets with Adverse Selection, working paper Maskin, E., and J. Tirole (1992): The Principal-Agent Relationship with an Informed Principal, II: Common Values, Econometrica, 60, McLean, R., and A. Postlewaite (2002): Informational Size and Incentive Compatiility, Econometrica, 70, McLean, R., and A. Postlewaite (2004): Informational Size and Efficient Auctions, Review of Economic Studies, 71,

19 Mezzetti, C. (2004): Mechanism Design with Interdependent Valuations: Efficiency, Econometrica, 72, Prescott, E., and R. Townsend (1984): Pareto Optima and Competitive Equiliria with Adverse Selection and Moral Hazard, Econometrica, 52, Riley, J. (1979): Informational Equilirium, Econometrica, 47, Rothschild, M., and J. Stiglitz (1976): Equilirium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information, Quarterly Journal of Economics, 90, Rustichini Aldo, Mark A. Satterthwaite, Steven R. Williams (1994), Convergence to Efficiency in a Simple Market with Incomplete Information, Econometrica, 62, Rustichini, A., and P. Siconolfi (2008): General Equilirium in Economies with Adverse Selection, Economic Theory, 37, Schmeidler, D. (1973): Equilirium points of nonatomic games, Journal of Statistical Physics, 7, Wilson, C. (1977): A Model of Insurance Markets with Incomplete Information, Journal of Economic Theory, 16,