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 t<t' u(,t)<u(O,t), for any 0 within a sufficiently small neighborhood of Bo. 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 e>0 such that u(e, t)+E<u(e, t for any 8 in the neighborhood Let to be the best type in To(from the other side's point of view) and let ( 9, A, u be the equilibrium whose probability assessment obeys the restrictions ’(t)>p()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 (⊙)<p(6o), because v(0, to)>l, so that ()v(,t)≤(0)<(60)=H(0)v(6o,t),t∈T0,t≠to For 8 sufficiently close to Bo, u(e, to) N u(Bo, to)>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<t0 , u(θ, t) < u(θ, t0 ), for any θ within a sufficiently small neighborhood of θ0. 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, t0 ) for any θ in the neighborhood. Let t0 be the best type in T0 (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= 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) < µ(θ0), because v(θ, t0) > 1, so that µ(θ)v(θ, t) ≤ µ(θ) < µ(θ0) = µ(θ0)v(θ0, t), ∀t ∈ T0, t 6= t0. For θ sufficiently close to θ0, u(θ, t0) ≈ u(θ0, t0) > 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)< 1 or 2ta(e, t)<1.In that case, we say that the contract 6 is actively rationed. Now suppose that(, g) is a separating, stable allocation and that con- tract ] o is actively rationed. Suppose that g(eo, to)>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) < 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 contract θ0 is actively rationed. Suppose that g(θ0, t0) > 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