Asset Bubbles, Endogenous Growth, and Financial Frictions

Transcription

1 Gran-in-Aid for Scienific Research(S) Real Esae Markes, Financial Crisis, and Economic Growh : An Inegraed Economic Approach Working Paper Series No.36 Asse Bubbles, Endogenous Growh, and Financial Fricions Tomohiro Hirano and Noriyuki Yanagawa February, 2016 HIT-REFINED PROJECT Insiue of Economic Research, Hiosubashi Universiy Naka 2-1, Kuniachi-ciy, Tokyo , JAPAN Tel: hi-refined-sec@ier.hi-u.ac.jp hp://

2 Asse Bubbles, Endogenous Growh, and Financial Fricions Tomohiro Hirano and Noriyuki Yanagawa Firs Version, July 2010 This Version, December 2015 Absrac This paper analyzes he exisence and he effecs of bubbles in an endogenous growh model wih financial fricions and heerogeneous invesmens. Bubbles are likely o emerge when he degree of pledgeabiliy is in he middle range, implying ha improving he financial marke migh increase he poenial for asse bubbles. Moreover, when he degree of pledgeabiliy is relaively low, bubbles boos long-run growh; when i is relaively high, bubbles lower growh. Furhermore, we examine he effecs of a bubble burs, and show ha he effecs depend on he degree of pledgeabiliy, i.e., he qualiy of he financial sysem. Finally, we conduc a full welfare analysis of asse bubbles. Key words: Asse Bubbles, Endogenous Growh, and Financial Fricions This paper is a revised version of he our working paper, Hirano, Tomohiro, and Noriyuki Yanagawa July. Asse Bubbles, Endogenous Growh, and Financial Fricions, Working Paper, CARF-F-223, The Universiy of Tokyo. We highly appreciae valuable commens from Francesco Caselli and four anonymous referees hroughou revisions. We also hank helpful commens from Jean Tirole, Jose Scheinkman, Joseph Sigliz, Fumio Hayashi, Michihiro Kandori, Hioshi Masushima, Kiminori Masuyama, Alber Marin, Jaume Venura, and seminar paricipans a Economeric Sociey Word Congress 2010, and 2011 Midwes Macro Meeing, Bank of Canada. A par of his work is financially suppored by JSPS KAKENHI Gran Number Corresponding Auhor: Faculy of Economics, The Universiy of Tokyo, omohih@gmail.com Faculy of Economics, The Universiy of Tokyo, yanagawa@e.u-okyo.ac.jp 1

3 1 Inroducion Many counries have experienced large movemens in asse prices, called asse bubbles, which are associaed wih significan flucuaions in real economic aciviy. A noable example is he recen global economic upurn and downurn before and afer he financial crisis of Many economiss and policy makers wan o undersand why bubbles emerge and how hey affec real economies. 1 However, i is sill no clear how financial marke condiions affec he exisence condiion of bubbles. In his sudy, we firs examine he relaionship beween he emergence of asse bubbles and financial condiions, in oher words, wheher bubbles are more likely o occur in financially developed or less-developed economies. Empirically, here is a complicaed relaionship beween financial marke condiions and asse bubbles. Emerging marke economies, such as hose in Souh Eas Asia, ofen experience bubble-like dynamics. Caballero (2006) and Caballero and Krishnamurhy (2006) found ha financial imperfecion is a key elemen in bubbles in emerging marke economies. However, no all counries wih less developed financial markes experience bubble-like dynamics. For example, he financial sysems in some African counries are less developed han in Asia (UNECA, 2006), ye hey have no experienced bubble-like macro dynamics. This may sugges ha financial qualiy below a cerain hreshold canno susain asse bubbles. In fac, counries in he Souh Eas Asia began o develop heir financial markes in he 1980s, and his was one of he reasons for heir high growh raes (World Bank, 1993). On he oher hand, improving financial condiions migh promoe asse bubbles. For example, Allen (2001) poined ou ha he financial liberalizaion resuling from financial sysem developmen in hese counries was a facor in he emergence of bubbles in he 1990s. 2 Addiionally, advanced economies like he U.S. experienced informaion fricions problems in financial markes such as subprime problems, suggesing ha advanced economies may also face financial imperfecions (see Campello e al, 2010; Brunnermeier and Sannikov, 2014). From hese observaions, i seems ha financial marke condiions and he emergence of bubbles may have a non-linear relaionship. In oher words, bubbles may 1 See, for example, Akerlof and Shiller (2009). 2 The Japanese economy experienced asse bubbles in he 1980s, bu he srucural reforms of he Japanese financial sysem and subsequen financial liberalizaion maerialized before he rise in asse prices. See Shigemi (1995) for a more deailed discussion. 2

4 no occur in financially underdeveloped or in well-developed economies. They end o occur in counries wih an inermediae level of financial developmen. The firs purpose of his sudy is o formulae his non-linear relaionship heoreically. For his purpose, we use an endogenous growh model wih heerogeneous invesmens and financial marke imperfecions. In our model, enrepreneurs swich beween producive and unproducive saes. In he producive sae, enrepreneurs invesmens yield high reurns, while hey yield low reurns in he unproducive sae. In addiion, enrepreneurs can pledge only a fracion of he reurns from heir invesmens. The endogenous growh model wih heerogeneous invesmens is crucial o formulaing an inuiive undersanding of he non-linear relaionship. For example, Farhi and Tirole (2012) recenly examined he exisence of bubbles and found ha hey can exis when he pledgeabiliy level is low, alhough heir main focus was on he effecs of ouside liquidiy. However, hey assumed homogeneous invesmen opporuniies. Hence, if pledgeabiliy is very low, he ineres rae becomes very low and he growh rae, which equals zero in he seady-sae in heir sudy, becomes relaively high compared o he ineres rae. Thus, based on hese assumpions, bubbles can exis despie very poor financial marke condiions. 3 On he oher hand, if here are heerogeneous invesmens, he marke ineres rae may no decrease much, even if he financial marke is very poor. Because he reurn from low-yield invesmens becomes he lower bound for he ineres rae. Thus, he growh rae becomes very low compared o he ineres rae, and bubbles canno exis in very poor financial marke condiions. This resul suggess ha improving financial marke condiions migh increase he emergence of bubbles if he financial marke sars from a sae of underdevelopmen. 4 Based on he exisence condiion of bubbles, we can also examine he relaionship beween echnological progress and he condiions leading o asse bubbles. Scheinkman (2014) recenly poined ou he imporance of his relaionship. Since echnological progress is a facor for promoing economic growh raes, i seems o increase he exisence of bubbles. However, if i also increases he ineres rae, 3 Caballero (2006) and Caballero and Krishnamurhy (2006) boh assume a high exogenously given growh rae in emerging counries. Thus, even heir model canno capure he non-linear relaionship. 4 In his sense, our model is relaed o Masuyama s (2007, 2008) model showing ha a beer credi marke may be more prone o financing wha he calls bad invesmens ha do no have posiive spillover effecs on fuure generaions. 3

5 echnological progress may no lead o bubbles. Moreover, bubbles may in urn affec invesmen financing wih echnological progress, suggesing ha here is a wo-way feedback relaionship beween echnological progress and bubbles. We will show ha he ype of echnological progress affecs his relaionship, and will derive resuls consisen wih Scheinkman s (2014) sylied facs. Moreover, i is no ye obvious how bubbles affec economic growh. The second purpose of his sudy is o invesigae he macroeconomic effecs of bubbles. Here we examine wheher bubbles enhance or impair growh, as well as he relaionship beween hese macroeconomic effecs and financial condiions. In he process, we analyze how financial condiions deermine he effecs of bubbles collapse on he economic growh rae. We will show ha he effec of bubbles on economic growh depends on financial marke condiions. Bubbles have boh crowd-ou and crowd-in effecs on invesmen and growh raes. Since bubbles crowd savings away from invesmens, bubbles decrease he economic growh rae. On he oher hand, bubbles increase he rae of reurn on savings and improve borrowers neworh, which in urn crowds in heir fuure invesmens. Tha is, bubbles endogenously generae he balance shee effec emphasized by Bernanke and Gerler (1989). Our main finding is ha he relaive impac of hese effecs depends on he degree of pledgeabiliy. When he pledgeabiliy level is relaively low, he crowd-in effec dominaes he crowd-ou effec and bubbles enhance he economic growh rae. On he oher hand, if he pledgeabiliy level is relaively high, he crowd-ou effec dominaes, and bubbles decrease he economic growh rae. This examinaion also has imporan implicaions for he effecs of bubbles afer hey burs, which our resuls sugges is no uniform. A counry s financial condiion has a significan effec on he growh pah afer he collapse of bubbles. If he imperfecion of he financial marke is relaively high (i.e., if pledgeabiliy is relaively low), he bubble burs decreases he growh rae permanenly. This implies ha in economies wih low pledgeabiliy, bubbles can emporarily mask low economic growh raes due o he poor financial marke condiions. On he oher hand, economies wih high pledgeabiliy will experience a decline in he economic growh rae immediaely afer he bubble burss, bu will recover and achieve a high growh rae. Tha is, he burs may enhance he long-run growh rae if he financial markes are in relaively good condiion. 4

6 Moreover, his resul implies ha if he emporary negaive produciviy shock is sufficienly large, he level of oal oupu becomes permanenly lower han he prebubble rend level, despie recovery in he economic growh. This resul is consisen wih empirical evidence on he effecs on growh of various ypes of financial crises. For example, Cerra and Saxena (2008) show ha mos financial crises are associaed wih a decline in growh ha leaves oupu permanenly below is pre-crisis rend. Finally, we conduc a rigorous full welfare analysis of asse bubbles in an infinielylived agen model wih heerogeneous invesmens and financial marke imperfecions. In our framework, we assume ha bubbles will collapse wih posiive probabiliy and ha enrepreneurs are risk-averse. Enrepreneurs care abou increased volailiy in consumpion arising from he collapse of a bubble. We consider he welfare effecs of his increased volailiy from he bubble s burs. We find analyically ha bubbles increase welfare, regardless of wheher hey increase or decrease he long-run economic growh rae and even if hese are expeced o collapse. The economic inuiion for his resul lies in he consumpion-smoohing effecs of bubbles. In his economy, enrepreneurs face borrowing consrains and canno consume smoohly agains idiosyncraic shocks o he produciviy of invesmen. In his siuaion, he circulaion of bubble asses serves as an insurance device agains idiosyncraic produciviy shocks, hereby increasing welfare. The res of his paper is organized as follows. Subsecion 1.1 provides a lieraure review. In secion 2, we presen our basic model, boh wih and wihou bubbles. In secion 3, we presen he dynamics of bubbles. In secion 4, we examine he exisence condiion of bubbles, and in secion 5, we examine he effecs of bubbles on economic growh raes. In secion 6, we show how he effecs of he bubbles burs are relaed o financial marke condiions. In secion 7, we conduc a full welfare analysis of bubbles, and secion 8 concludes he paper. 1.1 Relaed Work in he Lieraure Our sudy considers he exisence of bubbles in an infiniely lived agens model. Wih regard o he exisence of bubbles in infinie horizon economies, i is commonly hough ha bubbles canno arise in deerminisic sequenial marke economies wih a finie number of infiniely lived agens (Tirole, 1982). The Tirole model assumes a perfec financial marke, ha is, agens can borrow and lend freely. Tirole showed ha in such an environmen, no equilibrium wih bubbles exiss. Our resul is con- 5

7 sisen wih he Tirole resul. Tha is, when he financial marke is perfec in he poin ha pledgeabiliy is equal o one, bubbles canno arise even in our seing. We show ha bubbles can arise even in an infiniely lived agens model if he financial marke is imperfec. Of course, he possibiliy of bubbles in infinie horizon economies wih borrowing consrains has been recongnized in seminal papers on deerminisic fia money (deerminisic bubbles) (Bewley, 1980; Townsend, 1980; and Scheinkman and Weiss, 1986). These seminal papers proved he exisence of a moneary equilibrium in an endowmen economy where no borrowing and lending are allowed. 5 Given hese sudies, imporan sudies by Kocherlakoa (1992) and Sanos and Woodford (1997) more expliciy examined he (necessary) condiions for he exisence of deerminisic bubbles. Addiionally, he recen imporan paper by Hellwig and Lorenzoni (2009) proved ha he resuling se of equilibrium allocaions wih self-enforcing privae deb is equivalen o he allocaions susained wih raional bubbles. All of hese sudies are, however, based on an endowmen economy. Our paper is in line wih research examining bubbles in an infiniely lived agens model. Our paper s conribuion is ha we develop a full-blown macroeconomic model wih heerogeneous invesmens and financial fricions, and provide a full characerizaion on he relaionship beween he exisence of bubbles and financial fricions in a producion economy. There are many papers examining he relaionship beween bubbles and invesmen. However, in he lieraure, he crowd-ou and crowd-in effecs are examined separaely. The convenional wisdom (Samuelson, 1958; Tirole, 1985) suggess ha bubbles crowd invesmen ou and lower oupu. According o he radiional view, he financial marke is perfec and all savings in he economy flow o invesmen. In his siuaion, bubbles crowd savings away from invesmen once hey appear in he economy. Sain-Paul (1992), Grossman and Yanagawa (1993), and King and Ferguson (1993) exend he Samuelson-Tirole model o economies wih endogenous growh, and show ha bubbles reduce invesmen and lower long run economic growh. 67 Recenly, however, some sudies such as Woodford (1990), Caballero and 5 As Kocherlakoa (1992) poins ou, alhough Scheinkman and Weiss (1986) implicily provide examples of bubbles in an infiniely lived agens model, hey do no explicily give he necessary condiions for he exisence of bubbles. Kocherlakoa provided he condiions. 6 This crowd-ou effec of bubbles has been criicized because i seems inconsisen wih he hisorical evidence ha invesmen and economic growh raes end o surge when bubbles arise, and hen sagnae when hey burs. 7 Olivier (2000) shows ha he conclusions reached by Sain-Paul (1992), Grossman and Yana- 6

8 Krishnamurhy (2006), Kiyoaki and Moore (2008), Kocherlakoa (2009) developed a model wih financial fricions, and showed ha bubbles crowd invesmen in and increase oupu. 8 These sudies demonsrae ha financial marke imperfecions preven he ransfer of enough resources o hose wih invesmens from hose wihou invesmens, resuling in underinvesmen. Bubbles help o ransfer resources beween hem. One novel poin of our sudy is ha we have combined hese wo effecs and shown he degree of financial imperfecion, i.e., he degree of pledgeabiliy, is crucial for undersanding which of hese effecs is dominan. Marin and Venura (2012) also invesigaed wheher bubbles are expansionary. There are some significan differences. Firs, Marin and Venura (2012) assume ha no agen can borrow or lend hrough financial markes because none of he reurns from invesmen can be pledgeable. Tha is, hey consider a siuaion where financial markes are compleely shu down. 9 On he oher hand, in our model, enrepreneurs are allowed o borrow as long as hey offer pledgeable asses (collaeral) o secure debs. Our main focus is o invesigae he relaionship beween he degree of pledgeabiliy and bubbles. We show ha boh he emergence and he effecs of bubbles are significanly dependen on he degree of pledgeabiliy, ha is, he degree of financial imperfecion. Second, Marin and Venura (2012) use a wo-period overlapping generaions model assuming ha young agens wih invesmen opporuniies canno borrow a all because financial markes are compleely shu down, bu hey can creae new bubble asses in every period. This assumpion of a new bubble creaion in every period direcly produces wealh effecs for he young and is crucial for crowd-in effecs of bubbles. Tha is, here are no crowd-in effecs and only Tirole s (1985) crowd-ou effecs wihou his assumpion. They invesigaed he condiions of new bubble creaions for he exisence of bubbles. On he oher hand, our model absracs from such new bubble creaion, and insead assumes ha agens live infiniely, and heir ype changes sochasically in each period. Enrepreneurs buy bubbles for speculagawa (1993), and King and Ferguson (1993) crucially depend on he ype of asse being speculaed on. Bubbles in equiy markes can be growh-enhancing while bubbles in unproducive asses are growh-impairing. 8 Hirano and Yanagawa (2010), Marin and Venura (2011), Miao and Wang (2011), Wang and Wen (2012), and Aoki and Nikolov (2013) also show he crowd-in effec of bubbles. 9 In Woodford (1990), no reurns from invesmen can be pledgeable. In Kocherlakoa (2009), agens can borrow agains bubbles in land prices. However, wihou such bubbles, here is no borrowing or lending. 7

9 ive purposes when hey have low produciviy, and sell hem when hey are high produciviy. Since bubbles increase he rae of reurn on savings, his speculaive aciviy endogenously improves borrowers ne worh and generaes crowd-in effecs. Third, financial fricions are crucial for he exisence of bubbles in our model wih infiniely-lived agens, while in OLG models, as Tirole (1985) shows, bubbles can arise even in a perfec finanical marke if an economy is dynamically inefficien. Addiionally, our paper uses an infiniely lived agens model, while Farhi and Tirole (2012) and Marin and Venura (2012) are based on overlapping generaions models. As Farhi and Tirole (2012) poin ou, he poenial benefi of using an infiniely lived agens model would be ha i is in principle more suiable for realisic quaniaive exploraions which he recen macroeconomic lieraure emphasizes. Caballero and Krishnamurhy (2006) developed a heory of sochasic bubbles in emerging markes using an overlapping generaions model, hough wih exogenously given growh raes and inernaional ineres raes. They implicily assume a low pledgeabiliy level, and ha wihou bubbles, he domesic ineres rae was lower han he inernaional ineres rae. Hence, our argumen is a generalizaion of heir argumen. Kiyoaki and Moore (2008) is also releed o our sudy. In heir heory, since deerminisic fia money faciliaes exchange for is high liquidiy, people hold money despie is low rae of reurn, emphasizing he role of money as a medium of exchange. In our model, however, we emphasize he role of bubbles as a sore of value. Enrepreneurs buy and sell bubble asses for speculaive purposes because hey have a high reurn. Our paper is also relaed o he growh lieraure. As Levine (1997) and Beck e al. (2000) show empirically, i is widely acceped ha improving financial marke condiions enhances long-run economic growh. However, he effec on growh volailiy is no ye clear. In our sudy, sochasic bubbles end o occur when financial markes have an inermediae level of financial developmen. This suggess ha growh volailiy ends o be high in he middle range of financial developmen, which can offer an explanaion for empirical findings from Easerly e al. (2000) and Kunieda (2008) ha growh volailiy is high when financial developmen is an inermediaed level. In erms of welfare effecs of bubbles, our resul ha bubbles enhance he consumpion-smoohing effec shares similariy wih Bewley (1980) who examined deerminisic fia money as a means of self-insurance agains idiosyncraic income 8

10 risk. There are some significan differences. Firs, our model is based on a producion economy wih invesmen opporuniies and focuses on idiosyncraic shocks o produciviy of invesmen, while he Bewley s model is based on an endowmen economy and focuses on income shocks. Second, we consider an economy where borrowing and lending are allowed (i.e., we consider he whole range of pledgeabiliy of collaeral), while Bewley considered an economy where financial markes are compleely shu down. Third, in our model, enrepreneurs can employ oher means o save besides bubble asses, i.e., hrough lending or by invesing in heir own invesmen projecs, while in Bewley s model, fia money is he only means of saving. Fourh, we examine he welfare effecs of sochasic bubbles, while Bewley s model deals wih deerminisiic fia money. 2 The Model Consider a discree-ime economy wih one homogeneous good and a coninuum of enrepreneurs. A ypical enrepreneur has he following expeced discouned uiliy: [ ] E 0 β log c i, (1) =0 where i is he index for each enrepreneur, and c i is he enrepreneur s consumpion a dae. β (0, 1) is he subjecive discoun facor and E 0 is he expecaion operaor condiional on dae 0 informaion. A each dae, each enrepreneur mees high-produciviy invesmen projecs (hereinafer H-projecs) wih probabiliy p, and low-produciviy ones (L-projecs) wih probabiliy 1 p. 10 The invesmen echnologies are as follows: y i +1 = α i z i, (2) where z( i 0) is he invesmen level a dae, and y+1 i is he oupu a dae + 1. α i is he marginal produciviy of he invesmen a dae. α i = α H if he enrepreneur has H-projecs, and α i = α L if he/she has L-projecs. We assume 10 Gerler and Kiyoaki (2010), Kiyoaki and Moore (2008), and Kocherlakoa (2009) use a similar seing. In Woodford (1990), he enrepreneurs have invesmen opporuniies in alernaing periods. 9

11 α H > α L. 11 The probabiliy p is exogenous, and independen across enrepreneurs and over ime. A he beginning of each dae, enrepreneurs know wheher hey have H-projecs or L-projecs. We call enrepreneurs wih H-projecs (L-projecs) H-ypes ( L-ypes ). In his economy, we assume ha because of fricions in a financial marke, he enrepreneur can pledge a mos a fracion θ of he fuure reurn from invesmen o crediors (See Har and Moore (1994) and Tirole (2006) for he foundaions of his seing.). Thus, in order for deb conracs o be credible, deb repaymen canno exceed he pledgeable value. Tha is, he borrowing consrain becomes: r b i θαz i, i (3) where r and b i are he gross ineres rae, and he amoun of borrowing a dae, respecively. The parameer θ [0, 1], which is assumed o be exogenous, can be naurally aken o be he degree of imperfecion of he financial marke. In his paper, we consider an economy wih asse bubbles, called a bubble economy. We define bubble asses as hose producing no real reurn, ha is, he asse s fundamenal value is zero. Aggregae supply of bubble asses is assumed o be consan over ime X. Here, following Weil (1987), we consider sochasic bubbles, in he sense ha hey may collapse. In each period, bubble prices become zero (i.e., bubbles burs) a a probabiliy of 1 π condiional on survival in he previous period. A lower π means riskier bubbles because hey have a higher probabiliy of collapsing. In line wih he lieraure, once bubbles collapse, hey do no arise again (heir reappearance is no expeced ex-ane.). This implies ha bubbles persis wih a probabiliy π(< 1) and ha heir prices are posiive unil hey rever o zero. Le P x be he per uni price of bubble asses a dae. P x = P > 0 if bubbles survive a dae wih probabiliy π, and P x = 0 if hey collapse a dae wih probabiliy 1 π. As we will show, P is endogenously deermined in equilibrium. Le x i be he level of bubble asses purchased by ype i enrepreneur a dae. Each enrepreneur has he following hree consrains: flow of funds consrain, he borrowing consrain, 11 We can also consider a model where capial goods are produced hrough he invesmen echnology. For example, le k+1 i = αz i i be he invesmen echnology, where k is capial goods. Capial fully depreciaes in one period. Consumpion goods are produced by he following aggregae producion funcion: Y = K σ N 1 σ k 1 σ, where K and N are he aggregae capial and labor inpu, and k is he economy s per-labor capial, capuring he exernaliy o generae endogenous growh. In his ype of model, we can obain he same resuls as his sudy. 10

12 (3), and he shor-sale consrain: c i + z i + P x x i = y i r 1b i 1 + b i + P x x i 1, (4) x i 0, (5) where represens he bubble economy. Boh sides of (4) include bubbles. P x x i 1 on he righ hand side is he sales of bubble asses, and P x x i on he lef hand side is he new purchase of hem. We define he ne worh of he enrepreneur in he bubble economy as e i y i r 1b i 1 + P x x i 1. We assume ha (3) is he borrowing consrain, ha is, bubbles do no conribue o pledgeable value or collaeral. Even so, bubbles can lead o increased invesmens by improving he borrowers ne worh, as we will explain in deail in secion We should add a few remarks abou he shor-sale consrain (5). As Kocherlakoa (1992) showed, he shor-sale consrain is imporan for he exisence of bubbles in deerminisic economies wih a finie number of infiniely lived agens. Wihou he consrain, bubbles always represen an arbirage opporuniy for an infiniely lived agen, who can gain by permanenly reducing holdings of he asse. However, i is well known ha in such economies, equilibria can only exis if agens are consrained no o engage in Ponzi schemes. Kocherlakoa (1992) demonsraed ha he shor-sale consrain is one of no-ponzi-game condiions and hence, i can suppor bubbles by eliminaing he agen s abiliy o permanenly reduce his holdings of he asse (see Kocherlakoa (1992) for deails.). In our model, wihou he shor sale consrain, enrepreneurs can obain funds infiniely by shor-selling bubble asses. As a resul, he ineres rae rises sufficienly in he credi marke and bubbles grow faser han he growh rae of he economy. Therefore, bubbles canno be susained. In oher words, wihou he shor-sale consrain, bubbles canno 12 We can relax he assumpion concerning he borrowing consrain. For example, we can consider a case where he enrepreneur can use boh a fracion θ of he reurn from invesmen and a fracion θ x of he expeced reurn from bubble asses as collaeral. In his case, he borrowing consrain can be wrien as: r b i θαz i i + θ x πp +1 x i. I is shown ha if θ x is sufficienly small, H-ypes do no purchase bubble asses in equilibrium. For deerminisic bubbles, i.e., π = 1, unless θ x = 1, H-ypes do no purchase bubble asses in equilibrium. Kocherlakoa (2009) analyzes his special case wih θ x = 1 under coningen deb conracs where even H-ypes buy bubble asses. In our model, we focus on he case where θ x is sufficienly small so ha H-ypes do no purchase bubble asses in equilibrium. We explore his poin in greaer deail in he Technical Appendix. 11

13 arise in equilibrium. 2.1 Opimal Behavior of Enrepreneurs We now characerize he equilibrium behavior of enrepreneurs in he bubble economy. We consider he equilibrium where α L r < α H. In equilibrium, he ineres rae mus be a leas as high as α L, since no agen lends o projecs if r < α L. For H-ypes a dae, he borrowing consrain (3) binds since r < α H and he invesmen in bubbles is no aracive, ha is, (5) also binds. We will verify his resul in he Technical Appendix. Since he uiliy funcion is log-linear, each enrepreneur consumes a fracion 1 β of he ne worh in every period, ha is, c i = (1 β)(y i r 1b i 1 + P x x i 1). 13 Then, by using (3), (4), and (5), he invesmen funcion of H-ypes a dae can be wrien as: z i = β(y i r 1b i 1 + P x 1 θαh r x i 1). (6) This is a popular invesmen funcion in financial consrain problems. 14 We see ha he invesmen equals he leverage, 1/ [ 1 (θα H /r ) ], imes savings, β(y i r 1b i 1 +P x x i 1). Leverage increases wih θ and is greaer han one in equilibrium, implying ha when θ is larger, H-ypes can finance more invesmen, z i. We also learn ha he presence of bubble asses increases enrepreneurs ne worh. In our model, enrepreneurs buy bubble asses for speculaive purposes when hey have L- projecs, and sell hose asses when hey have opporuniies o inves in H-projecs. For L-ypes a dae, since c i z i = (1 β)e i, he budge consrain (4) becomes + P x x i b i = βe i. Each L-ype allocaes savings, βe i, o hree asses, i.e., z i, x i, and ( b i ). Each L-ype chooses opimal amouns for b i, x i, and z i such ha he expeced marginal uiliy from invesing in hese hree asses is equalized. By solving he uiliy maximizaion problem explained in he Technical Appendix, we can derive he L-ype s 13 See, for example, chaper 1.7 of Sargen (1988). 14 See, for example, Bernanke and Gerler (1989), Bernanke e al. (1999), Holmsrom and Tirole (1998), Kiyoaki and Moore (1997), and Masuyama (2007, 2008). 12

14 demand funcion for bubble asses: P x i = π P +1 P r r P +1 P βe i, (7) From (7), we learn ha an enrepreneur s porfolio decision depends on he survival probabiliy of bubbles, π. When π is high, he bursing probabiliy is low, and he demand for bubble asses increases. The remaining fracion of savings is spli across z i z i + ( b i ) = P+1 (1 π) P r P +1 P βe i. and ( b i ): Since invesing in L-projecs (z i ) and secured lending o oher enrepreneurs ( b i are boh safe asses, z i 0 if r = α L, and z i ) = 0 if r > α L. Tha is, he following condiions mus be saisfied: (r α L )z i = 0, z i 0, and r α L 0. Moreover, when r = α L, invesing in L-projecs and secured lending o oher enrepreneurs are indifferen for L-ypes. 2.2 Equilibrium We denoe he aggregae consumpion of H-and L-ypes a dae as C H and C L, respecively. Similarly, le Z H, Z L, B H, and B L be he aggregae invesmen and he aggregae borrowing of each ype, respecively, and X be he aggregae invesmen in bubbles. Then, he marke clearing condiions for goods, credi, and bubbles are: C H + C L B H + Z H + Z L = Y, (8) + B L = 0, (9) X = X, (10) where Y is he aggregae oupu a dae. The compeiive equilibrium is defined as a se of prices {r, P x } =0 and quaniies { c i, b i, z i, y+1, i C H, C L, B H, B L, Z H, Z L, X, Y+1}, such ha (i) he =0 marke clearing condiions, (8), (9), and (10) are saisfied, and (ii) each enrepreneur chooses consumpion, borrowing, bubble asses, and invesmens o maximize he 13

15 expeced discouned uiliy (1) under he consrains (2), (3), (4), and (5). 2.3 Bubbleless Economy: To examine he effecs of bubbles, we firs examine an economy wihou bubbles as a benchmark case. Our model wihou bubbles is based on Kiyoaki (1998). Le he economy wihou represen he bubbleless economy, in which P x = 0 for any. The enrepreneur s ne worh in he bubbleless economy is defined as e i y i r 1 b i 1. Obviously, if θ is sufficienly high, all oal savings are used only for H-projecs and r = α H. Hence, we focus on he case where he ineres rae is sricly lower han α H and he borrowing consrain binds for H-ypes, α L r < α H. Since here are no bubbles, he invesmen funcion for H-ypes a dae can be wrien as: By aggregaing (11), we have: z i = β(yi r 1 b i 1). (11) 1 θαh r Z H = βeh 1 θαh r = βpy 1 θαh r, (12) where E H is he aggregae ne worh of H-ypes a dae. Since every enrepreneur has he same opporuniy o inves in H-projecs wih probabiliy p in each period, he aggregae ne worh of H-ypes a dae is a fracion p of he aggregae oupu a dae, i.e., E H = py. For L-ypes, if r = α L, lending and borrowing o inves are indifferen. Thus, how much hey inves in heir own projecs is indeerminae a an individual level. However, heir aggregae invesmen level is deermined by he goods marke clearing condiion, (8): Z H + Z L = βy. (13) βy is he oal savings. If r > α L, Z L mus be zero. Thus, he following condiions mus be saisfied: Z L (r α L ) = 0, Z L 0, r α L 0. The aggregae oupu is Y +1 = α H Z H + α L Z L. 14

16 In he bubbleless economy, Y equals he aggregae wealh of enrepreneurs, A, i.e., Y = A. The growh rae of Y = A becomes: g Y +1 Y = βα H β(α H α L )l, (14) where l Z L /βy is he raio of low-produciviy invesmens o oal invesmens. As long as he amoun of L-projecs, l, is zero, oal savings are allocaed only o H-projecs, and he economic growh rae becomes βα H, which is he same as he growh rae under θ = 1. If l > 0, however, he difference in produciviy beween H-projecs and L-projecs, α H α L, decreases he growh rae and g βα H β(α H α L )l. becomes Nex, we examine he equilibrium level of l and r. The key poin is he size of Z H relaive o oal savings βy. Since Z H is an increasing funcion of θ, Z H > βy a r = α L if θ is sufficienly high. Tha is, if he possible borrowing level of H- projecs is sufficienly high, r becomes greaer han α L in equilibrium due o he ighness of he credi marke. Thus, L-ypes have no incenives o inves in heir L- projecs, and l becomes zero in equilibrium. g becomes βα H and r should saisfy Z H = βy. On he oher hand, if θ is low and Z H < βy a r = α L, hen r equals α L and l becomes 1 p/(1 θαh α L ) > 0 in equilibrium. In summary, we can derive he following Proposiion. Proposiion 1 When bubbles do no exis, he equilibrium ineres rae, r, and he equilibrium growh rae, g, are he following increasing funcions of θ: r = r(θ) = α L, if 0 θ < (1 p) αl α H, θα H αl, if (1 p) 1 p α θ < 1 p, H α H, if 1 p θ 1. where L(θ) = Max[1 g = g(θ) = βα H β(α H α L )L(θ), (15) p 1 θαh α L, 0]. In he bubbleless economy, once he iniial oupu, Y 0, is given, hen he economy achieves he balanced growh pah immediaely, i.e., here are no ransiionary 15

17 dynamics. Figure 1 depics Proposiion 1. We ake θ on he horizonal axis, and g and r on he verical axis. As we will show laer, he necessary condiion for he exisence of sochasic bubbles is g > r under he bubbleless economy. Hence, he relaionship beween g and r is imporan for our resuls. Figure 1 shows ha boh he relaion beween g and θ and he relaion beween r and θ are non-linear. Hence, i is shown ha under some parameer condiions, only in he middle range of θ is g greaer han r. The inuiive reason for his resul is as follows. The growh rae generaed by L-projecs, βα L, is lower han he rae of reurn of L-projecs α L. When θ is sufficienly low, H-ypes canno gaher sufficien funds and mos are invesed in L-projecs. Consequenly, he growh rae becomes sufficienly low and close o (bu higher han) βα L, and he growh rae is lower han he ineres rae, α L, i.e., g(θ = 0) < r(θ = 0). In he middle range of θ, he ineres rae is sill α L since H-projecs are no enough o absorb all oal savings, bu he growh rae can be higher han α L since mos of he savings are invesed in H-projecs, leading o high economic growh, i.e., g(θ) > r(θ) for he middle range of θ. If θ becomes sufficienly high, however, all oal savings are invesed in H-projecs and he growh rae becomes βα H, bu he ineres rae becomes high and equal o α H if θ is close o 1, i.e., g(θ) < r(θ) for sufficienly high θ. Hence, only in he middle range of θ, g(θ) > r(θ). From his inuiive explanaion, we can undersand ha heerogeneous invesmen opporuniies are crucial for his resul. In he middle range of θ, mos resources are allocaed o H-projecs, which leads o high economic growh bu he ineres rae remains low a α L. In he Appendix, we provide more discussion abou he heoreical characerisics wherein g ends o be greaer han r only when θ is in he middle range in he bubbleless economy. Moreover, we can see ha o derive his resul, wo ypes of echnologies are no crucial and we can exend his argumen o a more general environmen wih a coninuum of produciviy. Le us suppose, for example, ha here are coninuously disribued invesmen opporuniies wih differen produciviies from α o α and he populaion share of enrepreneurs who have lower echnology is sufficienly high. In his case, when θ is almos 0, mos resources are allocaed o he lowes echnology and he growh rae becomes close o βα, which is lower han α, i.e., g(θ) < r(θ) around θ = 0. When θ goes up o he middle range, he ineres rae becomes higher han α because here are a coninuum of echnologies. However, he ineres 16

18 rae is deermined by he rae of reurn of he marginal ype of echnology. On he oher hand, resources can be allocaed o he echnologies wih higher produciviy han he marginal echnology. Hence, he growh rae can be higher han he rae of reurn of he marginal ype, ha is, g(θ) > r(θ) in he middle range of θ. On he oher hand, if θ becomes close o 1, almos all resources are allocaed o he projec wih α, and he ineres rae becomes close o α, which is higher han he economy s growh rae βα under θ = 1, i.e., g(θ) < r(θ) for sufficienly high θ. In boh he Appendix and he Technical Appendix, we explain his coninuum case more rigorously, and show ha under some condiions, g is greaer han r only in he middle range of θ even in he coninuum case. 2.4 Economy wih Bubbles We are now in a posiion o derive he dynamics of he bubble economy. Since we assume ha raional bubbles are sochasic, ha is, bubbles persis wih probabiliy π < 1, we focus on he dynamics unil bubbles collapse, i.e., P x = P > 0. (8) can be rewrien as Z H + Z L + P X = βa, (16) or Z H + Z L = βy (1 β)p X, where A Y +P X is he enrepreneur s aggregae wealh in he bubble economy a dae. Compared o (13), we can see ha he resources allocaed o he real invesmens Z H + Z L becomes from βy o βy (1 β)p X by he exisence of bubbles, P X > 0. This reducion in resources is he crowd-ou effec of bubbles, which is similar o he effec in he radiional lieraure such as Tirole (1985). Since a par of he oal savings is invesed in he bubble asses, he resources allocaed o real invesmens should be crowded ou. On he oher hand, bubbles have anoher effec because he invesmen level of H-projecs is deermined as: Z H = βpa 1 θαh r = βpy 1 θαh r + βpp X 1 θαh r, (17) 17

19 where pa is he aggregae wealh of H-ypes a dae. (More deails abou he aggregaion of each variable will be explained in he Technical Appendix). This second erm is he crowd-in effec of bubbles on invesmen. Inuiively, since he possible borrowing level is an increasing funcion of A, H-ypes can gaher more funds by he exisence of bubbles and hus increase heir invesmens. Moreover, he invesmens expand more han he direc increase in he ne worh because of he leverage effec. Bubbles endogenously generae balance shee effecs. In oher words, bubbles work o reallocae he resource oward producive invesmens. I is worh noing why A can be higher han Y and he posiive balance shee effecs work hough we exclude he possibiliy of bubble creaions in every period. As we will describe in more deail below, he equilibrium rae of he reurn on bubble asses becomes higher han he rae of reurn on low-produciviy invesmens, α L. Hence, he exisence of bubbles can improve ne worh, A, by increasing he rae of reurn on savings a 1. This is why he crowd-in effec works even wihou bubble creaions in every period, and he balance shee effecs are generaed endogenously. 15 In summary, alhough he bubbles crowd ou he resource allocaed o real invesmens, hey reallocae resources oward high-produciviy invesmens hrough he crowd-in effec. Moreover, when θ is low, a high share of resources are allocaed o low-produciviy invesmens if here are no bubbles. Hence, he crowd-in effec may dominae he crowd-ou effec when θ is low. We will rigorously prove his inuiion in he laer secion. Nex, we examine he equilibrium ineres rae. When P X > Max βa βpa 1 θαh α L, 0 ϕ > L(θ), only H-ypes inves and he equilibrium ineres rae r (> α L ) is deermined o saisfy ϕ = 1 p 1 θαh r r = θαh (1 ϕ ) 1 p ϕ, where ϕ P X/βA is he size of he bubbles (he share of he value of he bubble 15 We provide an analysis abou he effecs of bubble creaions wihin our framework in he Technical Appendix. 18

20 asses). I follows hen ha r is an increasing funcion of ϕ because of he ighness of he credi marke. On he oher hand, if ϕ L(θ), he ineres rae becomes r = α L and boh L-ypes and H-ypes inves in equilibrium. Thus, in he bubble economy, he equilibrium ineres rae is: [ ] r = Max α L, θαh (1 ϕ ). (18) 1 p ϕ This means ha as long as he size of he bubbles is small, he ineres rae says low a α L, bu when he bubbles become large enough, hen he ineres rae sars o rise. In his paper, we will examine he relaionship beween he economic growh rae and asse bubbles wih hree ypes of examinaion: (i) he relaionship beween he economic growh rae and ϕ in he bubble economy, (ii) he relaionship beween he economic growh rae and θ in he bubble economy, and (iii) a comparison beween he economic growh rae in he bubble economy and ha in he bubbleless economy for each θ. We will firs examine (i). Togeher wih (17), we have he evoluion of aggregae oupu: Y +1 = α H βpa 1 θαh α L + α L α H βpa 1 θαh r ( βy (1 β)p X βpa 1 θαh α L ) if ϕ L(θ), = α H (βy (1 β)p X) if ϕ L(θ). When he bubbles are small, boh L-ypes and H-ypes inves in equilibrium. The firs and second erms in he firs line represen aggregae oupu a dae + 1 produced by H-and L-ypes, respecively. When he bubbles are large, hen only H-ypes inves. Y +1 Y By rearranging (19), we can derive he economic growh rae: = where P X Y βα H β(α H α L )L(θ) + ( (α H α L )β(1 L(θ)) (1 β)α L) P X Y = βϕ 1 βϕ and βϕ 1 βϕ βα H (1 β)α H P X Y (19) is an increasing funcion of ϕ. The dynamic sysem if ϕ L(θ), if ϕ L(θ), (20) 19

21 of his economy is mainly characerized by (20), alhough we have no ye derived p he equilibrium ϕ. By he exisence of bubbles P X, he amoun of βp 1 θαh X = α L β(1 L(θ))P X shifs from L-projecs o H-projecs by he crowd-in effec and he ne conribuion o Y +1 of his effec is (α H α L )β(1 L(θ))P X. Conversely, P X prevens (1 β)p X resources from allocaion o real invesmens by he crowd-ou effec of bubbles, and he negaive impac on Y+1 is (1 β)α L P X. Hence, (α H α L )β(1 L(θ)) (1 β)α L P X shows he crowd-in and crowd-ou effecs of bubbles, Y and we will derive in a laer secion ha ( (α H α L )β(1 L(θ)) (1 β)α L) is posiive as long as bubbles saisfy he exisence condiion. In oher words, he crowd-in effec dominaes he crowd-ou effec, and he growh rae Y +1 in he Y bubble economy is an increasing funcion of he size of he bubbles ϕ as long as ϕ L(θ). On he oher hand, if ϕ L(θ), only H-ypes are producing, and he growh rae Y +1 in he bubble economy is a decreasing funcion of he size of he Y bubbles ϕ. In oher words, he relaionship beween he economic growh rae and bubble size is invered U-shaped and L(θ) = Max[1 bubbles ha maximizes he economic growh rae. p, 0] is he size of he 1 θαh α L 3 Dynamics of Raional Bubbles Nex, we examine he dynamics of raional bubbles and derive he equilibrium ϕ. From he definiion of ϕ P X/βA, ϕ evolves over ime as ϕ +1 = P +1 P ϕ A. (21) +1 A The evoluion of he size of he bubbles depends on he relaionship beween he growh rae of aggregae wealh and ha of he bubbles. When we aggregae (7), and solve for P +1 /P, we obain he required rae of reurn on bubble asses: P +1 P = r (1 p ϕ ) π(1 p) ϕ > r α L, if π < 1. (22) (1 p ϕ )/[π(1 p) ϕ ] capures he risk premium on bubble asses, which is greaer han one as long as π < 1; he required rae of reurn is sricly greaer han 20

22 he ineres rae. From his and he relaionship r r, we learn ha bubbles increase he rae of reurn on savings compared o he bubbleless economy as long as bubbles persis. This high rae of reurn on bubble asses increases enrepreneurs ne worh. 16 Using (19) and A +1 = Y +1 + P +1 X = Y +1 + (P +1 /P ) βϕ A, he growh rae of he aggregae wealh in he bubble economy can be wrien as: A +1 A = β{α H (1 L(θ)) + α L (L(θ) ϕ ) + P +1 P ϕ } if ϕ L(θ), β{α H (1 ϕ ) + P +1 P ϕ } if ϕ L(θ). From (18), (22), and he definiion of enrepreneurs aggregae wealh, (21) can be rewrien as: (1 p ϕ ) (23) ϕ +1 = (1 + αh α L α L θα H p π(1 p) ϕ ) β + (1 π)(1 p) π(1 p) ϕ βϕ ϕ if ϕ L(θ), (24) θ 1 ϕ if ϕ L(θ). β π(1 p) (1 θ)ϕ Using (24), we examine he susainable dynamics of ϕ. For sochasic bubbles o be susainable, he following condiion mus be saisfied for any : 0 < ϕ < 1. If his condiion is violaed, he bubbles explode, i.e., bubbles do no exis. As examined in he previous sudies (Tirole 1985; Farhi and Tirole 2012), here is a coninuum of saring values for he share of bubbles in oal savings ha are consisen wih equilibrium. The dynamics of bubbles ake hree paerns. The firs is ha bubbles become oo large and explode o ϕ 1. The economy canno susain his dynamic pah, and hus, bubbles canno exis in his paern. The second paern is ha ϕ becomes smaller over ime and converges o zero as long as bubbles persis. This pah is referred o as asympoically bubbleless, where as 16 For deerminisic bubbles, i.e., π = 1, we have P +1 /P = r α L in equilibrium. Even in his case, bubbles affec he long-run economic growh rae on he balanced growh pah if and only if P +1 /P = r > α L. See our working paper version (he firs submied version o RESud, Hirano and Yanagawa July, CARF Working Paper) for deails. 21

23 long as bubbles persis, heir effecs decrease, evenually becoming small. The hird paern is ha ϕ converges o a posiive value for as long as bubbles survive. From (24), we can derive ha ϕ mus be consan over ime, unless ϕ is asympoically bubbleless. Following Weil (1987), we refer o his equilibrium wih consan ϕ as he sochasic seady-sae, where enrepreneurs wealh, he bubbles, and he oupu grow a he same consan rae as long as he bubbles persis, A +1/A = P +1 /P = Y +1/Y. 4 Exisence Condiion of Sochasic Bubbles In his secion, we examine he exisence condiion of sochasic bubbles. In oher words, we invesigae wheher a dynamic pah wih bubbles does no explode. Mahemaically, we will check wheher he dynamic sysem (24) has a non-negaive seady-sae, ϕ = ϕ. As we show below, he financial marke condiion, θ, is crucial o he exisence of bubbles. (Hereafer, proofs of all Proposiions are given in Appendix). Proposiion 2 Sochasic bubbles wih survival probabiliy π can exis if and only if θ saisfies he following condiion, [ ] α L [1 πβ(1 p)] pβπα H θ Max, 0 < θ < θ πβ(1 p). α H (1 πβ) Moreover, we can use he srucure of he bubbleless economy o characerize he exisence condiion. The exisence condiion for bubbles (boh sochasic and deerminisic bubbles) is ha he growh rae is no lower han he ineres rae under he bubbleless economy. This condiion is consisen wih he exisence condiion saed in Tirole s (1985) sudy, alhough he Tirole model is based on an exogenous growh model wih overlapping generaions, while ours is based on an endogenous growh model wih infiniely-lived agens. Proposiion 3 The necessary condiion for he exisence of a bubble is ha he equilibrium growh rae is no lower han he equilibrium ineres rae under he bubbleless economy. From Proposiion 2, we can see ha bubbles end o exis when he degree of financial imperfecion, θ, is in he middle range. 22 In oher words, improving

24 condiions in he financial markes migh increase he exisence of bubbles when he iniial condiion of θ is low. 17 This resul is in sharp conras o he resuls from previous sudies, such as Farhi and Tirole (2012), in which bubbles are more likely o emerge when he financial marke is more imperfec (i.e., when pledgeabiliy is more limied). 18 The inuiion for his resul is as follows. If θ is low, H-ypes canno borrow sufficienly, and he economic growh rae mus be low, even wih bubbles. On he oher hand, he ineres rae canno be lower han α L, since here is an opporuniy o inves in L-projecs, even if θ is low. Hence, under a very low level of θ, bubbles grow a a higher rae han he economy, and hence canno exis. Since we assume heerogeneous invesmen opporuniies, he ineres rae has he lower bound, and we hus obain a differen resul from ha of Farhi and Tirole (2012). Figure 1 is a ypical case represening he relaionship beween θ and bubble regions. 19 The Figure shows ha wih sufficienly high or low financial fricion, bubbles canno exis, suggesing ha in financially underdeveloped or well-developed economies, bubbles are no likely o arise. They are likely o emerge in counries in an inermediae sage of financial developmen. 20 As we explained in he Inroducion, based on he experiences in advanced economies like he U.S., θ in he real world may be away from θ = 1, i.e., perfec pledgeabiliy. 17 Researchers such as Kaminsky and Reinhar (1999) and Allen and Gale (1999) poin ou ha financial liberalizaion causes bubbles. Based on our model, an inerpreaion of his effec is as follows. For insance, before financial liberalizaion, he economy is in non-bubble regions. Afer liberalizaion, θ increases, and he borrowing consrain is relaxed, causing he economy o ener bubble regions. 18 Noe ha if αl [1 πβ(1 p)] pβπα H α H (1 πβ) < 0, hen sochasic bubbles can arise even a θ = Though he growh rae is sricly greaer han he ineres rae, bubbles canno arise in he economy unless agens expec o be able o pass bubbles on o oher agens. This expecaion is a sufficien condiion for bubbles o exis. Here, we assume ha he condiion is saisfied when bubbles appear. 20 Readers may wonder why phenomena ha look like bubbles occur repeaedly in he real world where he financial sysem is coninually developing over ime, hough our model suggess ha bubbles do no appear in high θ regions. We propose one inerpreaion from our model. In our sudy, we assume a common θ for boh high and low invesmens. However, we can use differen values of θ for hose projecs. In such a case, he imporan facor for he exisence of bubbles is θ H, which is placed only on high-profi invesmens. Taking his ino accoun, consider he siuaion in which exising projecs wih α L disappear. Then, new invesmen opporuniies appear in he economy ha are more profiable han he exising α H. In such a siuaion, he θ for hese new projecs is imporan for he exisence of a bubble. If he θ is low, he economy will again ener bubble regions, even if i was previously in non-bubble regions wih a high θ. In he real world, his process migh repea iself. 23

25 As we also explained in he inroducion, his heoreical resul is consisen wih empirical observaions, and implies ha growh volailiy ends o be high when θ is in he middle range because sochasic bubbles end o occur. Hence, his resul can be an explanaion abou he recen empirical resuls by Easerly e al. (2000) or Kunieda (2008), who show ha growh volailiy is high when financial developmen is a an inermediae level Discussion: Technological Progress and Asse Bubbles Based on he exisence condiion of bubbles and he impacs of asse bubbles on he long-run economic growh rae, we can clarify he relaionship beween echnological progress and asse bubbles. Scheinkman (2014) highlighed he imporance of his relaionship. 22. Given θ < πβ(1 p), when we solve he exisence condiion θ for α H /α L, we ge: α H [1 π(1 p)β] > αl (1 πβ)θ + pβπ. Tha is, bubbles are more likely o arise as inequaliy in produciviy, α H /α L, rises. This resul suggess ha we need o clarify he ypes of echnological progress o undersand he relaionship beween he exisence condiion and echnological progress. Proposiion 4 Effecs of Technological Innovaion on Asse Bubbles: (i) Suppose a echnological innovaion ha increases α H, ermed high-ech specific progress. This high-ech specific progress increases he exisence of bubbles. (ii) Suppose echnological innovaion ha increases boh α H and α L by simulaneously, ermed general progress. This echnological progress is neural in erms of he exisence of bubbles. From hese resuls, we can draw a leas wo poins. Firs, i is possible ha echnological progress, such as high-ech specific progress, increases he exisence of 21 Aghion e al. (1999) and Masuyama (2007, 2008) show ha macroeconomic volailiy is high when financial marke developmen is a an inermediae level, hough we have a raher differen source of high volailiy. In our sudy, volailiy occurs because bubbles appear. In hese oher sudies, volailiy comes from he ineres rae or he qualiy of invesmens. 22 We hank an anonymous referee who poined ou his implicaion. We also hank Jose Scheinkman for houghful commens on his poin. 24