International transmission of bubble crashes

Transcription

1 Inernaional ransmission of bubble crashes Lise Clain-Chamosse-Yvrard and Takashi Kamihigashi Preliminary Version Absrac We are ineresed in he impac and he ransmission of a bubble burs arising in a counry o anoher counry. We exend he overlapping generaions model wih sochasic bubbles developed by Weil (1987 in which he burs of he bubble is based on self-fulfilling beliefs (sunspo o an open economy wih wo large idenical counries, informaion symmery and homogeneous beliefs. In each counry, a bubble represened by an asse wih no fundamenal value is supplied and raded across counries. We show ha even hough he realizaion of sunspos is specific o one counry, he sunspo is spread o oher counry hrough he porfolio choice of agens. When only he foreign counry is subjec o sunspos, he burs of he foreign bubble generaes a change in confidence in he domesic bubble involving is immediae collapse. When boh counries are subjec o sunspos, agens ake ino accoun he risk of he domesic bubble collapse in heir decisions. Therefore, he burs of he foreign bubble can generae asymmeric effecs on he domesic one. The domesic bubble can persis a a higher value jus afer he burs of Foreign bubble, bu may deflae over ime. JEL classificaion: D91, E44, F30. Keywords: Sochasic bubble, Sunspo, Collapse, Conagion. 1 Inroducion Hisory of financial markes overflows wih examples of financial crises saring wih he he burs of Duch ulip bubble during 17h. The crash of a speculaive bubble are ofen a he roo of financial crises. As illusraed during he subprime crisis, financial crises are conagious phenomena. Sared in he U.S, i propagaed o oher counries. From Kindleberger (1978, we could consider he burs of Sea Souh bubble and Mississippi Bubble occurring in 1720 an example of conagion. 1 Because of globalizaion, privae insiuions hold inernaional diversified porfolio making financial inerdependen, and promoing hus he ransmission of financial crises across counries. Our paper focus on ransmission of bubble burs across asse markes in a simple dynamic general equilibrium model. Aix-Marseille Universiy (Aix-Marseille School of Economics, CNRS-GREQAM and EHESS. lise.clainchamosse-yvrard@univ-amu.fr. Research Insiue for Economics and Business Adminisraion, Kobe Universiy. kamihig@rieb.kobeu.ac.jp. 1 For furher deails abou bubbles, financial crisis and conagion in hisory, he reader can refer o Kindleberger and Aliber (

2 Some conribuions on financial crises and conagion provide an explanaion based on informaion asymmery (among ohers, Calvo and Mendoza (2000 and Kodres and Prisker (2002 (To be developped?. In our paper, agens expecaions and beliefs are only he key deerminans underlying inernaional ransmission of financial crises across counries. We exend he overlapping-generaions exchange economy wih sochasic bubbles developed by Weil (1987 in which bubbles have a consan probabiliy o burs ino a wo-counry model wih informaion symmery and homogeneous beliefs. 2 The wo counries are perfecly symmeric in all respecs. There is a unique perishable good, and a bubble represened by an asse wih no fundamenal value is supplied by each counry and raded across borders. Following Weil (1987, all agens believe ha he he foreign (domesic bubble burss if a sunspo occur in he foreign (domesic counry. We derive condiions for boh bubbles o occur and analyze he response of he domesic bubble o he burs of he foreign bubble. When only he foreign counry is subjec o sunspos, a saionary equilibrium wih domesic and foreign bubbles exiss before he burs of he foreign bubble provided ha he foreign bubble is no oo risky. We find he similar condiion as in Weil (1987, namely a sufficienly high probabiliy for he foreign bubble o persis. In such an economy, he burs of he foreign bubble generaes a change in confidence in he domesic bubble involving is immediae collapse. Even hough he realizaion of sunspos is specific o one counry, he financial crisis arising in he foreign counry propagaes o oher counry hrough he porfolio of agens. Inernaional porfolio diversificaion makes asse markes inerdependen. The risk of foreign bubble o crash is hus spread o he domesic bubble. When boh counries are subjec o sunspos, he scope for he exisence of a saionary equilibrium wih boh bubbles before he bursing of he foreign bubble is reduced if agens expec ha he value of he domesic bubble is zero jus afer he burs of he foreign bubble. Only sufficienly safe bubbles can exis. This equilibrium is likely o occur in a dynamically inefficien economy only if he probabiliy for Foreign and Home bubbles o persis is closed o 1. Furhermore, we show ha he burs of he foreign bubble does no necessarily cause he immediae crash of he domesic bubble. Since agens inernalize he risk of he domesic bubble o explode, he domesic bubble can persis even a a higher value jus afer he burs of he foreign bubble, hen deflae over ime. Our paper is closely relaed o he lieraure on raional bubbles (Samuelson (1958, Tirole (1985, Weil (1987, and Sanos and Woodford (1997. Since Tirole s seminal papers (1982, 1985, he overlapping generaion model is a useful framework o sudy raional bubble. However, Tirole (1985 sudies raional bubbles which never burs. To ackle his problem, Weil (1987 exends Tirole s framework by inroducing a probabiliy for he bubble o burs. Referring o sunspo heory, agens coordinae heir beliefs in he bubble wih respec o he realizaion of sunspos which occurs wih a non-zero probabiliy. In his paper, he shows ha sochasic bubbles are likely o occur if agens have sufficienly confidence in heir exisence. 3 Building on Weil (1987, our paper shows ha only safe bubble can exis in an open economy. Recen papers analyze he effec of bubble and is burs on he real economic aciviy (Kocherlakoa (2009, Farhi and Tirole (2010, Hirano and Yanagawa (2010, Marin and Venura (2012, Miao and Wang (2012. Inroducing financial fricions, hey show ha he burs of he bubble generaes a recession wih a drop of invesmen and oupu. Furhermore, hese analyzes are done in a closed economy. In conras o hese conribuions, we consider only he financial sphere of 2 We absrac for he producion side of he economy o highligh conagion effecs arising in asse markes. 3 Wigniolle (2014 exends he condiions for he exisence of sochasic bubbles in an overlapping generaions model by considering rank-dependen uiliy funcion. 2

3 he economy sudying he inernaional ransmission of a bubble crash arising in one counry o anoher bubble. To he bes of our knowledge, few paper has joinly analyzed he exisence of sochasic bubbles in open economy and he inernaional ransmission of bubble crashes. Tandon and Wang (2003 analyze bubblesà la Weil in a small open economy. In heir paper, only small open economy is affeced by sunspos. Therefore, he burs of he bubble has no impac on he bubble issued by he res of he world. In our paper, we reverse his resul by considering a wo-counry model. Considering a world economy of several counries, Venura (2008 sudies he inernaional ransmission of a bubble crash beween counries wih heerogeneous fundamenals. He shows ha a bubble crash arising in an developing counry affecs emergen and developed economies. In our paper, we argue ha an inernaional ransmission of bubble crash can occur beween counries which are perfecly idenical in erms of fundamenals. This paper is organized as follows. In he nex secion, we presen he closed economy model and analyze he dynamic properies. Secion 3 exends he model o a wo-counry model. In secion 4, we analyze he ransmission of Foreign bubble burs o Home bubble when Home counry is immune agains sunspo. Secion 5 sudies a world economy when neiher Home counry nor Foreign counry is immune agains sunspo. Concluding remarks are provided in Secion 6, and all he proofs are gahered in a final Appendix. 2 A Closed Economy Framework The world economy consiss of wo large counries i = {H, F } perfecly idenical wih respec o preferences, endowmens and populaion size. H refers o Home counry, and F o Foreign counry. Each counry is characerized by a closed overlapping generaions exchange economy à la Weil (1987. For simpliciy, we assume no populaion growh, and a each dae, a generaion of uni size is born in each counry. There is a single perishable good, and each agen is endowed wih y 1 unis of his good when young and y 2 when old. In heir firs period of life, hey consume an amoun c i of he good, and d i +1 in heir second period of life. Furhermore, agens can save hrough inrinsically useless asse papers supplied by heir counry in a consan amoun normalized o one. Since hese asses have no fundamenal value, a posiive price Q i > 0 depics a bubble. The value of he bubble Q i follows a random process depending on he realizaion of an exrinsic random variable (a sunspo in counry i. There are wo saes of naure: sunspo and no sunspo. A sunspo occurs wih probabiliy 1 π [0, 1, and no sunspo wih probabiliy π. Suppose ha he economy of counry i is in a sae of naure wih no sunspos in period, all agens living in counry i expecs ha he bubble in counry i persiss (Q i +1 > 0 in period + 1 if no sunspo occurs, and ha he bubble burss (Q i +1 = 0 if a sunspo occurs. Afer he burs, he bubble canno reappear, and he economy of counry i behaves as an economy wihou bubble. Sricly speaking, we suppose ha he bubble Q i follows a Markov process wih he following ransiion marix: As underlined by Weil (1987, π can be seen as a measure of he confidence in he bubble Q i. In our paper, i is exogenous and consan hrough ime. For insance, he burs of Foreign bubble arising in Foreign counry does no modify he confidence of agens living in Home counry in he value of heir bubble Q i. 3

4 Q i = > 0 Q i = 0 Q i +1 = q i +1 > 0 π 0 Q i +1 = 0 1 π 1 The represenaive agen of a generaion born a period in counry i faces he following problem: where max c,d +1 0 E [ u(c i + v(d i +1 ] (1 s.. c i + Q i x i = y 1 (2 d i +1 = y 2 + Q i +1x i (3 y 1 = endowmen in he firs period of life y 2 = endowmen in he second period of life c i = consumpion by he agen in her firs period of life d i +1 = consumpion by he agen in her second period of life x i = amoun of asse held by he agen Assumpion 1 u ( c i and v ( d i are coninuous funcions defined on [0, +, C 2 on (0, +, sricly increasing (u > 0, v > 0 and concave (u < 0, v < 0. Moreover, we assume ha lim c 0 u ( c i = +, lim d i 0 v ( d i = +, and ε v (d i dv (d/v (d 1. When none of he bubble is no valued in period, Q i = 0, he economy of counry i is in he auarky regime. There are no inergeneraional rades. When he bubble in counry i is valued in period, Q i = q i > 0. The opimal behavior of he agen living in counry i is summarized by he following ineremporal rade-off: u (y 1 q i x i v (y 2 + q i +1 xi = π qi +1 q i (4 Asse marke clearing condiions imply x i = 1. We assume ha he economy iniially sars wih a bubble, and sudy equilibrium dynamics up o he random dae on which a sunspo occur in counry i involving he burs of he bubble Q i. 4 An equilibrium is a sequence { } q i + =0 which saisfies, for all, he following condiions: q i u (y 1 q i = πq i +1v (y 2 + q i +1 (5 0 q i < y 1 The dynamic equaion (5 admis wo saionary equilibria: a seady sae wihou bubble q i = 0 and a seady sae wih a posiive bubble (q i > 0. A saionary sochasic bubble q i > 0 is a soluion of he following equaion: 4 Since he probabiliy of he bubble o persis is lower han 1, he burs of he bubble will occur. 4

5 u (y 1 q i = πv (y 2 + q i (6 Lemma 1 (Weil 1987 Le r be he auarky implici ineres rae such ha 1 + r = u (y 1 /v (y 2. Under Assumpion 1, here exiss a unique saionary sochasic equilibrium q i (0, y 1 if and only if π > 1 + r. Proof. See Appendix. A sochasic saionary bubble exiss if and only if he bubble is no oo risky (π > 1 + r. This also means ha when he economy of counry i wihou bubble is dynamically efficien, ha is r 0, no saionary sochasic bubble can exis. 5 The global dynamics of he closed economy of counry i is given by he dynamic equaion (5. 6 Lemma 2 (Weil 1987 Under Assumpion 1, we obain he following condiions: 1. When π 1 + r, here exiss no sochasic bubble. 2. When π > 1 + r, hen here exiss a sochasic bubble such ha:. if q i 0 < q i, hen q i 0 as + ;. if q i 0 = q i, hen q i = q i ;. if q i 0 > q i, he resuling pahs are infeasible. Proof. See Appendix. Since here are no connecions beween counries, he economic dynamics of counry i before and afer he burs of counry j bubble remains he same, namely he dynamics of an economy wih a bubble. Suppose ha a ime = 0 agens living in Home counry expecs q 0 = q H, hey will save jus enough o keep consan he value of Home bubble regardless of he burs of Foreign bubble. Inerdependence beween asse markes are necessary for conagion o arise. In he nex secion, we describe he open economy, and sudy he ransmission of a bubble burs arising in one counry o anoher counry. 5 Since counries are perfecly idenical, boh economies wihou bubble are eiher dynamically efficien or inefficien. 6 As he lef-hand side of (5 is a sricly increasing and coninuous funcion of q i under Assumpion 1, we can rewrie he dynamics of he model as follows: q i = f(q i +1 (7 0 < q i < y 1 (8 (36 depics a backward perfec foresigh dynamics which is well defined. f is coninuously differeniable under our assumpions. 5

6 3 Open Economy Framework Financial markes are now open o inernaional rade wihou any ransacion coss. In conras o Secion 2, households living in counry i = {H, F } can hold a diversified porfolio of Home and Foreign asses, and each counry issues is own asse in a fixed amoun normalized o 2. 7 Our wo-counry overlapping generaions model is a one-good pure exchange economy wih wo asses. Furhermore, informaion are symmeric and beliefs are homogeneous across counries. If a sunspo occurs in counry i, all agens perfecly receive his informaion and believe ha he bubble issued by counry i burss. The represenaive agen of a generaion born a period living in counry i = {H, F } faces he following problem: max [ E u(c i + v(d i +1 ] (9 c i,di +1 0 s.. c i + Q H x i,h d i +1 = y 2 + Q H +1x i,h + Q F x i,f = y 1 (10 + Q F +1x i,f (11 where x i,j (i, j = {H, F } is he demand for counry j asse of an individual living in counry i. Since counries are perfecly idenical wih respec preferences, endowmens and populaion size, and agens share same beliefs in asse prices, we can claim ha a household born in counry i and a household born in counry j wih i j hold he same porfolio, herefore x H,j = x F,j = x j. Along he open economy equilibrium, he marke clearing condiions for Home asse and Foreign asse a period are such ha: This implies x H,j x H,H + x F,H = 2 and x H,F + x F,F = 2 (12 = x F,j = x j = 1. The world goods marke clearing condiions a period is saisfied by he Walras law. For he res of he paper, we analyze wo ypes of world economy. In he firs world economy, he bubble on Home asse is immune agains sunspos; whereas he bubble on Foreign asse is condiional o he realizaion of sunspo. In his siuaion, agens have full confidence in Home asse bubble and a parial one in Foreign asse bubble. In he second world economy, boh bubbles are no immune agains sunspos. 3.1 Home counry immune agains sunspos In his secion we are ineresed in how he burs of Foreign bubble affecs he value of Home bubble when sunspos only occur in Foreign counry. Only he exisence of Foreign asse bubble is condiional o he realizaion of a sunspo. Agens in boh counries believe ha Foreign bubble persiss Q F = q f > 0 when no sunspo occurs (wih probabiliy π, and burss Q F = 0 when a sunspo occurs in Foreign counry (wih probabiliy 1 π. In conras o he closed economy sudied in Secion 2, afer he burs of Foreign bubble, all agens living in Foreign or in Home 7 By normalizing he supply of asses o 2, he dynamics of he world economy afer he burs of one of he wo bubbles coincides wih he dynamics sudied in Secion 2. If we normalize o 1, we obain qualiaively he same resuls. 6

7 counry can save hrough Home asse. Therefore, he dynamics of he economy afer he crash of Foreign bubble is he same as in an economy wih one asse and wihou uncerainy. We will show ha he sunspo affecing Foreign bubble is spread o Home bubble hrough he porfolio of agens due o he financial inegraion. Home and Foreign bubble processes a each period are summarized by he following equaions: Q F = q F > 0 wih probabiliy π, Q F = 0 wih probabiliy 1 π; Q H = q H 0 if Q F = q F > 0, Q H = q H 0 if Q F = 0. (13 In his framework, here are only wo saes of naure in he world economy such ha: π (Q F = q F > 0, Q H = q H 0 (No sunspo in Foreing counry (Q F, Q H 1 π (Q f = 0, Q H = q H 0 (Sunspo in Foreign counry When Foreign bubble is no valued in period, Q F = 0, hen he opimal behavior of an agen born in counry i is summarized by he following equaion: ( u y 1 q H xi,h ( = qh +1 v y 2 + q H +1 xi,h q H (14 This ineremporal rade-off is similar o he one found in he moneary model by Samuelson (1958. In his case, he economy of counry i behaves as an economy wih one asse and wihou uncerainy along an equilibrium. When Foreign bubble is valued in period, Q F = q F > 0, he opimal behavior of an agen born in counry i is given by he ineremporal rade-off and he no-arbirage condiion beween boh bubbles: (15 u ( y 1 q F x i,f v (y 2 + q F +1 xi,f q H x i,h + q H +1 xi,h π qf +1 q F = π qf +1 q F = π qh +1 q H + (1 π qh +1 q H ( v y 2 + q H +1 xi,h v ( y 2 + q F +1 xi,f + q H +1 xi,h (16 (17 From he no-arbirage condiion (17, we deduce ha agens inves in boh asses as long as he expeced marginal reurn on Foreign bubble is equal o he expeced marginal reurn on Home bubble weighed by he marginal uiliies. Since we are ineresed in he effec of Foreign bubble burs on he value of Home bubble, we resric our focus on he case where Foreign bubble is valued a ime. We solve by backward 7

8 inducion o deermine he equilibrium wih Foreign and Home bubble. Afer Foreign bubble crash, he economy of counry i moves o he equilibrium wih one asse and wihou uncerainy saisfying he following dynamic equaion: ( u y 1 q H ( = qh +1 v y 2 + q H q H ( q H y 1 (19 Before he crash of Foreign bubble, an equilibrium is a sequence { q F, q H all he following condiions: } + which saisfies for =0 u ( y 1 q F q H v (y 2 + q+1 F + qh +1 = π qf +1 q F ( q F π +1 q F qh +1 q H = (1 π qh +1 q H ( v y 2 + q H +1 (20 v ( y 2 + q+1 F + (21 qh +1 where q H is an equilibirum value of Home bubble saisfying equaion ( To highligh he effec of Foreign bubble burs on he dynamics of Home bubble, we simplify he analysis considering only he saionary equilibrium. In his equilibrium, Foreign and Home bubble are consan before he burs if Foreign bubble such ha q F = q F and q H = q H for all. We can show from (21 ha a saionary equilibrium exiss before he crash of Foreign bubble only if q H = 0. As counries are symmeric in erms of fundamenals, he expeced reurn on +1 Foreign bubble is equal o he he expeced reurn on Home bubble a he saionary equilibrium. Therefore, he expeced marginal gain o inves in Foreign bubble is zero. Because of he noarbirage condiion, he marginal loss o inves in Foreign bubble mus be zero. This implies ha he value of Home bubble afer he burs of Foreign bubble is zero. Oherwise, an arbirage opporuniy would exis. Le Q = q F + q H. Given q H = 0, a saionary equilibrium +1 Q (wih q F > 0 and q H > 0 is a soluion of he following equaion: u (y 1 Q = πv (y 2 + Q (22 Proposiion 1 Under Assumpion 1 and given q H +1 = 0, a saionary sochasic equilibrium Q (0, y 1 wih sochasic bubbles exiss if and only if π > 1 + r. Proof. See Appendix. Proposiion 1 means ha upon o he random dae on which Foreign bubble burss, boh counries experience a sochasic saionary bubble provided ha bubbles are sufficienly safe. This equilibrium may occur when he economy wihou bubble are dynamically inefficien. The condiion for exisence of sochasic saionary bubbles in open economy is he same as for he exisence a sochasic bubble in closed economy. As corollary o Proposiion 1 we obain he following: (23 8

9 Corollary 1 Le π > 1 + r. Home bubble jumps o zero and does no reappear in he fuure afer he burs of Foreign bubble. When Foreign bubble explodes, Home bubble explodes as well in he same ime. In oher words, afer he realizaion of a sunspo in Foreign counry, no bubbles persis in boh counries. Figure 1 give a qualiaive illusraion of Proposiion 1 and Corollary 1. We assume ha he economy experiences Home and Foreign bubble and is in he saionary equilibrium before period T. A period T, a sunspo occurs in Foreign counry implying he burs of Foreign bubble. A he same period T, Home bubble jumps o zero. Q F Q H q F q H 0 T 0 Figure 1: T In his framework, we show ha even hough he realizaion of sunspos is specific o Foreign counry, a sunspo is spread o Home counry hrough he porfolio choice of agens under financial marke inegraion. Following he burs of Foreign bubble, agens lose confidence in Home bubble. All old agens would like o sell Home bubble and no young agens would like o buy i. Thus, Home bubble burss as well. In he nex secion, we sudy he second case in which Home and Foreign counry are no immune agains a sunspo shock. 3.2 Home and Foreign counries no immune agains sunspos In his secion, we sudy he effec of Foreign bubble burs on Home bubble when boh bubbles are affeced by he realizaion of a counry-specific sunspo. More precisely, he exisence of Home and Foreign bubble is condiional o he realizaion of differen random evens: Home bubble is condiional o sunspo H and Foreign bubble o sunspo F. To highligh he role of financial inegraion in he ransmission, we suppose ha sunspo H and sunspo F are wo exrinsic independen and idenical disribued random variables. Sunspo H and sunspo F have same probabiliy 1 π o appear. In oher words, agens as much confidence in Home bubble as in Foreign bubble. 8 The price processes a each period are summarized by he following equaions: 8 If he wo random variables are no idenical disribued, here canno exis an equilibrium. Suppose ha Foreign bubble has a higher probabiliy o persis han Home bubble, bu Foreign bubble burss in he firs place. In such a siuaion, all agens who hold Home bubble would like o sell i and no one would like o buy i. Therefore, here would be an excess supply, an equilibrium wih posiive price could no exis. 9

10 Q i = q i > 0 wih probabiliy π, = 0 wih probabiliy 1 π; Q j = q j > 0 if Q i = q i > 0, Q j = q j 0 if Qi = 0. (24 where i, j = {H, F } and i j. In conras o Secion 3.1, here are now four saes of naure in he world: (no sunspo H, no sunspo F, (sunspo H, no sunspo F, (no sunspo H, sunspo F and (sunspo H, sunspo F. (Q F, Q H π 2 π(1 π (1 ππ (1 π 2 (Q F = q F > 0, Q H = q H > 0 (Q f = q F 0, Q H = 0 (Q f = 0, Q H = q H 0 (Q f = 0, Q H = 0 When none of boh bubbles are valued in period, Q F = 0 and Q H = 0, he economy of counry i is in he auarky regime. There are neiher inernaional rades nor inergeneraional rades. When only one of hese wo bubbles is valued a period, Q i = 0 and Q j = q j > 0 wih i j, he opimal behaviour of an agen born in Counry i is given by he following equaion: ( u y 1 q j xi,j v (y 2 + q j = π qj xi,j q j (25 Since he economy of corresponds o a single-good exchange economy wih one asse in his case, we find he same ineremporal rade-off as he closed economy sudied in Secion 2. Lasly, when boh bubbles are valued a period Q H = q H > 0 and Q F = q F > 0, he opimal behaviour of an agen born in Counry i is summarized by: u ( y 1 q H x i,h v ( y 2 + q H +1 xi,h q F x i,f + q F +1 xi,f = π π qh +1 q H + (1 π qh +1 q H ( v y 2 + q H +1 xi,h v (y 2 + q H +1 xi,h + q F +1 xi,f (26 ( π qf +1 v y q F 2 + q H +1x i,h ( π qh +1 v y q H 2 + q H +1 xi,h + q F +1x i,f + q F +1 xi,f + (1 π qf +1 q F + (1 π qh +1 q H v ( y 2 + q F +1 xi,f = v ( y 2 + q H +1 xi,h (27 Since boh counries are no immune agains sunspos, we obain a modified ineremporal rade-off and a modified he no-arbirage condiion beween Foreign and Home bubbles compared o he case in which only Foreign counry is affeced by sunspos. 10

11 To simplify he analysis, we consider ha a sunspo occur in Foreign counry in firs place. Therefore, he bubble issued by Foreign counry burss before he bubble of Home counry. 9. To highligh he effec of he burs of Foreign bubble on he value of Home bubble, we resric our focus on he case in which boh bubbles are valued a ime, Q F = q F > 0, Q H = q H > 0. We solve by backward inducion o deermine he equilibrium wih Foreign and Home bubble. Afer he burs of Foreign bubble, he economy of each counry moves o he equilibirum wih one asse saisfying he following dynamic equaion: ( u y 1 q H ( = π qh +1 v y 2 + q H q H ( q H y 1 (29 The economy behaves as he closed economy sudied in Secion 2 afer he collapse of Foreign bubble. From Lemma 2, we know ha a saionary equilibrium q H (0, y 1 exiss if π > 1 + r. Before he crash of Foreign bubble, an equilibrium is a sequence { q F, q H all he following condiions: } + which saisfies for =0 u ( y 1 q H q F v (y 2 + q +1 H + qf +1 = π π qh +1 q H + (1 π qh +1 q H ( v y 2 + q H +1 v ( y 2 + q +1 H + qf +1 π ( q F ( +1 qh +1 v y q F q H 2 + q H +1 + q +1 F = (1 π [ q F +1 q F v ( y 2 + q F +1 (30 qh ( +1 v y q H 2 + q +1 ] H (31 where q F and +1 qh are respecively an equilibirum value of Foreign bubble and Home bubble +1 saisfying equaion (28. Noe ha in period, agens do no known which bubble will burs in period + 1. Hence, he presence of q F in (31. For he res of he analysis, we ake +1 qh = +1 a (0, q H and q F +1 = b (0, qf as given. Noe ha as counries are perfecly idenical q H = q F = q. We sudy now he exisence of a saionary sochasic equilibrium wih Foreign and Home bubble before he crash of Foreign bubble. A saionary sochasic equilibrium ( q F, q H wih q F > 0 and q H > 0 is a soluion of he following equaion: u ( y 1 ( q F + q H [ = π πv (y 2 + q F + q H + (1 π a ] q H v (y 2 + a (32 b q F v (y 2 + b = a q H v (y 2 + a (33 Proposiion 2 Le Q = q F + q H. Under Assumpion 1, he following generically holds before he burs of counry i = {H, F } bubble: 9 Since counries are idenical, we obain same resul, if we assume ha Home bubble is he firs o explode. 11

12 1. If π 1 + r, hen here does no exis a saionary equilibrium wih sochasic bubbles. 2. If 1+ r < π (1+ r 1/2, hen a unique saionary equilibrium Q (0, y 1 (a, b ]0, q ] exiss wih sochasic bubbles q F = Q bv (y 2 + b/ [av (y 2 + a + bv (y 2 + b] and q H = Q av (y 2 + a/ [av (y 2 + a + bv (y 2 + b]. 3. If π > (1 + r 1/2, hen a unique saionary equilibrium Q (0, y 1 (a, b [0, q ] exiss wih sochasic bubbles. Proof. See Appendix. Proposiion 2 shows ha if agens expec a posiive value of Home bubble afer he burs of Foreign bubble, hen boh counries can experience a sochasic saionary bubble, as long as he economy wihou bubble are dynamically inefficien. We find he similar requiremen for he exisence of a saionary equilibrium as in he closed economy (Weil (1987. If agens expec ha he value of Home bubble is equal o zero afer he burs of Foreign bubble, hen he scope for he exisence of a sochasic saionary equilibrium is reduced. Only safe bubble can exis. This equilibrium is likely o occur in a dynamically inefficien economy only if he probabiliy for Foreign and Home bubbles o persis is sufficienly closed o 1. To highligh he impac of he burs of Foreign bubble on he exisence of Home bubble, we assume ha he economy experiences Home and Foreign bubble and is in he saionary equilibrium before period T. A period T, a sunspo occurs in Foreign counry implying he burs of Foreign bubble. Proposiion 3 Under Assumpion 1, he saionary value of Home bubble q H before he burs of Foreign bubble is sricly lower han q H, where q H is he saionary value of Home bubble afer he burs of Foreign bubble. Proof. See Appendix. Proposiion 3 relies on an argumen of marke size. For he res of he paper, we resric our focus only on safe bubble assuming ha: Assumpion 2 π > (1 + r 1/2 When q H = q F before he crash of Foreign bubble, hree cases emerge according o agens expecaions when boh counries are subjec o sunspos in conras o secion 3.1. From he analysis of Home bubble dynamics afer Foreign bubble crash (see Lemma 2 and Proposiion 2, we derive hree corollaries. Corollary 2 Under Assumpions 1 and 2, if a = q H = 0, hen Home bubble jumps o zero afer T he burs of Foreign bubble and does no reappear in he fuure. Following he burs of Foreign bubble in period T, all agens expec ha he value of Home bubble will be zero in period T, and beliefs are self-fulfilling. This resul is similar o he one fund in Secion 3.1 when Home counry is immune agains sunspos (see Figure 1 for a qualiaive illusraion. 12

13 Q F Q H q H T = qh q F q H 0 T 0 Figure 2: T Corollary 3 Under Assumpions 1 and 2, if a = q H = T qh, hen Home bubble jumps o q H afer he burs of Foreign bubble, and says a his value unil he occurrence of a sunspo in Home counry. If agens expec ha he value of Home bubble immediaely afer he burs of Foreign bubble in period T q H is equal o he saionary value T qh, hen hey save enough o keep he value of Home bubble consan unil he burs of Home bubble. In his case, he effec of Foreign bubble burs is posiive (see Figure 2. Afer he burs of Foreign bubble, young agens can save only hrough Home bubble. Savings in erms of Home bubble increases. Therefore, he value of Home bubble afer Foreign bubble crash (q H = T qh is greaer han before he burs (q H.[Noe ha he probabiliy for his case o occur is small. Since he seady sae afer Foreign bubble burs is unsable, he probabiliy for q H o be equal o T qi is small.] The nex corollary depics he las case: Corollary 4 Under Assumpions 1 and 2, if a = q H ]0, T qh [, hen Home bubble jumps o q H T afer he burs of Foreign bubble, and converges o zero in he long run. Q F Q H q H T = qh q F q H 0 T 0 Figure 3: T In he las case, agens expec ha he value of Home bubble immediaely afer he crash of Foreign bubble q H is posiive bu less han he saionary value T qh. The reurn of Home bubble falls. Agens flee financial marke and save less, he Home bubble collapse (see Figure 3 for a 13

14 qualiaive illusraion. The effec of Foreign bubble burs is negaive in long erm. Home bubble ends o collapse afer he Foreign bubble crash even before he occurrence of a sunspo in Home counry. In conras o Secion 3.1, he burs of Foreign bubble does no necessarily imply he immediae crash of Home bubble. When boh counries are subjec o sunspos, agens inernalize he risk of Home bubble burs, and hus coninue o inves in Home bubble afer he crash of Foreign bubble. [To be developped]. When q H > q F before he crash of Foreign bubble, Corollaries 2 and 3 hold. When q H < q F before he crash of Foreign bubble, Corollary 3 hold. (See Appendix 4 Concluding remarks In his paper, we develop a simple neoclassical framework o analyze he inernaional ransmission of bubble crashes. We exend Weil s (1987 overlapping generaions model in which he burs of he bubble is based on self-fulfilling beliefs (sunspo o a wo-counry model wih informaion symmery and homogeneous beliefs. Each counry issues is own bubble, and each agens can hold a porfolios of boh asses. We show ha even hough he realizaion of sunspos is specific o one counry, he sunspo is spread o oher counry hrough he porfolio choice of agens. When only he foreign counry is subjec o sunspos, he burs of he foreign bubble generaes a change in confidence in he domesic bubble involving is immediae collapse. When boh counries are subjec o sunspos, agens inernalize he risk of he domesic bubble burs. Therefore, he domesic bubble can persis jus afer he burs of Foreign bubble, bu may deflae over ime. Our framework can be considered as a firs sep for fuure research. For insance, we could exend i o analyze he impac of governmen policies (like ransacion coss on he inernaional ransmission of bubble crashes. 5 Appendix Proof of Lemma 1 This proof is similar as he one given by Weil (1987. A seady sae q i is a soluion of A(q i = B(q i, wih: A ( q i u ( y 1 q i (34 B ( q i πv ( y 2 + q i (35 One has A ( q i > 0 and B ( q i < 0. As c > 0 implies q i < y 1, all he saionary soluions q i belong o (0, y 1. To prove he exisence of a saionary soluion b, we use he coninuiy of A ( q i and B ( q i. 14

15 lim q i 0 A ( q i = u (y 1 lim B ( q i = πv (y 2 q i 0 lim A ( q i = u (0 = + q i y 1 lim B ( q i = πv (y 2 + y 1 q i y 1 We have lim qi y 1 A ( q i > lim qi y 1 B ( q i. If π u (y 1 / [v (y 2 ], hen lim qi 0 A ( q i > lim q i 0 B ( q i. No sochasic bubble seady sae exiss. If π > u (y 1 / [v (y 2 ], hen lim q i 0 A ( q i < lim q i 0 B ( q i. A sochasic bubble saionary equilibrium q i (0, y 1 exiss. Proof of Lemma 2 As he lef-hand side of (5 is a sricly increasing and coninuous funcion of q i, we can rewrie he dynamics of he model as follows: q i = f(q i +1 (36 0 < q i < y 1 (37 (36 depics a backward perfec foresigh dynamics which is well defined. f is coninuously differeniable and f(0 = 0 under Assumpion 1. Moreover, le ε v (d v (dd/v (d and ε u (c u (cc/u (c, f (q is given by: f (q i = 1 ɛ v(d q / (y 2 + q 1 + ɛ u (c q / (y 1 q (0, 1 (38 wih c = y 1 q i and d = y 2 + q i (39 Hence, f(q i > q i when q i (0, q i, f(q i = q i when q i = q i, and f(q i < q i when q i (q i, y 1. Under Assumpion 1, he seady sae q i is unsable in he forward dynamics. Furhermore, when q 0 > q i, here exiss a dae + s wih s > 0 during which q i +s > y 1. This implies c +s < 0 which is infeasible. Proof of Proposiion 1 The proof is equivalen o he proof of Lemma 1. Proof of Proposiion 2 To prove he exisence of a sochasic saionary equilibrium (q F, q H before he burs of counry i = {H, F } bubble, we have o solve by backward inducion: firs, we analyze he economy of counry i afer he crash of counry j, hen before he crash. Afer he burs of counry j bubble, he economy is equivalen o he one sudied in Secion 2. From Weil (1987, we deduce ha when π u (y 1 /v (y r, no saionary sochasic counry i bubble q i exiss afer he burs of counry j bubble under our Assumpion 1. When π > 1 + r, 15

16 a saionary sochasic Home bubble exiss in he economy afer he burs of Foreign bubble under Assumpion 1. The dynamics of Home bubble afer he burs of Foreign bubble is given in Lemma 2. Now, we analyze he exisence of he sochasic saionary equilibrium (q F, q H before he burs of Foreign bubble. A saionary sochasic equilibrium ( q F, q H wih q F > 0 and q H > 0 is a soluion of he following equaion: u ( y 1 ( q F + q H [ = π πv (y 2 + q F + q H + (1 π a ] q H v (y 2 + a (40 b q F v (y 2 + b = a q H v (y 2 + a (41 where a = q H +1 (0, q H and b = q F +1 (0, q F are respecively he equilibrium value afer he bubble crash of counry F and afer he bubble crash of counry H. Le Q = q F + q H. From 41, we ge: q H = Q av (y 2 + a av (y 2 + a + bv (y 2 + b q F = Q bv (y 2 + b av (y 2 + a + bv (y 2 + b (42 (43 A sochasic saionary equilibrium Q is a soluion of he following equaion: u ( y 1 Q [ = π πv ( y 2 + Q + (1 π av (y 2 + a + bv ] (y 2 + b Q (44 Le G( Q u ( y 1 Q and H( Q π { πv ( y 2 + Q + (1 π [av (y 2 + a + bv (y 2 + b] / Q }. One has G ( Q > 0 and H ( Q < 0. As c = y1 Q > 0 implies Q < y 1, all he saionary soluions Q belong o (0, y 1. To prove he exisence of a saionary soluion Q, we use he coninuiy of G ( Q and H ( Q. We consider firs he case in which π 1 + r. When π 1 + r, we know from Lemma 2 ha a = 0 and b = 0. Therefore, lim G ( Q Q 0 lim H ( Q Q 0 lim G ( Q Q y 1 lim H ( Q Q y 1 = u (y 1 = π 2 v (y 2 = u (0 = + = π 2 v (y 2 + y 1 We have lim Q y1 G ( Q > lim Q y1 H ( Q. When π r, lim Q 0 G ( Q > lim Q 0 H ( Q. Therefore, if π r, hen no sochasic saionary bubble exiss under Assumpion 1. When π 2 > 1+ r, lim Q 0 G ( Q < lim Q 0 H ( Q. Thus, if π 2 > 1+ r, hen a unique sochasic saionary 16

17 bubble Q (0, y 1 could exis under Assumpion 1. However, as π < 1 and π 1+ r, he condiion π 2 > 1 + r does no hold. Therefore, no sochasic saionary bubble exiss in he open economy when π 1 + r. Nex, we analyze he case where π > 1+ r. Firs, when a = 0 (b = 0, hen saionary sochasic bubbles q F > 0 and q H > 0 exis if and only if b = 0 (a = 0. Furhermore, we know ha when a = 0 and b = 0, a sochasic saionary equilibrium exiss if and only if π 2 > 1 + r. Now, we analyze he exisence of a sochasic saionary equilibrium when (a, b ]0, q ]. Firs, we can noe ha a < Q and b < Q. We make a proof by conradicion. Suppose ha a Q. As Q > q H, we deduce from (40 ha a saionary equilibrium Q saisfies he following inequaliy: u ( y 1 Q > π[πv (y 2 + Q + (1 πv (y 2 + a] Because of uiliy concaviy, we obain: u ( y 1 Q > πv (y 2 + a Since a q H, we ge u ( y 1 Q > πv (y 2 + q H We recall ha afer he burs of Foreign bubble, a saionary equilibrium q H is such ha: u ( y 1 q H = πv (y 2 + q H (45 Hence, Q > q H a. This resuls in a conradicion. Therefore, a < Q. We can provide a similar proof for b < Q. We deermine he boundary values of G( Q and H( Q a > 0 and b > 0. lim G ( Q Q 0 lim H ( Q Q 0 lim G ( Q Q y 1 lim H ( Q Q y 1 = u (y 1 [ = π πv (y 1 + (1 π av (y 2 + a + bv ] (y 2 + b = + 0 = u (0 = + [ = π πv (y 2 + y 1 + (1 π av (y 2 + a + bv ] (y 2 + b y 1 We have lim Q 0 G ( Q < lim Q 0 H ( Q and lim Q y1 G ( Q > lim Q y1 H ( Q. We deduce ha for given equilibrium values a ]0, q ] and b ]0, q ], here exiss a unique sochasic saionary equilibrium Q (0, y 1. Proof of Proposiion 3 17

18 Differeniaing (44 and (42 (a he saionaryy equilibirum, we obain: Q a q H a = = π (1 π [v (y 2 + a + av (y 2 + a] u (y 1 Q π 2 v (y 2 + Q + π (1 π [av (y 2 + a + bv (y 2 + b] / Q 2 (46 av (y 2 + a Q av (y 2 + a + bv (y 2 + b a + Q bv (y 2 + b v (y 2 + a + av (y 2 + a av (y 2 + a + bv (y 2 + b (47 Under Assumpion 1, Q / a > 0 and q H / a > 0. To prove ha q H < q H, we make a proof by conradicion. Suppose ha a = q H and q H q H. (40 is wrien as follows: u ( y 1 Q = π [ πv (y 2 + Q + (1 π qh ( ] q H v y 2 + q H As Q > q H, he concaviy of he uiliy funcion implies: π [ πv (y 2 + Q + (1 π v ( y 2 + q H ] u ( y 1 Q < πv ( y 2 + q H q H saisfies he following equaion u ( y 1 q H = πv (y 2 + q H. Hence, u ( y 1 Q < u ( y 1 q H Because of uiliy concaviy, we obain Q < q H. Hence, q H < q H, i resuls in a conradicion. Therefore, when a = q H, q H < q H. Since q H / a > 0, we deduce ha q H < q H a (0, q H. If q H > q F before he crash of Foreign bubble, hen 0 < b < a q H (see (41. Therefore, according o Lemma 2 and Proposiion 2, Corollaries 2 and 3 hold. If q H < q F before he crash of Foreign bubble, hen 0 < a < b q H. Hence, Corollary 3 only holds. References [1] Farhi, E. and J. Tirole (2012, Bubbly liquidiy, Review of Economic Sudies 79, [2] Hirano, T. and N. Yanagawa (2010, Asse bubbles, endogenous growh and financial fricions, working paper, Universiy of Tokyo. [3] Kindleberger, C. P. (1978, Manias, Panics, and Crashes: A Hisory of Financial Crises. Basic Books. [4] Kindleberger, C. P. and R. Z. Aliber (2005, Manias, Panics, and Crashes: A Hisory of Financial Crises, Palgrave Macmillan, 5h ediion. 18

19 [5] Kocherlakoa, N. (2009, Bursing bubbles: Consequences and cures, Minneapolis Fed. [6] Marin, A. and J. Venura (2012, Economic growh wih bubbles bubbles, American Economic Review 102, [7] Miao, J. and P. Wang (2012, Banking bubbles and financial crisis, working paper, Boson Universiy and HKUST. [8] Samuelson, P. (1958, An exac consumpion-loan model of ineres wih or wihou he social conrivance of money, Journal of Poliical Economy 66, [9] Tandon, A., and Y. Wang (2003, Confidence in domesic money and currency subsiuion, Economic Inquiry 41, [10] Tirole, J. (1985, Asse bubbles and overlapping generaions, Economerica 53, [11] Venura, J. (2012, Bubbles and capial flows, Journal of Economic Theory 147, [12] Weil, P. (1987, Confidence and he real value of money in overlapping generaions models, Quarerly Journal of Economics 102, l-22. [13] Wigniolle, B. (2014, Opimism, pessimism and financial bubbles,journal of Economic Dynamics and Conrol 41,