Economic Growh wih Bubbles AlberoMarin,andJaumeVenura March 2010 Absrac We develop a sylized model of economic growh wih bubbles. In his model, financial fricions lead o equilibrium dispersion in he raes of reurn o invesmen. During bubbly episodes, unproducive invesors demand bubbles while producive invesors supply hem. Because of his, bubbly episodes channel resources owards producive invesmen raising he growh raes of capial and oupu. The model also illusraes ha he exisence of bubbly episodes requires some invesmen o be dynamically inefficien: oherwise, here would be no demand for bubbles. This dynamic inefficiency, however, migh be generaed by an expansionary episode iself. JEL classificaion: E32, E44, O40 Keywords: bubbles, dynamic inefficiency, economic growh, financial fricions, pyramid schemes Marin: CREI and Universia Pompeu Fabra, amarin@crei.ca. Venura: CREI and Universia Pompeu Fabra, jvenura@crei.ca. CREI, Universia Pompeu Fabra, Ramon Trias Fargas 25-27, 08005-Barcelona, Spain. We hank Vasco Carvalho for insighful commens. We acknowledge suppor from he Spanish Minisry of Science and Innovaion(grans ECO2008-01666 and CSD2006-00016), he Generalia de Caalunya-DIUE(gran 2009SGR1157), and he Barcelona GSE Research Nework. In addiion, Venura acknowledges suppor from he ERC (Advanced Gran FP7-249588), and Marin from he Spanish Minisry of Science and Innovaion (gran Ramon y Cajal RYC- 2009-04624) and from he Lamfalussy Fellowship Program sponsored by he ECB. Any views expressed are only hose ofheauhorsanddononecessarilyrepresenheviewsofheecborheeurosysem.
1 Inroducion Modern economies ofen experience episodes of large movemens in asse prices ha canno be explained by changes in economic condiions or fundamenals. I is commonplace o refer o hese episodes as bubbles popping up and bursing. Typically, hese bubbles are unpredicable and generae subsanial macroeconomic effecs. Consumpion, invesmen and produciviy growh all endosurgewhenabubblepopsup,andhencollapseorsagnaewhenhebubbleburss. Here, we address he following quesions: Wha is he origin of hese bubbly episodes? Why are hey unpredicable? How do bubbles affec consumpion, invesmen and produciviy growh? In a nushell, he goal of his paper is o develop a sylized view or model of economic growh wih bubbles. The heory presened here feaures wo idealized asse classes: producive asses or capial andpyramidschemesor bubbles. Bohassesareusedasasoreofvalueorsavingsvehicle,bu hey have differen characerisics. Capial is cosly o produce bu i is hen useful in producion. Bubbles play no role in producion, bu iniiaing hem is cosless. 1 We consider environmens wih raional, informed and risk neural invesors ha hold only hose asses ha offer he highes expeced reurn. The heoreical challenge is o idenify siuaions in which hese invesors opimally choose o hold bubbles in heir porfolios and hen characerize he macroeconomic consequences of heir choice. Our research builds on he seminal papers of Samuelson(1958) and Tirole(1985) who viewed bubbles as a remedy o he problem of dynamic inefficiency. Their argumen is based on he dual role of capial as a producive asseand a sore of value. Tosaisfy he need for a sore of value, economies someimes accumulae so much capial ha he invesmen required o susain i exceeds he income ha i produces. This invesmen is inefficien and lowers he resources available for consumpion. In his siuaion, bubbles can be boh aracive o invesors and feasible from a macroeconomic perspecive. For insance, a pyramid scheme ha absorbs all inefficien invesmens in each period is feasible and is reurn exceeds ha of he invesmens i replaces. This explains he origins and he effecs of bubbles. Since bubbles do no have inrinsic value, heir size depends on he marke s expecaion of heir fuure size. In a world of raional invesors, his opens he door for 1 Iisdifficulofindheseidealizedasseclassesinfinancialmarkes,ofcourse,asexisingassesbundleorpackage ogehercapialandbubbles. Yewehinkhamuchcanbelearnedbyworkingwihhesebasicasses. Toprovide a simple analogy, we have surely learned much by sudying heoreical economies wih a full se of Arrow-Debreu securiies, even hough only a few bundles or packages of hese basic securiies are raded in he real world. 1
self-fulfilling expecaions o play a role in bubble dynamics and accouns for heir unpredicabiliy. The Samuelson-Tirole model provides an elegan and powerful framework o hink abou bubbles. However, he picure ha emerges from his heory is hard o reconcile wih hisorical evidence. In his model, bubbly episodes are consumpion booms financed by a reducion in inefficien invesmens. During hese episodes boh he capial sock and oupu conrac. In he real world, bubbly episodes end o be associaed wih consumpion booms indeed. Bu hey also end o be associaed wih expansions in boh he capial sock and oupu. A successful model of bubbles mus come o grips wih hese correlaions. Thispapershowshowobuildsuchamodelbyexendingheheoryofraionalbubblesohe case of imperfec financial markes. In he Samuelson-Tirole model, fricionless financial markes eliminae rae-of-reurn differenials among invesmens making hem eiher all efficien or all inefficien. Inroducing financial fricions is crucial because hese creae rae-of-reurn differenials and allow efficien and inefficien invesmens o coexis. Our key observaion is quie simple: bubbles no only reduce inefficien invesmens, bu hey also increase efficien ones. In our model, bubbly episodes are booms in consumpion and efficien invesmens financed by a reducion in inefficien invesmens. If he increase in efficien invesmens is sizable enough, bubbly episodes expand he capialsockandoupu. Thisurnsouobehecaseunderawiderangeofparameervalues. Toundersandheseeffecsofbubblyepisodes,iisusefuloanalyzeheseofransfersha bubbles implemen. Remember ha a bubble is nohing bu a pyramid scheme by which he buyer surrenders resources oday expecing ha fuure buyers will surrender resources o him/her. The economy eners each period wih an iniial disribuion of bubble owners. Some of hese owners bough heir bubbles in earlier periods, while ohers jus creaed hem. When he marke for bubbles opens, on he demand side we find invesors who canno obain a reurn o invesmen abovehaofbubbles; whileonhesupplysidewefindconsumersandinvesors whocanobain a reurn o invesmen above ha of bubbles. When he marke for bubbles closes, resources have been ransferred from inefficien invesors o consumers and efficien invesors, leading o an increase in consumpion and efficien invesmens a he expense of inefficien invesmens. Akeyaspecofheheoryishowhedisribuionofbubbleownersisdeermined. Asinhe Samuelson-Tirole model, our economy is populaed by overlapping generaions ha live for wo periods. The young inves and he old consume. The economy eners each period wih wo ypes ofbubbleowners: heoldwhoacquiredbubblesduringheiryouh,andheyoungwhoarelucky enough o creae hem. Since he old only consume, bubble creaion by efficien young invesors 2
plays a crucial role in our model: i allows hem o finance addiional invesmen by selling bubbles. Inroducing financial fricions also solves a nagging problem of he heory of raional bubbles, which was firs poined ou by Abel e al. (1989). In he Samuelson-Tirole model, bubbles can only exis if he invesmen required o susain he capial sock exceeds he income ha i produces. Abel e al. (1989) examined a group of developed economies and found ha, in all of hem, invesmen falls shor of capial income. This finding has ofen been considered evidence ha raional bubbles canno exis in real economies. Inroducing financial fricions ino he heory shows ha his conclusion is unwarraned. The observaion ha capial income exceeds invesmen only implies ha, on average, invesmens are dynamically efficien. Bu his does no exclude he possibiliy ha he economy conains pockes of dynamically inefficien invesmens ha could suppor a bubble. Nor does i exclude he possibiliy ha an expansionary bubble, by lowering he reurn o invesmen, creaes iself he pockes of dynamically inefficien invesmens ha suppor i. Insuchsiuaions,heesofAbeleal. wouldwronglyconcludehabubblesarenopossible. Besides building on he seminal conribuions of Samuelson (1958) and Tirole (1985), his paper is closely relaed o previous work on bubbles and economic growh. Sain-Paul (1992), Grossman and Yanagawa(1993), and King and Ferguson(1993) exend he Samuelson-Tirole model o economies wih endogenous growh due o exernaliies in capial accumulaion. In heir models, bubbles slow downhe growhraeof heeconomy. Olivier (2000) uses asimilar model o show how, if ied o R&D firms, bubbles migh foser echnological progress and growh. The model developed in his paper sresses he relaionship beween bubbles and fricions in financial markes. Azariadis and Smih(1993) were, o he bes of our knowledge, he firs o show ha conracing fricions could relax he condiions for he exisence of raional bubbles. More recenly, Caballero and Krishnamurhy (2006), Kraay and Venura (2007), and Farhi and Tirole (2009) conain models in which he exisence and economic effecs of raional bubbles are closely linked o financial fricions. Whereas we derive our resuls in a sandard growh model, hese papers sudy economies wih linear producion funcions or sorage echnology. The paper is organized as follows: Secion 2 presens he Samuelson-Tirole model, provides condiions for he exisence of equilibrium bubbles and discusses heir macroeconomic effecs. Secion 3 inroduces financial fricions and conains he main resuls of he paper. Secion 4 exends hese resuls o an economy wih long-run growh. Finally, Secion 5 concludes. 3
2 The Samuelson-Tirole model Samuelson (1958) and Tirole (1985) showed ha bubbles are possible in economies ha are dynamically inefficien, i.e. ha accumulae oo much capial. Bubbles crowd ou capial and raise he reurn o invesmen. We re-formulae his heory in erms of bubbly episodes and provide a quick refresher of is macroeconomic implicaions. 2.1 Basic seup Consider a counry inhabied by overlapping generaions of young and old, all wih size one. Time sarsa=0andhengoesonforever. Allgeneraionsmaximizeheexpecedconsumpionwhen old: U =E c +1 ;whereu andc +1 arehewelfareandheold-ageconsumpionofgeneraion. The oupu he counry is given by a Cobb-Douglas producion funcion of labor and capial: F(l,k )=l 1 k wih (0,1),and l andk arehecounry slaborforceandcapialsock, respecively. All generaions have one uni of labor which hey supply inelasically when hey are young,i.e. l =1. Thesockofcapialinperiod+1equalsheinvesmenmadebygeneraion duringisyouh. 2 Thismeansha: k +1 =s k, (1) where s is he invesmen rae, i.e. he fracion of oupu ha is devoed o capial formaion. Markes are compeiive and facors of producion are paid he value of heir marginal produc: w =(1 ) k and r = k 1, (2) wherew andr arehewageandherenalrae,respecively. Tosolvehemodel, weneedofindheinvesmenrae. Theolddonosaveandheyoung save all heir income. Wha do he young do wih heir savings? A his poin, i is cusomary o assume ha hey use hem o build capial. This means ha he invesmen rae equals he savings of he young. Since he laer equal labor income, which is a consan fracion 1 of oupu, he invesmen rae is consan as in he classic Solow(1956) model: s =1. (3) 2 Thais,weassumeha(i)producingoneuniofcapialrequiresoneuniofconsumpion,andha(ii)capial fully depreciaes in producion. We also assume ha he firs generaion found some posiive amoun of capial o workwih,i.e. k 0 >0. 4
Therefore,helawofmoionofhecapialsockis: k +1 =(1 ) k. (4) Equaion(4) consiues a very sylized version of a sandard workhorse of modern macroeconomics. A lo of progress has been made by adding more sophisicaed formulaions of preferences and echnology, various ypes of shocks, a few marke imperfecions, and a role for money. We shall nodoanyofhisherehough. 2.2 Equilibria wih bubbles Insead, we follow he pah-breaking work of Samuelson(1958) and Tirole(1985), and assume he young have he addiional opion of purchasing bubbles or pyramid schemes. These are inrinsically uselessasses,andheonlyreasonopurchasehemisoresellhemlaer. Leb behesockof old bubbles in period, i.e. already exising before period or creaed by earlier generaions; and leb N behesockofnewbubbles,i.e. addedinperiodorcreaedbygeneraion. Weassume hahereisfreedisposalofbubbles. Thisimplieshab 0andb N 0. Wealsoassumeha bubbles are creaed randomly and wihou cos. This implies ha new bubbles consiue a pure profiorrenforhosehacreaehem. 3 Raionaliy imposes wo resricions on he ype of bubbles ha can exis. Firs, bubbles mus grow fas enough or oherwise he young will no be willing o purchase hem. Second, he aggregae bubblecannogrowoofasoroherwiseheyoungwillnobeableopurchasehem. Therefore, ifb >0,hen E { b+1 b +b N } = k 1 +1, (5) b (1 ) k. (6) Equaion(5)saysha,forbubblesobearacive,heymusdeliverhesamereurnascapial. 4 Thereurnohebubbleconsissofisgrowhoverheholdingperiod. Thepurchasepriceofhe bubbleisb +b N,andhesellingpriceisb +1. Thereurnocapialequalsherenalraesinceeach 3 Noe ha new bubbles canno be linked o he ownership of objecs ha can be raded before he dae he bubbles appear. Oherwise he bubble would have already apppeared in he firs dae in which he objec can be raded. The reason issimple: a raionalindividual would be willing o pay a posiiveprice for an objec if hereis some probabiliy ha his objec commands a posiive price in he fuure. See Diba and Grossman(1987). 4 Wecanruleouhepossibiliyhabubblesdeliverahigherreurnhancapial. Asssumenoandlehereurn ohebubbleobesriclyhigherhan hereurnocapial. Then, nobodywouldinvesandhereurn ocapial wouldbeinfiniy. Buhismeanshahebubblemusgrowaaraeinfiniyandhisisnopossible. 5
uni of capial coss one uni of consumpion and i fully depreciaes in one period. Equaion(6) says ha, for bubbles o be feasible, hey canno ougrow he economy s savings. The savings of he youngconsisoflaborincomeandhevalueofnewbubblescreaedbyhem,i.e. (1 ) k +b N. Sinceheolddonosave,heyoungmusbepurchasinghewholeaggregaebubble,i.e. b +b N. The invesmen rae equals he income lef afer purchasing he bubbles as a share of oupu: and his implies he following law of moion for he capial sock: s =1 b k, (7) k +1 =(1 ) k b. (8) Equaion(8) shows he key feaure of he Samuelson-Tirole model: bubbles crowd ou invesmen and slow down capial accumulaion. Foragiveniniialcapialsockandbubble,k 0 >0andb 0 0,acompeiiveequilibriumisa sequence { k,b,b N } =0 saisfyingequaions(5),(6)and(8). Theassumpionhaheyoungonly buildcapialisequivalenoaddingheaddiionalequilibriumresricionhab =b N =0forall. This resricion canno be jusified on a priori grounds, bu we noe ha here always exiss oneequilibriuminwhichiissaisfied. 5 There are a couple of imporan differences beween he model described here and he original ones of Samuelson (1958) and Tirole (1985). Unlike us, Samuelson analyzed an economy wih a linear producion funcion or sorage echnology. Tirole analyzed insead a sandard growh model like he one we sudy. Unlike us, however, he made weak assumpions on preferences and echnology. Inparicular, heonly assumedheexisenceofuiliyandproducionfuncions, U =u(c,c +1 ) andf(l,k )wihsandardproperies. Unlikeus,bohSamuelsonandTiroleresricedheanalysis o he subse of equilibria ha are deerminisic and do no involve bubble creaion or desrucion. Thais,heyimposedheaddiionalresricionshaE b +1 =b +1 andb N =0forall. 6,7 Despie hese differences, we label he model described above as he Samuelson-Tirole model o give due credi o heir seminal conribuions. 5 Thisequilibriummighnoexisinhepresenceofrens. SeeTirole(1985)andCaballeroeal. (2010). 6 Underheseresricions,any bubblemushaveexised from heverybeginning of imeand ican neverburs, i.e. isvaluecanneverbezero. 7 Tohebesofourknowledge,Weil(1987)washefirsoconsidersochasicbubblesingeneralequilibrium. 6
2.3 Bubbly episodes An imporan payoff of analyzing sochasic equilibria wih bubble creaion and desrucion is ha his allows us o rigorously capure he noion of a bubbly episode. Generically, he economy flucuaes beween periods in which b = b N = 0 and periods in which b > 0 and/or b N > 0. We say ha he economy is in he fundamenal sae if b = b N = 0. We say insead ha he economyisexperiencingabubbly episode ifb >0and/orb N >0. Abubblyepisodesarswhen he economy leaves he fundamenal sae and ends he firs period in which he economy reurns o he fundamenal sae. The following proposiion provides he condiions for he exisence of bubbly episodes in he Samuelson-Tirole model: Proposiion1 Bubblyepisodesarepossibleifandonlyif<0.5. Theproofofhisproposiionexploisausefulrickhamakeshemodelrecursive. Lex be heaggregaebubbleasashareofhelaborincome,i.e. x (1 ) k Then,wecanrewrieEquaions(5)and(6)assayinghaifx >0,hen b b N andx N (1 ) k. E x +1 = 1 x+x N 1 x, (9) x 1. (10) Equaions(9) and(10) describe bubble dynamics. There are wo sources of randomness in hese dynamics: shocksobubblecreaion,i.e. x N ;andshocksohevalueofheexisingbubble,i.e. x. Anyadmissiblesochasicprocessforx N andx saisfyingequaions(9)and(10)isanequilibrium of he model. By admissible, we mean ha he sochasic process mus ensure ha x 0 and x N 0 for all. Conversely, any equilibrium of he model can be expressed as an admissible sochasicprocessforx N andx. To prove Proposiion 1 we ask if, among all sochasic processes for x N and x ha saisfy Equaion (9), here is a leas one ha also saisfies Equaion (10). I is useful o examine firs hecaseinwhichhereisnobubblecreaionaferabubblyepisodessars. Figure1plosE x +1 againsx, usingequaion(9)wihx N =x N s andx N =0forall >s, where sis heperiodin whichheepisodesars. Thelefpanelshowshecaseinwhich 0.5andheslopeofE x +1 aheoriginisgreaerhanorequaloone. Anyiniialbubblewouldbedemandedonlyifiwere 7
expecedoconinuouslygrowasashareoflaborincome,i.e. E x +1 >x inallperiods. Buhis meanshainsomescenarioshebubbleougrowshesavingsofheyounginfinieime, i.e. i violaes Equaion(10). Therefore, bubbly episodes canno happen if 0.5. The righ panel of Figure1showsinseadhecaseinwhich<0.5. Anyiniialbubblex s+1 > 1 2 1 canberuled ouwihhesameargumen. Buanyiniialbubblex s+1 1 2 1 canbeparofanequilibrium asiispossibleofindaprocessforx hasaisfiesequaions(9)and(10)simulaneously. 0.5 0. 5 E x 1 E x 1 x 1 2 1 x Figure 1 Allowing for bubble creaion does no relax he condiions for exisence of bubbly episodes. To seehis,noehabubblecreaionshifsupwardsheschedulee x +1 infigure1. Theinuiionis clear: new bubbles compee wih old bubbles for he income of nex period s young, reducing heir reurn and making hem less aracive. This complees he proof of Proposiion 1. This proof is insrucive and helps us undersand he connecion beween bubbles and dynamic inefficiency. To deermine wheher a bubbly episodes can exis, we have asked: Is i possible o consruc a pyramid scheme ha is aracive wihou exploding as a share of labor income? We haveseenhaheanswerisaffirmaiveifandonlyifhereexissochasicprocessesforx N and x suchha: E x +1 <x. Theseprocessesexisifandonlyif<0.5. To deermine wheher he economy is dynamically inefficien, we ask: Is he economy accumulaingoomuchcapial? Theanswerisaffirmaiveifandonlyifheinvesmenrequiredosusainhe 8
capialsockexceedsheincomehahiscapialproduces. Invesmenequals(1 x ) (1 ) k, while capial income is given by k. Therefore, he economy is dynamically inefficien if and only if (1 x ) (1 ) k > k. (11) Sraighforward algebra shows ha his condiion is equivalen o asking wheher here exis sochasicprocessesforforx N andx suchha: E x +1 <x +x N. These processes exis if and only if < 0.5. Therefore, in he Samuelson-Tirole model he condiions for he exisence of bubbly episodes and dynamic inefficiency coincide. 2.4 The macroeconomic effecs of bubbles To deermine he macroeconomic consequences of bubbly episodes, we rewrie he law of moion of hecapialsockusinghedefiniionofx : k +1 =(1 x ) (1 ) k. (12) Equaion(12) describes he dynamics of he capial sock for any admissible sochasic process for hebubble, i.e. x N andx ; saisfyingequaions(9)and(10). This consiues afull soluiono he model. Ineresingly, bubbly episodes can be lierally inerpreed as shocks o he law of moion of he capial sock of he Solow model. To beer undersand he naure of hese shocks, consider he following example: Example 1((n, p) episodes) Consider he subse of bubbly episodes ha are characerized by(i) a consan probabiliy of ending, i.e. Pr (b +1 =0 b >0)=p and (ii) an iniial bubble x N s and henaconsanraeofnew-bubblecreaion,i.e. x N =n x. ThelefpanelofFigure2showsa(n,p)episode. ThesolidlinerepresensEquaion(9),i.e. he valueofx +1 haleavesheyoungindifferenbeweenbuyinghebubbleorinvesingincapial. Afeaureof(n,p)episodesishahebubbledeclinesasashareoflaborincomehroughouhe episode,i.e. x +1 x forallandx 0. Onlyifheiniialbubbleismaximal,i.e. x s+1 x 1, 9
his rae of decline becomes zero. This paern of behavior is no generic, however, as he following example shows: Example 2((x N,p)episodes) Consider he subse of bubbly episodes ha are characerized by (i)aconsanprobabiliyofending,i.e. Pr (b +1 =0 b >0)=pand(ii)aniniialbubblex N s and henaconsanamounofbubblecreaionx N =x N. The righ panel of Figure 2 shows an (x N,p) episode. Any iniial bubbleconverges o x 2. If x s+1 <x 2,hebubblegrowshroughouheepisode. Ifx s+1>x 2,hebubbledeclineshroughou he episode. Once again, if he iniial bubble is maximal, i.e. x s+1 x 2, his rae of decline becomeszeroandhebubbleneverconvergesox 2. x 1 x 1 x 1 x 2 x 2 x x Figure 2 The only randomness in hese examples refers o he periods in which hey sar and end. Throughou he bubbly episode, he bubble moves deerminisically unil he episodes ends. This need no be, of course. Assume, for insance, ha bubble creaion randomly swiches beween being a fracion of he exising bubble, i.e. x N = n x ; and being a consan amoun, i.e. x N = x N. Thais,helawsofmoionofhebubbleinherighandlefpanelsofFigure2operaeadifferen (andrandom)imesduringagivenbubblyepisode. Then,x willconvergeoheinerval(0,x 2 ), and hen randomly flucuae wihin i unil he episode ends. Figure 3 shows he macroeconomic effecs of one of hese bubbly episodes. Assume iniially ha heeconomyisinhefundamenalsaesohaheappropriaelawofmoionisheonelabeled k F +1. SinceheiniialcapialsockisbelowheSolowseadysae,i.e. k <k F (1 ) 1 1,he 10
economy is growing a a posiive rae. When a bubbly episode sars, he invesmen rae falls and helawofmoionshifsbelowhefundamenalone. Inhefigure,k B +1 represenshelawofmoion when he bubbly episode begins. The picure has been drawn so ha he capial sock is above heseadysaeassociaedok B +1,i.e. k >k B. Asaresul,growhurnsnegaive. Throughou heepisode,k B +1 mayshifupordownashebubblegrowsorshrinks,alhoughialwaysremains below he original law of moion k+1 F. Evenually, he episode ends and he economy reurns o k F +1. k 1 F k 1 B k 1 k B k k F k Figure 3 Afirssigh,onecouldhinkofbubblyepisodesasakinonegaiveshocksoheinvesmen rae. Bu his would no be quie righ. Bubbles also affec consumpion direcly as passing he bubble across generaions increases he share of oupu ha he old receive and consume: c =[+(1 ) x ] k. (13) The relaionship beween bubbles and consumpion has herefore wo differen aspecs o i. Pas bubbles reduce he capial sock and, ceeris paribus, his lowers consumpion. Bu presen bubbles raiseheshareofoupuinhehandsofheoldand,ceerisparibus,hisraisesconsumpion. 2.5 Discussion Bubbles affec allocaions hrough wo channels: (i) by implemening a se of inergeneraional ransfers, and (ii) by creaing wealh shocks. The firs channel is a cenral feaure of a pyramid scheme, by which buyers surrender resources oday expecing fuure buyers o surrender resources o 11
hem. These inergeneraional ransfers are feasible because he economy is dynamically inefficien. The second channel are he wealh shocks associaed wih bubble creaion and desrucion. When bubbles appear, hose lucky individuals who creae hem receive a windfall or ransfer from he fuure. This is anoher cenral feaure of a pyramid scheme whereby he iniiaor claims ha, bymakinghim/herapaymennow, heoherparyearns herighoreceiveapaymenfrom a hird person laer. By successfully creaing and selling a bubble, young individuals have assigned hemselvesandsoldhe righs oheincomeofageneraionlivinginheveryfarfuureor,o be more exac, living a infiniy. This appropriaion of righs is a pure windfall or posiive wealh shock for he generaion ha creaes hem. Naurally, he opposie happens when bubbles burs since his consiues a negaive wealh shock for hose ha are holding hem and see heir value collapse. Once exended o allow for random bubbles and random bubble creaion and desrucion, he Samuelson-Tirole model provides an elegan and powerful framework o hink abou bubbly episodes. Unforunaely, he macroeconomic implicaions of he Samuelson-Tirole model are a odds wih he facs along wo key dimensions: 1. The model predics ha bubbles can only appear in dynamically inefficien economies, i.e. 0.5. However, Abeleal. (1989)examinedagroupofdevelopedeconomiesandfound ha,inallofhem,aggregaeinvesmen,i.e. (1 x ) (1 ) k,fallsshorofaggregae capialincome,i.e. k. 2. The model predics ha bubbles lead o simulaneous drops in he sock of capial and oupu. Hisorical evidence suggess however ha bubbly episodes are associaed wih increases in he capial sock and oupu. We nex show ha hese discrepancies beween he heory and he facs res on one imporan assumpion: financial markes are fricionless. 3 Inroducing financial fricions We exend he model by inroducing a moive for inrageneraional rade and a financial fricion ha impedes his rade. We show ha his relaxes he condiions for he exisence of bubbly episodes. Moreover, hese episodes can lower he reurn o invesmen and lead o expansions in he capial sock. 12
3.1 Seup wih financial fricions Assumehaafracionε [0,1]ofheyoungofeachgeneraioncanproduceoneuniofcapial wih one uni of he consumpion good. We refer o hem as producive invesors. The remaining young are unproducive invesors, as hey only have access o an inferior echnology ha produces δ<1unisofcapialwihoneuniofheconsumpiongood. Thisheerogeneiycreaesgainsfrom borrowing and lending. If markes worked well, unproducive invesors would lend heir resources o producive ones and hese would inves on everyone s behalf. This would bring us back o he Samuelson-Tirole model. We shall however assume ha his is no possible because of some unspecified marke imperfecion. The goal here is o analyze how his financial fricion affecs equilibrium oucomes. Now he evoluion of he capial sock depends no only on he level of invesmen bu also on is composiion. Le A be heaverageefficiency of invesmen. Then, Equaion (1)mus be replaced by k +1 =s A k. (14) For insance, inhe benchmark case in which he young use all heir savings obuild capial we have ha: A =ε+(1 ε) δ A. (15) Since all individuals inves he same amoun, he average efficiency of invesmen is deermined by he populaion weighs of boh ypes of invesors. The invesmen rae is sill deermined by Equaion(3)andhedynamicsofhecapialsockaregivenby k +1 =(1 ) A k. (16) SinceA<1,financialfricionslowerhelevelofhecapialsockbuheydonoaffechenaure ofisdynamics. Thisresuldoesnogohroughonceweallowforbubbles. 3.2 Equilibria wih bubbles The inroducion of financial fricions forces us o make an assumpion abou he disribuion of rens from bubble creaion. In he Samuelson-Tirole model, all invesmen is carried ou by producive invesors and he disribuion of rens is inconsequenial. Wih financial fricions, his isnolongerhecasesincehedisribuionofwealh andhenceofheserens affecsheaverage 13
efficiency of invesmen. We use b NP and b NU o denoe he sock of new bubbles creaed by produciveandunproduciveinvesors,respecively. Naurally,b NP +b NU =b N. Recall ha raionaliy imposes wo resricions on he ype of bubbles ha can exis. Firs, bubbles mus grow fas enough or oherwise he young will no be willing o purchase hem. Second, heaggregaebubblecannogrowoofasoroherwiseheyoungwillnobeableopurchasehem. While he second of hese resricions sill implies Equaion(6), he firs of hem now implies ha ifb >0,hen E { b +1 b +b NP +b NU =δ k 1 +1 if } [ δ k+1 1 ], k 1 +1 = k 1 +1 if if b +b NP (1 ε) (1 ) k b +b NP (1 ε) (1 ) k b +b NP (1 ε) (1 ) k <1 =1 >1. (17) Equaion (17) is nohing bu a generalizaion of Equaion (5) ha recognizes ha he marginal buyerofhebubblechangesashebubblegrows. Ifhebubbleissmall,hemarginalbuyerisan unproducive invesor and he expeced reurn o he bubble mus equal he reurn o unproducive invesmens. If he bubble is large, he marginal buyer is an producive invesor and he expeced reurn o he bubble mus be he reurn o producive invesmens. Bubbles affec boh he level of invesmen and is composiion. As in he Samuelson-Tirole model, he bubble reduces he invesmen rae and Equaion(7) sill holds. Unlike he Samuelson- Tirole model, he bubble now affecs he average efficiency of invesmen as follows: (1 ) A k +(1 δ) b NP δ b (1 ) k A = b if 1 if b +b NP (1 ε) (1 ) k b +b NP (1 ε) (1 ) k <1 1. (18) To undersand Equaion(18), noefirs ha inhe fundamenal saeb =b NP =b NU =0and he average efficiency of invesmen equals he populaion average A. Bubbles raise he efficiency of invesmen hrough wo channels. Firs, exising bubbles displace a disproporionaely high share of unproducive invesmens. This is why A is increasing in b. Second, bubble creaion byproduciveinvesorsraisesheirincomeandexpands heirinvesmen. Thisiswhy A isalso increasinginb NP. When all unproducive invesmens have been eliminaed, he average efficiency of invesmen reaches one. 14
Wecanhusre-wriehedynamicsofhecapialsockasfollows: (1 ) A k +(1 δ) bnp δ b if k +1 = (1 ) k b if b +b NP (1 ε) (1 ) k b +b NP (1 ε) (1 ) k <1 1. (19) Bubbles now have conflicing effecs on capial accumulaion and oupu. On he one hand, exising bubbles reduce he invesmen rae. On he oher hand, new bubbles raise he efficiency of invesmen. Ifhefirseffecdominaes,i.e. b NP < 1 δ b δ,bubblesareconracionaryandcrowd oucapial. Ifinseadhesecondeffecdominaes,i.e. b NP > δ 1 δ b,bubblesareexpansionary and crowd in capial. We are ready o define a compeiive equilibrium for he modified model. For a given iniial capial sock and bubble, k 0 > 0 and b 0 0, a compeiive equilibrium is a sequence { k,b,b NP saisfying Equaions (6), (17) and (19). As we show nex, here are many,b NU } =0 such equilibria. 3.3 Bubbly episodes wih financial fricions The following proposiion provides he condiions for he exisence of bubbly episodes in he model wih financial fricions: Proposiion 2 Bubbly episodes are possible if and only if: A < A+δ { } A max A+δ, 1 1+4 (1 ε) δ ifa>1 ε ifa 1 ε. Proposiion 2 generalizes Proposiion 1 o he case of financial fricions. Once again, we use he rickofmakinghemodelrecursivehroughachangeofvariables. Definenowx NP b NP (1 ) k and x NU b NU (1 ) k. Then, we can rewrie Equaions (6) and (17) as saying ha if x >0, 15
hen E x +1 = 1 δ (x +x NP +x NU ) [ A+(1 δ) x NP δ x δ (x +x NP +x NU ) 1 A+(1 δ) x NP δ x, +x NU = 1 x+x NP 1 x 1 x+x NU 1 x +x NP ] if x+xnp 1 ε if x +x NP 1 ε if x +x NP 1 ε <1 =1 >1, (20) x 1. (21) Equaions(20) and(21) describe bubble dynamics in he modified model. Any admissible sochasic process for x NP, x NU and x saisfying Equaions (20) and (21) is an equilibrium of he model. Conversely, any equilibrium of he model can be expressed as an admissible sochasic process for x NP,x NU andx. To prove Proposiion 2 we ask again if, among all sochasic processes for x NP, x NU and x ha saisfy Equaion (20), here is a leas one ha also saisfies Equaion (21). Consider firs hecaseinwhichhereisnobubblecreaionaferabubblyepisodesars. Figure4plosE x +1 againsx,usingequaion(20)wihx NP =x NP s,x NU =x NU s andx NP =x NU =0forall>s, wheresisonceagainheperiodinwhichheepisodesars. Thelefpanelshowshecaseinwhich A A+δ andheslopeofe x +1 aheoriginisgreaerhanorequaloone. Thismeansha anyiniialbubblewouldbedemandedonlyifiwereexpecedoconinuouslygrowasashareof laborincome,i.e. ifiviolaesequaion(21),andhiscanberuledou. TherighpanelofFigure 4showshecaseinwhich< A A+δ. NowheslopeofE x +1 aheoriginislesshanoneand, asaresul,e x +1 muscrosshe45degreelineonceandonlyonce. Lex behevalueofx a hapoin. Anyiniialbubblex s+1 >x canberuledou. Buanyiniialbubblex N s x canbe parofanequilibriumsinceiispossibleofindasochasicprocessforx hasaisfiesequaions (20) and(21). 16
E x 1 E x 1 1 x x 1 x Figure 4 Is i possible ha bubble creaion relaxes he condiions for he exisence of bubbly episodes? Consider firs bubble creaion by unproducive invesors, i.e. x NU. As in he Samuelson-Tirole model,hisypeofbubblecreaionshifsheschedulee x +1 upwards. Theinuiionishesameas before: new bubbles compee wih old bubbles for he income of nex period s young, reducing heir reurn and making hem less aracive. Therefore, allowing for bubble creaion by unproducive invesors does no relax he condiions for he exisence of bubbly episodes. Considernexbubblecreaionbyproduciveinvesors,i.e. x NP. Thisypeofbubblecreaion shifs he schedule E x +1 upwards if x (0,A] (1 ε,1], bu i shifs i downwards if x (A,1 ε]. Toundersandhisresul,iisimporanorecognizehedoubleroleplayedbybubble creaion by producive invesors. On he one hand, new bubbles compee wih old ones for he income of nex period s young. This effec reduces he demand for old bubbles and shifs he schedulee x +1 upwards. Onheoherhand,produciveinvesorssellnewbubblesounproducive invesors and use he proceeds o inves, raising average invesmen efficiency and he income of nexperiod syoung. ThiseffecincreaseshedemandforoldbubblesandshifshescheduleE x +1 downwards. Thissecondeffecoperaeswheneverx 1 ε,andidominaeshefirseffeconly ifx A. Hence,ifA>1 ε,bubblecreaionbyproduciveinvesorscannorelaxhecondiion for he exisence of bubbly episodes. If A 1 ε, bubble creaion { does relax he condiion } for he exisence of bubbles. Namely, A hiscondiionbecomes<max A+δ, 1. Figure 5 provides some inuiion for 1+4 (1 ε) δ his resul by ploing E x +1 againsx, using Equaion (20)and assuminghax NU =0while 17
x NP =x NP s if=sand x NP = 0 ifx (0,A] (1 ε,1] 1 ε x ifx (A,1 ε], forall>s. Thelefpanelshowshecaseinwhichbubblecreaionbyproduciveinvesorsdoes no affec he condiions for he exisence of bubbly episodes. The righ panel shows insead he case in which bubble creaion by producive invesors weakens he condiions for he exisence of bubbly episodes. This complees he proof of Proposiion 2. E x 1 E x 1 A 1 x A 1 x Figure 5 Wih financial fricions, he connecion beween bubbles and dynamic inefficiency becomes more suble. To deermine wheher a bubbly episodes can exis, we asked again: Is i possible o consruc apyramidschemehaisaracivewihouexplodingasashareoflaborincome? Wehaveshown haheanswerisaffirmaiveifandonlyifhereexissochasicprocessesforx NP, x NU andx such ha: E x +1 <x. These processes exis if and only if saisfies he resricion in Proposiion 2. To deermine wheher he economy is dynamically inefficien, we ask again: Is he economy accumulaing oo much capial? This quesion is now ricky because here are wo ypes of invesmens. The condiion ha aggregae invesmen be higher han aggregae capial income, i.e. (1 x ) (1 ) k > k, 18
asks wheher he average invesmen exceeds he income i produces. Even if his were no he case, he economy migh sill be dynamically inefficien since i migh conain pockes of invesmens ha exceed he income hey produce. We need o check for his addiional possibiliy. Invesmens byunproduciveinvesorsequal ( 1 ε x x NP ) (1 ) k,whileheircapialincomeisgiven δ (1 ε x x NP ) by A+(1 δ) x NP k δ x. Therefore, hese invesors consiue a pocke of dynamic inefficiency if and only if: ( 1 ε x x NP ) (1 ) k δ (1 ε x x NP ) > A+(1 δ) x NP δ x. k (22) Sraighforward algebra shows ha his condiion holds if and only if here exis sochasic processes forforx NP andx suchha: E x +1 <x +x NP. A When A+δ < 1 1+4 (1 ε) δ hisresricionisweakerhanhecondiionforheexisenceof bubbly episodes in Proposiion 2. The inuiion for his resul is ha someimes bubbly episodes can only exis if here is enough bubble creaion. This requires he economy o be no only dynamically inefficien, bu o be sufficienly so o suppor bubble creaion. This discussion sheds some ligh on he analysis of Abel e al. (1989). The finding ha aggregae invesmen falls shor of aggregae capial income sill implies ha > 0.5. Under his parameer resricion, financial fricions are crucial for bubbly episodes o exis and heir removal would eliminae hese episodes a once. Bu i does no follow ha, under his parameer resricion, bubblyepisodescannoexis. Thisisforworeasons: (i)if0.5<< A A+δ,inhefundamenal saeherearepockesofdynamicinefficiencyhawouldsupporabubbleifiwereopopup; A and (ii) if A+δ < 1, here are no pockes of dynamic inefficiency in he 1+4 (1 ε) δ fundamenal sae bu an expansionary bubble ha lowers he reurn o invesmen would creae such pockes iself. This second case brings a simple bu powerful poin home: wha is required for bubbly episodes o exis is ha he economy be dynamically inefficien during hese episodes and no in he fundamenal sae. Bubbles ha crowd in capial can conver a dynamically efficien economy ino a dynamically inefficien one. 3.4 The macroeconomic effecs of bubbles revisied We have shown ha financial fricions weaken he condiions for bubbly episodes o exis. We show nex ha hey also modify he macroeconomic effecs of bubbly episodes. To do his, we rewrie 19
helawofmoionofhecapialsockusinghedefiniionofx andx NP : [ A+(1 δ) x NP δ x ] (1 ) k if k +1 = (1 ) k x if x +x NP 1 ε x +x NP 1 ε <1 1. (23) Equaion (23) describes he dynamics of he capial sock for any admissible sochasic process forhebubble, i.e. x NP, x NU andx ; saisfyingequaions(20)and(21). Onceagain, noeha bubbly episodes are akin o shocks o he law of moion of capial. As in he Samuelson-Tirole model, bubbles can grow, shrink or randomly flucuae hroughou hese episodes. The macroeconomic effecs of bubbly episodes depend on wheher x NP is smaller or greaer δ han x. If smaller, bubbly episodes are conracionary and hey lower he capial sock 1 δ and oupu. If greaer, bubbly episodes are expansionary and hey raise he capial sock and oupu. 8 Conracionary episodes are similar in all regards o hose analyzed in he Samuelson- Tirole model. As for expansionary episodes, heir macroeconomic effecs are illusraed in Figure 6. Assume iniially ha he economy is in he fundamenal sae so ha he appropriae law of moion is he one labeled k+1 F. This law of moion for he fundamenal sae is ha of he sandard Solow model. Assume ha, iniially, he capial sock is equal o he Solow seady sae 1 sohak =k F [A (1 )] 1. Whenanexpansionarybubblepopsup,ireducesunproducive invesmens and uses par of hese resources o increase producive invesmens. As can be seen fromequaion(23),helawofmoionduringhebubblyepisodeliesabovek F +1 : inhefigure,kb +1 represens he iniial law of moion when he episode begins. As a resul, growh urns posiive. Throughou he episode, k B +1 may shif as he bubble grows or shrinks. The capial sock and oupu, however, unambiguously increase relaive o he fundamenal sae. Evenually, he bubble burssandheeconomyreurnsoheoriginallawofmoionk F. 8 Evenwihinasingleepisode,heeffecsonhecapialsockandoupumighvaryhroughimeoracrosssaes of naure. 20
k 1 B k 1 F k 1 k F k Figure 6 Expansionary and conracionary episodes differ also in heir implicaions for consumpion. In his model, consumpion is sill expressed by Equaion (13). Therefore, regardless of heir ype, allpresenbubblesraiseheincomeinhehandsofheoldandhusconsumpion. However,he effec of pas bubbles depends on heir ype. If hey were conracionary, he curren capial sock and herefore consumpion are lower. If insead hey were expansionary, he curren capial sock and herefore consumpion are higher. 3.5 Discussion The Samuelson-Tirole model has been criicized because he condiions for he exisence of bubbly episodes and heir macroeconomic effecs seem boh unrealisic. We have shown ha hese criicisms do no apply o he model wih financial fricions. In paricular, (i) bubbly episodes are possible even if aggregae invesmen falls shor of capial income and; (ii) bubbly episodes can be expansionary. The criiques o he Samuelson-Tirole model herefore sem from he assumpion ha financial markes are fricionless and raes of reurn o invesmen are equalized across invesors. We can summarize our findings on he connecion beween bubbles and financial fricions wih hehelpoffigure7. Thelinelabeled C provides,foreachδ,helargeshaisconsisenwih heexisenceofconracionaryepisodes. 9 Thelinelabeled E provides, foreachδ, helarges 9 Theseepisodesarepossibleifandonlyifheeconomyisdynamicallyinefficieninhefundamenalsae. Therefore, C = A A+δ. 21
ha is consisen wih he exisence of expansionary episodes. 10 These lines pariion he (,δ) spaceinofourregions. 11 1 0.5 IV I C III II E 1 Figure 7 BubblyepisodesarepossibleinRegionsII-IV,bunoinRegionI.InRegionsIIandIII,< C andconracionaryepisodesarepossible. InRegionIIIandIV,< E andexpansionaryepisodes arepossible. Wecanhinkofδasameasureofhecossoffinancialfricions. Thehigherisδ,he smaller are he gains from borrowing and lending and he smaller are he coss of financial fricions. In he limiing case of δ 1, he Samuelson-Tirole model applies: only conracionary episodes arepossibleandhisrequires<0.5. Asδ decreases,hecondiionsforexisencearerelaxedfor bohypesofbubblyepisodes. Inhelimiingcaseofδ 0,financialfricionsareverysevereand all ypes of bubbly episodes are possible regardless of. 10 Fomally,heexisenceofaexpansionarybubblyepisoderequiresheexisenceofariple { x,x NP } saisfyinge x +1 <x andx δ 1 δ <xnp,wheree x +1 isasinequaion(20). Thismeansha: 1 1+4 (1 ε) δ E = A (1 δ) δ+a (1 δ) when δ 0.5 ε 1 ε when δ> 0.5 ε 1 ε,x NU 11 Figure7hasbeendrawnunderheassumpionhaε<0.5. ThisguaraneeshaRegionIVexiss. 22
4 Bubbles and long-run growh The Samuelson-Tirole model predaes he developmen of endogenous growh models. To maximize comparabiliy, he model wih financial fricions used he same producion srucure. No surprisingly, i has lile o say abou he relaionship beween bubbles and long-run growh. We now generalize he producion srucure and allow for he possibiliy of consan or increasing reurns o capial. We show ha he condiions for exisence of bubbly episodes do no change. However, even ransiory episodes have permanen effecs and can even lead he economy ino or ou of negaive-growh raps. 4.1 Seup wih long-run growh We assume ha he producion of he final good consiss of assembling a coninuum of inermediae inpus,indexedbym [0,m ]. Thisvariable,whichcanbeinerpreedashelevelofechnology in period, will be obained endogenously as par of he equilibrium. The producion funcion of he final good is given by he following symmeric CES funcion: m 1 y =η µ q m dm 0 µ, (24) where q m denoes unis of he variey m of inermediae inpus and µ > 1. The consan η is a normalizaion parameer ha will be chosen laer. Throughou, we assume ha final good producersarecompeiive,andwenormalizehepriceofhefinalgoodoone. Producion of inermediae inpus requires labor and capial. In paricular, each ype of inermediaeinpum [0,m ]isproducedaccordingohefollowingproducionfuncion, q m =(l m,v ) 1 (k m,v ) a, (25) wherel m,v andk m,v respecivelydenoeheuseoflaborandcapialocoverhevariablecossof producing variey m. Besides his use of facors, he producion of any given variey requires he paymenofafixedcosf m givenby 1=(l m,f ) 1 (k m,f ) a ifq m >0 f m = 0 ifq m =0, (26) 23
where l m,f and k m,f respecively denoe he use of labor and capial o cover he fixed coss of producing variey m. To simplify he model, we assume ha inpu varieies become obsolee in one generaion and, as a resul, all generaions mus incur hese fixed coss. I is naural herefore o assume ha he producion of inermediae inpus akes place under monopolisic compeiion and free enry. Thisproducionsrucureisaspecial caseofhaconsideredby Venura(2005). 12 Heshows ha, under he assumpions made, i is possible o rewrie Equaion(24) as y =k µ, (27) where we have chosen unis such ha η = (µ) µ (1 µ) µ 1. Mainaining he assumpion of compeiive facor markes, facor prices can now be expressed as follows w =(1 ) k µ and r = k µ 1. (28) Equaion(27) shows ha here are wo opposing effecs of increasing he sock of physical capial. On he one hand, such increases make capial abundan and hey have he sandard effec of decreasing is marginal produc. The srengh of his diminishing-reurns effec is measured by. On he oher hand, increases in he sock of capial expand he varieies of inpus produced in equilibrium, which has an indirec and posiive effec on he marginal produc of capial. The srengh of his marke-size effec is measured by µ. If diminishing reurns are srong and markesizeeffecsareweak( µ<1)increasesinphysicalcapialreducehemarginalproducofcapial. Ifinseaddiminishingreurnsareweakandmarke-sizeeffecsaresrong( µ 1)increasesin physical capial raise he marginal produc of capial. 4.2 Equilibria wih bubbles How does his generalizaion of he producion srucure affec he dynamics of bubbles and capial? Forabubbleobearacive,isexpecedreurnmusbealeasequalohereurnoinvesmen. 12 TheAppendixconainsaformalderivaionofheequaionshafollow. 24
Formally, his requiremen mus now be wrien as follows, E { b +1 b +b NP +b NU } =δ k µ 1 [ δ k µ 1 = k µ 1, k µ 1 ] if if if b +b NP (1 ε) (1 ) k µ b +b NP (1 ε) (1 ) k µ b +b NP (1 ε) (1 ) k µ <1 =1 >1, (29) which is a generalizaion of Equaion (17). The dynamics of he capial sock, in urn, are now given by, k +1 = (1 ) A k µ +(1 δ) b NP δ b if (1 ) k µ b if b +b NP (1 ε) (1 ) k µ b +b NP (1 ε) (1 ) k µ <1 1, (30) whichisageneralizaionofequaion(19). Foragiveniniialcapialsockandbubble,k 0 >0and b 0 0,acompeiiveequilibriumisasequence { k,b,b NP } =0 saisfyingequaions(6),(29) and(30).,b NU 4.3 Bubbly episodes wih long-run growh This generalizaion of he producion srucure expands he se of economies ha can be analyzed wih he model. I does no, however, affec he condiions for he exisence of bubbly episodes. Thais, Proposiion 2 applies for any valueof µ. Toseehis, definex b NP b NU b (1 ) k µ, x NP (1 ) k µ andx NU (1 ) k µ. OnceweapplyourrecursiverickoEquaion(29), we recover Equaion(20). Hence, bubble dynamics are sill described by Equaions(20) and(21). I is useful a his poin o provide an addiional characerizaion of he condiion for dynamic inefficiency. Combining Equaions(22) and(30) we find ha he economy has pockes of dynamic inefficiency if and only if he growh rae exceeds he reurn o unproducive invesmens, G +1 ( k+1 k ) µ δ k µ 1 +1 R U +1. (31) This condiion is quie inuiive. Remember ha, for a bubble o exis, i mus grow fas enough obearacivebunooofasoougrowisdemand,i.e. isgrowhmusbebeweeng +1 and R U +1. WhenhecondiioninEquaion(31)failshereisnoroomforsuchabubble. Considerfirshecaseinwhich µ<1. ConracionarybubblesreduceG +1 andincreaser U +1. 25
This is why conracionary episodes areonly possibleif, in hefundamenal sae, G +1 >R U +1. Expansionarybubbles,however,increaseG +1 andlowerr U +1. Thisiswhyexpansionaryepisodes aresomeimespossibleevenif,inhefundamenalsae,g +1 <R U +1. Consider nex he case in which µ 1. Now, marke-size effecs dominae diminishing reurns and he relaionship beween he capial sock and he reurn o invesmen is reversed. Conracionary bubbles sill reduce G +1 bu now hey also reduce R U +1. Despie his, we sill find ha conracionary episodes are only possible if, in he fundamenal sae, G +1 > R U +1. The reason, of course, is ha he decrease in R U +1 is small relaive o he decrease in G +1. Expansionary bubbles sill increase G +1 bu now hey also increase R+1 U. Despie his, we sill findhaexpansionaryepisodesmighbepossibleevenif,inhefundamenalsae,g +1 <R U +1. Thereason,onceagain,ishaheincreaseinR U +1 issmallrelaiveoheincreaseing +1. 4.4 The macroeconomic effecs of bubbles Wecanrewriehelawofmoionofhecapialsockusinghedefiniionsofx andx NP : k +1 = [ A+(1 δ) x NP δ x ] (1 ) k µ (1 ) k µ x if if x +x NP 1 ε x +x NP 1 ε <1 1. (32) Equaion(32) describes he dynamics of he capial sock for any admissible sochasic process for hebubble,i.e. x NP,x NU andx ;saisfyingequaions(20)and(21). Onceagain,noehabubbly episodes are akin o shocks o he law of moion of capial. As in previous models, bubbles can grow, shrink or randomly flucuae hroughou hese episodes. If diminishing reurns are srong andmarke-sizeeffecsareweak,i.e. µ<1,allheanalysisofsecion3applies. Therefore,we resric he analysis o he new case in which diminishing reurns are weak and marke-size effecs aresrong,i.e. µ 1. An ineresing feaure of bubbly episodes when µ 1is ha, even if hey are ransiory, hey can have permanen effecs on he levels and growh raes of capial and oupu. We illusrae hiswihhehelpoffigure9. Thelefpaneldepicshecaseofanexpansionarybubble. Iniially, heeconomyisinhefundamenalsaesohaheappropriaelawofmoionisheonelabeled k F +1. Since he iniial capial sock is below he seady sae, i.e. k < k F [(1 ) A] 1 1 µ, growh is negaive. We hink of his economy as being caugh in a negaive-growh rap. When an expansionary bubble pops up, i reduces unproducive invesmens and uses par of hese re- 26
sources o increase producive invesmens. During he bubbly episode, he law of moion of capial lies above k+1 F : in he figure, kb +1 represens he iniial law of moion when he episode begins. Throughouheepisode,k+1 B mayshifashebubblegrowsorshrinks. Growhmaybeposiive if, during he bubbly episode, he capial sock lies above is seady-sae value as shown in he figure. Evenually, he bubble burss bu he economy migh keep on growing if he capial sock aheimeofbursingexceedsk F. Thebubblyepisode,houghemporary,leadsheeconomyou of he negaive-growh rap and i has a permanen effec on long-run growh. k 1 B k 1 F k 1 k 1 kf 1 kb 1 k B k k F k k F k k B k Figure 8 Naurally, i is also possible for bubbles o lead he economy ino a negaive growh rap hereby having permanen negaive effecs on long-run growh. The righ panel of Figure 9 shows an example of a conracionary bubble ha does his. This secion has exended he model o allow for endogenous growh. This does no affec he condiions for he exisence of bubbly episodes. I does, however, affec some of heir macroeconomic effecs. Firs, he behavior of he reurn o invesmen during bubbly episodes is reversed. Second, bubbly episodes can have long-run effecs on economic growh. 5 Furher issues and research agenda We have developed a sylized model of economic growh wih bubbles. In his model, financial fricions lead o equilibrium dispersion in he raes of reurn o invesmen. During bubbly episodes, unproducive invesors demand bubbles while producive invesors supply hem. Because of his, bubbly episodes channel resources owards producive invesmen raising he growh raes of capial and oupu. The model also illusraes ha he exisence of bubbly episodes requires some 27