Chapter 7 Markets with Adverse Selection 7.1 A market model These notes introduce some ideas for modeling markets with adverse selec- tion. Thisframeworkwasoriginallyintendedtodealwithmarketsthat cannot be easily accommodated by the standard signaling game e.g., be- cause there is two-sided adverse selection. For present purposes, however, it is enough to deal with the simplest case in which there is adverse selection on one side of the market only. The use of the competitive paradigm to analyze markets with adverse selection goes back to Spence (1973). The ideas presented here were devel- oped in a series of papers Gale (1991, 1992, 1996). There are two mutually exclusive classes of individuals (agents). We can think of them as buyers and sellers, but nothing depends on this interpretation. The agents on one side of the market have private information, so we call them the informed agents. The agents on the other side of the market are the uninformed.Anagent’s private information is represented by his type. There is a finite set of types T. Each type consists of a (non-atomic) continuum of identical agents and the measure of agents of type t is denoted by N(t) > 0.ThereareM>0 uninformed agents. There is a finite set of contracts Θ. (Later the theory is extended to an infinite set). Each contract involves one agent from each side of the market. If an uninformed agent exchanges a θ contract with an informed agent of type t, the uninformed agent’s payo? is u(θ,t) and the informed agent’s payo? is v(θ,t). Each agent has a reservation utility, the utility he gets if a contract is 1 2 CHAPTER 7. MARKETS WITH ADVERSE SELECTION not exchanged and he has to take his next best option. With an appropriate normalization of the payo? functions, the reservation utility is 0 for every type. The equilibrium choices made by the agents are described by an alloca- tion, that describes the number of agents of each type that chooses a given contract. An allocation of agents consists of a pair of functions (f,g) where f : Θ → R + and g : Θ × T → R + .Weinterpretf(θ) as the measure of uninformed agents choosing contract θ and g(θ,t) as the measure of informed agents of type t choosing contract θ. The allocation (f,g) is attainable if X θ f(θ) ≤ M and X θ g(θ,t) ≤ N(t),?t. The attainability condition contains an inequality because it is possible that some agents will choose their outside option (i.e., no trade) When an agent selects a contract θ he does not know the probability of trade or the type of agent from the other side of the market that he will be matched with. Let λ(θ,t) denote the probability that an uninformed agent choosing contract θ will exchange the contract with a type-t agent and let μ(θ) denote the probability that an informed agent choosing contract θ will exchange the contract with an uninformed agent. Note that all agents on a given side of the market have the same beliefs about their trading possibilities, that is, the same probability assessment of trading with a given type on the other side of the market. The probability assessment (λ,μ) is consistent with the allocation (f,g) if λ(θ,t)= g(θ,t) m(θ) ,?t μ(θ)= f(θ) m(θ) , for any contract θ such that m(θ) > 0,wherem(θ) measures the long side of themarketforcontractθ,thatis, m(θ)=max ( f(θ), X t g(θ,t) ) . 7.2. STABILITY 3 If m(θ)=0consistency is automatically satisfied. Then a market equilibrium consists of an (attainable) allocation (f,g) and a probability assessment (λ,μ) such that the allocation maximizes the payo? of each type, that is, f(θ) > 0 implies X t u(θ,t)λ(θ,t)=u ? =max θ ( X t u(θ,t)λ(θ,t) ) ,?θ; and g(θ,t) > 0 implies v(θ,t)μ(θ)=v ? (t)=max θ {v(θ,t)μ(θ)},?θ,t. The equilibrium (f,g,λ,μ) is said to be orderly if at most one side of the market for any contract is rationed, that is, max ( X t λ(θ,t),μ(θ) ) =1. Without this requirement there exist many trivial equilibria. 7.2 Stability A perturbation is an allocation (f,g) such that f(θ)= X t g(θ,t) > 0,?θ. Define an equilibrium for the (ε, ? f,?g)-perturbed market by replacing the allocation (f,g) by (1?ε)(f,g)+ε( ? f,?g) in the equilibrium conditions above. Then a perfect market equilibrium (f,g,λ,μ) is definedtobethelimitof a sequence of equilibria (f ε g ε ,λ ε ,μ ε ) of the (ε, ? f,?g)-perturbed market as ε converges to 0. Note that in a perturbed market the probability assessment is uniquely determined by the allocation and the consistency condition. Call (f,g) an equilibrium allocation if (f,g,λ,μ) is a market equilib- rium for some (λ,μ) and call (f,g) an equilibrium allocation for the (ε, ? f,?g)- perturbedmarketif(f,g,λ,μ) isamarket equilibriumof the (ε, ? f,?g)-perturbed market for some (λ,μ). An attainable allocation (f,g) is stable if, for any 4 CHAPTER 7. MARKETS WITH ADVERSE SELECTION perturbation ( ? f,?g) there is a sequence of equilibrium allocations (f ε ,g ε ) such that (f ε ,g ε ) is an equilibrium allocation of the (ε, ? f,?g)-perturbed market and lim ε→0 (f ε ,g ε )=(f,g). Note that the probability assessments do not necessarily converge to a unique limit. It is easy to see that (f,g) must be an equilibrium allocation if (f,g) is stable. Let (f,g) be a stable allocation and let t 0 be a fixed but arbitrary type. For any type t,letu ? denote the equilibrium payo? of the uninformed and v ? (t) the equilibrium payo? of type t. Then there exists an equilibrium (f,g,λ,μ) such that for any contract θ and any type t 6= t 0 , [v ? (t) >μ(θ)v(θ,t)] =? λ(θ,t)=0. To see this, let (f k,n ,g kn ) be the perturbation defined by f kn (θ)=(k + |T|?1)/n g kn (θ,t)= ? 1/n t 6= t 0 k/n t = t 0 . By choosing k and n appropriately, we can ensure that (f kn ,g kn ) is an at- tainable allocation. By stability, there exists a sequence (f ε ,g ε ,λ ε ,μ ε ) con- verging to (f,g,λ kn ,μ kn ) as ε → 0,where(f ε ,g ε ,λ ε ,μ ε ) is an equilibrium for the (ε,f kn ,g kn )-perturbed game. By compactness, the sequence ? (λ kn ,μ kn a has a limit point μ 0 and it is clear that if (f,g,λ kn ,μ kn ) is an equilibrium, then so is (f,g.λ 0 ,μ 0 ). Consider t 6= t 0 .Foreachθ,if μ kn (θ)v(θ,t) <v ? (t) then, for all ε>0 su?ciently small, μ ε (θ)v(θ,t) <v ε (t), where v ε (t) is the equilibrium payo? in (f ε ,g ε ,λ ε ,μ ε ).Thenfort 6= t 0 λ ε (θ,t)= g ε (θ,t) m ε (θ) = 1/n m ε (θ) ≤ 1/n (k + |T|?1)/n = 1 (k + |T|?1) . 7.3. ROBUSTNESS OF SEPARATING EQUILIBRIUM 5 So, in the limit, λ kn (θ,t)=1/(k + |T|?1) and, taking limits as k,n →∞ , it follows that λ 0 (θ,t)=0. Since this is true for any θ and t, the desired property holds. Note that if (f,g,λ,μ) is a perfect market equilibrium then, by construc- tion, (λ,μ) is orderly. The reason is that for any perturbation, the definition of consistency implies that every (λ,μ) is orderly and it remains so in the limit. 7.2.1 A continuum of contracts The assumption of a finite number of contracts is convenient. It simplifies the description of an equilibrium and makes the existence of equilibrium a technically straightforward matter. For some purposes, it is more convenient to have a continuum of contracts. In particular, when it comes to charac- terizing the degree of separation in an equilibrium it is nice to be able to consider “neighboring” contracts. So let us assume that Θ is a subset of some finite-dimensional Euclidean space and suppose that u(·,t) and v(·,t) are continuously di?erentiable functions of θ on some open superset of Θ. The theory can be extended from a finite subset of Θ to the entire space by taking limits, but for simplicity I shall assume that the definition of equi- librium and the restrictions on beliefs, derived above, can be applied directly to the limit market. With the assumption that (f,g) has a finite support, the definition of equilibrium extends in the obvious way. A stable allocation (f,g) is defined to be an equilibrium allocation such that for any type t 0 we can find an equilibrium probability assessment (λ,μ) having the properties derived in the proposition above. 7.3 Robustness of separating equilibrium Let (f,g) be a stable allocation and suppose that there exists a contract θ 0 belongingtotheinteriorofΘ such that two or more types “pool” at θ 0 .We can assume without loss of generality that there exists a pair t 6= t 0 and that g(θ 0 ,t) > 0 and g(θ 0 ,t 0 ) > 0. Let u ? and v ? (t) denote the equilibrium payo?s to the uninformed and type t, respectively, for the equilibrium allocation (f,g) and let T 0 = {t ∈ T|v ? (t)=μ(θ 0 )v(θ 0 ,t)} 6 CHAPTER 7. MARKETS WITH ADVERSE SELECTION Note that the definition of T 0 is determined by (f,g) independently of the particular choice of (λ,μ). We assume that the types are ranked in the following sense: for any pair t<t 0 , u(θ,t) <u(θ,t 0 ), for any θ within a su?ciently small neighborhood of θ 0 . Because the number of types is fixed and the utility functions are continuous, we can assume that these inequalities hold within a fixed, compact neighborhood. Then there exists an ε>0 such that u(θ,t)+ε<u(θ,t 0 ) for any θ in the neighborhood. Let t 0 be the best type in T 0 (from the other side’s point of view) and let (f,g,λ,μ) be the equilibrium whose probability assessment obeys the restrictions: [v ? (t) >μ(θ)v(θ,t)] =? λ(θ,t)=0,?t 6= t 0 . To avoid some pathological cases, we assume that every t ∈ T 0 has a positive equilibrium payo? v ? (t) > 0. Without loss of generality we can normalize the payo? functions so that v(θ 0 ,t)=1for all t ∈ T 0 . Now suppose that there exists a contract θ arbitrarily close to θ 0 satisfying v(θ,t 0 ) > 1 >v(θ,t),?t ∈ T 0 ,t6= t 0 . Then it follows that λ(θ,t)=0for any t 6= t 0 . Toseethis,notethatthe equilibrium condition for t 0 implies that μ(θ)v(θ,t 0 ) ≤ μ(θ 0 )v(θ 0 ,t 0 )=μ(θ 0 ) which in turn implies that μ(θ) <μ(θ 0 ), because v(θ,t 0 ) > 1,sothat μ(θ)v(θ,t) ≤ μ(θ) <μ(θ 0 )=μ(θ 0 )v(θ 0 ,t),?t ∈ T 0 ,t6= t 0 . For θ su?ciently close to θ 0 , u(θ,t 0 ) ≈ u(θ 0 ,t 0 ) > P t λ(θ 0 ,t)u(θ 0 ,s) so P t λ(θ,t)=λ(θ,t 0 ) < 1. Then orderliness implies that μ(θ)=1,contradict- ing the equilibrium condition μ(θ)v(θ,t 0 ) ≤ μ(θ 0 )v(θ 0 ,t 0 ) ≤ 1. 7.4. EQUILIBRIUM RATIONING 7 Proposition 1 Let (f,g) be a stable allocation and (λ,μ) an equilibrium probability assessment. For any contract θ 0 let T 0 = {t : μ(θ 0 )v(θ 0 ,t)= v ? (t)}.Supposethatv ? (t) > 0 for any t ∈ T 0 and that for any ε>0 there is a contract θ that is ε-close to θ 0 such that v(θ,t 0 ) > 1 >v(θ,t),?t ∈ T 0 ,t6= t 0 , where t 0 is the best type in T 0 and v(θ 0 ,t 0 )=1=v(θ 0 ,t). Then there is at most one type t such that g(θ 0 ,t) > 0. Note that this proposition does not imply that θ 0 is only optimal for one type. Typically, there will be another type t 0 6= t such that μ(θ 0 )v(θ 0 ,t 0 )= v ? (t 0 ) although g(θ 0 ,t 0 )=0. 7.4 Equilibrium rationing We want the set of contracts to be “large”, to allow for “all possible con- tracts”. This means that some contracts will not be traded in equilibrium, in fact, some cannot be traded. For example, since contracts include the terms of trade and it cannot be the case that contracts requiring di?erent “prices” for the same “good” are available in equilibrium, some contracts must be “rationed”. This kind of “rationing” is analogous to missing markets or the e?ect of the classical budget constraint in ruling out the availability of some commodity bundles. There is a narrower sense in which rationing occurs in equilibrium. Sup- pose that a contract is traded by some agents but the probability of trade is less than one. This kind of rationing is di?erent from missing markets. A market clearly exists for this contract but some individuals who attempt to trade the contract will find themselves constrained ex post. Suppose that m(θ) > 0 and that either μ(θ) < 1 or P t λ(θ,t) < 1.In that case, we say that the contract θ is actively rationed. Now suppose that (f,g) is a separating, stable allocation and that con- tract θ 0 is actively rationed. Suppose that g(θ 0 ,t 0 ) > 0 and consider first the case where g(θ 0 ,t 0 ) >f(θ 0 ).Let T 0 = {t ∈ T|v ? (t)=μ(θ 0 )v(θ 0 ,t)} where (λ,μ) is the equilibrium probabilitity assessment satisfying [v ? (t) >μ(θ)v(θ,t)] =? λ(θ,t)=0,?t 6= t 0 , 8 CHAPTER 7. MARKETS WITH ADVERSE SELECTION for any contract θ 6= θ 0 su?ciently su?ciently close to θ 0 . As usual, we can assume that t 0 is the best type, i.e., the type preferred by the other side of themarketforanyθ near θ 0 . If v ? (t 0 )=0, thenwithout lossof generalitywecaneliminatethe rationing of that type by reducing g(θ 0 ,t 0 ) until f(θ 0 )=g(θ 0 ,t 0 ).So,considerthecase where v ? (t 0 ) > 0. To rule out di?cult cases, we assume that v ? (t) > 0 for all t ∈ T 0 . Then we can normalize v(θ 0 ,t)=1for all t ∈ T 0 . Suppose there exists a contract θ in the neighborhood of θ 0 such that v(θ,t 0 ) >v(θ 0 ,t 0 )=1, and v(θ,t 0 ) >v(θ,t),?t ∈ T 0 ,t6= t 0 . The equilibrium condition requires that μ(θ) < 1 so P t λ(θ,t)=1.But λ(θ,t) > 0 for t 6= t 0 only if μ(θ)v(θ,t)=v ? (t). Suppose that this is true. Then v(θ,t)= μ(θ 0 ) μ(θ) ≥ v(θ,t 0 ), contradicting our hypothesis. Thus, λ(θ,t)=0for all t 6= t 0 and λ(θ,t 0 )=1, contradicting the equilibrium condition for the long side. Suppose now that the uninformed side of the market is actively rationed at θ 0 ,thatis,g(θ 0 ,t 0 ) <f(θ 0 ).Ifλ(θ 0 ,t 0 )u(θ 0 ,t 0 )=0then we can eliminate the rationing by changing the allocation (f,g).Ifλ(θ 0 ,t 0 )u(θ 0 ,t 0 ) > 0 and λ(θ 0 ,t 0 ) < 1 then μ(θ 0 )=1. Suppose there exists a nearby contract θ such that v(θ,t 0 ) >v(θ 0 ,t 0 )=1and v(θ,t 0 ) >v(θ,t),?t ∈ T 0 ,t6= t 0 . Then μ(θ) < 1 and λ(θ,t 0 )=1by the usual argument. But for θ su?ciently close to θ 0 , continuity implies that u(θ,t 0 ) >λ(θ 0 ,t 0 )u(θ 0 ,t 0 ), contradicting the equilibrium condition. Thus, we have proved the following result. Proposition 2 Suppose the allocation (f,g) is stable and separating. Let θ 0 be a contract and T 0 the set of types for which θ 0 is optimal; suppose that v ? (t) > 0 for all t ∈ T 0 and v(θ 0 ,t)=1for all t ∈ T 0 .Lett 0 be the unique type that chooses θ 0 ,thatis,g(θ 0 ,t 0 ) > 0. Then there is no active rationing at θ 0 if the following conditions are satisfied: for any ε>0, there exists a contract θ that is ε-close to θ 0 and satisfies v(θ,t 0 ) >v(θ 0 ,t 0 )=1, 7.5. MORAL HAZARD AND CREDIT RATIONING 9 and v(θ,t 0 ) >v(θ,t),?t ∈ T 0 ,t6= t 0 . 7.5 Moral hazard and credit rationing The phenomenon of credit rationing is important for several reasons. At the empirical level, it has been argued that firms face constraints on their ability to raise external finance,whichinturnraisesquestionsaboutthee?ciency of the allocation of investment. At the theoretical level, the possibility that “prices” do not adjust to clear markets goes to the heart of the perfect compe- tition paradigm. Understanding credit rationing may help us to understand phenomena such as unemployment, under-insurance, and so forth. 7.5.1 The Stiglitz-Weiss model There is a continuum of entrepreneurs, each of whom has a single project. The entrepreneur can choose one of two development strategies i =1,2 characterized by the parameters (θ i ,y i ),whereθ i is the probability of success and y i is the payo? to the project in the event of success. An unsuccessful project has a payo? ofzero.Weassumethat θ 1 y 1 = θ 2 y 2 ,θ 2 >θ 1 ,y 2 <y 1 . Each project requires an invesment of one unit of the numeraire good. There is a continuum of investors each of whom has an initial endowment of one unit of the numeraire and wants to invest in one project. All agents are risk neutral. The entrepreneurs have a reservation utility of zero and the investors have a reservation utility of one (suppose, for example, that their alternative to investing in a risky project is to invest in a safe asset that has azeronetreturn). Investors and entrepreneurs are matched at random. Projects are financed using a standard debt contract: in exchange for the investment of one unit in his project, the entrepreneur promises to pay the investor r units after the project pays o?. When the contractual payment is r, the entrepreneur’s payo?from a type- i project is θ i max{y i ?r,0}. There is a unique contract 0 <r ? <y 2 such that the entrepreneur prefers type 2 if r<r ? and prefers type 1 if r ? <r<y 2 . For r ≥ y 2 both projects earn zero for the entrepreneur. 10 CHAPTER 7. MARKETS WITH ADVERSE SELECTION The investors expected payo? from the project is θ 2 r for r ≤ r ? and θ 1 r for r ? <r<y 2 . NotethatIassumetheentrepreneurischoosing2 when r = r ? . This is not the only optimal choice for the entrepreneur, but it is the only equilibrium choice: if the entrepreneur chooses 1 when r = r ? then the investor will prefer r = r ? ? ε for all ε>0 su?ciently small, which is not possible in equilibrium. Suppose, for example, that θ 2 r ? >θ 1 y 1 ,θ 2 r ? ≥ 1. Then it is an equilibrium for all investors and entrepreneurs to choose the contract r ? . This is the global optimum for the investors and it is the only contract that can be traded in equilibrium by the entrepreneurs. If there are more entrepreneurs than investors, there will have to be active rationing at r ? . An entrepreneur facing the prospect of rationing would gladly pay r ? +ε and get finance for sure. But a higher interest rate is not attractive to the investors, who perceive that it violates the incentive constraint, leading to the choice of the high risk project and lowering the investor’s payo?. 7.5.2 The Bester model One of the reasons why we observe active rationing is that there is only one variable in the contract, the interest rate, which cannot be used simultane- ously to clear the market and provide incentives to choose the better project. Bester (1985) shows how collateral can be used to improve the incentives in the contract (relax the incentive constraint) and allow the interest rate to be used to clear the market. Suppose that eachentrepreneurhas an asset that can beused as collateral. The total value of the collateral is K and the value surrendered in the event of the failure of the project is 0 ≤ k ≤ K. Then a contract is an ordered pair (r,k) and the entrepreneur prefers type-2 projects if and only if θ 2 (y 2 ?r)+(1?θ 2 )k>θ 1 (y 1 ?r)+(1?θ 1 )k. Given the assumption of equal expected outcomes θ 1 y 1 = θ 2 y 2 ,type-2 are preferred if and only if r<k. The value of the collateral may be less to the investor than to the en- trepreneur, in which case the use of collateral involves a deadweight loss. Suppose that the value of the collateral to the investor is a fraction 0 <γ<1 7.6. ADVERSE SELECTION AND CREDIT RATIONING 11 of its worth to the entrepreneur. Then it will be optimal to minimize the use of collateral and put r = k (putting r>kwill have no incentive e?ect and consequently it will be of no value at all). Then as long as K>y 2 it will be possible to achieve the no-rationing outcome. The equilibrium contract will be (r ? ,k ? )=(y 2 ,y 2 ). At this contract, entrepreneurs are indi?erent between undertaking a project and remaining inactive and investors are all willing to accept the contract. It is easy to see that investors prefer this contract to any other contract of the form (r,r) such that r<r ? andthatinvestorspreferittoanyother contract of the form (r,r) such that r>r ? . Since contracts of the form (r,r) and (r,0) are the only Pareto e?cient contracts, we know that any contract (r ? ,k ? ) is worse for either the entrepreneurs or the investors and so we can use orderly rationing to support an equilibrium in which (r ? ,k ? ) is chosen. 7.6 Adverse selection and credit rationing In the moral hazard model described above, entrepreneurs are identical and choose projects of di?erent types. In a model of adverse selection, en- trepreneurs are endowed with di?erent types of projects. Consider the ana- logue of the two-type model above. There are two types of entrepreneurs i =1,2 with projects (θ i ,y i ),whereθ 1 y 1 = θ 2 y 2 ,θ 2 >θ 1 and y 2 <y 1 .Each project requires an invesment of one unit of the numeraire good. There is a continuum of investors each of whom has an initial endowment of one unit of the numeraire and wants to invest in one project. All agents are risk neutral. The entrepreneurs have a reservation utility of zero and the investors have a reservation utility of one. We assume that there are more entrepreneurs than investors, so rationing must occur in equilibrium unless entrepreneurs are at their reservation level. Suppose the face value of the debt is r>0.Thepayo?to an entrepreneur of type i is θ i max{y i ?r,0} and the payo? to the investor is θ i min{y i ,r}. 12 CHAPTER 7. MARKETS WITH ADVERSE SELECTION 7.6.1 Pooling equilibrium There exists a pooling equilibrium in which all agents choose the contract r ? . Rationing occurs in this equilibrium because, although entrepreneurs would be willing to accept a higher interest rate r = r ? + ε and get finance for sure rather than face the prospect of rationing, investors are not attracted to a higher interest rate because they believe that they are more likely to be matched with a high-risk type at the higher interest rate. Let μ(r) denote the probability that an entrepreneur can trade a contract r and let μ i (r) denote the probability that an investor can trade a contract r with an entrepreneur of type i =1,2. In the pooling equilibrium, the entrepreneurs’ payo?sare μ(r ? )θ i (y i ?r ? ) ≥ 0,i=1,2, and the investors’ payo?sare X i μ i (r ? )θ i r ? = ˉ θr ? ≥ 1, where ˉ θ is the average probability of success. To support this equilibrium we have to choose the probabilities μ(r),μ 1 (r) and μ 2 (r) so that no one wants to deviate. Let ˉr denote the value of r satisfying θ 1 ˉr = ˉ θr ? and assume that θ i max{y i ? ˉr,0} <μ(r ? )θ i (y i ?r ? ),i=1,2, whichwillbetrueifr ? is not “too” low. Then set μ(r)=0,μ 1 (r)=1,μ 2 (r)=0,?r ? <r≤ ˉr and μ(r)=1,μ 1 (r)=0,μ 2 (r)=0,?r>ˉr. For r<r ? we simply assume that the investors believe that only the bad types are forthcoming: μ(r)=0,μ 1 (r)=1,μ 2 (r)=0,?r<r ? . 7.6. ADVERSE SELECTION AND CREDIT RATIONING 13 It is easy to check that under the maintained assumptions the agents will not want to deviate to a contract r 6= r ? with these probability assessments. This equilibrium is not “stable”, however. In particular, if we perturbed the equilibrium by assigning a small measure of the good types to every contract r, there is no way that the types could endogenously re-allocate themselves to o?set the e?ect of the perturbation. The problem does not arise with the higher interest rates r>r ? ; it arises with lower interest rates r<r ? . In an equilibrium of the perturbed model, contracts r<r ? must be more heavily rationed than r ? in order to discourage defections from r ? .But rationing hurts the high-risk entrepreneurs more than low-risk entrepreneurs, because they have higher payo?s. Thus, if the low-risk entrepreneurs weakly prefer r ? to r, the high-risk entrepreneurs will strictly prefer r ? to r. Formally, μ(r ? )θ 2 (y 2 ?r ? ) ≤ μ(r)θ 2 (y 2 ?r) implies that μ(r) <μ(r ? ), and together with y 1 >y 2 this implies that μ(r ? )θ 1 (y 1 ?r ? ) <μ(r)θ 1 (y 1 ?r). Thus, in an equilibrium of the perturbed model, the investors’ beliefs must assign probability one to the high type, conditional on trading r<r ? : μ(r) <μ(r ? ),μ 1 (r)=0,μ 2 (r)=1,?r<r ? . But this is clearly inconsistent with equilibrium, since X i μ i (r)θ i r = θ 2 r> ˉ θr ? for r<r ? su?ciently close to r ? .Thus,r ? cannot be a stable outcome. 7.6.2 Separating equilibrium There exists a separating equilibrium in which each type of entrepreneur gets a distinct contract. However, since every entrepreneur would prefer a lower interest rate to a higher one, other things being equal, the only way two constracts can be traded in equilibrium is if one of them is rationed. As usual, the stability criterion will select the best (Pareto-dominant) separating equilibrium. In this equilibrium, the high-risk types are not ra- tioned and have issue debt with a high face value r 1 . The low-risk types issue debt with a lower face value r 2 and accept a probability of trade μ(r 2 ) < 1. 14 CHAPTER 7. MARKETS WITH ADVERSE SELECTION To make the necessity of rationing very clear, we can assume that there are more investors than entrepreneurs. Then the investors must trade with probability one in equilibrium and they can do this only if they are at their reservation utility. This implies that θ i r i =1. Thentheprobabilityoftradeforthelow-risktypeisdeterminedbythe condition that the high-risk type is indi?erent between r 1 and r 2 : θ 1 (y 1 ?r 1 )=μ(r 2 )θ 1 (y 1 ?r 2 ), or μ(r 2 )=(y 1 ?r 1 )/(y 1 ?r 2 ) < 1. To support this equilibrium we define the functions μ(r),μ 1 (r) and μ 2 (r) as follows: μ(r)= ? ? ? 0 r<r 2 (y 1 ?r 1 ) (y 1 ?r) r 2 <r<r 1 1 r 1 <r; (μ 1 (r),μ 2 (r)) = ? ? ? ? ? (0,1) r<r 2 3 θ 2 r?1 (θ 2 ?θ 1 )r , 1?θ 1 r (θ 2 ?θ 1 )r ′ r 2 <r<r 1 (0,0) r 1 <r. With these beliefs, the investors are unwilling to lend at r<r 2 and just willingtolendatr 2 <r<r 1 . They are rationed at r>r 1 . The entrepreneurs are rationed at r<r 2 and unwilling to borrow at r>r 1 . The high-risk type is just willing to borrow at r 2 <r<r 1 and the low-risk type is unwilling to borrowinthesamerange. 7.6.3 Collateral Now suppose that each type of entrepreneur has an asset that can be used as collateral. The asset is worth K to the entrepreneur and γK to the investors, where 0 <γ<1. A contract now specifies an ordered pair (r,k) where r is the repayment in the event of success and k is the value of collateral surrendered by the entrepreneur in the event of failure. (It does not pay to surrender collateral in the event of success because this weakens the incentive to avoid risk). 7.6. ADVERSE SELECTION AND CREDIT RATIONING 15 Because surrendering collateral is ine?cient ex post, the contract should minimize the use of collateral. The best (Pareto-dominant) separating equi- librium will be one in which the high-risk type uses no collateral and the low-risk type uses just enough to distinguish itself from the high-risk type. Thus, if the equilibrium contracts are denoted by (r 1 ,k 1 ) and (r 2 ,k 2 ) we must have (r 1 ,k 1 )=(1/θ 1 ,0) and (r 2 ,k 2 )=(1/θ 2 ,k 2 ) where k 2 satisfies θ 1 (y 1 ?r 1 )=θ 1 (y 1 ?r 2 )?(1?θ 1 )k 2 . References Bester, H. “Screening vs. Rationing in Credit Markets with Imperfect In- formation,” American Economic Review 75 (1985) 850-55. Banks, Je?rey and Joel Sobel. “Equilibrium Selection in Signaling Games,” Econometrica 55 (1987) 647-61. H. Bester, “Screening vs Rationing in Credit Markets with Imperfect Infor- mation,” American Economic Review 75 (1985) 850-855. Cho, In Koo and David Kreps. “Signaling Games and Stable Equilibria,” Quarterly Journal of Economics 102 (1987) 179-221. Engers, Maxim. “Signalling with Many Signals,” Econometrica 55 (1987) 663-74. Engers, Maxim and Luis Fernandez. “Market Equilibrium with Hidden Knowledge and Self-selection,” Econometrica 55 (1987) 425-39. Gale, Douglas. “Incomplete Mechanisms and E?cient Allocation in Labour Markets,” Review of Economic Studies 58 (1991) 823-51. Gale, Douglas. “A Walrasian Theory of Markets with Adverse Selection,” Review of Economic Studies 59 (1992) 229-55. 16 CHAPTER 7. MARKETS WITH ADVERSE SELECTION Gale, Douglas. “Equilibria and Pareto Optima of Markets with Adverse Selection,” Economic Theory 7 (1996) 207-35. Kohlberg, Elon and Jean-Francois Mertens. “On the Strategic Stability of Equilibria,” Econometrica 54 (1986) 1003-37. Quinzii, Martine and Jean-Charles Rochet. “Multidimensional Signalling,” Journal of Mathematical Economics 14 (1985) 261-84. Ramey, Garey. “D1 Signaling Equilibria with Multiple Signals and a Con- tinuum of Types,” Journal of Economic Theory 69 (1996) 508-31. Rothschild, Michael and Joseph Stiglitz. “Equilibrium in Competitive In- surance Markets: An Essay on the Economics of Imperfect Informa- tion,” Quarterly Journal of Economics 90 (1976) 630-49. Spence, Michael. “Job Market Signaling,” Quarterly Journal of Economics 87 (1973) 355-74. Stiglitz, J. and A. Weiss. “Credit Rationing in Markets with Imperfect Information,” American Economic Review 71 (1981) 393-410. Stiglitz, J. and A. Weiss. “Incentive E?ects of Terminations: Applications to the Credit and Labor Markets,” American Economic Review 73 (1983) 912-27.