Endogenous Growth: Innovation, Credit Constraints, and Stock Price Bubbles

Transcription

1 Endogenous Growh: Innovaion, Credi Consrains, and Sock Price Bubbles Sicheng He Iowa Sae Universiy November 8, 2018 Absrac We sudy he poenial for raional bubbles in he innovaion secor o affec long erm economic growh. We show ha sock marke prices of R&D firms could include a bubble componen when credi consrains are presen. Bubbles are self-susained in equilibrium by a "liquidiy" premium ha originaes when credi consrains are relaxed. Bubbles expand borrowing and producion capaciy of R&D firms, simulae innovaion and increase he growh rae. Bubbles are magnified by igher credi consrains and scarce invesmen opporuniies. In conras o Hirano and Yanagawa (Resud, 2017), in our model: (i) bubbles are incorporaed as par of he sock price raher han providing value o an oherwise unproducive asse; (ii) bubbles can arise a any level of financial developmen. Finally, we show ha bubbles can creae permanen reallocaion effecs benefiing he innovaion secor over oher secors. hesich@iasae.edu. I hank Juan Carlos Cordoba for his valuable guidance. I also hank Joydeep Bhaacharya, Rajesh Singh, Quinn Weninger, Gary Lyn, Oavio Baraloil and all paricipans in ISU Macro-Developmen-Environmen reading group for heir valuable advice. All errors are mine. 1

2 1 Inroducion Innovaion drives modern economic growh (Romer 1990, Grossman and Helpman 1991, Aghion and Howi 1992). A he same ime, innovaion and echnological progress ofen correlae wih bubbles. For example, in his classic book, Shiller (2015) finds ha here was rapid economic growh and widespread disseminaion of echnological innovaions in he 1920s which led o bubbles burs laer in he Grea Depression. Similarly, Sorescu e al. (2018) sudy 51 major innovaions inroduced beween 1825 and 2000, from seam engine rain o smarphone. They deec bubbles in approximaely 73% of he innovaion. A well-known insance where innovaion and bubbles coincided was he so called do-com boom. In he lae 1990s, beween 7,000 o 10,000 new Inerne companies were founded seeking o ake advanage of he new possibiliies open by he inerne. This was a period of rapid innovaion and expanded variey of inerne producs (Wang 2007). A he same ime, he Nasdaq Composie sock marke index rose 400%. A second feaure of innovaion, or research and developmen (R&D), aciviies is ha hey ofen face credi consrains (Brown, Marinsson and Peersen 2012). Sudies have found ha his is paricularly imporan for small and medium size firms (Beck and Demirguc-Kun 2006). Credi consrains are likely due o asymmeric informaion and lack of collaeral. Informaion asymeries in urn arise from he underlying characerisics of innovaion since insiders have beer informaion abou he real chances of succees. This paper develops a heory of economic growh driven by innovaion, innovaors facing collaeral consrains, households acing as venure capialiss, and sock prices of R&D firms deermining he exen of R&D aciviies. As we show, our model can deliver raional bubbles ha are susained in equilibrium by a "premium" ha arises when collaeral consrains are relaxed. Our model is able o explain why bubbles can exis as an essenial par of a growing economy. We use he model o show he poenial effecs of bubbles on innovaion and long erm growh. Our baseline follows endogenous growh models wih expanding varieies firs developed by Romer (1990). Romer s model is widely used o sudy 2

3 issues of innovaion and endogenous growh. I has a final goods secor wih a represenaive firm, a monopolisic compeiive inermediae goods secor and a R&D secor. The final goods secor and inermediae goods secor in his paper are sandard. Compeiive final goods producers use inermediae goods, each producer of an inermediae good is a monopolis who produces a differeniaed variey which righs of producion are purchased from he R&D secor. Our major difference wih sandard variey models of endogenous growh is in he R&D secor which is subjec o credi consrains as in Kiyoaki and Moore s (2005) and Miao and Wang (2018). We assume here are a coninuum of R&D firms, owned by households, which use boh capial and labor o creae new varieies ha are sold o inermediae producers. R&D firms have random invesmen opporuniies which allow hem o ransform oupu ino capial which is useful for R&D producion. Firms use heir revenue from selling heir paens and inraemporal deb from firms wihou invesmen opporuniies o fund heir invesmen. R&D firms face credi consrains which are relaed o heir equiy value (feaure 2). In he even of defaul, lenders ake over he firm bu some capial is loss in he process. Thus, debs canno be larger han he aking over value. We show ha he model exhibi muliple equilibria. In paricular, here are wo possible balanced growh pahs (BGP), a bubbleless one and a bubbly one. Bubbles exis in R&D firms sock price for one of he wo BGP (feaure 1). In paricular, here may be wo componens exising in sock prices. One is relaed o he fuure revenue of innovaion and invesmen aciviies which deermines he sock fundamenal value. The oher one is no relaed o fuure income direcly. Even a firm wihou any capial can sill have high sock price. This par is defined as bubble. Our baseline model no only reflecs wo feaures in innovaion in an endogenous growh model, bu also provides a mechanism abou how bubbles affec credi consrains and innovaion. The mos imporan resul of he model is bubbles simulae innovaion when here are credi consrains. There are wo effecs when here are bubbles. The direc one is posiive while he indirec one is negaive. The direc effec is he exisence of bub- 3

4 bles increases he value of collaeral direcly so firms can borrow more from lenders when hey have invesmen opporuniies hus he R&D indusry has more invesmen and produce more blueprins. This is similar wih crowd in effecs which have been sudied in oher lieraure (Hirano and Yanagawa 2017, Miao and Wang 2018). A he same ime, since sock value is also relaed wih capial which we define as fundamenal value of sock, looseness of credi consrains reduce he demand of capial. Thus capial price decreases and he value of collaeral decreases. Wha is more, he decrease of capial price also decreases he capial revenue from producing and selling blueprins so i is a negaive effec on invesmen. These wo negaive effecs offse some of he crowd in effec. However, he direc effec is he dominan one hus bubbly BGP has higher growh rae han bubbleless BGP. I is worh o menion ha bubbles size are relaed wih credi consrains. The igher credi consrains are binding, he bigger he bubbles are. The effecs of bubbles will also be bigger when credi consrains are igher. Since here are effecs in differen direcions, bubbles can affec he inside value of he firm differenly from he ouside value. We find ha bubbles ypically increase he ouside value of he firm, he value o he lenders in he even of defaul. Thus, lenders would like o lend more o he borrowers. Bu bubbles may increase or decrease he inside value of he firm, he equiy value o he owners. Figure 1 explains he resuls. In our model, here is cos for lenders aking over he firm when borrowers defaul so lenders only ge he firm wih ξ of capial. k 1 and k 2 are borrowers capial in wo cases. V bb and V nb are bubbly and bubbless value of he firm given capial k. In boh cases, he sock price he lenders can ge is higher wih bubbles. However, i is undeermined wheher bubbles increase he value (k 2 case) or decrease he value (k 1 case). This is differen wih Miao and Wang (2018). In heir paper, bubbles invariably increase he sock price while in our model i mainly increase he value, o he lenders, in he even of defaul. The reason why raional bubbles exis and are susained in one of he equilibria is due o heir role relaxing borrowing consrains which provides he underlying liquidiy premium. Even hough he revenue from growh of he bubbles 4

5 Figure 1: The relaionship beween equiy value and capial is less han ineres rae, people sill accep he bubbles because hey ge addiional invesmen revenue since bubbles increase collaeral. We also sudy wha happens when bubbles burs. We find ha invesmen is decreasing afer he burs of bubbles due o he credi consrains so capial gradually converges o he bubbleless BGP. However, capial price acs much more rapidly. The mechanism is jus he same as wha we have described before. We hen exend our model o sudy he reallocaion effecs of bubbles. We show ha besides he effecs in he baseline model, sock price bubbles in R&D secors also arac more labor ino R&D secors which furher help he innovaion and economic growh. Our model suggess bubbles in innovaion secor is good for he growh so governmen shall no make hem burs wihou careful condiions. The exisence of bubbles means innovaion secor may face igh financial consrains. Thus, he righ hing governmen shall do is o reduce he financial fricions and help innovaion firm o ge enough funds when here are bubbles in innovaion secor. 5

6 In he remaining par of his secion, we review relaed lieraure and discuss our conribuion. In secion 2, we inroduce he baseline model. Secion 3 is abou analysis of equilibria. In secion 4, we derive BGPs for boh bubbly economy and bubbleless economy and compare he difference. We hen give mechanism how bubbles work. In secion 5, we sudy he dynamics of bubbles. We sudy he dynamics around he BGP and wha happens when bubbles burs. Exensions are in secion 6 where we have sochasic bubbles and reallocaion effecs. Secion 7 is conclusion. 1.1 Relaed lieraure This paper is mosly relaed wih lieraure seeking o undersand he connecion beween growh and bubbles. Tirole (1982) finds ha bubbles do no exis in sandard infinie period complee marke models because he exisence of bubbles lead o he violaion of ranseversaliy condiions. However, economiss find ha bubbles exis in some incomplee marke models. Overlapping Generaions (OLG) models arac a lo of aenion among such models. Samuelson (1958) is an early sudy using OLG model wih money. In his sudy, money is pure bubble helping o solve lacking of deb marke. Tirole s work (1985) is a fundamenal sudy for bubbles in OLG models and has inspired a large lieraure. Among hem are some papers using OLG models o sudy growh wih bubbles (e.g., Caballero, Farhi and Hammour 2006, Marin and Venura, 2012). They find bubbles can crowd savings away from invesmen bu hey also find bubbles may also provide addiional asse and encourage invesmen. However, These OLG papers have some disadvanages. The marke incompleeness relies on he lack of marke beween generaions which are no he reasons lead o he exisence of bubbles. They are also no suiable o do realisic quaniaive exploraions as Hirano and Yanagawa (2017) poin ou. Besides hese sandard OLG models. Olivier (2000) uses a coninuousime OLG model. Olivier s sudy is he one which is close o our paper. He finds ha bubbles in R&D secor can benefi he growh while bubbles in oher ype of asses may harm he economic growh. However, Olivier s 6

7 R&D secor is very simple and does no reflec he feaures of innovaion. Also, bubbles are no generaed by any properies of R&D secor. Bubbles in R&D secors and oher secors are generaed by demographic reasons and do no have any difference. In recen years, here are some papers using infiniely lived agens model o sudy bubbles. Hirano and Yanagawa (2017) build a model wih financial fricions and heerogeneous invesmens. However, here is no innovaion secor so he echnology is exogenous and we canno know he relaionship beween bubbles and innovaion. Secondly, hey use a useless asse as bubbles. Alhough his kind of fia bubbles have long radiion in lieraure, i does no reflec wha happens in innovaion. For example, do-com bubbles happen in sock price and has no relaionship o any useless asse. This useless asse also leads o srong crowd ou effecs as in Olivier (2000). Thirdly, heir model can only be used o sudy bubbles in counries wih inermediae level of financial fricions. Bubbles do no exis in financially underdeveloped or well-developed counries. This is clearly no rue in innovaion because we have already seen some bubbles in he R&D secor of Unied Saes which is one of he mos financially developed counries in he world. Miao and Wang (2014) build a model using sock value as collaeral and have wo secors. One secor has exernaliy while he oher one does no. In heir model, bubbles can relax collaeral consrains. However, i does no have innovaion secor so he echnology of he model is exogenous. Our paper, however, solve all he problems we discuss before. Our model follows he work by Miao and Wang (2018). Alhough i is no an endogenous growh model, Miao and Wang provide a novel way o hink abou bubbles. They use an infiniely lived agens model wih credi consrains and heerogeneous invesmen opporuniies o show ha bubbles can exis in sandard infiniely lived agens model wih incomplee marke and no violae ranseversaliy condiions by reducing liquidiy mismach when here are invesmen opporuniies. Our paper has he similar seing in R&D secor and find ha bubbles no only affec seady sae, bu also increases BGP of an endogenous growh model by simulaing innovaion. To he exen of our knowledge, here is no paper sudying bubbles in an endogenous model wih 7

8 a well defined R&D secor before. Since bubbles in our paper exis in sock price raher han on an useless asse, our paper follows Oliver (2000) and bubbles will always help innovaions raher han hur he economic growh. Bu our paper shows ha even bubbles may no increase sock price direcly, i sill simulae innovaions. The mechanism in our paper are relaed more wih innovaion raher han demographics which is a exogenous variable in innovaion. Also, our paper find ha alhough Miao and Wang s mechanism does exis, bubbles have more complex effecs han Miao and Wang s mechanism. Besides heir posiive effecs, here are also some negaive effecs o offse i. Failing o consider negaive effecs resuls in overesimae he benefi of bubbles. Thus, bubbles increase sock value which can be acquired by lenders raher han borrowers. In our exension secion, we also find social resource has been reallocaed o R&D secor when here are bubbles. This is a new effec which help he growh of he economy. Our model is raher robus wih financial developmen. For underdeveloped counries, bubbles are always able o exis. Even for he mos developed counries, bubbles are sill able o exis if he invesmen opporuniies do no come oo ofen. Thus, our model can be used in sudy bubbles in differen counries. Besides he papers in growh and bubbles. Our paper is relaed wih papers in differen fields. Firs of all, his paper is relaed wih papers sudying endogenous growh wih innovaion. Relaionship beween economic growh and innovaion have been sudied by economiss for a long ime boh in empirical way and heoreical way. Economiss find economic daa provides evidence ha innovaion and growh are posiively relaed (Griliches and Lichenberg 1984, Zachariadis 2003). There are also a large number of papers focus on sudying growh and innovaion in heoreical way. Romer (1990) builds a model wih a R&D secor where echnological innovaion is in he form of expanding varieies creaed by labor in R&D secors and exising knowledge. I is he R&D keeps he economy growh in long erm. Grossman and Helpman (1991) and Aghion and Howi (1992) boh build model o sudy how R&D which improve producs qualiy have effecs on growh. Boh empirical and heoreical sudies find ha R&D has srong effecs on economic growh. Our paper exend sudies in his field by inroduce 8

9 credi consrains and bubbles ino R&D secor. Our paper is also relaed o papers sudying credi consrains. The seminal work of Kiyoaki and Moore (1997) inroduces collaeral consrains ino general equilibrium and finds collaeral consrains have significan effec on he whole economy. Numerous sudies follow Kiyoaki and Moore o sudy he effec of collaeral consrains (e.g., Cordoba and Ripoll 2004, Iacoviello 2005 and Liu, Wang and Zha 2013). However, mos of hese sudies focus on business cycles. There are few heoreical papers sudy innovaion wih credi consrains. Amable, Chaelain and Ralf (2010) is one rying o sudy credi consrains wih R&D. They find ha paens creaed from R&D process can be used as collaeral o reduce he negaive effec of collaeral consrains. Our sudy provides a noval way o hink of credi consrains and R&D. 2 The baseline model Since our model is raher complicaed, we use figure 2 o help us inroduce our model before we describe i in deail. Arrows in figure 2 indicae flow of resource and goods. The represenaive household hold shares of firms in R&D secors and provides labor o R&D firms. The household also ge income by receiving dividends and wages. R&D firms use capial and labor o produce new paens and sell paens o inermediae goods. Afer ha, a firm in R&D secor has invesmen opporuniy wih probabiliy π. Those who have invesmen opporuniies borrow from hose wihou invesmen opporuniy and inves bu hey are consrained by credi consrains. Final goods are ransformed ino new capial when firms inves and firms rade capial afer invesmen sage. Afer buying paens from R&D firms, inermediae goods producers produce inermediae goods by using final goods. They sell inermediae goods o final goods producer who use inermediae goods o produce final goods. Besides he flow of resource, figure 2 also poin ou R&D firm j canno borrow more han firm s discouned value. When here are bubbles, firm j s discouned value is greaer han wihou bubble. 9

10 Figure 2: Bird s eye view of he baseline model (Arrows are flow of resource and goods) 2.1 Households There is a represenaive household in our model who has a sandard uiliy funcion β ln C =0 where β is he discoun rae and C is he consumpion in period. Household provides all is labor inelasically every period and aggregae labor supply is normalized o 1. Household rades socks of firms in R&D secors every period and also receive dividends from socks i holds. Household uses wages and income from rading socks o buy consumpion and do no have any oher way o save. Thus, Household faces budge consrains C + ( V j ) D j ψ j +1dj = V j ψ j dj + W 10

11 where W is he wage rae, V j, D j are R&D firm j s cum-dividend equiy value and dividend and ψ j is household s holdings of firm j s shares. Transversaliy condiions are V j lim T βt T ψj T = 0 C T Thus, he represenaive household maximizes is uiliy funcion while budge consrains and ranseversaliy condiions are saisfied. We define he growh rae of consumpion g c +1 and g c +1 = C +1 C 1 ρ +1 = β C = β ( ) 1 + g c 1 +1 C +1 where ρ +1 is he sochasic discoun facor in asse pricing lieraure. 2.2 Final goods producer In our model, here is only one kind of final goods. Le he final goods are numeraire and all consumpions, invesmen and inpus are using final goods. For simpliciy we assume here is only one represenaive firm produce final goods and i is a price aker. The final goods producer uses inermediae goods o produce and he echnology is Y = A N n=1 (Xn ) σ dn, 0 < σ < 1 Here N is oal number of varieies in period and X n is he amoun of inermediae goods n he final goods producer uses. A denoes he echnology of final goods producer. We use P n o denoe he price of inermediae goods n. Profi maximizaion problem of he final goods producer is 11

12 max Y N n=1 P n X n dn subjec o he producion funcion. I is easy o solve profi maximizaion problem and we have he demand funcion for inermediae goods n P n = σa (X n ) σ Inermediae goods producers Inermediae good X n is produced in compeiive monopolisic markes. To produce an inermediae goods n, an inermediae goods producer has o pay a paen fee η n o he R&D firm who creaes blueprin n firs. Afer paying he paen fee, he inermediae goods producer can produce any amoun of inermediae goods a any periods. The echnology of inermediae goods producer is i can ransform one uni of final produc o one uni of X n. Thus his profi is (P n 1) X n Since we have already had inermediae goods n s demand funcion, we can find he price inermediae goods producer of goods n se P n = 1 σ and he amoun he producer produces X n = σ 2 1 σ A 1 1 σ (1) Then he profi of producing goods n every period is ( 1 σ σ ) σ 2 1 σ A 1 1 σ. Since we know ha i is compeiive monopolisic markes, he discouned oal profis from selling goods n mus be equal o he cos of buying paen 12

13 o produce goods n, which means ( ) 1 σ ρ (s, ) σ σ A 1 σ σ s= = ηn Here ρ (s, ) = s v=+1 ( ρv+1 ) if s, ρ (s, ) = 1 if s =. Since only variables in η n are ime variables, paens creaed in he same period have he same price. This resul gives us ( ) 1 σ ρ (s, ) σ σ A 1 σ = ηn = η σ (2) s= 2.4 R&D Secor There are a coninuum of firms j [0, 1] in R&D secor. In every period, here are hree sages. We firs briefly inroduce he hree sages and hen provide deails. A he firs sage, firms hire labor o creae new blueprins and sell hem o inermediae producers as paens. During he second sage, some firms have opporuniies o inves and ge new capial. They can use heir own fund or loans from oher firms o inves. A he hird sage, firms rade capial wih each oher. A he beginning of period, firm j in R&D secor has K j amoun of capial i accumulaed a he end of period 1. Thus capial a he firs sage is given. I hen hires L j amoun of labor. Technology for firm j uses boh capial and labor o creae new blueprins. T j is he amoun of new blueprins creaed by firm j in period. Curren echnology level (curren amoun of blueprins N ) also has effec on he innovaion process. producion funcion of R&D firm j is T j = Z ( ) K j α ( ) N L j 1 α The where Z is an exogenous parameer. This echnology of innovaion means ha echnology has spillover effecs. Every invenion benefis fuure invenion by increasing labor produciviy. This propery is common seing in endogenous growh models. Capial depreciaes a rae δ every period. Cap- 13

14 ial reurn of producing new blueprins a period is r j K j = max η Z ( ) K j α ( L j N L j 1 α ) W L j (3) I is worh o menion ha capial-labor raio for all firms in R&D secor are same. To see his, we jus solve firms profi maximizaion problem and have W N = (1 α) η Z ( K j N L j ) α (4). By using his resul and capial reurn formula above we find ha r j = r which means every firm has same capial reurn rae. Afer firm j sells is blueprins and ge he revenue comes he second sage. Every firm has a probabiliy of π o have invesmen opporuniy and hose firms have invesmen opporuniies can ransform final produc ino capial. The echnology is 1 uni of final produc a period can be ransformed ino 1 uni of capial. We assume he marke of capial is open afer he invesmen hus firm j has o use he profis i sells he blueprins and exernal source o inves. We assume he only source of exernal financing for j is inraemporal loans E j from oher firms. Those who borrow from oher firms have choice beween defaul or no defaul. There is no force o ensure borrowers from defauling so borrowers are required o provide enough collaeral o secure heir loans. Following Miao and Wang (2018), he value of firm is used as collaeral. If he owner of borrower chooses o defaul and escape wih he fund, lenders will ake over he firm o compensae heir loss. However, we assume he lender may be no familiar wih he borrower s firm. There may be a cos during he ake over process and he cos is 1 ξ of oal capial. Thus he credi consrains are E j ρ +1 V j +1(ξ (1 δ) K j ) 14

15 ( ) Here V K j is firm j s cum-dividend equiy value when here is capial K j. The credi consrains mean ha if borrowers defaul, he discouned value of he firm lef o lenders are no less han he loans so lenders do no have any loss. For borrower j, i is beer o pay back deb E j han defaul and lose he firm values ρ +1 V+1((1 j δ) K j ) so here is no defaul in his economy. Afer invesmen, all firms come o he hird sage a which hey can buy and sell capial o each oher and pay he dividends. Thus he profi of invesmen is q I j I j where q is he price of capial and I j are how many capial firm j plans o creae by invesmen. From he seing above, we can wrie R&D firm j s cum-dividend equiy value a period by using recursive form. V ( K j ) = (1 π) max K j +1,Bj +π max K j I+1,Ij,Bj [ ( )] D j + ρ +1 V +1 K j +1 [ D j I + ρ +1V +1 ( K j I+1 )] (5) Here D j and K j +1 are dividend and capial for nex period when here is no invesmen opporuniy while D j I and Kj I+1 are dividend and nex period capial when here is invesmen opporuniy. The cum-dividend equiy value now is equal o he expeced value of dividend plus discouned fuure cumdividend equiy value when firms make bes choice of deb, invesmen and fuure capial. Firms also face some consrains. There are budge consrains (6) and (7) D j + q K j +1 + E j = r j K j + q (1 δ) K j + E j (6) D j I + q K j I+1 + Ej + I j = r j K j + E j + q (1 δ) K j + q I j (7) (6) are budge consrains when here is no invesmen opporuniy while (7) are budge consrains where here is invesmen opporuniy. Invesmen is 15

16 consrained by available fund I j r j K j + E j (8) and deb canno violae credi consrains E j ρ +1 V j +1(ξ (1 δ) K j ) (9). Bellman equaion (5) alone wih (3), (6), (7), (8) and (9) consis of R&D firm j s dynamic programming problem. 2.5 Compeiive Equilibrium Afer we describe our model, we can define compeiive equilibrium. Le K = 1 0 Kj dj, I = 1 0 Ij dj, T = 1 T j 0 dj are aggregae capial, invesmen, new blueprins. Definiion 1 A compeiive equilibrium is defined as allocaions { } Y, K, C, I, N, E j, T, L j, I j, K j, T j, Y i, ψ j, X n and prices { } w, Pn, R, j q, η, r, V j such ha household maximize is uiliy and firms in all hree secors maximize heir profis and marke clearing condiions are saisfied which are sock marke is clearing ψ j clearing 1 0 Lj dj = 1, deb marke is clearing 1 = 1, labor marke is 0 Ej dj = 0, capial marke is 1 0 Xn dn+ clearing K +1 = (1 δ) K +I, goods marke are clearing C + N n=0 I = Y and he amoun of paen follows N +1 = N + T. 3 Analysis of Equilibria Similar wih oher endogenous growh model, many variables in our model increase o infiniy. resul of hese models. Balanced growh pah (BGP) is he mos imporan To find he BGP, we derend variables which are increasing wih ime. Le c = C N, k = K N, d = D N, = T N, b = B N, w = 16

17 N, 1 + g+1 N = N +1. Thus W N and capial reurn equaion can be wrien as c ρ +1 = β ( ) (10) c g c +1 r k = Zη (k ) α w (11) We firs consider he problem of R&D secion. This problem is no a conracion mapping and may have muliple soluions. Proposiion 2 Suppose q > 1, soluion of R&D firm j s problem is V ( K j ) = a K j + B (12) where a = r + q (1 δ) + π (q 1) ( r + ρ +1 a +1 ξ (1 δ) ) (13) BB = [1 + π (q 1)] ρ +1 B +1 (14) and q = ρ +1 a +1 (15). Proof. Assume soluion of R&D firm j s problem is (12). Subsiue (12), (6) and (7) ino (5) we have a K j + B = max r K j K j + q (1 δ) K j (16) +1,Kj I+1,Ij,Bj +ρ +1 B +1 + (1 π) [ ] q K j +1 + ρ +1 a +1 K j +1 +π [ (q 1) I j q K j I+1 + ρ ] +1a +1 K j I+1 and wo oher consrains (8) and (9) are combined o one consrain I j r K j + ρ +1 a +1 ξ (1 δ) K j + ρ +1 E +1 (17) 17

18 By aking firs order derivaive of K j +1 we have (15). Since q > 1, firm j invess as many as i can so (17) is binding. By subsiuing (17) ino (16) and compare he lef hand side and righ hand side we ge (13) and (14). (15) shows ha he price of capial is equal o he value i increases. This is relaed wih Tobin s Q heory. Tobin s Q heory saes ha if he replacemen cos of capial is less han he firm s value hen he firm will increase heir invesmen o have more capial while if he replacemen cos of capial is greaer han he firm s value hen he firm will no inves and decrease capial. Since (15) holds, firm j is indifferen beween buying and selling is exising capial. Hence K j I+1 and Kj I+1 are indeerminae. We know q 1 because he marginal cos of producing capial is 1. When q > 1, firms wih invesmen opporuniies inves as many as hey can so (17) is always binding. If q = 1, however, firms are indifferen in making more invesmen and credi consrains do no have o bind any more. This is he reason we resric our main analysis o q > 1. The wo equaions (13) and (14) play key roles in our model. I shows ha R&D firm j s cum-dividend equiy value is wrien as a K j + B. The firs erm a K j means ha capial affecs he equiy value while he second erm B does no relae wih any goods or producs. In lieraures abou bubbles, economiss define he firs erm as fundamenal value of a firm while he second erm is viewed as bubbles. To see why a K j is he fundamenal value of he firm, we rewrie (13) wih (15) and ge q = ρ +1 r +1 + ρ +1 q +1 (1 δ) + πρ +1 (q +1 1) (r +1 + q +1 ξ (1 δ)) (18) If we use ϕ +1 = π (q +1 1) (r +1 + q +1 ξ (1 δ)) as he expeced invesmen revenue from nex period by increasing one uni of capial, we rewrie i as q = ρ +1 ( r+1 + ϕ +1 ) + ρ+1 (1 δ) q +1 (19) 18

19 The soluion of (19) is q = i=+1 ρ (i, ) (1 δ) i 1 (r i + ϕ i ) + Υ ρ(, 0) (1 δ) Here Υ is a consan. r i is he revenue of one uni of capial a period i while ϕ i is expeced invesmen revenue for one uni of capial. By using ranseversaliy condiions, Υ = 0. Thus q = i=+1 ρ (i, ) (1 δ)i 1 (r i + ϕ i ) is oal fuure income if firm j buy one uni of capial in period. a K j = 1 ρ i= ρ(i, ) (1 δ) i 1 (r i + ϕ i ) K j reflecs he oal expeced revenue from a firm wih K j capial. I is jus fundamenal value of a firm. I is worh o menion ha since our paper uses discree model raher han coninuous ime model, a is more complicaed han Miao and Wang s (2018) model. In heir model, a = q because he reurn r +1 and depreciaion is omied in coninuous ime model. The second erm of equiy B, however, is no relaed wih any fundamenal fuure revenue direcly and are viewed as bubbles by economiss. Transiion of bubbles comes from (14). When here are invesmen opporuniies, bubbles can be used as collaeral o increase invesmen profi by π (q 1) B. Jus as he definiion given by Miao and Wang (2018), π (q 1) is liquidiy premium. Laer when we discuss he balanced growh pah, one can easily show ha bubbles grow like his do no violae ransversaliy condiions because he growh is smaller han one when discouned by he discoun facor. The reason why people bear he loss o accep such kinds of asse is i can reduce he liquidiy mismach when firms have invesmen opporuniies bu are resriced by credi consrains. This is consisen wih feaure 1 of R&D ha rapid echnological innovaions ofen correlae wih bubbles. When here are invesmen opporuniies, bubbles help firms face credi consrains. Thus, here are more paens creaed and echnological innovaions are faser. This effec is similar wih crowd in effec in mos lieraure abou bubbles. I is deserved o menion ha his is jus he direc 19

20 effec of bubbles. When we compare balanced growh rae beween bubbly equilibrium and bubbleless equilibrium we will find here are also indirec effecs of bubbles and hey may offse some of crowd in effec. Anoher observaion of (14) is bubbles eiher exis from he beginning or hey never appear. As we have discussed before, he dynamic programming problem is no a conracion mapping and may have muliple soluions. We have wo cases here, an equilibrium wih bubbles and an equilibrium wihou bubble. The exisence of bubbles is jus a consensus of he marke no relaing wih any fundamenal of a cerain firm. If all agree and believe oher will accep he exra values hen he bubbles exis. If hey do no accep or believe ohers will no accep he exra values here is no bubble. Since we are focusing on derended variables. We derend (14) ino b = [1 + π (q 1)] ρ +1 (1 + g +1 ) b +1 (20) Before we move on o sudy wo differen equilibria, we firs discuss a lile furher o he general case. Thus We have already known every invenor has he same capial labor raio. T = 1 0 T j dj = Z (K ) α N 1 α N +1 = N + Z (K ) α N 1 α 1 + g N +1 = 1 + Zk α so we have and g N +1 = (21) = Zk α (22) I = 1 0 I j dj = πr K + πρ +1 [a +1 (ξ (1 δ)) K + B +1 ] 20

21 Since K +1 = (1 δ) K + I, [ ] (1+g+1)k N +1 = (1 δ) k +πr k +πρ +1 a+1 (ξ (1 δ)) k + (1 + g+1)b N +1 (23) From he resul of firm j s R&D work, we can ge r = α (Zη ) 1 α ( w ) α 1 α 1 α (24) Goods marke clearing condiion implies c + X + πr k + πρ +1 [ a+1 (ξ (1 δ)) k + (1 + g N +1)b +1 ] = AX σ (25) Thus, equaions (1), (2), (10), (11), (13), (15), (20), (21), (22), (23), (24) and (25) alone wih ransversaliy condiions consis of a dynamic sysem which characerize he derended equilibria of our model. 4 Balanced Growh Pah In his secion, we derive and compare balanced growh pah (BGP) of wo cases. The firs one is he case when here is no bubble while he second one is he case wih bubbles. We have derended all variables in las secor so variables should be a seady sae alone BGP. We use derended variables wihou ime subscrip o denoe he seady sae of hese variables. Alone BGP, we know ha g N = g C. We will use g N as a subsiue when here is g C for convenience. We firs show ha in boh cases, given q and r, capial k and growh rae g N are deermined in same way. Proposiion 3 k is deermined implicily by equaion ( 1 σ ) 2 1 σ σ 1 σ A 1 σ rk = Z 1 β (1 + Zk α ) 1 kα (1 α) r α 21 α 1 α α 1 α [ ( 1 σ ) 2 1 σ σ 1 σ A 1 σ Z 1 β (1 + Zk α ) 1 ] 1 1 α (26)

22 and g N = Z (k) α (27) Proof. (27) is he direc resul of (21) and (22). We only need o ge (26) hen growh rae is deermined by q, a and r. From (10), 1 ρ = β (1 + g N ) = β (1 + Zkα ) 1 This resul alone wih (2) give From (24) we have η = ( 1 σ ) 2 1 σ σ 1 σ A 1 σ 1 ρ = ( 1 σ ) 2 1 σ σ 1 σ A 1 σ 1 β (1 + Zk α ) 1 w = (1 α) r α α 1 α α 1 α Subsiue hese resuls ino (11) we ge (26). [ ( 1 σ ) 2 1 σ σ 1 σ A 1 σ Z 1 β (1 + Zk α ) 1 ] 1 1 α 4.1 Bubbleless BGP In bubbleless equilibrium we know b = 0. From (20) we know ha if here is no bubble in one period, here is no bubble for all periods. Thus equaion (20) becomes an ideniy. A he same ime, (23) becomes (1 + g+1)k N +1 = (1 δ) k + πr k + πρ +1 [a +1 (ξ (1 δ)) k ] (28) Proposiion 4 When credi consrains are binding, capial price q l capial reurn r l, derended capial k l and growh rae g N l q(1 + g N ) β alone wih (26) and (27). is deermined by 1 + g N = (1 δ) + πr + πq (ξ (1 δ)) (29) = r + q (1 δ) + π (q 1) (r + qξ (1 δ)) (30) 22

23 Proof. We ge (29) by subsiuing (15) ino (28). By (15) and (13) we have (30). (29), (30) alone wih (26) and (27) we derive from proposiion 2, we ge a four variables equaions sysem which give us q l, r l, k l and gl N. Unforunaely, i is impossible o derive he analyical soluion of he variables. Thus, laer we canno compare he resuls beween bubbleless BGP and bubbly BGP direcly. However, we can use numerical mehod o check he resuls. 4.2 Bubbly BGP We now sudy bubbly BGP. Here b 0 and we canno omi he (20). Jus like wha we have discussed in Bubbleless BGP, nex proposiion gives us he resul of capial price q capial reurn r, derended capial k and growh rae g N. We use q b, r b, k b and g N b as denoaion. Proposiion 5 When here are credi consrains, q b = 1 β + πβ πβ r bb, k bb and g N bb are deermined by q(1 + g N ) (1 β + πβ) (1 β + πβ) (1 δ) πβ 2 = r + πβ ( + 1 β β r + 1 β + πβ ξ (1 δ) πβ ) (31) alone wih (26) and (27). Proof. Consider he case alone BGP. (20) gives us q b. By (15) and (13) we have q(1 + g N ) β = r + q (1 δ) + π (q 1) (r + qξ (1 δ)). Plug q b ino i we have (31). (31), (26) and (27) are a hree variables hree equaions sysem. I is easy o check ha q b > 1 which means when collaeral consrains are binding, bubbles always exis and he exisence of bubbles never oally 23

24 eliminae he effecs of collaeral consrains. 4.3 Compare Bubbly BGP wih Bubbleless BGP According o proposiion 2, 3 and 4, we know how o ge values of variables on BGP. Unforunaely, i is impossible o ge mos resuls analyically for he sysem is oo complicaed. We use numerical mehod alone wih equaions we derive before o discuss how bubbles affec R&D and how credi consrains affec bubbles. The parameers are repored in able 1. parameer β δ π ξ α σ A Z value Table 1: Values of Parameers We can ge boh bubbly BGP and bubbleless BGP. Some imporan variables alone he BGP are repored below in able 2. Here e is derended deb. Compared wih BGP wihou bubble, bubbly BGP has higher derended variable b/v(k) g N q a r k e v(k) bubbly 57.59% 4.0% bubbleless 0 3.5% Table 2: Variables values alone bubbly and bubbleless BGPs capial level. Thus, he growh rae of bubbly BGP is greaer. A he same ime, capial price q, relaionship beween capial and equiy value a and capial reurn rae r decrease. The derended sock value of bubbly BGP, however, is less han he bubbleless one. To undersand hese phenomena, we firs review he derended capial ransiion (23) (1+g N +1)k +1 = (1 δ) k +πr k +πρ +1 [ a+1 (ξ (1 δ)) k + (1 + g N +1)b +1 ] From ransiion funcion, we know ha capial increases if revenue increases which means πr k increases or firms have more access o exernal funding [ which means ρ +1 a+1 (ξ (1 δ)) k + (1 + g+1)b ] N increases. +1 Bubbles increase discouned sock price if lenders ake over he firms hus provide more collaeral o help reduce liquidiy mismach. This direc effec 24

25 increases invesmen in R&D secor so i is he posiive effec on growh rae. This posiive effec can be called crowd in effec like oher lieraures (Hirano and Yanagawa 2017, Miao and Wang 2018) abou bubbles. If we only consider his direc effec, bubbles cerainly help R&D and growh. However, here is also some indirec effecs in general equilibrium which offse some of he posiive effecs. Firs of all, capial in our model no only be used o creae new paens, bu also be used o increase cum-dividend equiy value so ha when hey have invesmen opporuniies hey can borrow more. Since bubbles in sock price have he same effec, demand of capial decreases which decreases price of capial q. Tha s why we see in boh examples q and a drop significanly when here are bubbles. This negaive effec offses some posiive effecs especially when credi consrains are no binding very much. I is worh menion ha his direc effec does no ensure higher derended sock price of bubbly BGP. This is because indirec effecs reduce he sock price by reducing a a he same ime bubbles increase he sock price. Someimes he indirec effecs are no oo big so he sock price sill increases while someimes he indirec effecs are big enough so he sock price may decrease. However, collaeral consrains are relaed wih sock price if lenders ake over and his sock price always increases. This is he jus we show in figure 1 in inroducion. Tha is no he end of he sory. The effecs we discuss above only ensure alone bubbly BGP firms ge more loans. Since capial price q decreases, reurn of capial r also decreases which means ha even wih same amoun of capial firm ge less revenue hrough innovaion aciviies and has o decrease he invesmen. Though here are hese wo negaive effecs which offse some of he posiive effec, he posiive effec is always he dominan one so sock price bubbles always encourage invesmen in innovaion secor and increase growh rae. We can see his resul when we do robus check. I is worh o menion ha hese direc and indirec effecs are very similar wih Oliver s finding. 25

26 4.4 Robusness of our model In some lieraure abou bubbles, bubbles may only exis in some economies saisfy some cerain condiions. Hirano and Yanagawa (2017) find ha bubbles only exis when an economy had inermediae financial fricions in heir model. Thus, his kinds of bubble region resric he usefulness of he model. Alhough i is impossible o derive he condiions under which bubbles exis analyically, we can use numerical way o show ha bubbly BGP is raher robus so under mos siuaions bubbles may exis. We also show ha he resul bubbly BGPs always have higher growh rae is robus. Wha is more, he igher credi consrains are, he more benefi bubbles bring. Our sudy focuses on wo parameers ξ and π. ξ reflecs he aking over cos when defaul. The smaller he ξ is, he more cosly he aking over is. Thus, credi consrains are igher. During he robus es, we assume oher parameers will have values he same as he sudy in previous subsecion. ξ b/v (k) 49.54% 54.69% 60.15% 62.39% g 4.1% 4.0% 3.9% 3.8% Table 3: Robusness check when credi consrains are igher Table 3 is he resul showing how differen ξ affec bubbles. b/v (k) characerize he average size of bubbles compare wih firms value. When ξ decreases, credi consrains are igher and igher. Growh rae decreases and bubbles increase. Even when ξ = 0.1 which means lenders can only ake over 10% of original capial when borrowers defaul, bubbles sill exis. This es means our model is very robus on financial condiions. Bubbles is possible o exis even in an economy wih exremely igh credi consrains. Besides ξ, π is anoher parameer we have ineresed in. π is he probabiliy a firm find an invesmen opporuniy in one period. If he probabiliy is higher, more firms have invesmen opporuniies hus he economy is more effi cien in reallocaing social resource. Thus he credi consrains are no binding so igh as before. For his reason, bubbles are shrinking quickly wih he increasing of π. This can be seen in numerical analysis. 26

27 π b/v (k) 70.37% 57.59% 37.94% 11.89% g 1.9% 4.6% 4.7% 4.9% Table 4: Robusness check when probabiliy of invesmen opporuniies increase Wha is he relaionship when boh ξ and π change? We give a region in which bubbly BGP are possible o exis. As usual, oher parameers are he same as before. As shown in figure 3, he shade area is he region Figure 3: Region in which bubbly BGPs are possible o exis. where bubble are possible o exis. Wih he decreasing of ξ, bubbles may exis in economy wih higher chance of invesmen opporuniies. This is because decreasing of ξ makes credi consrains igher. Even here are a large number of firms can inves, hey sill wan o borrow more. From figure 3, we find ha bubbles are possible o exis in many differen cases which 27

28 Figure 4: Bubbles wih differen π and ξ means our model is more useful han Hirano and Yanagawa (2017). Figure 4 gives us how π and ξ affec bubbles. Generally speaking, if π decreases of ξ decreases, credi consrains are igher, hus value of bubbles are greaer. Figure 5 gives us he informaion on he difference of growh rae beween bubbly BGPs and bubbleless BGPs. Bubbly BGPs always have higher growh rae. A he same ime, he smaller he π and ξ are, he bigger he difference i is. The reason is very simple, smaller π and ξ increase he ighness of credi consrains so bubbles play more imporan role in he economy. 5 Dynamics In his secion, we sudy he dynamics of he model. We firs sudy he ransiion around bubbly BGP. We hen sudy wha happens when bubbles burs unanicipaedly. The sochasic burs of bubbles will be sudied in nex 28

29 Figure 5: Difference of growh rae beween bubbly and bubbleless BGPs wih differen parameers secion where we exend our baseline model ino a sochasic model. 5.1 The dynamics around bubbly BGP Since we have a big dynamic sysem, we are unable o derive analyical resuls for local dynamics. However, we can solve i numerically. We find rank condiions are saisfied for our examples. As in figure 6, we sar from he poin where derended capial is abou 10% more han alone BGP pah. A his poin, derended bubbles is abou 2.8% smaller han he BGP bubbles. Wih ime going on, derended capial is decreasing while derended bubbles are increasing. In he end, hey converge o he level of hose alone BGP wih bubbles. 29

30 Figure 6: Dynamics when here are 10% more capial han alone bubbly BGP 5.2 Unanicipaed burs of bubbles One of he mos obvious feaure of bubbles are bubbles end o burs. For example, when Do-com bubbles burs suddenly, Nasdaq Composie index fell 25% in one week. The price of Bicoin dropped from around $19000 o around $6000 in less han wo monhs. (He 2018) Very few people realize he bubbles is going o burs before i really happens. There are wo ways o deal wih burs of bubbles. The firs approach is bubbles will burs unanicipaedly. However, many economiss believe ha alhough people do no know when bubbles burs, hey expec bubbles will burs sooner or laer. Thus he second approach is sochasic bubble. We sudy how unanicipaed burs of bubbles have effecs on he economy in our model. In nex secion, we sudy wha happens when bubbles burs sochasically. We assume he economy is growing alone bubbly BGP when here is an unanicipaed shock a period 2. The shock changes he consensus ha people believe ohers will no accep he overvaluaion of equiy. Thus, bubbles burs and here is no bubble from ha periods on. Figure 7 is he resul of wha happens when bubbles burs. From figure 7 we can see wha happens wih oher variables when bubbles 30

31 Figure 7: Unanicipaed burs of bubbles burs. Since here is no bubble any more, credi consrains bind igher and firms canno ge so much loans as before. Firms have o reduce heir invesmen which leads o he decreasing of derended capial from period 3. Capial gradually converges o he level of bubbleless BGP. Growh is also slower because innovaion is slowed wih he limi of capial. Capial price, however, jump a he ime of bubbles burs and hen grows slowly. The jump of price is due o he jump of capial demand since capial is now he only insrumens o be used o increase he value of collaeral. Afer ha, amoun of capial is decreasing which leads o he scarciy of capial which drives he price up gradually and also increases capial reurn rae. Reurn of capial r increases. When bubbles burs, he jump of capial price increases capial reurn rae immediaely. Afer ha, capial reurn rae goes up gradually wih he increasing of capial and capial price. a increases following he same paern of capial price. Thus, economy will converge o bubbleless 31

32 BGP gradually if bubbles burs unanicipaedly. 6 Exensions In his secion, we sudy wo exensions of our model. In he firs exension, we inroduce sochasic bubbles ino our model. Up o now, bubbles are deerminisic and hey only burs when here is an unanicipaed shock. People do no believe bubbles may burs unless i really happens even hough hey admi here are bubbles in sock marke. These assumpions are no very realisic. Though i is hard o know when bubbles burs, people believe bubbles will burs in he fuure and he probabiliy of burs affecs people s decision. To reflec his, we assume bubbles may burs every periods wih a probabiliy. The second exension is sudying how bubbles affec resource allocaion. In he baseline model, we focus on how bubbles affec R&D secor direcly. However, bubbles in one secor may have reallocaion effecs of resource. In he second exension, we assume boh final produc secor and R&D secor mus use labor o produce and labor flow from one secor o he oher secor freely. Thus, bubbles in R&D secors can reallocae labor. 6.1 Sochasic bubbles We assume if here are bubbles, hey may burs a probabiliy θ every period. The burs will happen before R&D aciviies. Oher seing are he same as baseline model. If bubbles burs, everyhing works like he bubbleless equilibrium in he baseline model. We only need o consider he case when here are bubbles. Everyhing is he same as baseline model excep (5) becomes V ( K j ) = (1 π) (1 θ) max K j +1,Bj +π (1 θ) max K j I+1,Ij,Bj [ ( )] D j + ρ +1 V +1 K j +1 [ D j I + ρ +1V +1 ( K j I+1 )] + θv # ( ) K j (32) 32

33 ( ) where V # K j is he bubbleless cum-dividend equiy value we have derived before. We show he nex proposiion in Appendix Proposiion 6 When here are bubbles, V ( K j ) = a K j + B where a = (1 θ) r +(1 θ) q (1 δ)+(1 θ) π (q 1) [ r + ρ +1 a +1 ξ (1 δ) ] +θa # (33) B = (1 θ) [1 + π (q 1)] ρ +1 B +1 (34) We can see he sochasic bubbles model has he similar resul wih baseline model. The only difference beween (33), (34) and (13), (14) are he probabiliy of burs. There is 1 θ chance ha bubbles sill exis a he beginning of ha period hus we have erms similar like before. However, here is probabiliy θ ha bubbles burs. If bubbles do burs, firms will operae as he firms in bubbleless equilibrium so we have he erm θa #. We hen use numerical mehod o sudy he sochasic model. Here we have θ = 0.05 and all oher parameers are same as he baseline model. The resuls are in figure 8. Afer he burs of bubbles, he pah o he bubbleless equilibrium is similar wih he unanicipaed shock. The mechanism is he same as baseline model. 6.2 Reallocaion effecs model We assume he household has labor supply _ L. represenaive household is Budge consrain of he C + ( V j ) D j ψ j +1dj = V j ψ j _ dj + W L Final goods producer now has he echnology Y = A N n=1 (Xn ) σ (L Y ) 1 σ dn, 0 < σ < 1 Here L Y is he labor hired by final goods producer. Thus, only _ L L Y labor works in R&D secor. Since mos derivaion and resuls are similar wih 33

34 Figure 8: Burs of sochasic bubbles baseline model. We pu all he derivaion ino appendix. We only provides he numerical resuls here. Given he parameers in baseline model and _ L = 2, we have bubbly BGP and bubbleless BGP in able 5. The resuls are similar like we have discussed b/v (k) g N q a r k L Y bubbly % bubbleless 0 1.4% Table 5: Values of variables alone bubbly and bubbleless BGPs in reallocaion effecs model in baseline model. The only difference is ha labor works in final producs secor in bubbly economy is less han he labor in bubbleless economy. Which means labor flow from final producs secor o R&D secor. The inuiion is simple. We have discussed ha here are more capial in R&D secor when 34

35 here are bubbles hus he marginal produciviy of labor in R&D secor is higher. R&D firms would like o hire more labor o produce new blueprins. This effec alone wih he effecs we discussed before increases he posiive effecs of bubbles in R&D secor and helps he economic growh. We also sudy he burs of bubbles. As he baseline model, he paern of burs boh in unanicipaed shock and sochasic bubbles show he similar resul. We will only repor he unanicipaed shock here in figure 9. The sochasic bubbles are repored in appendix. Figure 9: Unanicipaed burs of bubbles in reallocaion effecs model Mos of he pah are similar wih baseline model. The only significan difference is when bubbles burs, growh rae firs increases and hen decreases 35

36 gradually in his model while growh rae decreases wih he decreasing capial in baseline model. This may be a surprising resul. Why does he economy grow faser when he bubbles burs and capial sar decreasing? This is because he change of sochasic discoun rae ρ. Since ρ increases generally afer he burs of bubbles, he price of paen η also increases. Alhough capial decreases soon afer he burs of bubbles, increasing of paen price increases he marginal produciviy of labor in R&D secor hus i aracs more labor flows ou of final goods secor and works for R&D secors. This effec compensae he decreasing of capial. A he beginning, his reallocaion effec is srong enough so here are more paens are produced. Wih less and less capial, marginal produciviy of labor is decreasing and people flow ou of R&D secor and he growh rae is smaller and smaller unil i reaches he bubbleless BGP. 7 Conclusion In his paper, we inroduce credi consrains ino a sandard endogenous growh model wih innovaion. We show ha here are muliple equilibria and sock price bubbles exis in one of he equilibria. To he exen of our knowledge, his is he only sudy abou endogenous growh wih bubbles by using a well defined R&D growh model and i is he only model in which bubbles are generaed by feaures of innovaion secors. Our sudy finds ha sock price bubbles in innovaion secor encourage innovaion and increase growh rae by reducing liquidiy mismach. Economy wih igher credi consrains benefi more from bubbles. Thus, i may be wise for governmens no o make bubbles in innovaion secor burs bu use policy insrumens o reduce financial fricions in innovaion secor. This paper can be a bridge beween radiional growh model and bubbles. Besides he economic phenomenon we discussed in his paper. There are some exension which can be done in he fuure. Our paper has R&D secor and bubbles. A he same ime, household buy shares of R&D firms. These are close o he siuaion he venure capialiss face. In he fuure, we can exend he R&D secor and his model may be used o sudy venure 36