Chapter 7 Markets with adverse selection 7.1 A market model These notes introduce some ideas for modeling markets with adverse selec- tion. This framework was originally intended to deal with markets that 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 deve oped in a series of paper 6). 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. An agent'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. There areM>0 informed agents There is a finite set of contracts e. 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 0 contract with an informed agent of type t, the uninformed agent's payoff is u(8, t) and the informed agent's payoff v(0, t). Each agent has a reservation utility, the utility he gets if a contract is
Chapter 7 Markets with Adverse Selection 7.1 A market model These notes introduce some ideas for modeling markets with adverse selection. This framework was originally intended to deal with markets that cannot be easily accommodated by the standard signaling game e.g., because 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 developed 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. An agent’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. There are M > 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 payoff is u(θ, t) and the informed agent’s payoff is v(θ, t). Each agent has a reservation utility, the utility he gets if a contract is 1
CHAPTER 7. MARKETS WITH ADVERSE SELECTION not exchanged and he has to take his next best option. With an appropriate normalization of the payoff functions, the reservation utility is 0 for every ty 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(, g) where f:e→R+andg:×T→R+. We interpret f() as the measure of uninformed agents choosing contract 0 and g(e, t) as the measure of informed agents of type t choosing contract 8. The allocation(, g) is attainable if ∑f(0)≤M ∑9(,1)≤N(t,Mt The attainability condition contains an inequality because it is possible that some agents will choose their outside option (i.e, no trade e. When an agent selects a contract 0 he does not know the probability of rade or the type of agent from the other side of the market that he will be matched with. Let A(8, t) denote the probability that an uninformed agent choosing contract 6 will exchange the contract with a type-t agent and let u(e) denote the probability that an informed agent choosing contract 0 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(, u) is consistent with the allocation ( 9 A(,t) (6,t) (6) f() for any contract such that m(0)>0, where m(0) measures the long side of the market for contract 0, that is () f()∑9(0,1)
2 CHAPTER 7. MARKETS WITH ADVERSE SELECTION not exchanged and he has to take his next best option. With an appropriate normalization of the payoff functions, the reservation utility is 0 for every type. The equilibrium choices made by the agents are described by an allocation, 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+. We interpret f(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, where m(θ) measures the long side of the market for contract θ, that is, m(θ) = max ( f(θ), X t g(θ, t) )
7. 2. STABILITY If m(0)=0 consistency is automatically satisfied Then a market equilibrium consists of an(attainable) allocation ( g and a probability assessment(, u) such that the allocation maximizes the payoff of each type, that is, f(e)>0 implies u(0,1)N(O,t)=t'= max(,t)(0,t)},v; and g(0, t)>0 implies v(6,t)p(6)=t(t) max 3u (6,t)p(6)},v6,t The equilibrium (, 9, A, u) is said to be orderly if at most one side of the market for any contract is rationed. that max A(6,+,;1()}=1 Without this requirement there exist many trivial equilibria 7. 2 Stability A perturbation is an allocation (, g) such that f()=∑9(,1)>0, Define an equilibrium for the (E, f, 9-perturbed market by replacing the allocation(, g)by(1-e(, g)+e(, g in the equilibrium conditions above Then a perfect market equilibrium(, g, A, u) is defined to be the limit of a sequence of equilibria(fg, A, us)of the (E, f, g-perturbed market as a converges to 0. Note that in a perturbed market the probability assessment is uniquely determined by the allocation and the consistency condition Call ( g) an equilibrium allocation if (f, g, A, u)is a market equilib- rium for some(A, u) and call ( g) an equilibrium allocation for the(e, f, 9)- perturbed market if(, g, A, p)is a market equilibrium of the(E, f, 9-perturbed market for some(A, p). An attainable allocation ( g) is stable if, for any
7.2. STABILITY 3 If m(θ)=0 consistency 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 payoff 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) = maxθ {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, ˆf, gˆ)-perturbed market by replacing the allocation (f,g) by (1−ε)(f,g)+ε( ˆf, gˆ) in the equilibrium conditions above. Then a perfect market equilibrium (f, g, λ, µ) is defined to be the limit of a sequence of equilibria (f εgε, λε , µε) of the (ε, ˆf, gˆ)-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 equilibrium for some (λ, µ) and call (f,g) an equilibrium allocation for the (ε, ˆf, gˆ)- perturbed market if (f, g, λ, µ) is a market equilibrium of the (ε, ˆf, gˆ)-perturbed market for some (λ, µ). An attainable allocation (f, g) is stable if, for any
CHAPTER 7. MARKETS WITH ADVERSE SELECTION perturbation(, g)there is a sequence of equilibrium allocations(f, g)such that(f, g)is an equilibrium allocation of the(e, f, g)-perturbed market and lin(f,92)=(f,9) Note that the probability assessments do not necessarily converge to a unique limit. It is easy to see that(, g) must be an equilibrium allocation if ( 9) is stable Let (, g) be a stable allocation and let to be a fixed but arbitrary type For any type t, let u* denote the equilibrium payoff of the uninformed and u(t) the equilibrium payoff of type t. Then there exists an equilibrium ( 9, A, u) such that for any contract 0 and any type tf to t’(t)>p()v(,切=A(O,t)=0 To see this, let(fk, n,gn)be the perturbation defined by +一 9(e, t) 1/nt≠to By choosing k and n appropriately, we can ensure that(fKm, gn) tainable allocation. By stability, there exists a sequence(f, 9, i, u)con- verging to(, g, A", ukin)as e-0, where(f, g, A, u)is an equilibrium for the(e, fkn, gkn)-perturbed game. By compactness, the sequence (kn, u has a limit point uo and it is clear that if (, g, A m, ukn )is an equilibrium then so is(f,g.A,p° Consider tf to. For each 0, if (6)v(6,t)0 sufficiently small b)v(6,t)<t2(t), where v (t) is the equilibrium payoff in(f, g, A, u). Then for tf to (,t) g(,t)1 (0)m(0)(k+r-1)/n-(k
4 CHAPTER 7. MARKETS WITH ADVERSE SELECTION perturbation ( ˆf, gˆ) there is a sequence of equilibrium allocations (f ε, gε ) such that (f ε, gε) is an equilibrium allocation of the (ε, ˆf, gˆ)-perturbed market and limε→0 (f ε , gε )=(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 t0 be a fixed but arbitrary type. For any type t, let u∗ denote the equilibrium payoff of the uninformed and v∗(t) the equilibrium payoff of type t. Then there exists an equilibrium (f, g, λ, µ) such that for any contract θ and any type t 6= t0, [v∗ (t) > µ(θ)v(θ, t)] =⇒ λ(θ, t)=0. To see this, let (f k,n, gkn) be the perturbation defined by f kn(θ)=(k + |T| − 1)/n gkn(θ, t) = ½ 1/n t 6= t0 k/n t = t0. By choosing k and n appropriately, we can ensure that (f kn, gkn) is an attainable allocation. By stability, there exists a sequence (f ε, gε, λε , µε) converging to (f, g, λkn, µkn) as ε → 0, where (f ε, gε, λε , µε) is an equilibrium for the (ε, f kn, gkn)-perturbed game. By compactness, the sequence © (λkn, µknª has a limit point µ0 and it is clear that if (f, g, λkn, µkn) is an equilibrium, then so is (f, g.λ0 , µ0). Consider t 6= t0. For each θ, if µkn (θ) v(θ, t) 0 sufficiently small, µε (θ) v(θ, t) < vε (t), where vε(t) is the equilibrium payoff in (f ε, gε, λε , µε). Then for t 6= t0 λε (θ, t) = gε(θ, t) mε(θ) = 1/n mε(θ) ≤ 1/n (k + |T| − 1)/n = 1 (k + |T| − 1)
7.3. ROBUSTNESS OF SEPARATING EQUILIBRIUM So, in the limit, An(e, t)=1/(k+r-1) and, taking limits as ki, n-o it follows that A(8, t)=0. Since this is true for any 0 and t, the desired property holds Note that if(f, 9, A, u)is a perfect market equilibrium then, by construc ion,(A, p)is orderly. The reason is that for any perturbation, the definition of consistency implies that every(A, p) is orderly and it remains so in the unit 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 e is a subset of some finite-dimensional Euclidean space and suppose that u(, t )and v(, t) are continuously differentiable functions of 0 on some open superset of The theory can be extended from a finite subset of e 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 ( g) is defined to be an equilibrium allocation such that for any type to we can find an equilibrium probability assessment(A, u) having the properties derived in the proposition above 7. 3 Robustness of separating equilibrium Let(, g) be a stable allocation and suppose that there exists a contract Bo belonging to the interior of e such that two or more types "pool"at Bo. We can assume without loss of generality that there exists a pair tt and that 9(60,t)>0andg(6o,t)>0 Let u' and u*(t)denote the equilibrium payoffs to the uninformed and type t, respectively, for the equilibrium allocation(, g) and let To=ItE Tlv(t)=u(Bo)v(8o, t))
7.3. ROBUSTNESS OF SEPARATING EQUILIBRIUM 5 So, in the limit, λkn(θ, t)=1/(k + |T| − 1) and, taking limits as k, n → ∞, it follows that λ0 (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 construction, (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 characterizing 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 differentiable 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 equilibrium 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 t0 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 θ0 belonging to the interior of Θ such that two or more types “pool” at θ0. We can assume without loss of generality that there exists a pair t 6= t 0 and that g(θ0, t) > 0 and g(θ0, t0 ) > 0. Let u∗ and v∗(t) denote the equilibrium payoffs to the uninformed and type t, respectively, for the equilibrium allocation (f,g) and let T0 = {t ∈ T|v∗ (t) = µ(θ0)v(θ0, t)}
CHAPTER 7. MARKETS WITH ADVERSE SELECTION Note that the definition of To is determined by ( g) independently of the particular choice of (A, u). We assume that the types are ranked in the following sense: for any pair t0 such that u(e, t)+Ep()v(,t=A(,t)=0,t≠to To avoid some pathological cases, we assume that every t e To has a positive equilibrium payoff v*(t)>0. Without loss of generality we can normalize there exists a contract 0 arbitrarily close to Bo satisfying ow suppose that v(,to)>1>v(,1),t∈T0,t≠to Then it follows that A(0, t)=0 for any tf to. To see this, note that the equilibrium condition for to implies that 1()v(,to)≤p(6o0(60,t0)=p(6o) which in turn implies that (⊙)l, so that ()v(,t)≤(0)A(Bo, t)u(eo, s)so Lta(e, t)=A(e, to)< 1. Then orderliness implies that u(0)=l, contradict ing the equilibrium condition 1()v(,to)≤p(6)(60,to)≤1
6 CHAPTER 7. MARKETS WITH ADVERSE SELECTION Note that the definition of T0 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 0 such that u(θ, t) + ε µ(θ)v(θ, t)] =⇒ λ(θ, t)=0, ∀t 6= t0. To avoid some pathological cases, we assume that every t ∈ T0 has a positive equilibrium payoff v∗(t) > 0. Without loss of generality we can normalize the payoff functions so that v(θ0, t)=1 for all t ∈ T0. Now suppose that there exists a contract θ arbitrarily close to θ0 satisfying v(θ, t0) > 1 > v(θ, t), ∀t ∈ T0, t 6= t0. Then it follows that λ(θ, t)=0 for any t 6= t0. To see this, note that the equilibrium condition for t0 implies that µ(θ)v(θ, t0) ≤ µ(θ0)v(θ0, t0) = µ(θ0) which in turn implies that µ(θ) 1, so that µ(θ)v(θ, t) ≤ µ(θ) P t P λ(θ0, t)u(θ0, s) so t λ(θ, t) = λ(θ, t0) < 1. Then orderliness implies that µ(θ)=1, contradicting the equilibrium condition µ(θ)v(θ, t0) ≤ µ(θ0)v(θ0, t0) ≤ 1
7.4. EQUILIBRIUM RATIONING Proposition 1 Let ( g be a stable allocation and (A, p) an equilibrium probability assessment. For any contract o let To=t: u(eo)u(0o, t) u*(t)). Suppose that u*(t)>0 for any t E To and that for any e>0 there is a contract o that is E-close to bo such that v(,to)>1>v(,t),tt∈To,t≠to, ere to is the best type in To and v(0o, to)=1=v(0o, t). Then there is at st one type t such that g(8o, t)>0 Tote that this proposition does not imply that bo is only optimal for one type. Typically, there will be another type t'f t such that u(bo)u(0o, t') u*(t although g(0o, t)=0 7.4 Equilibrium rationing We want the set of contracts to be "large", to allow for "all possible con- ed 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 different "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 effect 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 different 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(0)>0 and that either u(e)0 and consider first the case where g(eo, to)>f(8o). Let To=ItE Tlu(t)=u(eo)o(00, t )) where(, u) is the equilibrium probabilitity assessment satisfying t’(t)>p()v(,切)=入(6,t)=0,wt≠to
7.4. EQUILIBRIUM RATIONING 7 Proposition 1 Let (f, g) be a stable allocation and (λ, µ) an equilibrium probability assessment. For any contract θ0 let T0 = {t : µ(θ0)v(θ0, t) = v∗(t)}. Suppose that v∗(t) > 0 for any t ∈ T0 and that for any ε > 0 there is a contract θ that is ε-close to θ0 such that v(θ, t0) > 1 > v(θ, t), ∀t ∈ T0, t 6= t0, where t0 is the best type in T0 and v(θ0, t0)=1= v(θ0, t). Then there is at most one type t such that g(θ0, t) > 0. Note that this proposition does not imply that θ0 is only optimal for one type. Typically, there will be another type t 0 6= t such that µ(θ0)v(θ0, t0 ) = v∗(t 0 ) although g(θ0, t0 )=0. 7.4 Equilibrium rationing We want the set of contracts to be “large”, to allow for “all possible contracts”. 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 different “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 effect 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. Suppose that a contract is traded by some agents but the probability of trade is less than one. This kind of rationing is different 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) 0 and consider first the case where g(θ0, t0) > f(θ0). Let T0 = {t ∈ T|v∗ (t) = µ(θ0)v(θ0, t)} where (λ, µ) is the equilibrium probabilitity assessment satisfying [v∗ (t) > µ(θ)v(θ, t)] =⇒ λ(θ, t)=0, ∀t 6= t0
CHAPTER 7. MARKETS WITH ADVERSE SELECTION for any contract 0+ o sufficiently sufficiently close to ]o. As usual, we can assume that to is the best type, i. e, the type preferred by the other side of the market for any 0 near 0o Ifu*(to)=0, then without loss of generality we can eliminate the rationing of that type by reducing g(0o, to)until f(0o)=g(0o, to). So, consider the case where u(to)>0. To rule out difficult cases, we assume that u*(t)>0for all tE To. Then we can normalize u(0o, t)=l for all t E To. Suppose there exists a contract 0 in the neighborhood of ]o such that (6,to)>v(6o,to)=1 and v(6,t0)>v(6,t),t∈T0,t≠ The equilibrium condition requires that u(0)0 for tf to only if u(0)o(0, t)=u*(t). Suppose that this is true n(,t)=2(020(to) contradicting our hypothesis. Thus, A(0, t)=0 for all t+ to and A(0, to)=1 contradicting the equilibrium condition for the long side Suppose now that the uninformed side of the market is actively rationed at 0o, that is, g(0o, to)0 and A(0o, to)v(0o, to)=1 v(8,to)>v(6,t),t∈To,t≠to Then u(0)A(Bo, to)u(eo, to), contradicting the equilibrium condition. Thus, we have proved the following result Proposition 2 Suppose the allocation(, g) is stable and separating. Let 8o be a contract and To the set of types for which Bo is optimal; suppo u(t>0 for all t E To and v(Bo, t)=l for all t E To. Let to .ique type that chooses Bo, that is, g(8o, to)>0. Then there is no active rationing at ]o if the following conditions are satisfied: for any E>0, there exists a contract e that is E-close to Bo and satisfies (6,to0)>v(6,to)
8 CHAPTER 7. MARKETS WITH ADVERSE SELECTION for any contract θ 6= θ0 sufficiently sufficiently close to θ0. As usual, we can assume that t0 is the best type, i.e., the type preferred by the other side of the market for any θ near θ0. If v∗(t0)=0, then without loss of generality we can eliminate the rationing of that type by reducing g(θ0, t0) until f(θ0) = g(θ0, t0). So, consider the case where v∗(t0) > 0. To rule out difficult cases, we assume that v∗(t) > 0 for all t ∈ T0. Then we can normalize v(θ0, t)=1 for all t ∈ T0. Suppose there exists a contract θ in the neighborhood of θ0 such that v(θ, t0) > v(θ0, t0)=1, and v(θ, t0) > v(θ, t), ∀t ∈ T0, t 6= t0. The equilibrium condition requires that µ(θ) 0 for t 6= t0 only if µ(θ)v(θ, t) = v∗(t). Suppose that this is true. Then v(θ, t) = µ(θ0) µ(θ) ≥ v(θ, t0), contradicting our hypothesis. Thus, λ(θ, t)=0 for all t 6= t0 and λ(θ, t0)=1, contradicting the equilibrium condition for the long side. Suppose now that the uninformed side of the market is actively rationed at θ0, that is, g(θ0, t0) 0 and λ(θ0, t0) v(θ0, t0)=1 and v(θ, t0) > v(θ, t), ∀t ∈ T0, t 6= t0. Then µ(θ) λ(θ0, t0)u(θ0, t0), contradicting the equilibrium condition. Thus, we have proved the following result. Proposition 2 Suppose the allocation (f, g) is stable and separating. Let θ0 be a contract and T0 the set of types for which θ0 is optimal; suppose that v∗(t) > 0 for all t ∈ T0 and v(θ0, t)=1 for all t ∈ T0. Let t0 be the unique type that chooses θ0, that is, g(θ0, t0) > 0. Then there is no active rationing at θ0 if the following conditions are satisfied: for any ε > 0, there exists a contract θ that is ε-close to θ0 and satisfies v(θ, t0) > v(θ0, t0)=1
7.5. MORAL HAZARD AND CREDIT RATIONING (6,t0)>v(6,t),t∈To,t≠to 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, which in turn raises questions about the efficiency 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(0i, yi), where Bi is the probability of success and yi is the payoff to the project in the event of success. An unsuccessful project has a payoff of zero. We assume that 611=62y2,62>61,< 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 a zero net return 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 off When the contractual payment is r, the entrepreneur's payoff from a type- i project is 0; maxyi-r, 0. There is a unique contract 0<r*<y2 such that the entrepreneur prefers type 2 if r< r*and prefers type 1 if r*<r<y2 For r 2 y2 both projects earn zero for the entrepreneur
7.5. MORAL HAZARD AND CREDIT RATIONING 9 and v(θ, t0) > v(θ, t), ∀t ∈ T0, t 6= t0. 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, which in turn raises questions about the efficiency 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 competition 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 (θi, yi), where θi is the probability of success and yi is the payoff to the project in the event of success. An unsuccessful project has a payoff of zero. We assume that θ1y1 = θ2y2, θ2 > θ1, y2 < y1. 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 a zero net return). 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 off. When the contractual payment is r, the entrepreneur’s payoff from a typei project is θi max{yi−r, 0}. There is a unique contract 0 < r∗ < y2 such that the entrepreneur prefers type 2 if r<r∗ and prefers type 1 if r∗ <r<y2. For r ≥ y2 both projects earn zero for the entrepreneur
CHAPTER 7. MARKETS WITH ADVERSE SELECTION The investors expected payoff from the project is B2r for r0 sufficiently small, which is not possible in equilibrium Suppose, for example, that 02r>61y,b2r≥ 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 atoning at r*. An entrepreneur facing the prospect of rationing would gladly pay r*+E 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 payoff: 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 each entrepreneur has an asset that can be used as collateral The total value of the collateral is K and the value surrendered in the event of the failure of the project is061(y1-r)+(1-61)k Given the assumption of equal expected outcomes 0191=0292, 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<y<1
10 CHAPTER 7. MARKETS WITH ADVERSE SELECTION The investors expected payoff from the project is θ2r for r ≤ r∗ and θ1r for r∗ 0 sufficiently small, which is not possible in equilibrium. Suppose, for example, that θ2r∗ > θ1y1, θ2r∗ ≥ 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 payoff. 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 simultaneously 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 each entrepreneur has an asset that can be used 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 θ2(y2 − r) + (1 − θ2)k>θ1(y1 − r) + (1 − θ1)k. Given the assumption of equal expected outcomes θ1y1 = θ2y 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 entrepreneur, 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