Online Appendix for Bailout Stigma

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1 Online Appendix for Bailout Stima Yeon-Koo Che Chonwoo Choe Keeyoun Rhee Auust 1, 2018 A Technical Preliminaries In this section, we derive technical results that will be used frequently in the proofs. Definition A.1. Define the followin: (i) m(a, b) := E[θ θ [a, b] ] 0 a b 1; (ii) γ(a) := max{θ a θ m(a, θ) + S} a [0, 1]; (iii) m(a, b, c, d) := E[θ θ [a, b] [c, d] ] 0 a b < c d 1; (iv) γ(a, b, c) := max{θ a θ m(a, b, c, θ) + S} 0 a b < c 1. Given that θ is a continuous random variable, m and m are well-defined and differentiable with respect to every arument. Next, recall our lo concavity assumption: Assumption A.1 (Lo-concavity). d 2 dθ 2 lo f(θ) < 0 for all θ [0, 1]. Based on the above, we obtain the followin results. Lemma A.1. (i) m(a, b), m(a, b) 1 0 a b 1; (ii) γ is a well-defined function and a b γ(a) 1 a [0, 1]; (iii) m(a, b, c, d) 1 0 a b < c d 1; (iv) γ is a well-defined a d function. Proof. See Banoli and Berstrom (2005) for the proof of part (i). θ a. For part (ii), note that (i) implies θ m(a, θ) is increasin and continuous in θ for all Thus γ(a) defined in Definition A.1-(ii) exists and is uniquely determined for every 1

2 a [0, 1] as the minimum of the unique solution to θ m(a, θ) = S and 1. To prove the second part of (ii), first suppose γ(a) = m(a, γ(a))+s. Applyin the implicit function theorem, we have d γ(a) = da a m(a,γ(a)) 1 γ m(a,γ(a)). From Corollary 1 of Szalay (2012), we have m(a, b) + m(a, b) 1 for a b a m(a,γ(a)) all 0 a b 1 if f( ) is lo-concave. This implies d γ(a) = 1. If γ(a) = 1, da 1 m(a,γ(a)) γ then γ(x) = 1 for all x > a, implyin that the riht derivative of γ(a) is 0 for all a [0, 1] such that γ(a) = 1. To prove part (iii), define ˇF (θ) := F (θ) [F (c) (F (b) F (a))] for every θ c. ˇF (θ) is lo-concave as shown below: d 2 lo ˇF (θ) dθ 2 = d ( ) f(θ) dθ F (θ) [F (c) (F (b) F (a))] = d ( ) f(θ) dθ F (θ) F (θ) F (θ) [F (c) + (F (b) F (a))] = d ( ( )) f(θ) dθ F (θ) F (c) + (F (b) F (a)) 1 + F (θ) [F (c) + (F (b) F (a))] 0. F (c)+(f (b) F (a)) In the above, the last inequality follows since 1 + is positive and decreasin F (θ) [F (c)+(f (b) F (a))] in θ, and f(θ) is decreasin in θ by the lo-concavity f(θ). Usin the lo-concavity of ˇF (θ), we F (θ) now show m 1. Differentiatin m with respect to d, we have d m d = f(d) [(F (d) F (c)) + (F (b) F (a))] 2 < ( 1 d (F (d) F (c)) + (F (b) F (a)) To prove m d ˇδ : [c, 1] R as b 1, it suffices to show d ˇδ(θ) := θ a d θdf (θ) + ( 1 d (F (d) F (c)) + (F (b) F (a)) ) θdf (θ). c ( 1 (F (d) F (c))+(f (b) F (a)) 1 θ udf (u), (F (θ) F (c)) + (F (b) F (a)) c d c ) θdf (θ) d c θdf (θ) ) 1. Define a map which is a variant of δ(θ) introduced by Banoli and Berstrom (2005). Since m(a, b, c, θ) = 1 θ udf (u), we need to show ˇδ(θ) is increasin in θ. Since ˇF (θ) = (F (θ) (F (θ) F (c))+(f (b) F (a)) c F (c)) + (F (b) F (a)), we have df = d ˇF for all θ c. Thus ˇδ(θ) can be rewritten as ˇδ(θ) = θ 1 ˇF (θ) θ c ud ˇF (u). Interatin θ c ud ˇF (u) by parts yields ˇδ(θ) = θ θ ˇF (θ) θ ˇF (u)du c ˇF (θ) = θ c ˇF (u)du. ˇF (θ) 2

3 From Theorem 1-(ii) in Banoli and Berstrom (2005), θ ˇF (u)du is lo-concave for all θ c if ˇF (u) is lo-concave for all θ c. Thus increasin in θ. θ c ˇF (u)du ˇF (θ) c is increasin in θ, which implies ˇδ(θ) is also Part (iv) follows from part (iii): since θ m(a, b, c, θ) is increasin and continuous in θ, γ(a, b, c) is the minimum of the unique solution to θ m(a, b, c, θ) = S and 1. Q.E.D. For future reference, we reproduce our reularity conditions below. Assumption A.2 (Assumption 2 in the main text). (i) 0 < θ < θ < 1, (θ; θ, S) := m( θ, θ) + S θ (m(θ, γ(θ)) m( θ, θ)) is decreasin in θ; (ii) If (0; θ, S) 0, then (0; θ, S ) 0 for every S > S and every θ (0, 1); (iii) For every 0 < θ < θ < 1, 2m( θ, θ) m(0, θ) is decreasin in θ; (iv) θ0 m(θ0, γ(θ0)) + S > 0 for every S > 0, where θ0 = γ(0); (v) θ 2m(0, θ) for all θ [0, 1]. B Proofs for Section 3 Lemma 1. In any equilibrium without overnment interventions, there is a cutoff 0 θ 1 1 such that all types θ θ 1 sell in each of the two periods at price m(0, θ 1 ) and all types θ > θ 1 hold out in t = 1 and are offered price m(θ 1, γ(θ 1 )) in t = 2, which types θ [θ 1, γ(θ 1 )) accept. If θ 1 = 1, then S 2(1 E[θ]). Proof of Lemma 1. Fix a ame with discount factor δ (0, 1) and fix an equilibrium of the required form. Let (q 1 (θ), q 2 (θ)) be the units of the asset a type-θ firm sells in each of the two periods and (t 1 (θ), θ 2 (θ)) be the correspondin transfers. Step 1. There exists 0 ˆθ ˇθ θ 1 such that q 1 (θ) = q 2 (θ) = 1 for θ < ˆθ; q 1 (θ) = 1, q 2 (θ) = 0 for any θ (ˆθ, ˇθ); q 1 (θ) = 0, q 2 (θ) = 1 for any θ (ˇθ, θ); and q 1 (θ) = q 2 (θ) = 0 for any θ > θ. Proof. In pure stratey equilibrium, we have q i (θ) {0, 1} for each θ, i = 1, 2. The expected discounted payoff for type-θ when imitatin type-θ is u(θ ; θ) := q 1 (θ )[S +t 1 (θ )]+[1 q 1 (θ )]θ + δ[q 2 (θ )(S+t 2 (θ ))+(1 q 2 (θ ))θ]. Let Q( ) := q 1 ( )+δq 2 ( ) and t( ) := q 1 ( )t 1 ( )+δq 2 ( )t 2 ( ). Since we must have u(θ; θ) u(θ ; θ) for all θ, θ in equilibrium, it follows that (S θ)[q(θ) Q(θ )] t(θ ) t(θ). Similarly we must have u(θ ; θ ) u(θ; θ ), which leads to (S θ )[Q(θ ) Q(θ)] 3

4 t(θ) t(θ ). Combinin these two inequalities, we have (θ θ )[Q(θ) Q(θ )] 0. This implies monotonicity: Q(θ) Q(θ ) for any θ θ. Since q i (θ) {0, 1} for each θ, i = 1, 2, the monotonicity implies the desired property. Step 2. If ˇθ < 1, then those that hold out in t = 1 must be offered m(ˇθ, γ(ˇθ)) in equilibrium, which is accepted by types θ (ˇθ, γ(ˇθ)) where γ(ˇθ) > ˇθ. Proof. The belief in t = 2 for the holdouts is the truncated distribution of F on [ˇθ, 1]. This is essentially a one-shot problem with a truncated support. Thus the stated result follows from the definition of γ. Step 3. ˆθ = ˇθ. Proof. Suppose to the contrary that ˆθ < ˇθ. Let p be the price offered in t = 1. By the zero profit condition, p must be a break-even price for the types that accept it. Since type-ˇθ firm must weakly prefer acceptin p in t = 1 to not sellin in either periods, we have p + S + δˇθ (1 + δ)ˇθ p + S ˇθ. (B.1) This means that all firms acceptin p in t = 1 will accept the same price p in t = 2 if that price were offered in t = 2, with strict incentive for all firms with θ < ˇθ. The fact that types θ (ˆθ, ˇθ) choose not to sell in t = 2 means that the price offered in t = 2 to those that accept p in t = 1, denoted by p, is strictly less than p. Suppose first ˇθ < 1. Then, by Step 2, an offer p 2 := m(ˇθ, γ(ˇθ)) must be made in equilibrium to those that hold out in t = 1, which is accepted by types θ (ˇθ, γ(ˇθ)). Since type-ˇθ must be indifferent between sellin at p in t = 1 only and sellin at p 2 in t = 2 only, we must have p + S + δˇθ = ˇθ + δ(p 2 + S). (B.2) In particular, p 2 +S > ˇθ, so p+s > ˇθ. This means that, if a buyer deviates by offerin p ε > p for sufficiently small ε > 0 in t = 2 to those that accepted p in t = 1, then all of them must accept that offer. The buyer makes a strictly positive profit with such a deviation since p is the break-even price. We thus have a contradiction. Suppose next ˇθ = 1, meanin that all types sell in t = 1. Here we invoke the D1 refinement to derive a contradiction. In the candidate equilibrium, the payoff for type θ > ˆθ is p+s +δθ, and the payoff for all types θ < ˆθ is some constant u, which must equal p + S + δˆθ since type-ˆθ must be indifferent. Let U (θ) be the equilibrium payoff type-θ enjoys when the candidate equilibrium is played: U (θ) := max{u, p + S + δθ}. Suppose a type θ firm deviates by refusin the bailout 4

5 in t = 1, and suppose a buyer offers p 2 to that deviatin firm in t = 2. Type-θ s deviation payoff is then θ +δ max{p 2 +S, θ}. Thus, when type-θ deviates by choosin holdout, the set of market s offers in t = 2 that dominate the candidate equilibrium for type-θ is D(holdout, θ) := {p 2 θ + δ max{p 2 + S, θ} max{u, p + S + δθ}}. Note that for fixed p 2, the payoff difference θ + δ max{p 2 + S, θ} max{u, p + S + δθ} is strictly increasin in θ, so the set D(holdout, θ) is nested in the sense that D(holdout, θ) D(holdout, θ ) for θ > θ. In other words, D(holdout, 1) is maximal, and more importantly, D(holdout, θ) is not maximal if θ < 1. Given this, the D1 refinement entails that the belief by the market must be supported on θ = 1 in case of holdout. Thus followin the deviation, the market s offer must be p 2 = 1. This means that, for the market s offer in the candidate equilibrium to satisfy D1, type ˇθ = 1 must enjoy the payoff of at least 1 + δ(1 + S) in case of deviation to holdout from the candidate equilibrium. Since type ˇθ = 1 chooses to sell in t = 1 in the candidate equilibrium, we must have p + S + δ 1 + δ(1 + S), (B.3) which implies p + S > 1. This in turn implies that in t = 2, a buyer can deviate by offerin a price slihtly below p and induce acceptance from all types that accepted p in t = 1. Once aain, the buyer makes a strictly positive profit with such a deviation, hence a contradiction. Step 4. All types θ < ˆθ = ˇθ (if is nonempty) are offered a sinle price in both periods equal to m(0, ˆθ). All types θ > ˆθ are offered price m(ˆθ, γ(ˆθ)), which is accepted by types θ (ˆθ, γ(ˆθ)). Proof. Suppose there are two distinct prices p, p that are accepted by positive measures of firms. By the zero profit condition, both prices must be breakin even for the types that accept them. But then, no type will accept the lower price, hence a sinle price is offered to all types θ < ˆθ. The market s break-even condition then pins down the price to m(0, ˆθ). The second statement follows from Step 2. Step 5. If ˆθ = 1, then S 2(1 E[θ]). Proof. Applyin the D1 arument as in Step 3, type-ˆθ s equilibrium payoff (1+δ)(p+S) should not be smaller than 1+δ(1+S) where p = m(0, 1) = E[θ] by Step 4. From (1+δ)(p+S) 1+δ(1+S) follows the stated condition as δ 1. Q.E.D. Theorem 2. 5

6 (i) There is an equilibrium in which firms with θ θ1 sell at price p 1 := m(0, θ1) in both periods, firms with θ (θ1, θ2) sell only in t = 2 at price p 2 := m(θ1, θ2), and firms with θ > θ2 never sell, where θ1 and θ2 are defined by (θ1; S) = 0 and θ2 = γ(θ1), respectively. We have θ1 θ0 θ2, hence p 1 p 0 p 2, where the inequalities hold strictly if the cutoff in the one-period model satisfies θ0 (0, 1). Given Assumption 1-(i), there is at most one such equilibrium with an interior θ1. (ii) Given Assumption 1-(ii), the t = 1 market in equilibrium is fully active if S S, suffers from partial freeze if S (S, S ), and full freeze if S < S, where S and S are defined by (0; S ) = 0 and (1; S ) = 0, respectively, and satisfy S > S 0 and S > max{s 0, S }. (iii) In addition, there is an equilibrium with full market freeze in t = 1 for any S. Proof of Theorem 2. To prove (i), note first that the existence of cutoffs θ 1 and θ 2 was established by Lemma 1. Thus it suffices to show θ 1 θ 0 θ 2 with strict inequalities if θ 0 (0, 1). Consider the t = 1 cutoff θ 1. If θ 1 < 1, then it satisfies (θ 1, S) 0. Thus θ 1 2m(0, θ 1 ) m(θ 1, γ(θ 1 )) + S < m(0, θ 1 ) + S since m(0, θ 1 ) < m(θ 1, γ(θ 1 )). Since θ 0 := sup{θ θ m(0, θ) + S}, we have θ 1 θ 0 with strict inequality if θ 0 < 1. If θ 1 = 1, then (θ 1, S) 0 by the D1 refinement, which in turn implies θ 0 = 1. Next, the t = 2 cutoff θ 2 = γ(θ 1 ) satisfies θ 2 m(θ 1, γ(θ 1 )) + S. Since θ 1 0, we have θ 2 = γ(θ 1 ) γ(0) = θ 0, where the inequality is strict for θ 0 < 1 and θ 1 > 0. For (ii), note that, by Assumption 1, θ 1 2m(0, θ 1 ) + m(θ 1, γ(θ 1 )) is strictly increasin in θ 1 for each S. Since θ 1 < 1 satisfies (θ 1, S) 0, a unique θ 1 can be found, which is increasin in S. From this and the first claim follows the second claim. For (iii), consider the candidate equilibrium in which the market in t = 1 completely freezes. This means that, in t = 2, we have one period equilibrium with cutoff iven by θ 0 and price p 0. In this equilibrium, the payoff for type θ θ 0 is θ + δ(p 0 + S) = θ + δθ 0 and the payoff for type θ > θ 0 is (1 + δ)θ. Thus the equilibrium payoff for type θ is U (θ) = max{θ + δθ 0, (1 + δ)θ}. Suppose a buyer deviates and offers p 1 in t = 1. Let p 2 be the market s offer in t = 2 to those that accept the deviation offer p 1. Then the payoff to type θ from acceptin p 1 is p 1 + S + δ max{p 2 + S, θ}. As in the proof of Lemma 1, define the set D(sell, θ) := {p 2 p 1 + S + δ max{p 2 + S, θ} max{θ + δθ 0, (1 + δ)θ}}. For fixed p 2, the payoff difference is strictly decreasin in θ, hence D(sell, 0) is maximal. Then by the D1 refinement, the market s belief must be supported on θ = 0 when a firm accepts a deviation offer p 1 in t = 1. Given this belief, no firm will accept p 1 if p 1 p 0. If p 1 > p 0, 6

7 then all types θ p 1 + S accept the deviation offer p 1. Then the buyer will lose money iven p 1 > p 0. Q.E.D. C Proofs for Section 4 In Section C.1, we establish necessary conditions for various equilibria. Section C.2 presents the main characterization of equilibria. To appreciate the main points, readers can skip Section C.1 and jump directly to Section C.2. C.1 Necessary Conditions for Equilibria We first characterize the various types of equilibria and derive necessary conditions for the existence of each type of equilibria. C.1.1 Equilibrium Cutoff Structure Lemma 2. In any equilibrium, there are four possible cutoffs 0 θ θ 1 θ ø θ 2 1 such that types θ Θ := [0, θ ) sell to the overnment in t = 1 and to the market in t = 2, types θ Θ 1 := (θ, θ 1 ) sell to the market in both periods, types θ Θ ø := (θ 1, θ ø ) sell only in t = 1 to the overnment, types θ Θ 2 := (θ ø, θ 2 ) sell only in t = 2 to the market, and types θ > θ 2 sell in neither period. Proof of Lemma 2. Similar to the proof of Lemma 1, fix a ame with discount factor δ (0, 1) and the probability of market collapse ε (0, 1). Also, fix any equilibrium of the required form. Let q (θ) be the unit of the asset a type-θ firm sells to the overnment, (q 1 (θ), q 2 (θ)) be the units of the asset the firm sells in each of the two periods, and (t (θ), t 1 (θ), t 2 (θ)) be the correspondin transfers. The expected payoff for a type-θ firm when playin as if it is type-θ is u(θ ; θ) = q (θ ) [ [S + t (θ )] + εδθ ] + (1 q (θ ))ε(1 + δ)θ + (1 ε){1 q (θ )} [ q 1 (θ ){S + t 1 (θ)} + {1 q 1 (θ )}θ ] + (1 ε)δ [ q 2 (θ ){S + t 2 (θ )} + {1 q 2 (θ )}θ ]. Let Q( ) := q ( ) + (1 ε){1 q ( )}q 1 ( ) + (1 ε)δq 2 ( ) and T ( ) := q ( )t ( ) + (1 ε){1 q ( )}q 1 ( )t 1 ( ) + (1 ε)δq 2 ( )t 2 ( ). Since u(θ; θ) u(θ ; θ) 0 and u(θ ; θ ) u(θ; θ ) 0 for every 7

8 θ θ in equilibrium, it follows that (S θ)[q(θ) Q(θ )] T (θ ) T (θ) and (S θ )[Q(θ ) Q(θ)] T (θ) T (θ ). Combinin these inequalities, we have (θ θ)[q(θ) Q(θ )] 0, which implies that Q(θ) is decreasin in θ. Since 1 > (1 ε) > (1 ε)δ and q j (θ) {0, 1} for every j =, 1, 2 in pure-stratey equilibrium, there exist cutoffs 0 θ θ 1 θ ø θ 1ø θ 2 1 such that: (i) (q (θ), q 1 (θ), q 2 (θ)) = (1, 0, 1) if θ [0, θ ]; (ii) (q (θ), q 1 (θ), q 2 (θ)) = (0, 1, 1) if θ (θ, θ 1 ]; (iii) (q (θ), q 1 (θ), q 2 (θ)) = (1, 0, 0) if θ (θ 1, θ ø ]; (iv) (q (θ), q 1 (θ), q 2 (θ)) = (0, 1, 0) if θ (θ ø, θ 1ø ]; (v) (q (θ), q 1 (θ), q 2 (θ)) = (0, 0, 1) if θ (θ 1ø, θ 2 ]; (vi) (q (θ), q 1 (θ), q 2 (θ)) = (0, 0, 0) if θ > θ 2. Applyin the same loic used for the proof of Step 3 in the proof of Lemma 1, it can be shown that θ ø = θ 1ø. Thus any equilibrium must be characterized by the cutoff structure 0 θ θ 1 θ ø θ 2 1. Q.E.D. In what follows, we describe the incentive compatibility constraints for firms in each type of equilibria. Since the equilibria are characterized by a set of cutoff types by Lemma 2, we derive conditions characterizin these cutoffs in each type of equilibria. Lastly, we find bailout terms compatible with each type of equilibria. C.1.2 Necessary Conditions for SBS Equilibrium In this equilibrium, there exists θ ø (0, θ 2 ) and θ 2 1 such that: types θ [0, θ ø ] sell to the overnment at price p in t = 1 but cannot sell in t = 2 due to m(0, θ ø ) < I; types θ (θ ø, θ 2 ] sell only in t = 2 at price m(θ ø, θ 2 ); the market fully freezes in t = 1 so no trade occurs. We derive necessary conditions for each type to play the above equilibrium strateies. The payoffs from playin the equilibrium strateies across the two periods are p + S + θ for all θ [0, θ ø ], and θ + m(θ ø, θ 2 ) + S for all θ (θ ø, θ 2 ]. Since type θ ø is indifferent between these choices, we must have m(θ ø, θ 2 ) = p. Meanwhile, type θ 2 must be the hihest type that will be induced to sell in t = 2. It must follow that θ 2 = γ(θ ø ), where γ is defined in Definition A.1-(ii). Thus we have the followin conditions that determine the marinal types θ ø and θ 2 : θ 2 = γ(θ ø ); p = m(θ ø, γ(θ ø )). (C.1) (C.2) By Lemma A.1, θ ø and θ 2 are uniquely determined by p. Moreover, since both γ(θ) and m(θ, γ(θ)) are differentiable and increasin in θ, we have d dp θ ø > 0. The case in which all firms sell to the overnment in t = 1 (i.e., θ ø = 1) can be supported if and only if p m(1, γ(1)) = 1. The reason is the followin. Applyin the same arument used to prove Lemma 1, one can easily see that the worst off-the-path belief consistent with D1 for a deviator (one who holds out) is 8

9 θ ø = 1. Given the belief, no firm deviates if and only if p m(1, γ(1)) = 1. Since we restrict p 1, such a case (θ ø = 1) can be observed only if p = 1. In addition, the SBS equilibrium requires that types θ [0, θ ø ] cannot sell in t = 2, hence m(0, θ ø ) < I. (C.3) Furthermore, either of the followin constraints on θ ø must hold: θ ø < I, θ ø m(θ ø, γ(θ ø )) + S 0. (C.4) (C.5) To see why, suppose the t = 1 market opens after the bailout and a buyer deviates and offers p 1 I. Since the belief in t = 1 is that only the types θ > θ ø are available for asset sales, the firms acceptin p 1 are assined the off-the-path belief as bein the worst of the available types (i.e., θ = θ ø ). First, suppose θ ø < I. Given the off-the-path belief, firms that deviate and sell at price p 1 cannot sell in t = 2, hence the total payoff from the deviation equals p 1 + S + θ. Clearly p 1 > m(θ ø, γ(θ ø )) for, otherwise, the deviatin firms will not sell at p 1. But if p 1 > m(θ ø, γ(θ ø )), then all types θ (θ ø, (p 1 + S) 1] will sell at p 1, and m(θ ø, (p 1 + S) 1) p 1 > 0 from the definition of γ( ). Thus no buyers in t = 1 deviate if θ ø < I. Second, suppose θ ø I. Given the off-the-path belief above, firms sellin at p 1 can sell at price θ ø in t = 2. Thus the total payoff to the deviatin firm is p 1 + S + max{θ, θ ø + S}. In the above, we showed that p 1 > m(θ ø, γ(θ ø )) is not possible. If p 1 m(θ ø, γ(θ ø )), then types θ > θ ø +S do not sell at p 1 since p 1+S+max{θ, θ ø +S} = p 1+S+θ < θ+max{m(θ ø, γ(θ ø ))+S, θ}. On the contrary, types θ (θ ø, θ ø +S] sell at p 1 if and only if p 1+θ ø +2S θ+m(θ ø, γ(θ ø ))+S, or equivalently p 1 θ + m(θ ø, γ(θ ø )) θ ø S. Let θ 1 := p 1 + (θ ø m(θ ø, γ(θ ø )) + S). Then types θ (θ ø, θ 1] sell to the deviatin buyer, so the deviatin buyer ets the expected payoff m(θ ø, θ 1) p 1. Since dθ 1 = 1 and hence d (m(θ dp 1 dp ø, θ 1) p 1) < 0, we have m(θ ø, θ 1) p 1 < 0 1 for any p 1 m(θ ø, γ(θ ø )) if and only if m(θ ø, θ 1) p 1 0 at p 1 = m(θ ø, γ(θ ø )) S. Since m(θ ø, θ 1) p 1 = θ ø m(θ ø, γ(θ ø )) + S, we must have θ ø m(θ ø, γ(θ ø )) + S 0. Let P SBS denote the rane of p s for which SBS equilibrium exists. Lemma C.1. If P SBS in non-empty, it is a convex set such that p > p 0 for all p P SBS. Proof. Since m(θ, γ(θ)) is increasin in θ, θø SBS (p ) is increasin in p if θø SBS is well-defined. 9

10 Hence there exists p SBS such that θ SBS ø since p 0 = m(0, θ 0) = m(0, γ(0)), we have p > p 0 for θ SBS ø is increasin in θ, there exists ˆp SBS θ SBS ø θ SBS ø (p ) satisfies (C.4) if and only if p < ˆp SBS (p ) satisfies (C.5) if and only if p ˆp SBS P SBS = (p 0, ˆp SBS (p ) satisfies (C.3) if and only if p < p SBS. Furthermore, (p ) > 0. Lastly, since θ m(θ, γ(θ))+s such that: (i) if (C.4) binds but (C.5) does not, then ; (ii) if (C.5) binds but (C.4) does not, then. Therefore, P SBS = (p 0, p SBS ) if (C.3) binds, ] if (C.4) binds, and P SBS = (p 0, ˆp SBS ) if (C.5) binds. Q.E.D. For every p P SBS, we let θø SBS (p ) denote the marinal type determined by (C.2). For expositional convenience, we may occasionally abbreviate θ SBS ø (p ) to θ SBS ø. C.1.3 Necessary Conditions for MBS Equilibrium In this equilibrium, there exist θ (0, θ 2 ) and θ 2 1 such that: types θ [0, θ ] sell to the overnment at price p in t = 1 and to the market at price m(0, θ ) in t = 2; types θ (θ, θ 2 ] sell only in t = 2 at price m(θ, θ 2 ); no asset trade occurs in the t = 1 market. In equilibrium, the total payoffs for the firms are p + m(0, θ ) + 2S for all θ [0, θ ], θ + m(θ, θ 2 ) + S for all θ (θ, θ 2 ], and 2θ for all θ > θ 2. For expositional ease, we treat the case in which θ 2 is interior (so it is characterized by an indifference condition). As one can see clearly, our characterization also works for the boundary case θ 2 = 1. From type θ s indifference condition, we have p +m(0, θ )+S θ m(θ, θ 2 ) = 0. Similarly, type θ 2 s indifference condition leads to θ 2 = m(θ, θ 2 ) + S = γ(θ ). Thus we have the followin conditions that determine θ and θ 2 : θ 2 = γ(θ ), p = θ m(0, θ ) S + m(θ, γ(θ )). (C.6) (C.7) Since θ m(0, θ) + m(θ, γ(θ)) is continuous and increasin in θ, the marinal type θ is uniquely determined by and increasin in p. If p is very lare in that p 1 m(0, 1) S +m(1, γ(1)) = 2 E[θ] S, then we assin θ = 1. Such an assinment is supported by the off-the-path belief at the t = 2 market that the holdouts in t = 1 are perceived as the hihest type θ = 1. Once aain, one can check this is the only belief that satisfies the D1 refinement. In addition, the bailout recipients also sell in t = 2. Thus, we have m(0, θ ) I. (C.8) Furthermore, the t = 1 market freezes completely. From the analysis of SBS equilibrium (recall 10

11 (C.4) and (C.5)), we showed that the t = 1 market freezes fully if and only if either θ < I or θ m(θ, γ(θ )) + S 0. Since θ > m(0, θ ) I from (C.8), the latter condition must hold: θ m(θ, γ(θ )) + S 0. (C.9) Lastly, all types θ [0, θ ] sell assets at price m(0, θ ) in t = 2. Thus we must have θ m(0, θ ) + S, or equivalently θ θ 0. (C.10) Lemma C.2. There exist p MBS (C.10) if and only if p [p MBS p MBS, p MBS ]. such that (C.7) admits a unique θ that satisfies (C.8) Proof. Since θ m(0, θ) + m(θ, γ(θ)) S is increasin in θ, θ MBS (p ) is increasin in p if welldefined. Moreover, since m(0, θ) is increasin in θ, there exists p MBS such that θ MBS (p ) is well-. Furthermore, since θ m(0, θ)+s is increasin defined and satisfies (C.8) if and only if p p MBS in θ, there exists p MBS if p p MBS by (C.7). such that θ MBS (p ) (if well-defined) satisfies (C.9) and (C.10) if and only. Puttin all these results toether, we have P MBS = [p MBS For every p [p MBS, p MBS, p MBS ]. Q.E.D. ], we let θ MBS (p ) denote the marinal type θ determined For expositional convenience, we may abbreviate θ MBS P MBS := [p MBS, p MBS ]. (p ) to θ MBS. We also let C.1.4 Necessary Conditions for MR Equilibrium In this equilibrium, a positive measure of firms sell to the market in t = 1 (in addition to a positive measure of firms sellin to the overnment). Bailout rejuvenates the t = 1 market. There are two possible types of MR equilibria: MR1 in which Θ, Θ 1, Θ 2, but Θ ø =, and MR2 in which Θ, Θ 1, Θ ø, Θ 2. MR1 equilibrium: In this equilibrium, there exist 0 < θ < θ 1 θ 2 1 such that: types θ [0, θ ] sell to the overnment in t = 1, and to the market in t = 2 at price m(0, θ ); types θ (θ, θ 1 ] sell to the market at price m(θ, θ 1 ) in both periods; types θ (θ 1, θ 2 ] sell to the market at price m(θ 1, θ 2 ) only in t = 2; types θ > θ 2 do not sell in either period. In equilibrium, firms total payoffs are p + m(0, θ ) + 2S for all θ [0, θ ], 2m(θ, θ 1 ) + 2S for all θ (θ, θ 1 ], θ + m(θ 1, θ 2 ) + S for all θ (θ 1, θ 2 ], and 2θ for all θ > θ 2. From these payoffs, 11

12 it is straihtforward to see the three marinal types must satisfy relevant incentive constraints (i.e., indifference in the case of an interior solution): p = 2m(θ, θ 1 ) m(0, θ ); θ 1 = max{θ [θ, 1] m(θ, θ) + S θ (m(θ, γ(θ)) m(θ, θ)) 0}; θ 2 = γ(θ 1 ). (C.11) (C.12) (C.13) (C.11) is the indifference condition for type-θ firm, (C.12) that for type-θ 1 firm, and (C.13) that that for type-θ 2 firm. Assumption A.2-(i) ensures that θ 1 is uniquely determined by θ. As before (C.12) allows for the possibility that θ 1 = 1. The D1 refinement suests that if θ 1 = 1, the worst off-the-path belief for a deviatin holdout firm is θ 1 = 1, so (C.12) ensures that iven that belief, no firm wishes to deviate. There are additional necessary conditions for the MR1 equilibrium. First, types θ [0, θ ] should be able to finance their projects in t = 2 (an implication of Lemma 2): m(0, θ ) I. (C.14) Second, types θ (θ 1, γ(θ 1 )] must prefer sellin only in t = 2 to either sellin to the market or sellin to the overnment in t = 1, which requires m(θ 1, γ(θ 1 )) m(θ, θ 1 ) and m(θ 1, γ(θ 1 )) p. Since the first inequality holds trivially, we only state the second condition: m(θ 1, γ(θ 1 )) p. (C.15) The conditions (C.11) (C.15) will be used later when characterizin the set of bailout terms that support the MR1 equilibrium. One can also see that the same conditions are also sufficient for MR1 to arise. Lemma C.3. (i) There exists ˇp MR1 such that (C.11) and (C.12) admit a unique (θ, θ 1 ) that satisfies 0 < θ < θ 1 1 if and only if p > ˇp MR1. (ii) There exists p MR1 and only if p p MR1 ˇp MR1. such that θ determined by (C.11) and (C.12) satisfies (C.14) if Proof. We first prove part (i). Given Assumption A.2-(iii), (C.11) defines θ 1 as a decreasin function of θ, labelled θ 1 (θ ), whenever well-defined. Furthermore, iven Assumption A.2-(i), 12

13 θ 1 θ 1 = max{θ > θ : 2m(θ, θ) θ m(θ, γ(θ)) + S 0} p > p θ MR1 1 p = 2m(θ, θ 1) m(0, θ ) p = 2m(θ, θ 1 ) m(0, θ ) no intersection if p is too low 0 θ θ MR1 Fiure 1 Characterization of θ MR1 and θ MR1 1 (C.12) defines θ 1 as an increasin function of θ, labelled θ I 1(θ ), whenever well-defined. In fact, there exists θ < 1 such that θ I 1(θ ) is well-defined if and only if θ > θ. 1 Therefore, (θ, θ 1 ) satisfyin (C.11) and (C.12), if well-defined, is characterized by a unique point of intersection between θ 1 (θ ) and θ I 1(θ ). Moreover, it can be shown that θ 1 (θ ) shifts up as p increases, which is also illustrated in Fiure 1. Lastly, one can find that there exists a p 1 such that θ 1 (θ ) and θ I 1(θ ) intersect. 2 Puttin all results toether, there exists ˇp MR1 such that two curves θ 1 (θ ) and θ I 1(θ ) intersect at a unique point if and only if p > ˇp MR1. We next prove part (ii). Since θ 1 (θ ) shifts up as p increases, θ determined by (C.11) and (C.12), is increasin in p for all p > p MR1. Therefore, there exists p MR1 ˇp MR1 such that θ determined by (C.11) and (C.12) satisfies m(0, θ ) I if and only if p p MR1. Q.E.D. For any p > ˇp MR1, let θ MR1 (p ) and θ1 MR1 (p ) denote the marinal types θ and θ 1 determined by (C.11) and (C.12). For expositional convenience, we may abbreviate θ MR1 (p ) and θ1 MR1 (p ) to θ MR1 and θ1 MR1, respectively. 1 From Assumption A.2-(i), θ I 1(θ ) is well-defined if and only if θ m(θ, γ(θ )) + S > 0. Since θ m(θ, γ(θ)) is increasin in θ and 1 m(1, γ(1)) + S = S > 0, there exists θ < 1 such that θ m(θ, γ(θ )) + S > 0 if and only if θ > θ. 2 To show this, we prove that there exists p at which θ 1 (θ ) = θ I 1(θ ). From the definitions of θ 1 ( ) and θ I 1( ), θ 1 (θ ) = θ I 1(θ ) is equivalent to p = (θ m(θ, γ(θ )) S) + m(θ, γ(θ )). (C.16) Since θ < θ 0 from Assumption A.2-(iv) and θ < 1, there exists p < 1 that solves (C.16). 13

14 MR2 equilibrium: In this equilibrium, there exist 0 < θ < θ 1 < θ ø θ 2 1 such that: types θ [0, θ ] sell to the overnment in t = 1 and to the market in t = 2 at price m(0, θ ); types θ (θ, θ 1 ] sell to the market at price m(θ, θ 1 ) in both periods; types θ (θ 1, θ ø ] sell to the overnment in t = 1 but do not sell in t = 2; types θ (θ ø, θ 2 ] sell at price m(θ ø, θ 2 ) only in t = 2; types θ > θ 2 do not sell in either period. As before, (θ, θ 1, θ ø, θ 2 ) must satisfy the followin conditions: p = 2m(θ, θ 1 ) m(0, θ ); θ 1 = m(0, θ ) + S; p = m(θ ø, γ(θ ø )); θ 2 = γ(θ ø ). (C.11) (C.17) (C.18) (C.19) Similar to the boundary case of the SBS equilibrium, the case in which all firms sell in t = 1 (i.e., θ ø = 1) can be supported if and only if p m(1, γ(1)) = 1. Indeed, the worst off-the-path belief consistent with D1 for a deviator (one who holds out) is θ ø = 1. Given this belief, no firm deviates if and only if p 1. Since we restrict p 1, such a case (θ ø = 1) can be observed only for p = 1. As before, the marinal type θ must also satisfy (C.14). Note that (C.14) implies m(θ ø, γ(θ ø )) > m(θ, θ 1 ) > I. Second, since Θ ø, we must have θ 1 < θ ø. (C.20) Lemma C.4. There exist p MR2 and p MR2 such that (C.11) and (C.17) admit a unique (θ, θ 1 ) that satisfies 0 < θ < θ 1 and (C.14) if and only if p [p MR2, p MR2 ). Proof. Similar to the proof of Lemma C.3, (C.11) defines θ 1 as a decreasin function of θ, labelled θ 1 (θ ), whenever well-defined. Moreover, (C.17) defines θ 1 as an increasin function of θ, labelled θ II 1 (θ ), whenever well-defined. Therefore, (θ, θ 1 ) satisfyin (C.11) and (C.17), if well-defined, is characterized by a unique point of intersection between θ 1 (θ ) and θ II 1 (θ ). Moreover, one can 14

15 θ 1 no intersection if p is too hih p > p θ 1 = m(0, θ ) + S θ MR2 1 p = 2m(θ, θ 1) m(0, θ ) p = 2m(θ, θ 1 ) m(0, θ ) no intersection if p is too low 0 θ θ MR2 Fiure 2 Characterization of θ MR2 and θ MR2 1 find that there exists p 1 at which θ 1 (θ ) and θ1 II (θ ) intersect. 3 Since θ 1 (θ ) shifts up as p increases, there exists ˇp MR2 such that (C.11) and (C.17) admit a unique (θ, θ 1 ) satisfyin θ > 0 if and only if p > ˇp MR2. Furthermore, as depicted in Fiure 2, θ determined by (C.11) and (C.17) is increasin in p for all p > ˇp MR2. Since θ1 II (θ ) θ for all θ θ0, there exists p MR2 > ˇp MR2 such that (θ, θ 1 ) determined by (C.11) and (C.17) satisfies 0 < θ < θ 1 if and only if p (ˇp MR2, p MR2 ). Lastly, there exists p MR2 > ˇp MR2 such that θ determined by (C.11) and (C.17) satisfies (C.14) if and only if p p MR2. Q.E.D. For any p > ˇp MR2, let θ MR2 (p ), θ1 MR2 (p ), and θø MR2 (p ) denote the marinal types θ, θ 1, and θ ø determined by (C.11), (C.17), and (C.18). For expositional convenience, we may abbreviate θ MR1 (p ), θ1 MR1 (p ), and θø MR2 (p ) to θ MR1, θ1 MR1, and θø MR2, respectively. Lemma C.5. Fix p. Suppose both (θ MR1 (p ), θ1 MR1 (p )) and (θ MR2 (p ), θ1 MR2 (p )) are welldefined at that p. If the former violates (C.15), then the latter satisfies (C.20). Conversely, if the former satisfies (C.20), then the latter violates (C.15). Proof. We first establish some technical results used in the proof. 3 We first show that if MR2 equilibrium exists, then it requires S < 1. Suppose to the contrary MR2 equilibrium exists for some S 1. Then θ1 II (θ ) = 1 for all θ 0, so any (θ, θ 1 ) satisfyin (C.17) violates (C.20), a contradiction. We next show that there exists p at which θ 1 (0) = θ1 II (0). From the definitions of θ 1 ( ) and θ1 II ( ), θ1 (0) = θ1 II (0) is equivalent to p = 2m(0, S). Since 1 > S > 2m(0, S) from Assumption A.2-(v), θ 1 (0) = θ1 II (0) at p = 2m(0, S) < 1. Thus, there is a unique point of intersection between θ 1 (θ ) and θ1 II (θ ) for any p (2m(0, S), 1), as is also seen in Fiure 2. 15

16 Claim 1. Suppose θ and θ 1 satisfy (C.11) and (C.12). Then θ and θ 1 satisfy (C.15) if and only if θ 1 m(0, θ ) + S. Proof. Suppose there exist θ and θ 1 determined by (C.11) and (C.12) that satisfy (C.15). Then we have θ 1 2m(θ, θ 1 ) m(θ 1, γ(θ 1 )) + S = p + m(0, θ ) m(θ 1, γ(θ 1 )) + S = m(0, θ ) + S + (p m(θ 1, γ(θ 1 ))) m(0, θ ) + S, where the first inequality follows from (C.12), the second equality follows from (C.11), and the last inequality follows from (C.15). Conversely, one can show that if θ and θ 1 satisfy (C.11) and (C.12), then θ 1 m(0, θ ) + S is sufficient for them to satisfy (C.15). Q.E.D. Claim 2. Suppose θ, θ 1, and θ ø satisfy (C.11), (C.17), and (C.18). Then θ 1 and θ ø satisfy (C.20) if and only if 2m(θ, θ 1 ) θ 1 m(θ 1, γ(θ 1 )) + S > 0. Proof. Suppose there exist θ, θ 1, and θ ø that satisfy (C.11) (C.18) and also (C.20). Then we have 0 = m(0, θ ) θ 1 + S = 2m(θ, θ 1 ) θ 1 p + S = 2m(θ, θ 1 ) θ 1 m(θ ø, γ(θ ø )) + S < 2m(θ, θ 1 ) θ 1 m(θ 1, γ(θ 1 )) + S, where the first equality follows from (C.17), the second equality follows from (C.11), the third equality follows from (C.18), and the last inequality follows from (C.20). Conversely, one can also show that if θ, θ 1, and θ ø satisfy (C.11) (C.18), then 2m(θ, θ 1 ) θ 1 m(θ 1, γ(θ 1 ))+S > 0 is sufficient for them to satisfy (C.20). Q.E.D. Fix a p at which both (θ MR1 (p ), θ1 MR1 (p )) and (θ MR2 (p ), θ1 MR2 (p )) are well-defined. Recall θ 1 (θ ), θ1(θ I ), and θ1 II (θ ) from the proofs of Lemma C.3 and Lemma C.4, which are the functions of θ correspondin to (C.11), (C.12), and (C.17), respectively. If (θ MR1 (p ), θ1 MR1 (p )) violates (C.15), then we have θ1 MR1 (p ) > m(0, θ MR1 (p ))+S from Claim 1. This implies θ 1 (θ MR1 (p )) > θ1 II (θ MR1 (p )). Furthermore, we have shown in Lemma C.4 16

17 θ 1 θ 1 = max{θ > θ : 2m(θ, θ) θ m(θ, γ(θ)) + S 0} θ 1 θ 1 = max{θ > θ : 2m(θ, θ) θ m(θ, γ(θ)) + S 0} θ MR1 1 θ MR2 1 θ 1 = m(0, θ ) + S p = 2m(θ, θ 1 ) m(0, θ ) θ 1 = m(0, θ ) + S θ1 MR2 θ1 MR1 p = 2m(θ, θ 1 ) m(0, θ ) 0 θ θ MR1 θ MR2 (a) θ 1 (θ MR1 ) > θ II θ 1 (θ MR2 ) < θ1 I(θMR2 1 (θmr1 ) ), or equivalently, 0 θ θ MR2 θ MR1 (b) θ 1 (θ MR1 ) θ II θ 1 (θ MR2 ) θ1 I(θMR2 1 (θmr1 ) ), or equivalently, Fiure 3 p P MR supports only one type of MR equilibria. that θ 1 (θ ) is decreasin in θ and θ1 II (θ ) is increasin in θ if they are well-defined. Thus we have θ MR1 (p ) < θ MR2 (p ) and θ1 MR1 (p ) > θ1 MR2 (p ) as illustrated by Fiure 3a. Moreover, we have shown in the proof of Lemma C.3 that θ1(θ I ) is increasin in θ if it is well-defined. Hence, we have θ I 1(θ MR2 (p )) > θ1(θ I MR1 (p )) = θ MR1 (p ) > θ1 MR2 (p ), which implies 2m(θ MR2 (p ), θ1 MR2 (p )) θ1 MR2 (p ) m(θ1 MR2 (p ), γ(θ1 MR2 (p ))) + S > 0. Therefore, (θ MR2 (p ), θ1 MR2 (p )) satisfies (C.20) by Claim 2. Conversely, If (θ MR1 (p ), θ1 MR1 (p )) satisfies (C.15), we have θ1 MR1 (p ) m(0, θ MR1 (p ))+ S from Claim 1. This implies θ 1 (θ MR1 (p )) θ1 II (θ MR1 (p )). Since θ 1 (θ ) is decreasin in θ and θ1 II (θ ) is increasin in θ if θ 1 and θ1 II are well-defined, we have θ MR1 (p ) θ MR2 (p ) and θ1 MR1 (p ) θ1 MR2 (p ) as illustrated by Fiure 3b. When θ1 MR1 (p ) < 1, we have 2m(θ MR1, θ1 MR1 ) θ1 MR1 m(θ1 MR1, γ(θ1 MR1 )) + S = 0. Then, by Assumption A.2-(i), we have 2m(θ MR2 (p ), θ1 MR2 (p )) θ1 MR2 (p ) m(θ1 MR2 (p ), γ(θ1 MR2 (p ))) + S 0, which implies that (θ MR2 (p ), θ1 MR2 (p )) violates (C.20) by Claim 2. When θ1 MR1 (p ) = 1, we 17

18 must have θ1 MR2 (p ) = 1: otherwise, we have θ1 MR1 (p ) > m(0, θ MR2 (p )) + S. This implies 1 = θ1 MR2 (p ) θø MR2 (p ). Then (θ1 MR2 (p ), θø MR2 (p )) violates (C.20). Q.E.D. The followin observation follows immediately from the above. Corollary C.1. At most one of MR1 and MR2 exists for any iven p. In what follows, we characterize the rane of bailout terms that admit either type of MR equilibria, denoted by P MR. Specifically, we show that P MR is a convex set. Lemma C.6. P MR is a convex set, whenever it is non-empty. Proof. Step 1. MR equilibrium cannot exist for any p p MR2. Fix p p MR2. Clearly, MR2 cannot exist by Lemma C.4. Suppose (θ MR1 (p ), θ1 MR1 (p )) exists and satisfies θ1 MR1 (p ) m(0, θ MR1 (p )) + S, which is equivalent to θ 1 (θ MR1 (p )) θ1 II (θ MR1 (p )), where θ 1 and θ1 II are functions of θ correspondin to (C.11) and (C.17) respectively, as in the proof of Lemma C.4. Recall also that θ 1 (θ ) is decreasin in θ and θ1 II (θ ) is increasin in θ. These properties imply that (θ MR2 (p ), θ1 MR2 (p )) also exists at the same p as depicted by Fiure 3b, a contradiction to Corollary C.1. Therefore, if (θ MR1 (p ), θ1 MR1 (p )) exists, we must have θ1 MR1 (p ) > m(0, θ MR1 (p )) + S. By Lemma C.5, the last inequality implies (θ MR1 (p ), θ1 MR1 (p )) violates (C.15), so MR1 cannot exist, a contradiction. Step 2. MR equilibrium exists for every p (p MR1 Fix p (p MR1, 1] [p MR2 (θ MR2 (p ), θ MR2 If θ1 MR1 (p ) m(0, θ MR1, p MR2 MR1 exists. Otherwise, by Lemma C.5, (θ MR2, 1] [p MR2, p MR2 ). ). By Lemma C.3 and C.4, both (θ MR1 (p ), θ1 MR1 (p )) and (p ) and θ MR2 (p ) satisfy (C.14). 1 (p )) are well-defined. Furthermore, both θ MR1 (p )) + S, then by Claim 1, (θ MR1 (p ), θ1 MR1 (p )) satisfies (C.15), and (p ), θ1 MR2 (p )) satisfies (C.20), and MR2 exists. Step 3. If p MR1 < p MR2, then MR1 exists for every p (p MR1, p MR2 ). Fix p (p MR1, p MR2 ). Since p > p MR1, it follows from Lemma C.3 that there exists (θ MR1 (p ), θ1 MR1 (p )) that satisfies m(0, θ MR1 (p )) I. Suppose θ1 MR1 (p ) > m(0, θ MR1 (p )) + S, or equivalently, θ 1 (θ MR1 (p )) > θ1 II (θ MR1 (p )). Since p < p MR2, the inequality θ 1 (θ MR1 (p )) > θ1 II (θ MR1 (p )) and Lemma C.4 ensure that (θ MR2 (p ), θ1 MR2 (p )) is well-defined. The same inequality also implies θ MR2 (p ) > θ MR1 (p ), as is seen in Fiure 3a. However, θ MR2 (p ) > θ MR1 (p ) 18

19 implies m(0, θ MR2 (p )) > m(0, θ MR1 (p )) I, which contradicts p < p MR2. We thus conclude that θ1 MR1 (p ) m(0, θ MR1 (p )) + S. By Lemma C.5, (θ MR1 (p ), θ1 MR1 (p )) satisfies (C.15), so MR1 exists. Step 4. Suppose p MR2 if and only if p MR1 < p MR2, then MR equilibrium exists for every p [p MR2, p MR1 ] 2m(θ MR2 (p MR2 ), θ MR2 1 (p MR2 )) θ MR2 1 (p MR2 ) m(θ MR2 1 (p MR2 ), γ(θ MR2 1 (p MR2 ))) + S 0 (C.21) The proof is tedious and therefore omitted, but it is available from the authors. Step 5. Suppose p MR1 one can find that every p p [p MR2 p MR1 if p MR1, p MR2 p MR2. Proceedin similarly as in Step 4 and applyin Step 1 and 2, [p MR2, p MR2 ) supports MR equilibrium if (C.21) holds, but no ) supports MR equilibrium otherwise. Combinin all the results above toether, we conclude that: P MR < p MR2 [p MR2 ; P MR = [p MR2, p MR2 holds; P MR = if p MR1, p MR2 ) if p MR1 [p MR2, p MR2 = (p MR1 ) and (C.21) holds; P MR = (p MR1, p MR2 ) if, p MR2 ) ) but (C.21) does not hold; P MR = [p MR2, p MR2 ) if p MR1 p MR2 and (C.21) p MR2 but (C.21) does not hold. Q.E.D. The followin lemma describes an important property of P MR, which will be used in the proof of Theorem 3. Lemma C.7. If P MBS and P MR, then sup P MBS = inf P MR. Proof. Recall from Lemma C.2 that sup P MBS p = p MBS, we have = p MBS. Since the condition (C.10) binds at It follows that θ MBS θ MBS (p MBS ) m(θ MBS (p MBS ), γ(θ MBS (p MBS ))) + S = 0. (C.22) (p MBS ) = θ, where θ, as defined in the proof of Lemma C.3-(i), is a lower (p ) satisfies (C.7), (C.22) also implies bound of θ such that (C.12) is well-defined. Since θ MBS p MBS = 2θ MBS (p MBS ) m(0, θ MBS (p MBS )) = 2m(θ MBS (p MBS ), θ MBS (p MBS )) m(0, θ MBS (p MBS )). (C.23) 19

20 Since θ MBS (p MBS ) = θ, (C.23) implies that (θ MR1 (p ), θ MR1 1 (p )), the unique intersection be-, and thus p MBS = ˇp MR1. tween two curves θ 1 (θ ) and θ I 1(θ ), exists if and only if p > p MBS Since θ1(θ I ), if well-defined, is increasin in θ, θ MR1 (p ) m(θ MR1 (p ), γ(θ MR1 0 for all p > ˇp MR1. Since θ m(θ, γ(θ)) is increasin in θ, we have θ MR1 (p ) > θ MBS all p > p MBS = ˇp MR1 which implies p MBS (p ), θ MR2 (θ MR2 from (C.22). By (C.8), we have m(0, θ MR1 = p MR1 1 (p )) exists for some p p MBS S 0. Thus by Lemma C.3. Furthermore, that p MBS, we have θ MR2 (p )))+S > (p MBS ) for (p )) > I for all p > ˇp MR1 = ˇp MR1 (p ) m(θ MR2 2m(θ MR2 (p ), θ1 MR2 (p )) θ1 MR2 (p ) m(θ1 MR2 (p ), γ(θ1 MR2 (p ))) + S < 0,, implies that even if (p ), γ(θ MR2 (p )))+ where the strict inequality follows from Assumption A.2-(i). Hence, (θ MR2 (p ), θ1 MR2 (p )) violates, and thus p MBS = inf P MR. (C.20) by Claim 2. Therefore, we have inf P MR = p MR1 = p MR1 Q.E.D. C.1.5 Necessary Conditions for Equilibria under the Market Shutdown in t = 1 In equilibrium, there exist θ (0, θ 2 ) and θ 2 1 such that types θ [0, θ ] sell to the overnment at price p in t = 1 and sell to the market at price m(0, θ θ0) in t = 2 if m(0, θ ) I; and types θ (θ, θ 2 ) sell only in t = 2 at price m(θ, θ 2 ). As before, we derive conditions for such an equilibrium to exist. There are three types of equilibria: (i) m(0, θ ) < I; (ii) m(0, θ ) I and θ θ0; (iii) θ > θ0. In case (i), types θ [0, θ ] cannot sell in t = 2. Thus, as in SBS, we must have: θ 2 = γ(θ ), p = m(θ, γ(θ )), s.t. m(0, θ ) < I, which are same the as (C.1) (C.3). However, the conditions (C.4) and (C.5) are no loner necessary since t = 1 market is shut down. With the D1 refinement, the case θ = 1 arises if and only if p m(1, γ(1)) = 1 and E[θ] = m(0, 1) < I. In case (ii), all types θ [0, θ ] sell in t = 2 at price m(0, θ ). Since this equilibrium has 20

21 the same structure as MBS equilibrium, θ and θ 2 satisfy subject to θ 2 = γ(θ ), p + m(0, θ ) + S = θ + m(θ, γ(θ )), m(0, θ ) I, θ θ 0, which are the same as (C.6) (C.8) and (C.10). Given that the t = 1 market is shut down, the condition (C.9) is no loner necessary. With the D1 refinement, the case θ = 1 can be supported if and only if p 1 + m(1, γ(1)) m(0, 1) S = 2 E[θ] S and θ 0 = 1. Lastly, consider case (iii), where types θ [0, θ ] sell to the overnment in t = 1, but only types θ [0, θ 0] sell at price m(0, θ 0) in t = 2. In this equilibrium, θ and θ 2 satisfy θ 2 = γ(θ ), p = m(θ, γ(θ )), s.t. θ > θ 0. With the D1 refinement, the case θ = 1 can be supported if and only if p m(1, γ(1)) = 1. Usin the conditions above and proceedin similarly as in the proof of Theorem 3-(ii) below, one can show that the above conditions on θ and θ 2 are also sufficient to support each type of the equilibria. For later use, we let θ sd (p ) denote the marinal types satisfyin the conditions for each alternative type of equilibria. To avoid expositional complexity, θ sd (p ) can be occasionally abbreviated to θ sd. Note that θ sd (p ) is continuous and increasin in p if θ sd (p ) is well-defined at such p. C.2 Proofs of Theorem 3 and Proposition 2 4 Theorem 3. There exists an interval of bailout terms P k R + that supports alternative equilibrium types k = NR, SBS, MBS, MR, described as follows: (i) No Response (NR): If p P NR, then there exists an equilibrium in which no firm accepts the overnment offer and the outcome in Theorem 2 prevails. 21

22 (ii) No Market Rejuvenation Severe Bailout Stima (SBS): If p P SBS, then there exists an equilibrium with Θ = Θ 1 =, Θ ø, Θ 2. Moderate Bailout Stima (MBS): If p P MBS, then there exists an equilibrium with Θ, Θ 2, Θ 1 = Θ ø =. (iii) Market Rejuvenation (MR): If p P MR, then there exists an equilibrium with Θ 1. Specifically, P NR = [0, p 2], inf P SBS = p 0, and sup P MBS inf P MR, meanin an MR equilibrium requires a strictly hiher p than does an MBS equilibrium. Proof of Theorem 3. We first state a lemma that will be used in the proof. Lemma C.8. Suppose buyers in t = 2 believe that types θ [a, b] offer assets for sale. Then buyers offer the price m(a, γ(a) b). If m(a, γ(a) b) I, then types θ [a, γ(a) b] sell their assets. If m(a, γ(a) b) < I, then the t = 2 market fully freezes. Proof. See Proposition 1 of Tirole (2012). Q.E.D. Proof of Theorem 3-(i). Suppose p p 2. Recall p 2 is defined in Theorem 2-(i). Consider the strateies specified in Theorem 3-(i). The beliefs on the equilibrium path are iven by the Bayes rule. The off-the-path belief for firms which accept the bailout in t = 1 is θ = 0, and such a belief satisfies D1. Given these beliefs, type θ s payoff from acceptin the bailout is p + S + θ. If θ [0, θ 1], then we have 2p 1 + 2S θ + p 2 + S p + S + θ. If θ (θ 1, 1], then we have θ + max{θ, p 2 + S} p + S + θ. In either case, it is not profitable to accept the bailout. Hence the NR equilibrium exists for all p [0, p 2]. Proof of Theorem 3-(ii). Consider first the SBS equilibrium. As we saw in Section C.1.2, iven p P SBS, there exist θ ø = θ SBS ø (p ) and θ 2 = γ(θø SBS (p )) that satisfy (C.3) (C.5). We show that the prescribed equilibrium strateies are optimal for all types of firms. Consider t = 2. For bailout recipients, the t = 2 market fully freezes since m(0, θ ø ) < I from (C.3). For t = 1 holdouts, the t = 2 market offer is m(θ ø, γ(θ ø )). Given this, types θ (θ ø, γ(θ ø )] sell in t = 2, but types θ > γ(θ ø ) do not since θ m(θ ø, γ(θ ø )) + S θ γ(θ ø ). Consider now t = 1. Type-θ firm receives payoff p + S + θ from acceptin the 22

23 bailout and θ + max{m(θ ø, γ(θ ø )) + S, θ} from rejectin it. Since p = m(θ ø, γ(θ ø )) from (C.2), playin the prescribed equilibrium stratey in t = 1 is optimal for every type θ [0, 1]. Given the equilibrium strateies chosen by firms, it is straihtforward to see that the equilibrium price offers are also optimal and buyers break even. Next consider the MBS equilibrium. As we saw in Section C.1.3, for every p P MBS, the marinal types θ = θ MBS (p ) and θ 2 = γ(θ MBS (p )) satisfy (C.8) (C.10). Usin these conditions and proceedin similarly as in the SBS equilibrium, it is easy to show that the prescribed equilibrium strateies are optimal for all types of firms, and the equilibrium price offers are optimal for buyers in the market. What remains to show is that it is optimal for types θ [0, θ ] to sell at price m(0, θ ) in t = 2 after acceptin the bailout, and for buyers in t = 2 to offer the price m(0, θ ) to the bailout recipients. But these follow immediately from (C.8), (C.10), and Lemma C.8. Proof of Theorem 3-(iii). Consider the MR1 equilibrium first. Consider p P MR such that there exist θ = θ MR1 (p ), θ 1 = θ1 MR1 (p ), and θ 2 = γ(θ1 MR1 (p )) that satisfy (C.14) and (C.15). We first show that it is optimal for each type of firms to play the prescribed equilibrium strateies. Consider t = 2 on the equilibrium path. Since m(0, θ ) I from (C.14), θ < θ 1 m(0, θ ) + S < m(θ, θ 1 ) + S from (C.15) and Claim 1, and γ(θ 1 ) m(θ 1, γ(θ 1 )) + S from Definition A.1-(ii), it is optimal for types θ [0, θ ] to sell at price m(0, θ ), types θ (θ, θ 1 ] to sell at price m(θ, θ 1 ), and types θ (θ 1, γ(θ 1 )] to sell at price m(θ 1, γ(θ 1 )). However, types θ (γ(θ 1 ), 1] do not sell since θ > m(θ 1, γ(θ 1 )) + S for all θ > γ(θ 1 ) from Lemma A.1-(i). Next consider t = 1. Acceptin the bailout is optimal for types θ [0, θ ] since p + m(0, θ ) + 2S = 2m(θ, θ 1 ) + 2S θ + m(θ 1, γ(θ 1 )) + S, (C.24) where the first equality follows from (C.11) and the second inequality is from (C.12). From (C.24), it is also optimal for types θ (θ, θ 1 ] to sell at price m(θ, θ 1 ). Lastly, since 2m(θ, θ 1 ) + 2S < θ + max{θ, m(θ 1, γ(θ 1 )) + S} for all θ > θ 1, it is optimal for types θ (θ 1, 1] not to sell in t = 1. Next, we show that the equilibrium price offers are optimal for buyers in each period. The optimality of t = 2 prices directly follows from Lemma C.8, so we consider only t = 1. We show below that no buyer benefits by deviatin from offerin m(θ, θ 1 ). In t = 1, buyers believe that only types θ > θ are available for asset sales to the market. Suppose a buyer deviates and offers p m(θ, θ 1 ). Any firm that accepts this offer is assined the off-the-path belief that it is the worst available type, i.e., θ = θ. This belief is consistent with D1. Given this belief, any such 23

24 firm will be offered price p 2 = θ in t = 2. Since θ > m(0, θ ) I, the deviatin firm enjoys the total payoff p + S + max{θ, θ + S}. There are two possibilities, either 2m(θ, θ 1 ) θ m(θ 1, γ(θ 1 )) or 2m(θ, θ 1 ) θ < m(θ 1, γ(θ 1 )). Consider the former case. If p m(θ 1, γ(θ 1 )) 2m(θ, θ 1 ) θ, we have 2m(θ, θ 1 )+2S θ 1 +p +S, 2m(θ, θ 1 )+2S p +θ +2S, and p +S +θ θ +m(θ 1, γ(θ 1 ))+S. Therefore, no type θ (θ, 1] will sell at such p. If p > m(θ 1, γ(θ 1 )), then we have 2m(θ, θ 1 ) + 2S < θ 1 + p + S. Lettin θ := 2m(θ, θ 1 ) + S p < θ 1, all types θ ( θ, (p + S) 1] sell at p. However, by definition of γ( ), we have m( θ, (p + S) 1) p < m(θ 1, (p + S) 1) p < 0, and thus the deviatin buyer will make a loss by offerin p m(θ, θ 1 ). Consider the latter case next. If p (2m(θ, θ 1 ) θ, m(θ 1, γ(θ 1 ))], then types θ (θ, θ 1] will sell at p, where θ := p m(θ 1, γ(θ 1 )) + θ + S. Since lim p 2m(θ,θ 1 ) θ θ = θ 1, we have lim p 2m(θ,θ 1 ) θ (m(θ, θ 1) p ) = m(θ, θ 1 ) (m(θ, θ 1 ) + (m(θ, θ 1 ) θ )) < 0. Moreover, since dθ = 1 and m(a, b) < 1 for any 0 a b 1 from Lemma A.1-(i), we dp b have d (m(θ dp, θ 1) p ) 0 for all p (2m(θ, θ 1 ) θ, m(θ 1, γ(θ 1 ))]. As a result, it is not profitable for the buyers to offer any p (2m(θ, θ 1 ) θ, m(θ 1, γ(θ 1 ))]. If p > m(θ 1, γ(θ 1 )), then all types θ (θ, (p + S) 1] will sell at p. Since p > m(θ 1, γ(θ 1 )) > m(θ, γ(θ )), we have m(θ, (p + S) 1) p < 0 for all p > m(θ 1, γ(θ 1 )). Hence, the deviatin buyer will make a loss by offerin p. If p 2m(θ, θ 1 ) θ, then, as shown before, types θ > θ will not sell at such p. Consequently, all buyers in t = 1 optimally offer m(θ, θ 1 ). We now turn to the MR2 equilibrium. Suppose MR2 exists at p P MR. Then, by Lemma C.6, there exist θ = θ MR2 (p ), θ 1 = θ MR1 satisfyin (C.14) and (C.20). 1 (p ), θ ø = θø MR2 (p ), and θ 2 = γ(θ MR2 ø (p )) First, we show that it is optimal for each type of firms to play the prescribed equilibrium strateies. Consider t = 2 first. Since θ 1 = m(0, θ ) + S from (C.17), we have θ < θ 1 = m(0, θ ) + S, θ 1 = m(0, θ ) + S < m(θ, θ 1 ) + S, and θ > m(0, θ ) + S for all θ > θ 1. Therefore it is optimal for types θ [0, θ ] to sell at price m(0, θ ), but types θ (θ 1, θ ø ] not to sell at that price. Furthermore, it is optimal for types θ (θ, θ 1 ] to sell at price m(θ, θ 1 ). Finally, from the definition of γ( ), it is optimal for types θ (θ ø, γ(θ ø )] to sell at price m(θ ø, γ(θ ø )), but types θ (γ(θ ø ), 1] not to sell at that price. Consider next t = 1. From (C.11), (C.17), and (C.18), 24