Online Appendix for The E ect of Diversi cation on Price Informativeness and Governance

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1 Online Appendix for The E ect of Diersi cation on Price Informatieness and Goernance B Goernance: Full Analysis B. Goernance Through Exit: Full Analysis This section analyzes the exit model of Section. in full. B.. Exit Under Concentration Lemma 5 characterizes the thresholds that emerge in any equilibrium under concentration. Lemma 5 (Concentration, exit): The unique threshold for each manager, c con;exit, is gien by the solution of c = exit (F (c )) where exit () =! + : (9) Prices and trading strategies are characterized by Lemma, where is gien by con;exit F c con;exit. B.. Exit Under Diersi cation Lemma 6 characterizes the most e cient equilibrium under diersi cation. Lemma 6 (Diersi cation, exit): The working threshold under the most e cient equilibrium is gien by c di;exit = if ( F ()) >< c i;exit the largest solution of c = exit (F (c )) if ( F ()) < c ii;exit the largest solution of c = exit (F (c )) if ii;exit < < >: c con;exit if ; (0) ii;exit

2 where ii;exit F and c ii;exit, exit () = exit () =! + : ()! + : () Prices and trading strategies are characterized by Lemma, where is gien by di;exit F c di;exit. Lemma 6 leads to Proposition in the main text. Note that, in the model of Section with exogenous asset alues, ( F ()) < < automatically led to a moderate liquidity shock, i.e., one that is large enough to force a shocked inestor to sell some good rms, in contrast to a small shock where she can sell bad rms only. Howeer, whether the shock is small or moderate depends on the size of the shock relatie to the mass of bad rms, and so when asset alues and thus the mass of bad rms are endogenous, we must consider two separate cases, gien by the second and third lines of equation (0). If ii;exit < < (the third line, case-(iii) ) there are su ciently few bad rms that a shocked inestor must sell good rms, and so we are in the moderate-shock case where price informatieness, and thus the working threshold, are strictly higher under diersi cation. If ii;exit, there are enough bad rms that a shocked inestor need not sell any good rms, and so we are in the smallshock case. Recall that a small-shock equilibrium can inole an unshocked inestor either selling all bad rms, or retaining some bad rms. The former inoles a greater punishment for shirking and thus leads to fewer bad rms in equilibrium. If ( F ()) (the rst line, case-(i) ), the shock is so small that a small-shock equilibrium can be sustained een if the unshocked inestor sells all bad rms. This will be the case in the most e cient equilibrium; price informatieness, and thus the threshold, are again strictly higher under diersi cation. If ( F ()) < ii;exit (the second line, case-(ii) ), the shock is su ciently large that a small-shock equilibrium requires an unshocked inestor to retain some bad rms. This reduces the punishment for shirking, and also the reward for working by lowering the price of a retained rm below, thus creating su ciently many bad rms that the inestor can satisfy a shock by selling only them. In this case, price informatieness and the threshold may be lower under diersi cation. The ambiguity of case-(ii) explains the formulation of Proposition that the threshold is strictly higher under diersi cation if l where

3 l ( F ()). The lowest l can be is ( F ()): if ( F ()), we are guaranteed to be in case-(i), where the threshold is unambiguously higher under diersi cation the ambiguity of case-(ii) is irreleant. Howeer, l could be as high as. This will occur if the threshold is higher under diersi cation een in case-(ii). Then, the threshold is higher not only in case-(i), but also in case-(ii) and case-(iii) also. Note that, een if the threshold is lower under diersi cation in case-(ii), the condition in Proposition is only su cient, not necessary, for diersi cation to be more e cient. Een if l = ( F ()), diersi cation is more e cient not only if l (case-(i)), but also in case-(iii) which inoles > l. While Lemma 6 focuses on the most e cient equilibrium, Proposition 5 considers all equilibria and show that, if is su ciently high, any equilibrium under diersi cation is strictly more e cient than the concentration benchmark. If is su ciently low, there exist less ef- cient equilibria these are the small-shock equilibria where the inestor retains some bad rms. In such equilibria, the incenties to work are decreasing in the frequency with which a bad rm is retained. This frequency is greater if is low, since a bad rm is always retained upon no shock, and if L is low, since a smaller shock allows the inestor to retain more bad rms upon a shock. Howeer, under the e ciency criterion, the most e cient equilibrium will be chosen and so goernance is always weakly stronger under diersi cation. Proposition 5 (Comparison of equilibria, exit): Suppose ( F ()). There is [0; ) s.t. if, any equilibrium under diersi cation is strictly more e cient than any equilibrium under concentration. B..3 Proofs Proof of Lemma 5. In equilibrium, the buyer and the inestor beliee the manager follows threshold c. Gien (3) and (4), the manager expects the price to be P con ( i ; F (c )) if he chooses i. Therefore, he works if and only if (!) +!P con (; F (c )) ec i (!) +!P con (; F (c )) ; where P con ( i ; ) is gien in equation (7). In equilibrium, c must be such that the aboe incentie constraint binds at ec i = c, i.e., c = exit (F (c )). Note that exit () is decreasing in and is bounded from aboe and below. Therefore, a solution always exists and is unique, 3

4 as required. The equilibrium is characterized by Lemma, where = con;exit. Proof of Lemma 6. We use type-(i) equilibrium, type-(ii) equilibrium, and type-(iii) equilibrium to refer to the equilibria gien in parts (i), (ii), and (iii) of Lemma, respectiely. First, suppose. Based on Lemma, any equilibrium is type-(iii). Therefore, c di;exit = c con;exit. Similar to the proofs of Lemma part (iii) and Lemma 5, such an equilibrium indeed exists. Second, suppose works if and only if ii;exit <. Consider a type-(ii) equilibrium. The manager (!) +! [p di ( ) + ( ) ] ec i (!) +! [ + ( ) p di ( )] : Using p di () = +, we obtain +( )( ) i =, ec i exit ( ). Therefore, c must sole c = exit (F (c )). Similar to the proof of Lemma part (ii), if = ii;exit an equilibrium with these properties indeed exists. By de nition of c ii;exit, such an equilibrium is more e cient than any other type-(ii) equilibrium. Moreoer, simple algebra shows that exit () > exit (), and so c ii;exit > c con;exit. That is, an equilibrium with = ii;exit is more e cient than any type-(iii) equilibrium. Finally, we show that an equilibrium with = ii;exit is more e cient than any type-(i) equilibrium. Based on part (i) of Lemma, the threshold of the alternatie equilibrium must satisfy ( ). Howeer, since by assumption > ii;exit, it follows that < ii;exit, i.e., the alternatie equilibrium is less e cient. Third, suppose < ii;exit. Note that type-(ii) or type-(iii) equilibria do not exist in this range. Indeed, by the de nition of ii;exit, any other type-(ii) equilibrium satis es ii;exit, and since < ii;exit, it also satis es < ( ), which contradicts it being a type-(ii) equilibrium. Any type-(iii) equilibrium must hae = F ccon;exit. Since c con;exit < c ii;exit and < ii;exit, it must be < ( ), which contradicts it being a type-(iii) equilibrium. Next, we argue that a type-(i) equilibrium exists in this range. Recall that there are only two kinds of type-(i) equilibrium: those in which the inestor sells bad rms if = 0 and those in which she retains them. We argue that the following strategies 4

5 are an equilibrium: the manager s working threshold is if ( F ()) >< c = c i;exit the largest solution if ( F ()) < >: of c! ii;exit ; = F (c ) + F (c ) the inestor s trading strategy is 0 if >< i = x ( i ; ) = if i = and = 0 >: if i = and = L; where and prices are < if ( F ()) = : 0 if ( F ()) < < ii;exit < F (c + ) if x p F (c i (x i ) = )+( )( )( F (c )) i = 0 : if x i > 0: ; To see that the aboe equilibrium indeed exists, rst note that the prices in this equilibrium follow from the inestor s trading strategy and Bayes rule. Second, gien these prices, the inestor s trading strategy is optimal. Indeed, note that ( F (c )), and so the inestor can satisfy her liquidity need by selling only bad rms. Since x i > 0 ) p i =, she has strict incenties to fully retain good rms, and weak incenties to sell bad rms. The manager works if and only if (!) +!p i (0) ec i (!) +! [ + ( ) (( )p i (0) + )], (!) +! ( ( ) ( )) (p i (0) ) ec i : 5

6 Using the explicit form of p i (0) as gien aboe, this yields (!) +! ( ( ) ( F (c ) )) F (c ) + ( ) ( ) ( F (c )) ec i, H ( ; c ) ec where H (; c) = F (c)! + F (c): Existence follows if there is f0; g such that H ( ; c ) = c and ( F (c )). Note that = requires c =. Therefore, a type-(i) equilibrium with = exists if and only if ( F ()). Since this equilibrium attains the rst best, it is the most e cient one. Suppose ( F ()) < ii;exit. The only candidate equilibrium is a type- (i) equilibrium with = 0, and so it must be the most e cient equilibrium. The working threshold must sole H (0; c ) = c ; a solution always exists. Also note that H (0; c ) = exit (F (c )) < exit () for any (0; ), and so c i;exit < c ii;exit. Since < ii;exit, it must be < i;exit, which shows that a type-(i) equilibrium indeed exists. Proof of Proposition 5. Suppose ( F ()). From Lemma 5, the unique working threshold in any equilibrium under concentration is strictly smaller than, i.e., c con;exit <. Moreoer, since c con;exit soles exit (F (c )) = c, lim! c con;exit = (!). From Lemma 6, a type-(i) equilibrium under diersi cation exists in which the working threshold is. From the proof of Lemma 6, the only other possible equilibrium is a type-(i) equilibrium in which the inestor retains bad rms if = 0. If such an equilibrium exists, the working threshold must satisfy exit (F (c )) = c. Note that as! any solution of exit (F (c )) = c conerges to. Therefore, if is su ciently close to, any equilibrium under diersi cation is strictly more e cient than any equilibrium under concentration. B. Goernance Through Voice: Full Analysis This section analyzes the oice model of Section. in full. 6

7 B.. Voice Under Concentration Lemma 7 characterizes the monitoring threshold in any equilibrium under concentration. Lemma 7 (Concentration, oice): In any equilibrium, the monitoring threshold, c con;oice, is gien by the solution of c =n = oice (F (c )), where oice () min + ( ( )) ; + : (3) Prices and trading strategies are characterized by Lemma, where is gien by con;oice F c con;oice. B.. Voice Under Diersi cation Lemma characterizes the monitoring threshold in any equilibrium under diersi cation for ( F ()). Lemma If ( F ()), then an equilibrium always exists under diersi cation, and the monitoring threshold is in any equilibrium. In the exit model, Lemma 6 held only for the most e cient equilibrium under diersi cation; Proposition 5 required an extra condition on to guarantee that goernance is stronger under all equilibria under diersi cation. In the oice model, part (ii) of Proposition 3 holds for all equilibria under diersi cation, without the need for an extra condition on. This is because, while multiple equilibria exist, they di er only in terms of the inestor s trading strategy, and not her monitoring strategy. For ( F ()), een if price informatieness is lower under diersi cation, per-security monitoring incenties remain higher. This is because the only way in which price informatieness can be lower is if the inestor retains bad assets upon a shock, and so being retained is not fully reealing that a rm has been monitored. Howeer, this does not a ect the inestor s incenties to monitor, since her payo of a monitored and retained rm is its fundamental alue of, regardless of the stock price. Thus, the threshold is in any equilibrium for which ( F ()). Proposition 3, in the main text, soles for the ownership structure that maximizes rm alue. Howeer, if the inestor could endogenously choose ownership structure, she would 7

8 select the one that maximizes her expected portfolio alue minus monitoring costs (expected trading pro ts are zero under both structures): she only internalizes the e ect of her monitoring on her z units rather than the entire rm. This result is gien in Proposition 6: Proposition 6 (Inestor s choice of equilibrium, oice): For any 0 < L L, there exists < n (L) n (L) such that, if < n < n (L), the inestor s expected payo net of monitoring costs under any equilibrium of diersi cation is strictly higher than under any equilibrium under concentration. ( Finally, while unnecessary for the main result in Section. (which holds for F ())), for completeness, Lemma 9 gies the most e cient equilibrium under diersi cation when > ( F ()). Lemma 9 Suppose > ( F ()). There are ( F ()) < y y such that the monitoring threshold under the most e cient equilibrium is gien by c di;oice = where >< >: c ii the largest solution of c = oice (F (c )) if ( F ()) < < y maxfc ii ; c c iiig if y < y iii the largest solution of c = oice (F (c )) if y; oice () Prices and trading strategies are characterized by Lemma. " (4) # ( ) + ( ( ) + ) + : (5) B..3 Proofs Proof of Lemma 7. monitors if and only if The proof of Lemma 4 shows that under concentration the inestor ec i =n x con ( ) n ( p con ( )) x con ( ) n (p con ( ) ) : (6) This holds if and only if ec i =n x con ( ) n +, ec i =n oice ( )

9 Thus, the cuto in any equilibrium must satisfy c =n = oice ( ). In equilibrium, = F (c ), and so c con;oice must sole c =n = oice (F (c )), as required. Note that as a function of c, oice (F (c )) is strictly positie and bounded from aboe. Therefore, a strictly positie solution always exists. If ( ) 0, then oice (F (c )) is decreasing in c and so the solution is unique. Note that since V con ( i ; ) is deried from Lemma, the equilibrium is characterized by Lemma, where is gien by con;oice. Proof of Lemma. Suppose ( F ()). Recall that in any equilibrium under diersi cation, c. Therefore, ( F ()) implies ( ). From Lemma, this implies that if an equilibrium exists, it must be type-(i). For any cuto c (which could di er from the cuto c anticipated by the buyers), she expects to sell x (c) [0; ] of a good rm upon a shock, where x (c) = min ; max 0; ( F (c)) F (c) is increasing in c. Indeed, she will sell good rms only if she cannot satisfy her liquidity need by selling all ( F (c)) bad rms. Since ( ) <, we can rewrite x (c) = max ( 0; ) : F (c) Let (c ; c) be the inestor s expected payo, including the possibility of trade, gien that the buyers set prices under the belief there are F (c ) monitored rms, and the inestor monitors a fraction F (c) of all rms, where we allow for c 6= c. Therefore, (7) (c ; c) =n = F (c) ( x (c) ) + ( F (c)) F (c) E [c i jc i < c] () < F (c) ( E [c i jc i < c]) if c F = + (9) : ( ) + F (c) ( ( ) E [c ijc i < c]) otherwise. Since ( F ()) ) F, and it is always sub-optimal to choose c > regardless of buyers beliefs, the inestor s optimal threshold soles c arg max c[0;] F (c) ( E [c i jc i < The rst deriatie with respect to c is f (c) ( c) and the second deriatie is f 0 (c) ( c) 9

10 f (c). Therefore, if an equilibrium exists, it must entail c =. Since ( by construction, there is a type-(i) equilibrium with such a threshold. F ()), Proof of Proposition 6. Gien a threshold c and number of rms n, the inestor s net payo s under concentration and diersi cation, respectiely, are con;oice (n; c) = n ( + F (c) ) F (c) E [c i jc i < c] di;oice (n; c) = n ( + F (c) ) nf (c) E [c i jc i < c] : Note that di;oice (; c) = con;oice (; c) for any xed c, that di;oice (n; c) and con;oice (n; c) hae a unique maximum at and n respectiely, and that in any equilibrium c di;oice and c con;oice < n. Moreoer, under the conditions of part (ii) of Proposition 3, c con;oice < c di;oice under any equilibrium under concentration and diersi cation. Below we state and proe an auxiliary lemma which is useful for the proof of Lemma 9. Lemma 0 Suppose > ( F ()). Consider an equilibrium under diersi cation in which each rm is good w.p. = F (c ) where > ( ). Then: (i) If the equilibrium is type-(ii), then oice (F (c )) ( x di (F (c ))) c oice (F (c )) ; (30) where oice () is gien by (5). (ii) If the equilibrium is type-(iii), then c = oice (F (c )) ; (3) where oice () is gien by (3). Proof. Let (c ; c) be as de ned in the proof of Lemma. Then, c = F ( ) is an equilibrium only if c arg max c0 (c ; c). Also, let x = x di ( ) and p = p di ( ) if the equilibrium is type-(ii) and let x = x con ( ) =n and p = p con ( ) if the equilibrium is type-(iii). 0

11 If = 0, the inestor obtains a payo of from a good rm. She can obtain a payo of x p + ( x ) from a bad rm by selling x of each bad rm, which generates the highest payo. Recall that, from Lemma, the inestor has no incenties to sell less than x of a bad rm when = 0. Thus, for any x 0 < x, x p + ( x ) > x 0 ip (x 0 i) + ( x 0 i) ) x p x 0 ip (x 0 i) > (x x 0 i) > 0: Therefore, in any equilibrium, the pricing function in the range [0; x ) must satisfy this condition. This condition also implies that the inestor cannot raise more reenue from a single deiation to selling x 0 < x from one particular bad rm. Without loss of generality and to simplify the exposition, hereafter we assume x i (; 0) = 0 and x 0 (0; x ) ) p (x 0 i) = p. These o -equilibrium prices presere monotonicity, and x p x 0 ip (x 0 i) > (x x 0 i). Moreoer, note that if the inestor found it optimal to monitor rms under the general pricing function (i.e., before specializing to p (x 0 i) = p ), deiating to sell x 0 < x from a gien rm cannot be su ciently bene cial to induce deiation from monitoring rms under the pricing rule x 0 (0; x ) ) p (x 0 i) = p. Indeed, since the pricing rule x 0 (0; x ) ) p (x 0 i) = p is the lowest that satis es monotonicity, a deiation from monitoring rms is less bene cial than under the general pricing function, and hence suboptimal as well. Intuitiely, if she sold less than x from bad rms, she would receie the same price as if she sold x, and so her incenties to monitor are no di erent. Suppose = L, and consider a type-(iii) equilibrium. We argue that the inestor has no incenties to deiate from selling x from each rm. We consider two cases.. Suppose. We rst argue x p. To see why, recall that in this case that x = minf p ; g. Therefore, either x p = or x =. Since p, x = ) x p. Second, since x p, the inestor raises more funds when she chooses x i = x rather than x i = (when x < ). Therefore, she will sell x from each bad rm and x p ( ) from each good rm. If x p = then x p ( ) = x, and p p if x = then p, which implies that she has to sell the entire portfolio to raise L. Either way, she will sell x from each rm in her portfolio.. Suppose <. Note that x = minf ; g and p > > imply x p =, and p

12 so x p <. If the inestor deiates from selling x from each rm, she would deiate to fully selling minf ; g bad rms, selling a fraction x of ( ) minf ; g bad rms, and selling a fraction ^x = maxf0; g of all good rms. Deiation is not strictly preferred if and only if ( ) p [^xp + ( ^x) ] + ( ) min ; ( ) [x p + ( x ) ] + min ; ( ) [x p + ( x ) ] + ( ) [x p + ( x ) ], max 0; ( ) (p p x min ; ) ( p ) (3) If Note that, then (3) holds if and only if, where we used p = p con ( ). <, + < which must hold if the equilibrium is type-(iii). If > then condition (3) holds if and only if +, which must hold if the equilibrium is type-(iii). Therefore, condition (3) holds, and deiation is suboptimal. Combining cases and aboe, the inestor has no incenties to deiate from selling x from each rm, and so her payo is gien by (c ; c) = F (c) [ x ( p )] + ( F (c)) [ + x (p )] F (c) E [c i jc i < c] :(33) = + F (c) + x ( F (c)) F (c) E [c + i jc i < c] ; and (c ; f (c) = x + c: Substituting x = x con ( ) into the rst-order = 0, yields c = oice (F (c )), as required. This completes part (iii). Suppose = L, and consider the type-(ii) equilibrium. Recall that we must hae <.

13 Also recall that, by construction, x p + ( ) = : Therefore, the inestor can raise L by fully selling ( ) n bad rms and selling a fraction x of n good rms. Also, since x p <, selling x from eery rm will not raise enough reenue to satisfy the shock. Note that, regardless of rm alue, the inestor has strict incenties to sell x rather. Indeed, in the latter case the payo is, the lowest possible. Therefore, regardless of the proportion of good rms (i.e., een if 6= ), she will fully sell exactly ( ) n rms and a fraction x of all other rms. The inestor will prefer fully selling a bad rm if there are su cient numbers. Therefore, her expected payo from choosing cuto c is: 3 ( ) [x p + ( x ) ] 0 n o (c ; c) = ( F (c)) 6 4 min ; 7 F (c) n o A 5 + [x p + ( x ) ] min ; F (c) 0 o 3 max n0; F (c) F (c) +F (c) 4( ) o A5 + [x p + ( x ) ] max n0; F (c) F (c) E [c i jc i < c] : F (c) Using x = x di ( ) and p = p di ( ), we obtain: (c ; c) = + F (c) + x ( + F (c)) + ( ) ( ) F (c) E [c i jc i < c] : ( x ) max ff (c) ; 0g Note (c ; f (c) = oice ( ) c < 0 if F (c) < : ( x ) if F (c) > ; which is a strictly decreasing function of c, and oice () is gien by (5). Since f is strictly 3

14 positie, f (0) > 0 ;c) f(c) c=0 > 0. Therefore, (c ; c) is maximized at arg max c0 ( ; c) = >< oice ( ) if F ( oice ( )) < F ( ) if F ( oice ( ) ( x) ) F ( oice ( )) >: oice ( ) ( x) if < F ( oice ( ) ( x) ) : In equilibrium, we require c ( ) = F ( ). Therefore, must satisfy oice ( ) ( x ( )) F ( ) oice ( ) ; as required. Proof of Lemma 9. We proe the result in three steps. First, suppose. Based on Lemma, the equilibrium must be type-(iii). Based on part (ii) of Lemma 0, the monitoring threshold must sole c = oice (F (c )). Note that oice (F (c)) is continuous, oice (F (0)) = ( ) and oice () = ( minf ; g), and hence, by the intermediate alue theorem, + a solution always exists. Gien a threshold that satis es c = oice (F (c )), by construction there is a type-(iii) equilibrium with this threshold. Second, we proe that if ( F ()) < <, there always exists a type-(ii) equilibrium where the monitoring threshold is gien by part (i) of Lemma 0, i.e., the largest solution of c = oice (F (c )). In particular, it is su cient to show that c = oice (F (c )) has a solution such that F < c (which is equialent to ( ) < ). Indeed, when c = F then oice (F (c )) =. Since ( F ()) <, then c = F ) oice (F (c )) > F. Furthermore, when F (c ) = then h oice (F (c )) = <, since F (c ) = =. Since oice (F (c )) is continuous i + in c, by the intermediate alue theorem, a solution strictly greater than F exists. By construction, there is a type-(ii) equilibrium with such a threshold. Third, suppose ( always F ()) < <. We compare the e ciency of the sustainable equilibria. First note that any type-(i) equilibrium is less e cient than a type-(ii) equilibrium. Indeed, in the former case the equilibrium threshold c i must satisfy ( in the latter case the equilibrium threshold c ii must satisfy > ( F (c i )), and F (c ii)). Therefore, 4

15 c ii > c i, as required. Next, consider type-(iii) equilibria. When <, such equilibria exhibit x p =, where p = + + Note that oice () > oice () if and only if Also note that. Therefore, wheneer these equilibria exist, oice () + ( ( )) ( ) ( ) ( ( )) : >, + ( ( )) < : (34) ( ( )), + : Therefore, if, then (34) always holds, which implies that the most e cient equilibrium + is type-(ii). In this case, y = y =. In other words, wheneer a type-(ii) equilibrium exists (i.e., ( Suppose < F ()) < < ), it is the most e cient equilibrium.. Note that (34) is equialent to + () = " + + # () < 0, where + + : Note that min () < 0. Also, recall ( ii ) <. Therefore, it is su cient to focus on ( ) <, <. It can be eri ed that < 0. Therefore, there is ^ > ^ = such that () 0, ^ where ^ is the largest root of (), gien by + +! + u t + +! : (35) 5

16 Note that a type-(iii) equilibrium requires where < + + <, + <. Also note that < F () in both a type-(ii) and type-(iii) equilibrium. Therefore, the releant range is F (). This interal is non-empty if + and only if + F () F () F () <, F () Note that ( F ()) < for all. Since < if < + F () + + F (), the most F () F () + e cient equilibrium is type-(ii). This establishes the existence of y, the threshold below which a type-(ii) equilibrium is most e cient. Suppose F () F () < : < < + : (36) If <, then + oice () is a decreasing function, and so iii, gien by the solution of = F ( oice ()), is unique. Therefore, the equilibrium with iii is most e cient if and only if We now proe that ^ + ( ) max + ; ^ ^ < x, < iii:. To do so, we rst proe that + + < x + x + x : (37) To see this, note that u ^ x, t + +! x + +!, 6

17 x x !! < 0 or 0 and x x + +!, x + x + < or and + + x + x + x Note x + > x + x + x, x + x + > 0; Cwhich proes (37). Using (37), we hae ^ > + +, > : + + This eentually yields + + > + ( ) ( ); which always holds. Since ^, + iii is most e cient only if ^ < iii and < + + < iii + iii + iii : +, i.e., 7

18 Note that lim! iii > 0 = lim! ^. By continuity, there is y [y; ) such that, if > y, the most e cient equilibrium is type-(iii). C Endogenous Information: Full Analysis Lemma characterizes the equilibrium leel of inestigation under concentration. Lemma (Information Acquisition, Concentration): In any equilibrium, the seller inestigates each asset with intensity con <, which is unique and de ned by the solution to: c 0 () = ( ) ( )n min ( ; [( )( )+]+ ( )( )+ ) : (3) The intuition behind the inestigation threshold, (3), is as follows. Up to a point, the greater the seller s number of units n, the more she can sell if she learns that the asset is bad, and thus the greater her incenties to gather information. Howeer, after a point, information acquisition becomes independent of n. The seller s stake n is now su ciently large that she is no longer forced to sell it in its entirety upon a shock. As a result, she can only partially sell her stake upon learning that the asset is bad, otherwise she is fully reealed, and so further increases in stake size do not increase her inestigation incenties (see also Edmans (009)). Lemma characterizes the equilibrium leel of inestigation under diersi cation. Lemma (Information Acquisition, Diersi cation): (i) If (c 0 ) (( )) ( ), then there exists an equilibrium in which the seller inestigates each rm with intensity di = ( ) : (ii) If < ( ) and (0; ], then in any equilibrium di > 0. (iii) If ( ) and =, then there exists an equilibrium with di = 0. Lemma shows that, since all assets are ex ante identical and the cost function is conex, the seller chooses a constant i = i. The intuition behind part (i) is as follows. Intuitiely, when the seller inestigates a certain measure of assets, by the law of large numbers she knows how many will be bad. Thus, when is small, she inestigates with just enough intensity

19 to reeal enough bad assets that she can exactly satisfy any shock by selling them all. She has no incentie to inestigate less, since if she su ers a shock, she will hae to sell some assets of unknown alue, some of which will be good. She has no incentie to inestigate more. Doing so will uncoer additional bad assets, but she does not need to sell these assets as she is already satisfying her liquidity need, and earns no pro t by oluntarily selling these assets, since she receies a price of. Part (ii) shows that seller always inestigates with strictly positie intensity when is small, and part (iii) shows that if = and is large then there always exists an equilibrium in which the seller does not inestigate. Intuitiely, if the shock is large and likely, she has little exibility to keep the good assets and sell the bad ones. C. Proofs Proof of Lemma. A rm s expected alue is now gien by i f; + ; g, where + is the alue of an unidenti ed asset, i.e., an asset of unknown alue. The equilibrium selling strategies and prices are gien by < 0 if i f + ; g and = 0 x i ( i ; ) = : x n minf (x); g otherwise, p i and + if x >< con+ i = 0 con p i (x i ) = + if x con( )( )+ i (0; x] >: if x i > x: Since c 0 () =, in any equilibrium <. Next, suppose (0; ). We rst show a set of results regarding the possible equilibrium selling strategies, similar to the proof of Lemma.. x i (; ) > 0 and x i ( + ; L) > 0. To see this, note that if = L, the seller will sell a positie amount. If = 0 and i =, suppose that x i = 0. Her payo in this case is. Since x i ( + ; L) > 0, p i (x i ( + ; L)) >, and so type- has a pro table deiation to x i ( + ; L). 9

20 . x i (; 0) 6 fx i (; L); x i (; 0); x i ( + ; L)g. Suppose not. Then, from point it must be that x i (; 0) > 0 and p i (x i (; 0)) <. Thus, type-(; 0) can deiate to x 0 i = 0 and receie a payo of. 3. x i (; ) fx i ( + ; L); x i (; L)g. Suppose not. Then, it cannot be x i (; ) 6 fx i ( + ; 0); x i (; 0)g since then p (x i (; )) = and the seller s payo is. Indeed, since x i ( +; L) > 0 and p i (x i ( + ; L)) >, the seller has a pro table deiation to x i (+; L) > 0. Howeer, if x i (; ) fx i ( + ; 0); x i (; 0)g, then based on point, x i (; ) = x i ( + ; 0) 6= x i (; 0). Note that if p i (x i ( + ; 0)) < +, type-( +; 0) would hae a pro table deiation to x = 0. Howeer, since x i (; ) = x i (+; 0) 6= x i (; 0), p i (x i ( + ; 0)) + requires x i (+; 0) = x i (; L), contradicting x i (; ) 6 fx i ( + ; L); x i (; L)g. Therefore, x i (; ) fx i ( + ; L); x i (; L)g as required. 4. x i (; 0) = x i (; L) = x > 0. Suppose on the contrary that x i (; 0) 6= x i (; L). Gien point 3, x i ( + ; L) 6= x i (; L). Moreoer, in following both strategies, the seller either obtains the highest reenue, or she obtains a reenue of at least L. Therefore, the liquidity constraint does not bind, and the strategies must generate the same payo. That is, letting x = x i (; ): x 0 p i (x 0 ) + ( x 0 ) = x L p i (x L ) + ( x L ), x 0 p i (x 0 ) x L p i (x L ) = (x 0 x L ): There are two cases. First, if x 0 = x i ( + ; L) (and so x L = x i (; L)), we must hae x 0 p i (x 0 ) x L p i (x L ) (x 0 x L ) ( + ) and, x 0 p i (x 0 ) x L p i (x L ) (x 0 x L ) which holds if and only if (x 0 x L ) (x 0 x L ) (x 0 x L ) ( + ) ; which can neer hold. Second, if x 0 = x i (; L) (and so x L = x i ( + ; L)), we must 0

21 hae x 0 p i (x 0 ) x L p i (x L ) (x 0 x L ) ( + ) and, x 0 p i (x 0 ) x L p i (x L ) (x 0 x L ) which holds if and only if (x 0 x L ) ( + ) (x 0 x L ) (x 0 x L ); which also can neer hold. 5. x i ( + ; L) = x. Suppose instead x i ( + ; L) = x 0 i 6= x. Since x 0 i 6= x, point 4 yields p i (x 0 i) +. Moreoer, note that either x 0 ip i (x 0 i) or x 0 i generates the highest reenue that can be obtained in equilibrium. There are two cases: (a) Suppose p i (x) < p i (x 0 i). If x 0 i x then type-(; 0) has a pro table deiation to x 0 i, since she can sell more shares for a higher price. If x 0 i < x then if x i (; L) = x then type-(; L) has a pro table deiation to x 0 i (which satis es her liquidity need). If instead x i (; L) 6= x then p i (x) =, and type-(; 0) has a pro table deiation to x 0 i. (b) Suppose p i (x) p i (x 0 i). Then, type-(; L) must play x with positie probability. By reealed preference, this means that xp i (x) x 0 ip i (x 0 i) (x x 0 i) : Since type- also weakly prefers x oer x 0 i, xp i (x) x 0 ip i (x 0 i) (x x 0 i) : Howeer, type-( + ) weakly prefer x 0 i oer x, xp i (x) x 0 ip i (x 0 i) (x x 0 i) ( + ) : The combination of the three conditions implies x x 0 i = 0, a contradiction.

22 6. p i (x) < +. Based on points -5 and the application of Bayes rule, p i (x) max + [0;] ( ) ( ) + ( ) [ ( ) + ( ) ] + Indeed, the highest possible alue of p i (x) arises if type-(; L) chooses x w.p., and type-( + ; 0) chooses x w.p.. Note that since (0; ) and (0; ), for eery [0; ] the RHS is strictly smaller than x i (; L) = x. Suppose instead x i (; L) = x 0 i 6= x. Based on points 4 and 6, p i (x 0 i) > + > p i (x). Therefore, type-( + ; L) has strict incenties to deiate to x 0 i since it leads to a trading pro t and also satis es her liquidity need.. x i ( + ; 0) 6= x. Suppose instead that x i ( + ; 0) = x. Based on point 6, p i (x) < +. Therefore, type-( +; 0) has strict incenties to deiate from x i = x to x i = x i (; 0) = 0. Suppose instead x i (; 0) = x 0 i > 0. Then, p i (x 0 i) = and so x 0 i 6= x i ( + ; 0). Based on points -, either x i ( + ; 0) = 0 or p i (x i ( + ; 0)) = +, and so the equilibrium payo of type-( + ; 0) is +. Then, type-( + ; 0) has strict incenties to deiate to x 0 i and receie a payo strictly higher than +. These points show that in any equilibrium with (0; ), we hae x i (; L) = x i ( + ; L) = x i (; L) = x i (; 0) = x > 0, x i (; 0) = 0, andx i ( + ; 0) 6= x. : Now, gien this, without loss of generality (for inestigation incenties) we consider the case where x i ( + ; 0) = 0. Note that the selling strategies in the proposition satisfy the conditions of the preious part of the proof. n Gien o these selling strategies, the prices satisfy Bayes rule. We can show x = n min p (x); using the same arguments made in the proof of Lemma. i The seller s expected payo from choosing [0; ] is " # [( )n + (xp () = i (x) + (n x))] +( )[xp i (x) + (n x)] " # ( )n( + ) + ( ) +[xp i (x) + (n x)( + )] c () :

23 The rst-order condition implies " [( )n + (xp i (x) + (n x))] +( )[xp i (x) + (n x)] # " ( ) ( ) n min ( )n( + ) +[xp i (x) + (n x)( + )] # = c 0 (), ( ) ( ) (p i (x) ) x = c 0 (), ( ) [ ( )( )+]+ ; ( )( )+ = c 0 () : Therefore, in equilibrium, if (0; ) then it must satisfy (3). Note that the RHS of (3) is decreasing in and the LHS is increasing in (since c 00 > 0 by assumption), and so if a solution exists, it is unique. Furthermore, if < then at = 0, the RHS is greater and as =, the LHS is greater. Thus, if <, there exists a unique satisfying (3). Suppose =. The analysis aboe shows that we cannot hae > 0 in equilibrium. Indeed, if > 0 then must satisfy (3). Howeer, = implies that if satis es (3), then = 0. Therefore, if = it must be = 0. Consider the prices and quantities in the proposition when con = 0. Clearly, gien = 0, the selling strategy is weakly optimal and the prices are consistent with these strategies. It is left to erify the seller has no incenties to deiate to > 0. Indeed, substituting = and p i (x) = + into () aboe yields () = n( + ) c (), which implies that no > 0 is optimal, as required. Proof of Lemma. First, note that the conexity of c () implies that, in any equilibrium, all assets are inestigated with equal intensity. Next, we proe part (i). Suppose in equilibrium is such that ( ), then the seller can satisfy the shock from selling only bad assets. Let < 0 if i f + ; g x i ( i ; ) = : if i = ; and < + if x p i (x i ) = + i = 0 : if x i > 0: Then, the proposed selling strategies are consistent with such an equilibrium gien prices, and prices are consistent with the proposed strategies by Bayes rule. 3

24 Suppose the seller chooses, where we allow for 6=. Let (; ) denote her payo. There are two cases. First, if ( ) then it is optimal for the seller to choose the same selling strategies as under the equilibrium, i.e., sell bad assets and retain all others. In this case, (; )=n = + c (). Second, suppose > ( ). If = 0, the seller strictly prefers to retain all assets not identi ed as bad, and is indi erent between selling and retaining bad assets. If = L then the highest payo (subject to meeting the liquidity need) ( ) is generated by a strategy in which the seller sells all bad assets, minf ( ) ; g from ( ) all unidenti ed assets, and minfmaxf ; 0g; g from all good assets. Since < ( ) ( ) ) < ( ), the former term is < and the last term is zero (i.e., ( ) the seller neer has to sell good assets). Then, (; )=n = c () + ( ) ( + ) ( + ) + ( ( ) ( ) + ( ) ) ( ) ( ) ( + ) 3 5 Combining the two cases we < = n : c 0 () + ( ) if < ( ) c 0 () if ( ) : De ne () (c 0 ) (( )). Since c 0 + ( ) 0, () ( ) ; ( ) (note that c 0 () = ) () < ) the rst-order condition implies < () if () ( ) < = : if () ( ) ( ) Therefore, if () ( ), there is an equilibrium with di =. ( ) Next, consider part (ii). Suppose on the contrary < ( ) and there is an equilibrium with = 0. Since > 0 and < ( ), and since the equilibrium is symmetric, there 4

25 is x > 0 on the equilibrium path such that xp (x). Since = 0, p (x) = +. Note that p (). Suppose the seller chooses > 0 and the following selling strategy: (i) if the seller does not need liquidity, she sells x units from all assets; (ii) if the seller needs liquidity, she sells x units from all unidenti ed assets, and none from good assets, x units from a fraction (0; ) of bad assets, and fully sells the remaining of bad assets. Using xp (x) and p (), her reenue is n ( ) [xp (x) + ( ) p ()] + n ( ) xp (x) n ( ) ( ) + [( ) + ( ) ] L Note that ( ) ( ) + [( ) + ( ) ], ; where < ( ) ) need. Her pro t is: (; 0) =n = " (0; ). Therefore, if =, the seller meets her liquidity ( ) ( + ) + ( + ) + ( ) [ (xp (x) + ( x) ) + ( ) p ()] + ( ) ( + ) c () : # Note (; = ( ) n [x + ( ) (p () )] nc 0 () : =0 > 0, which implies that the seller has a pro table deiation to > 0, a contradiction. This proes part (ii). Finally, consider part (iii). We argue that there exists an equilibrium with = 0. Let the seller s strategy in this equilibrium be selling x = are monotonic and gien by + < + if x p i x i (x i ) = : if x i > x: < unit from each asset, and prices 5

26 Suppose the seller deiates to > 0. To raise enough liquidity, the seller can follow the equilibrium strategy and sell x from each asset. This strategy generates an expected pro t of n ( + ). If so, the seller is better o choosing = 0. Any other strategy that generates an expected payo strictly higher than n ( + ) (and at least L of reenue) requires selling less than x from good assets and more than x from bad assets or unidenti ed assets. Since p i (x i ) < +, the strategy that maximizes the pro t from this deiation inoles fully selling all bad assets, which generates a reenue of n ( ). Consider a strategy that inoles selling x + " from each unidenti ed asset and x from each good asset, where "; > 0. The expected payo from this deiation is (; 0) = n ( ) + n [(x ) p i (x ) + ( (x )) ] +n ( ) [(x + ") p i (x + ") + ( (x + ")) ( + )] nc () Howeer, (; 0) > n ( + ) if and only if ( ) + [(x ) ( + ) + ( (x )) ( + )] + ( ) [(x + ") + ( (x + ")) ( + )] c ()! > +, [(x ) + ( (x ))] + ( ) ( (x + ")) > c () which neer holds. Therefore, a pro table deiation does not exist. D Robustness: Full Analysis D. Two Assets We de ne x (x i ; x j ) ; ( i ; j ), x T (x j ; x i ), T ( j ; i ), (; ), and (; ). We denote by e (; ) (e i (; ) ; e j (; )) the equilibrium strategy of type (; ). By symmetry, p i (x) = p j x T for all x i and x j, and e j (; ) = e i ; T. We therefore omit the subscript wheneer there is no risk of confusion. Let (; ) (e (; ) ; ) denote the equilibrium payo of type (; ), where (x; ) = x i p i (x) + (n= x i ) i + x j p i x T + (n= x j ) j : 6

27 We start by analyzing the case of separate buyers and then a single buyer. We focus on the case of small liquidity shocks ( =, so that a shock can be met by fully selling one bad asset) since this is where our results are strongest. D.. Separate Buyers Lemma 3 (Diersi cation, two assets, separate buyers): Suppose =. An equilibrium under diersi cation always exists and is unique. (i) If + then < 0 if x i = and = 0 di;i ( i ; j ; ) = : x di () = L= else p di () (39) and prices of asset i are if x >< i = 0 p i (x i ) = p di () = + if x + i (0; x di ()]; >: if x i > x di () : (ii) If < + then 0 if = (; ), or = and = 0 >< x di;i (; ) = x di () = L= if = (; ) and = 0, =, or = and = L p di () >: n= if = (; ) and = L; (40) and prices of asset i are if x >< i = 0 p i (x i ) = p di () = + if x ( )( )+( >: +( ) ) i (0; x di ()]; if x i > x di () : If = and = + p then the equilibria in part (i) and part (ii) coexist. 7 (4)

28 The intuition is as follows. If, then the seller retains an asset only if it is good + and she su ers no shock. If the asset is bad, or if she su ers a shock, she sells the asset to the same degree (x di () in this case), just as under concentration. In particular, een if asset i is good and asset j is bad, she still sells asset i upon a shock. Een though she could meet her liquidity need by selling only asset j, doing so would lead to the lowest possible price of. When, the probability of a good asset () and the probability of a liquidity shock + () are both high, and so the price of a partially sold asset p di () is also high as there is a high probability that this is a good asset sold due to a shock. Thus, the seller prefers to meet the shock by selling only x di () of the bad asset, een though doing so also requires her to sell x di () of the good asset to meet the shock. Put di erently, een though diersi cation gies the inestor the exibility to meet the shock by selling only bad assets, she chooses not to take adantage of this exibility. This issue does not arise with a continuum of rms since it is neer the case that all assets are good. Thus, a sold asset cannot be a good asset sold due to a shock, and so it receies the lowest possible price of. If <, then the price of a partially-sold asset is su ciently low that a seller with + exactly one bad asset does take adantage of her exibility to meet a liquidity shock by selling only the bad asset and retaining the good one. D.. Single Buyer Here, the buyer can condition p i on x j. We focus on equilibria with non-increasing prices in the following sense: If 0 x < x + " n= then p i (x) p i (x + ") and p j (x) p j (x + "). In other words, among balanced exit strategies, if a seller sells less of all rms, the prices of all rms must (weakly) increase. Lemma 4 (Diersi cation, two assets, single buyer): Suppose =. An equilibrium under diersi cation always exists. 3 (i) If > + p then the equilibrium is unique where: < 0 if = (; ) and = 0 x di;i ( i ; j ; ) = : x di () = L= else p di () (4) 3 If = and = + p then the equilibria in part (i) and part (ii) coexist.

29 and prices of asset i are if x >< i = 0 p +( )( ) i (x i ; x j ) = p di () = + if 0 < x +( )( >: ) i min fx di () ; x j g ; if min fx di () ; x j g < x i (ii) If < + p then in any equilibrium 0 if = and = 0, or = (; ) and = L >< x di;i (; ) = x di () = L= if = and = L, or = p di () >: n= if = (; ) and = L (43) and x di;i ((; ) ; 0) ; x di;j ((; ) ; 0) 6= (x di () ; x di ()) : Moreoer, p di () = + ( ) + + ( ) ( ) : The equilibrium with the lowest price informatieness satis es x di;i ((; ) ; 0) ; x di;j ((; ) ; 0) = (0; 0) and ( )+ p (0) = + if x >< ( )+( )+ i = 0 p i (x i ; x j ) = p di () if 0 < x i min fx di () ; x j g : >: if min fx di () ; x j g < x i The equilibria are similar to the separate buyer case, except for when the seller has exactly one bad asset and does not su er a shock. With separate buyers, the seller oluntarily sells x di () of the bad asset, to disguise the sale as being of a good asset and due to a shock, and retains the good asset. With a single buyer, such disguise is no longer possible, since the buyer 9

30 would see that the good asset is fully retained, infer that there has been no shock, and price the partially-sold asset at. She may thus choose to engage in balanced exit, i.e., sell both the bad and good asset to the same degree, to disguise their sale as being motiated by a shock. The lowest leel of price informatieness arises when the seller retains both assets upon no shock, because now being fully retained no longer fully reeals that an asset is good. D..3 Price Informatieness Proposition 7 (Price informatieness, two assets): Suppose =. then in any equilibrium under concentration and diersi ca- (i) Separate buyers: If < + tion P di;separate (; ) > P con (; ) and P di;separate (; ) < P con (; ) : (ii) Single buyer: There is (0; ) such that if < = and (; ] then in any equilibrium under concentration and diersi cation P di;single (; ) > P con (; ) and P di;single (; ) < P con (; ) : The intuition is as follows; we discuss the required parameters for the separate buyers case but the intuition is similar for the single buyer case. Price informatieness can be higher under diersi cation only if < + p, because only in this case does a seller with one bad asset take adantage of her exibility to meet a liquidity shock by selling only the bad asset. For price informatieness to be higher under diersi cation in any equilibrium, the price of a bad rm must be lower than under concentration. Under concentration, a bad rm s price is increasing in, since the higher the probability of a shock, the higher the probability that a partial sale is of a good rm due to a shock. Thus, a high (to increase the price of a bad rm under concentration) and a low (so that < + p ) ensure that price informatieness is higher under diersi cation in any equilibrium. D..4 Proofs Proof of Lemma 3. We start by proing seeral claims.. In any equilibrium, there is a unique x G (0; n=) such that e (L; ) = x G. 30

31 Proof: A symmetric equilibrium requires the seller to sell the same quantities from both assets if they hae the same alue. A pure strategy equilibrium requires x G to be unique. Since > 0, such x G exists. Since L > 0, if x G = 0 then the seller has a pro table deiation to selling n= units from each asset and raising more reenue. If x G = n=, since p (x G ), x G p (x G ) n= L > L=, which contradicts x G p (x G ) = L= that we proe in claim below.. In any equilibrium, (i) p i (x G ) (; ); (ii) x G p (x G ) = L=. Proof of part (i): Since > 0, p (x G ) >. Suppose on the contrary p (x G ) =. Since prices are non-increasing, there is x x G such that x x ) p (x) = and either x = n= or x > x ) p (x) <. Let x B = e i (L; ) = e j (L; ). Then, p (x B ) <. Moreoer, since x x ) p (x) =, it must be x B > x. If p (x B ) = then type- has a pro table deiation to (x; x). Therefore, p (x B ) >, which requires either e (L; (; )) = x B or e (0; (; )) = x B. Type-(; ) prefers x B oer x G only if x B p (x B ) + (n= x B ) ( + ) x G + (n= x G ) ( + ), p (x B ) x G + + x B which is strictly greater than +. Howeer, note that ( ) p (x B ) + ( ) + ( ) + ( ) i.e., the highest possible alue of p (x B ) arises when type-(; ) chooses x B regardless of her liquidity need and type- chooses x B only when there is a shock. In addition, ( ) + ( ) + ( ) + ( ) < +, 0 < which always holds. Therefore, type-(; ) neer chooses x B, a contradiction. Proof of part (ii): Suppose on the contrary x G p (x G ) > L=. Since prices are nonincreasing, there is " > 0 such that (x G ") p (x G ") L=. Since p (x G ) <, if = L then type- has a pro table deiation from (x G ; x G ) to (x G "; x G "). Suppose on the contrary x G p (x G ) < L=. Since =, the seller can sell n= from both rms, get a 3

32 price no lower than, and thus raise enough liquidity. This creates a pro table deiation. 3. In any equilibrium, e i (0; ) < x G and p (e i (0; )) =. Proof: Since the seller can fully retain both assets, p (e i (0; )) =. Suppose on the contrary e i (0; ) x G. Since p (e i (0; )) = and p (x G ) < (from claim ), it cannot be e i (0; ) = x G. Suppose e i (0; ) > x G. Since p (e i (0; )) = > p (x G ), if = L then type- has a pro table deiation from x G to e (0; ). 4. In any equilibrium, e i (0; (; )) < x G and p (e i (0; (; ))) =. Proof: the same as claim In any equilibrium, if the seller sells x B from a bad asset, either x B = x G or p (x B ) =. Proof: Suppose on the contrary the seller sells x B 6= x G from the bad asset w.p. > 0 in equilibrium, and p (x B ) >. Therefore, p (x B ) <. Based on claims -4, p (x B ) > requires the seller to sell x B from the good asset when = L and the other asset is bad. Let ^x be the quantity sold from the bad asset in this case. That is, e (L; (; )) = (x B ; ^x). Note that ^x x B. Otherwise, type-(; ) has a strictly optimal deiation from (x B ; ^x) to (^x; x B ), that is, selling more from the bad asset and still meeting her liquidity need. Moreoer, note that ^x = x B. To show this, if ^x 6= x B then based on claims -4, we must hae x B < ^x = x G or p (^x) =. (a) Suppose x B < ^x = x G. If type-(; ) prefers (x B ; x G ) oer (x G ; x G ) when = L, then type- must also prefer (x B ; x G ) oer (x G ; x G ) when = L. Howeer, since e (L; ) = (x G ; x G ), type- must be indi erent between the two: both strategies must generate the same payo, but also raise exactly L: otherwise, the seller can always sell less from the good asset and still meet her liquidity need. Howeer, this implies x B = x G, a contradiction. (b) Suppose p (^x) =. Then, type-(; ) has a strictly pro table deiation from (x B ; ^x) to (0; n=). Under both strategies, she raises enough reenue (recall n= L). Howeer, under the latter strategy her payo is n= ( + ), and under the former strategy her payo is strictly smaller: she is getting a payo for the bad asset, but since p (x B ) <, her payo for the good asset is strictly smaller than. 3

33 Therefore, e (L; (; )) = (x B ; x B ). By reealed preference, ((x B ; x B ) ; (; )) n= ( + ), p (x B ) + : Suppose type-(; ) chooses (x B ; x B ) w.p. > 0; then a contradiction. ( ) p (x B ) = + [ ( ) + ( ) ] + ( ) < + ; 6. In any equilibrium, if = 0 then the seller sells x G from a bad asset. Proof: Based on claim 5, if the seller sells x B from a bad asset, either x B = x G or p (x B ) =. Since p (x G ) >, if the seller does not need liquidity, she will maximize her payo by choosing x G. 7. In any equilibrium, e (L; ) = (x G ; x G ). Proof: By symmetry, the seller must sell in equilibrium the same quantity from both assets. Based on claim 5, if e (L; ) 6= (x G ; x G ), the seller must receie from the assets, and so her payo is. Since p (x G ) > and x G p (x G ) = L=, the seller maximizes her payo by choosing (x G ; x G ).. In any equilibrium, if p (x G ) > + then e (L; (; )) = (x G ; x G ), and if p (x G ) < + then e i (L; (; )) < x G < e j (L; (; )). Proof: Recall = implies that selling (0; n=) yields type-(; ) a payo of n= ( + ) and enough reenue to coer her liquidity need. Therefore, she prefers (x G ; x G ) if and only if ((x G ; x G ) ; (; )) n= ( + ), p (x G ) + : Suppose p (x G ) < +. If the seller chooses balanced exit (x; x) 6 f(x G; x G ) ; (0; 0)g, then by symmetry, p (x) = +. The reenue raised must be exactly L, else she can sell slightly less from the good asset and make a strictly higher pro t. Therefore, it cannot be x < x G, else type- will deiate from (x G ; x G ) to (x; x) when = L. Also, it cannot be x > x G. Indeed, if x > x G then ((x; x) ; (; )) ((x G ; x G ) ; (; )) implies 33