Capital Regulation in a Macroeconomic Model with Three Layers of Default

Transcription

1 Capial Regulaion in a Macroeconomic Model wih Three Layers of Defaul Lauren Clerc Alexis Derviz Caerina Mendicino Sephane Moyen Kalin Nikolov Livio Sracca Javier Suarez Alexandros Vardoulakis June 214 Absrac We develop a model which aims a providing a framework for he posiive and normaive analysis of macroprudenial policies. The basic model incorporaes opimizing financial inermediaries ( bankers ) who allocae heir scarce wealh ( inside equiy ) ogeher wih funds raised from saving households across wo lending aciviies, morgage lending and corporae lending. Exernal financing for all borrowers (including banks) akes he form of exernal deb which is subjec o defaul risk. The model shows he inerplay of hree inerconneced ne worh channels as well as disorions due o deposi insurance, and can be exended o analyse he implicaions of securiizaion and liquidiy risk. The seup allows an explici welfare analysis of macroprudenial policies. The projec was developed as par of he ESCB Macro-prudenial Research Nework (MaRs). We are graeful o seminar paricipans a he EABCN conference in Cambridge, he Second Conference of MaRs, ESSIM 214, he WGEM, and a he Sveriges Riksbank, Norges Bank, Banco de Porugal, Bank of England, Czech Naional Bank, Polish Naional Bank, Inernaional Moneary Fund, Offi ce of Financial Research a he US reasury, Banka Slovenije, Deusche Bundesbank for useful feedback on his projec. We also hank Michal Brzoza-Brzezina, Andrea Gerali, Philipp Harmann, Donghun Joo, Galo Nuño, Michael Kumhof and Rafael Wouers for valuable commens and suggesions and o Dominik Supera for superb research assisance. The opinions expressed in his paper belong o he auhors do no necessarily reflec he views of he Eurosysem or he Federal Reserve Sysem. Banque de France, Financial Sabiliy Deparmen. Czech Naional Bank Banco de Porugal, Economic and Research Deparmen. Deusche Bundesbank, Research Deparmen. European Cenral Bank, DG Research. European Cenral Bank, DG Inernaional. CEMFI and CEPR. Federal Reserve Board of Governors, Offi ce of Financial Sabiliy. 1

2 Keywords: Macro-prudenial policy; Financial fricions; Defaul risk. JEL: E3, E44, G1, G21 2

3 Non-echnical summary We develop a heoreical model which aims a providing a framework for he posiive and normaive analysis of macroprudenial policies. The basic model incorporaes opimizing financial inermediaries ( bankers ) who allocae heir scarce wealh ( inside equiy ) ogeher wih funds raised from saving households across wo lending aciviies, morgage lending o borrowing households and corporae lending o borrowing firms. The exernal financing for all borrowers (including banks) akes he form of no coningen deb which is subjec o defaul risk due o borrowers exposure o boh idiosyncraic and aggregae risk facors. Defauls are assumed o cause deadweigh losses as in he cosly sae verificaion seup of Gale and Hellwig (1985). Households and firms leverage is an endogenous muliple of heir ne worh. In conras, banks, which are assumed o obain heir ouside funding in he form of governmen-guaraneed deposis, have heir leverage limied by a regulaory capial requiremen. Imporanly, in spie of he presence of deposi insurance, we assume deposiors o suffer some ransacion coss if heir banks fail. This generaes a risk premium ha acs as an imporan source of amplificaion when bank solvency is weak. The model exhibis he operaion of hree inerconneced ne worh channels, which creae he poenial for amplificaion and propagaion noed in various srands of he exising lieraure (including he one operaing hrough he price of housing, which is he collaeral used by he borrowing households), as well as disorions due o deposi insurance. While limied ne worh ypically leads o under-invesmen, he risk subsidizaion linked o deposi insurance creaes he poenial for an excessive supply of bank credi. The basic model is hen suiable for a non-rivial welfare analysis of various forms of leverage regulaion (maximum loan-o-value or deb-o-income raios for morgages, capial requiremens imposed on banks lending aciviies, counercyclical capial charges, e ceera), ha are likely o be he core of macroprudenial policy. We use he basic model developed in his paper o analyze he effecs of capial requiremens on seady sae and on he ransmission of various ypes of shocks. Bu he basic 3

4 model can be exended o allow for he possibiliy of securiizaion and liquidiy risk (e.g. in he form of inerim funding shocks suffered by banks). These exensions may allow expanding he analysis o he regulaory reamen of securiizaion, liquidiy regulaion, and he assessmen of lending of las resor policies. While he basic model belongs o he class of non-moneary models, inroducing nominal rigidiies and a meaningful role for moneary policy consiues a naural hird possible exension ha would allow us o assess he ineracions beween macroprudenial policy and moneary policy. 4

5 1 Inroducion The financial crisis has clearly illusraed ha prudenial regulaion uniquely based on he soundness of individual insiuions is no suffi cien o ensure financial sabiliy. As a consequence, i has been proposed o develop a new approach which facors in he macroeconomic perspecive and akes ino accoun he connecions beween he financial and he nonfinancial secors, and he cenral role of financial inermediaries. So-called macro-prudenial policies should herefore be designed o address he conribuion of financial sabiliy, sysemic risk and he pro-cyclicaliy of he financial sysem o overall economic performance. However, lile is known abou he general equilibrium effecs of he several macroprudenial insrumens proposed so far. Policy makers demand analyical frameworks wih which o assess he implicaions and effeciveness of such insrumens, and address he opimal design of macro-prudenial policy. Beyond incorporaing financial fricions and disorions, a good model for policy analysis in his area should pu he banking sysem and financial inermediaion more generally a he cenre of he sage. A number of recen papers has focused on inroducing bank fricions ino oherwise mainsream macroeconomic models. Some of hese papers (e.g. Gerler and Kiyoaki 21, Meh and Moran 21) describe a bank ne worh channel ha operaes essenially along he same lines as he convenional enrepreneurial ne worh channel (noably in Bernanke, Gerler and Gilchris 1999, henceforh BGG) bu causing flucuaions in he availabiliy of bank credi raher han direcly on enrepreneurial invesmen. Mos papers eiher focus on one main bank fricion (like he bank ne worh channel) or oherwise capure several fricions in a reduced form manner, wihou explicily modeling he opimizing behavior of financial inermediaries and wihou explicily addressing he welfare analysis of macroprudenial policies. 1 The purpose of his paper is o encompass he mos relevan aspecs of macroprudenial 1 See he repor of he Mars research nework for a survey. See hp:// researcher_mars.en.hml. 5

6 concern in a single model wih enough microfoundaions o allow us o perform a welfare analysis of macroprudenial policy. The model is buil wih an eclecic perspecive, rying o provide a synhesis of he mos relevan inerlinkages beween he real and he financial secors idenified in he lieraure. Puing ogeher some of he mechanisms previously analyzed on a sand alone basis or from a more parial equilibrium perspecive may idenify relevan sources of feedback and clarify he relaive quaniaive relevance of each channel. This mulidimensional approach is essenial o provide coheren advice o he new macroprudenial auhoriies. While his paper is focused on bank capial regulaion, he key microprudenial policy ool and arguably one of he main ools for macroprudenial policy as well, he model presened here is rich enough o allow for he analysis of oher candidae macroprudenial insrumens, such as loan o value raios. The second main goal of he paper is o provide a model where defaul plays a cenral and maerial role. Up o he global financial crisis and even beyond i, he role of defaul has been largely overlooked in macroeconomics. A number of papers, following Kiyoaki and Moore (1997), allow for he possibiliy of defaul, bu rule i ou in equilibrium hrough appropriaely chosen financial conracs. Oher papers, following he seminal BGG model, do allow for defaul in equilibrium, bu consider sae coningen deb ha prevens defaul o unexpecedly flucuae due o aggregae shocks. Therefore, boh approaches absrac from some of he consequences of defaul for financial sabiliy and subsequenly for he real economy. In our work, defaul impinges on he balance shee of he lenders, influencing heir opimal behaviour and hereby macroeconomic oucomes. Moreover, overall defaul risks can arise from boh aggregae and idiosyncraic risk. For a discussion on he imporance o inroduce defaul in macro-models, see Geanakoplos (211) and Goodhar, Tsomocos and Shubik (213). Our model belongs o he class of dynamic sochasic general equilibrium (DSGE) models, bu several aspecs of is consrucion make i quie disan from he ypical DSGE model in use by cenral banks prior o 27, which some observers blame for having ignored he relevance of financial inermediaion and financial sabiliy. By he same oken, our 6

7 consrucion is far away from he classical framework of microprudenial supervision, which was very limied in is analysis of he impac of macroeconomic performance on financial inermediaion and lacked he mission and he analyical ools o properly consider he impac of prudenial policies on general macroeconomic performance. Hence, we ry o bridge he gap beween he micro and macro lieraure and build a framework which allows for he welfare analysis of he relevan policy insrumens. More in deail, he model developed in his paper ries o coherenly pu ogeher he following ingrediens: (i) non-rivial lending and borrowing decisions in he household secor, wih some households demanding bank loans for he purchase of housing, (ii) non-rivial borrowing decisions in he corporae secor, wih firms demanding bank loans for he funding of heir capial accumulaion, (iii) non-rivial defaul risk in all classes of borrowing, including he borrowing ha financial inermediaries obain from lending households in he form of bank deposis, (iv) a ne worh channel operaing a he level of each levered secor, indebed households, indebed enrepreneurs, and banks, and (v) a bank funding fragiliy channel which operaes hrough he premium requesed by risk averse deposiors who suffer some deadweigh losses if banks defaul. The raionale for macro-prudenial policies in our model arises from wo disorions. Firs, exernal financing fricions resuling from he possibiliy of defaul, which in he form of bankrupcy coss (or cosly sae verificaion, CSV), resric access o credi and may resul in oo lile borrowing compared wih a firs bes world wihou hese fricions (CSV disorion). Second, we assume ha he governmen guaranees he principal and ineres of bank deposis in full (hough no he ransacion coss suffered by he deposiors of a failed bank). This pushes banks o ake on risk a he expense of he deposi insurance agency (DIA), which may resul in more lending and borrowing in equilibrium han wha a social planner would find opimal when inernalizing he full coss of bank defaul (limied liabiliy disorion). The ineracion of hese wo disorions poins in opposie direcions and has a ne resul ha may imply seady sae levels of credi o he various secors above or below he ones ha a social planner would opimally choose. Indeed, we show ha seady sae 7

8 household welfare in our model is a hump-shaped funcion of credi availabiliy as deermined by bank capial regulaion. 2 We use he model o analyze he effecs of capial requiremens on seady sae and on he ransmission of various ypes of shocks. On op of ime-invarian capial requiremens, possibly differeniaed across classes of loans according o heir risk, we also consider couner-cyclical adjusmens o he capial raios (adjusmens ha, in pracice, migh be implemened in he form of a counercyclical capial buffer, as in Basel III). Three main resuls sand ou in our analysis. Firs, in he conex of our model, here is generally an opimal level of capial requiremens. In effec, capial requiremens reduce bank leverage, bank failure risk and he implici subsidies associaed wih deposi insurance. Simulaneously, hey force he banks o make a greaer use of bankers limied wealh. This second aspec makes capial requiremens have a poenial impac on he cos of equiy funding (due o is scarciy in he shor run) and on he paern of accumulaion of wealh by bankers (in he medium o long run). Lower leverage and, in he shor run, a larger cos of equiy funding lead banks o exend less credi and o be less fragile. However, oo high levels of capial requiremens may unduly resric credi availabiliy. Second, we find ha when bank leverage is high (because capial requiremens are low), he economy is more responsive o shocks. This shows ha limied liabiliy and he implici subsidies associaed wih deposi insurance, which allow banks o mee he required rae of reurn on equiy wih lower lending raes, consiue a poenially powerful channel of financial amplificaion. Third, he couner-cyclical adjusmen of capial raios may significanly improve he benefis of high capial requiremens, bu once again only up o a cerain level. Oherwise, he effecs due o he increase in bank fragiliy will backfire. This is, in fac, wha happens when he reference capial requiremens are ex ane oo low: heir furher relaxaion in 2 This hump-shaped welfare funcion is concepually similar o Benigno e al. (213), where i is no clear a priori wheher here is oo much or oo lile credi and wheher regulaion should resric or subsidize credi. However, heir model is subsanially differen from ours. 8

9 response o a negaive shock can be harmful. On one hand, a counercyclical reducion in he requiremens may allow he bank o charge lower loan raes on a larger amoun of loans. On he oher hand, if bank fragiliy ges furher deerioraed, banks funding coss will increase. This increase may hen well off-se he inended impac of he counercyclical adjusmen, causing long-lasing derimenal effecs on credi supply and GDP. The res of he paper is organized as follows. In Secion 2, we firs discuss how he paper fis in he exising lieraure. In Secion 3 we inroduce he key elemens of he core model, describing he equilibrium equaions ha direcly emanae from agens opimizaion. In Secion 4, we complee he se of equilibrium equaions wih hose referred o marke clearing. Secion 5 describes he calibraion of he model. Secion 6 conains he main posiive and normaive resuls for he model. Secion 7 concludes. The Appendix summarizes he main model equaions and provides a formal definiion of equilibrium. 2 Relaion o previous lieraure Our model builds on a large lieraure which includes financial fricions in general equilibrium models, including among ohers Carlsrom and Fuers (1997), BGG, and Kiyoaki and Moore (1997). Financial fricions are ypically found o increase he persisence of shocks and o amplify heir impac, hough his is no necessarily he case in all models. Brunnermeier e al. (212) provide a survey of he lieraure on he macroeconomic modelling of financial fricions. Brzoza-Brzezina e al. (213) compare he properies of models wih collaeral consrains like in Kiyoaki and Moore wih models wih an exernal finance premium as in BGG, concluding ha he business cycle properies of he exernal finance premium framework are more in line wih empirical evidence. A bunch of papers have incorporaed banking in DSGE models; hese include Curdia and Woodford (28), Goodfriend and McCallum (27), Gerali e al. (21), Meh and Moran (21), and Chrisiano, Moo and Rosagno (21). As in our paper, his lieraure mosly focuses on direc lending by banks, excluding securiizaion and invesmen banking 9

10 aciviies. In mos of his work he emphasis is on he role of bank lending in he propagaion of shocks (ypically moneary policy shocks) or in he opimal conduc of moneary policy, no on he ineffi ciencies associaed wih financial fricions ha creae a role for macro-prudenial policies. Moreover, in hese papers defaul normally does no feaure prominenly or a all. An excepion is Angeloni and Faia (213), who focus on capial regulaion in a model where banks are fragile and subjec o runs, and he main disorion arises from he fac ha he projecs funded by banks may be subjec o cosly early liquidaion. Our modeling of defaul coss hroughou he paper resembles he well-known seup of BGG. However, similar o Benes and Kumhof (211) bu in conras o BGG we do no allow deb repaymens o be coningen on he realizaion of aggregae shocks. This assumpion implies a resricion in he conracing space: conracs are incomplee in ha hey canno be made fully coningen on aggregae variables (perhaps due o verifiabiliy problems, publicaion lags, poenial manipulabiliy if conracually relevan, ec.). The emphasis on defaul is similar o several models ha have analyzed macroprudenial issues ouside he DSGE radiion, such as Goodhar, Sunirand and Tsomocos (26) and Goodhar e al. (212). Van den Heuvel (28) and Repullo and Suarez (213) analyze opimal bank capial regulaion in simple finie horizon models wih financial fricions. In a way, our paper provides a bridge beween his lieraure and he more radiional DSGE models wih financial fricions. Recen work by Mark Gerler and coauhors (Gerler and Karadi 211; Gerler and Kiyoaki 21; Gerler, Kiyoaki and Queralo 211) has also emphasized he presence of financing consrains for banks as a key facor in he propagaion of shocks. In heir work, he financing consrain arises from he fac ha bankers can choose o diver a fracion of available funds away from he funded projecs and in favour of he household of which hey are a member. To preven his, bankers need o have some posiive ne worh in he projec. In our work, he fricion comes from he possibiliy of overinvesmen (or excessive risk aking, from he sociey s poin of view) due o limied liabiliy, no from he possibiliy of diversion 1

11 (as also suggesed by Cole 211). 3 In addiion, we model explicily he inermediaion chain ha links saving households ha inves in bank deposis wih household and corporae borrowers ha obain bank loans. This allows us o keep rack of he ransmission of defaul risk across secors. Hirakaa e al. (213) also consider he full inermediaion chain and allow for borrowers and banks defaul. Our reamen differs from heirs in a leas wo imporan aspecs. Firs, hey consider a monopolis banker se up in which he conracs linking he bank wih deposiors and borrowers, respecively, are solved for joinly in order o maximize he uiliy of he banker. Secondly, Hirakaa e al. (as Angeloni and Faia, 212) consider uninsured deposis while we assume deposi insurance, and hink of is effecs on banks risk aking as one of he reasons for he need for macroprudenial policy. Collard e al. (212) also look a he socially excessive risk aking by banks due o limied liabiliy and deposi insurance and analyze he inerplay beween prudenial and moneary policy insrumens. In a similar vein, Marinez-Miera and Suarez (212) focus on he effec of capial requiremens on banks incenive o exend loans wih poenially highly correlaed defauls in case a so-called sysemic (rare bu poenially devasaing) shock occurs. Our model belongs o he class of DSGE models and we solve i wih sandard mehods, i.e. perurbaion mehods. This depars from Brunnermeier and Sannikov (214), who use a coninuous ime mehodology o solve for he full dynamics of a model wih financial fricions. Their model is very sylized and feaures producive expers and less producive households. Owing o financial fricions he wealh of expers is imporan for he expers abiliy o buy physical capial and use i producively. Brunnermeier and Sannikov find ha he sysem s reacion is highly nonlinear and he financial sysem is inherenly unsable; mos of he effecs are asymmeric and only arise in a downurn. The fac ha he analysis ha we perform does no allow o asses he imporance of non-lineariies in our seup should be aken as an imporan cavea. 3 In Iacoviello (213) bankers are more impaien and are subjec o a capial consrain so as o preven hem from borrowing in an unlimied way. 11

12 Our paper shares he goal of finding a raionale for macroprudenial policies wih papers ha have recenly pu he emphasis on pecuniary exernaliies, including Jeanne and Korinek (21), Bianchi (211) and Bianchi and Mendoza (21), Gersbach and Roche (212), and Chrisiano and Ikeda (213). The pecuniary exernaliies arise due o he fac ha agens do no inernalize he effec of heir acions (e.g. borrowing and invesmen decisions) on he prices of housing and physical capial, which in urn affec agens collaeral consrains. In our model he endogenous leverage of households and firms is also affeced by hese prices, so pecuniary exernaliies migh also play a role, bu i is hard o predic in which direcion (and o which exen) hey disor he allocaion of credi in our economy. In models wih pecuniary exernaliies, he bidirecional relaionship beween asse prices and financial consrains may produce feedback loops ha resul in excessive volailiy of hese prices and of macroeconomic variables more generally, calling for policies ha promoe financial sabiliy. Similar mechanisms may operae in our model bu assessing formally he welfare cos of such added volailiy is challenging (since requires assessing welfare ou of seady sae) and is beyond he scope of his firs exploraion of our model. In a less normaive vein, we analyze he poenial sabilizaion role of capial regulaion when he economy is hi by several ypes of shocks. 3 Model se-up We consider an economy populaed by households, enrepreneurs and bankers, whose main characerisics are as follows: Households. Households are risk-averse and infiniely lived and derive uiliy from a sorable consumpion good and from a durable good, housing, which provides housing services o is owners. The consumpion good acs as a numeraire in our economy and is price is normalized o one. The price of housing is denoed q H. Similar o Iacoviello (25) and subsequen lieraure here are wo ypes of households ha differ in heir discoun facor 12

13 (paien and impaien). Moreover, hey are grouped in wo disinc represenaive dynasies which provide risk-sharing o heir members: he saving dynasy and he borrowing dynasy. In equilibrium, he saving households consume ou of heir curren income and save he res for fuure use, while he borrowing households ake morgage deb from banks agains heir holdings of housing. Morgage deb is provided o he individual members of he dynasy agains heir individual housing unis on a limied-liabiliy non-recourse basis. This implies he possibiliy of defauling on morgage deb a an individual level wih he only implicaion for he borrower of losing he housing good wih which he morgage is secured. Thus, in conras o Iacoviello (25), morgage loans feaure defaul risk and, in case of defaul, he repossession of collaeral by he banks involves verificaion coss similar o hose considered in BGG. Boh ypes of households supply labour in a compeiive marke. Enrepreneurs. Enrepreneurs are risk neural agens specialized in owning and mainaining he sock of physical capial, which hey ren in each period o he firms involved in he producion of he consumpion good. 4 They live across wo consecuive periods and derive uiliy from ransferring a par of heir final wealh o he saving dynasy (o whom hey may be inerpreed o serve as an alruisic agen) and from leaving anoher par as a beques o he nex generaion of enrepreneurs (who can be inerpreed as heir heirs). Each of hese pars can be realisically inerpreed as dividends and reained earnings, respecively. Wih his OLG formulaion, which will also be posulaed for he bankers, we will be able o generae he same sor of dynamics for enrepreneurial and banking ne worh as in BGG. Is main virue is o allow us o keep he number of agens o care abou in he welfare calculaions o a minimum (he paien and he impaien households) wihou neglecing any of he consumpion capaciy generaed in he economy. Enrepreneurs finance heir iniial purchases of capial parly wih he inheried ne worh and parly wih corporae loans provided by banks. Similar o household loans, corporae loans are subjec o limied 4 A possible inerpreaion is ha physical capial would suffer prohibiively high depreciaion raes if owned and mainained by any oher class of agens. 13

14 liabiliy and defaul risk, and recovering residual reurns from bankrup enrepreneurs leads banks o incur verificaion coss. Bankers. Bankers are he providers of inside equiy o perfecly compeiive financial inermediaries ha we call banks. Like enrepreneurs, hey are risk neural agens who live across wo consecuive periods, and derive uiliy from making ransfers o he saving dynasy and leaving bequess o he nex cohor of bankers. 5 The acive bankers use he wealh inheried from he previous generaion of bankers o provide equiy funding o banks. A he end of each period, he gross reurn on such equiy is fully disribued o he bankers, who in urn disribue i o he paien households ( dividends ) and he nex cohor of bankers ( reained earnings ). To complee he model overview, we need o refer o banks, consumpion good producing firms, and capial good producing firms: Banks. Banks ouside funding is made up of fully insured deposis raised among he saving households. Banks operae under limied liabiliy and may defaul due o boh idiosyncraic and aggregae shocks o he performance of heir loan porfolios. Deposi insurance is funded wihin each period by levying lump-sum axes on paien households, if needed. From he sandpoin of savers, however, we assume ha recovering he fully insured principal and ineres of heir deposis in case of bank failure is cosly in erms of ime and effor, so ha deposis may sill pay a risk premium which will co-move wih bank s defaul risk. Bankers inside equiy conribuions are necessary for he banks o comply wih he exogenous regulaory capial requiremen. Banks are of wo ypes, who lend o impaien households (banks H) and o enrepreneurs (banks F ) respecively. 5 The induced dynamics of bankers ne worh is similar o ha in Gerler and Karadi (211), where a fracion of households become bankers a random in every period and remain as bankers in subsequen periods wih some probabiliy. 14

15 Producion of he consumpion good. There is a perfecly compeiive consumpion good producing secor made up of firms owned by he paien households. These firms combine capial rened from enrepreneurs wih household and enrepreneurial labour inpus in order o produce he consumpion good. This secor is no direcly affeced by financial fricions. Producion of he capial good and housing. There is a perfecly compeiive capial good and housing producing secor made up of firms owned by paien households. Like in Gerler, Kiyoaki and Queralo (211), hese firms opimize ineremporally in response o changes in he price of capial and are subjec o echnological illiquidiy in he form of invesmen adjusmen coss. Also his secor is no direcly affeced by financial fricions. We now urn o describe he model in deail. For he sake of breviy, he full se of model equaions, including he firs order condiions, is in he Appendix. 3.1 Households The economy is populaed by wo represenaive dynasies made up of a measure-one coninuum of ex ane idenical households each. Households are risk averse and maximize some ime-separable expeced uiliy funcions. One dynasy, idenified by he superscrip s, is made of relaively paien households wih a discoun facor β s. The oher dynasy, idenified by he superscrip m, is made of more impaien households wih a discoun facor β m β s. Thus, in equilibrium, he paien households save and he impaien households borrow. Dynasies provide consumpion risk sharing o heir members and are in charge of aking mos household decisions. 6 6 This laer feaure is convenien for he soluion of he model wih sandard echniques (i.e. avoiding kinks). 15

16 3.1.1 Saving households The dynasy of paien households maximizes [ E (β s ) [log +i ( c+i) s + v s +i log ( ] ) h s ϕ s ( ) ] +i +i l s 1+η 1 + η +i (1) i= where c s denoes he consumpion of non-durable goods and h s denoes he oal sock of housing held by he various members of he dynasy. l s denoes hours worked in he consumpion good producing secor, wih η he inverse of he Frisch elasiciy of labor supply. v s and ϕ s are preference parameers which can vary over ime (poenially causing flucuaions in, e.g., he equilibrium price of housing). Specifically, he housing preference parameer v s follows he saionary process ln v s = (1 ρ v ) ln v s + ρ v ln v 1 s + ε v, (2) where v s > is he consan long-erm average of he process, ρ v is he persisence parameer and ε v is an i.i.d. whie noise shock wih variance σ 2 v (housing demand shock). The disuiliy of labor parameer ϕ s follows he saionary process ln ϕ s = (1 ρ ϕ ) ln ϕ s + ρ ϕ ln ϕ s 1 + ε ϕ,where ϕ s > is he consan long-erm average of he process, ρ ϕ is he persisence parameer and ε ϕ is an i.i.d. whie noise process wih variance σ 2 ϕ (demand shock). The paien households dynamic budge consrains read as follows c s + q H h s + d w l s + q H (1 δ H )h s 1 + R 1d D 1 T + Π s (3) where q H is he price of housing, δ H is he (possibly ime-varying) rae a which housing unis depreciae, w is he wage rae, and R D 1 = R D 1(1 γp D b ) (4) where R D 1 is he fixed (gross) ineres rae received a on he savings deposied a banks a 1 in he previous period. The principal and ineres of bank deposis are fully guaraneed 16

17 by a deposi insurance agency (DIA) ha, for simpliciy, is assumed o ex pos balance is budge by imposing a lump sum ax T on paien households. However, we assume ha households incur a linear ransacion cos γ whenever hey have o recover he funds deposied in a failed bank. So P D b sands for he fracion of deposis in banks ha fail in period (which can be compued as he average deposi-weighed bank defaul rae realized in period ). 7 This ransacion cos inroduces a link beween he bank probabiliy of defaul and bank funding coss and a wedge beween he rae of reurn on deposis and he risk free rae. A he same ime, we model i in a way ha does no affec he incenives for any individual bank and, hus, preserves he usual disorions associaed wih limied liabiliy and deposi insurance (i.e. banks incenives o ake excessive risks a he expense of he DIA). Finally, Π s includes he profis accruing o he saving households from he ownership of he capial good and housing producing firms as well as he dividend ransfers received from enrepreneurs and bankers. The housing depreciaion rae is ime-varying and subjec o AR(1) shocks: δ H = δ H + ε δh (5) ε δh = ρ δh ε δh + v δh (6) where v δh is an i.i.d. shock Borrowing households The objecive funcion of he represenaive dynasy of impaien households has he same form and parameers as (1), excep for he discoun facor which for hem is β m < β s and will induce his dynasy o borrow raher han save in equilibrium. This explains he differences in heir dynamic budge consrains, ha read as follows: c m + q H h m b m w l m + max { ω m q H (1 δ H )h m 1 R m 1b m 1, } df m (ω m ) T m, (7) 7 P D b is specified furher in Secion 3.4. For evidence ha bank failure is cosly o deposiors even in he presence of deposi insurance, see Brown e al. (213). 17

18 where b m is he dynasy s aggregae borrowing from he banking sysem and R 1 m is he conracual gross ineres rae on he housing loan of size b m 1 agreed wih a bank in he previous period. The erm in he inegral reflec he fac ha he housing good and he deb secured agains i are assumed o be disribued across he individual households ha consiue he dynasy. Each impaien household experiences a he beginning of each period an idiosyncraic shock ω m o effi ciency unis of housing owned from he previous period and have he opion o (sraegically) defaul on he non-recourse housing loans associaed wih hose unis. 8 The shock ω m is assumed o be independenly and idenically disribued across he impaien households, and o follow a log-normal disribuion wih densiy and cumulaive disribuions funcions denoed by f m ( ) and F m ( ), respecively. The shock makes he effecive resale value of he housing unis acquired in he previous period be q H = ω m q H (1 δ H ) and, given ha defaul is cosless for households, makes defaul on he underlying loan ex pos opimal for he household whenever ω m q H (1 δ H )h m 1 < R 1b m m 1. 9 This explains he presence of he max operaor in he inegral in (7). Housing loans In each period, afer he realizaion of he idiosyncraic shock ω m, each individual household decides wheher o defaul on he individual loans aached o he housing held from he previous period and he residual ne worh is passed on o he dynasy, which is no liable for any unpaid deb. The dynasy hen makes he decisions on consumpion, housing, labor supply and deb for period and allocaes hem evenly across is members. Flucuaions in he ne worh of he dynasy (as capured by he las erm in he righ hand side of (7)) are driven by he changes in he ne worh of he loan-repaying households as well as he realizaion of zero ne worh from all housing unis owned by members ha 8 This shock is inended o capure idiosyncraic flucuaions in he value of houses and can be inerpreed as a reduced form represenaion of a sudden improvemen or worsening in he neighborhood, in he social equipmen available nearby or in he resource cos of mainaining he propery. See also Forlai and Lamberini (211) who use a similar formulaion. 9 See Geanakoplos (23) for a discussion of he ex pos opimaliy of his ype of behavior by he borrower, and Goodhar e al. (212) for an exension o he analysis of morgage conracs backed by housing collaeral. 18

19 defaul on heir housing loans. Defaul in period will occur for where ω m ω m = xm 1, R H R H qh (1 δ H ) q H 1 is he ex pos average realized gross reurn on housing, and x m Rm b m q H h m is a measure of a household leverage. The fracion of defauled morgages a period can hen be expressed as F m (ω m ) and he ne worh accruing o he dynasy ou of is aggregae housing invesmen in he previous period can be wrien as ( ) (ω m ω m )df (ω m ) R H q 1h H m 1. (8) Φ m ω m Now, using he same inermediae noaion as in BGG, we can more compacly wrie Φ m = (1 Γ m (ω m ))R H q H 1h m 1, (9) where Γ m (ω m ) = ω m ω m f (ω m ) dω m + ω m ω m f (ω m ) dω m. (1) The variable Φ m can be inerpreed as ne housing equiy afer accouning for repossessions of defauling households. Since he borrowing households will defaul on he loans aken a period according o a similar paern of behavior, he erms of such loans mus saisfy he following paricipaion 19

20 consrain for he lending bank: 1 (1 Γ H (ω H +1))(Γ m ( ω m +1) µ m G m ( ω m +1) )R H +1 q H h m ρ φ H b m. (11) Inuiively, his consrain says ha he bankers who conribue equiy φ H b m bank (where φ H o he lending is he capial requiremen on housing loans) should expec a gross expeced reurn on heir conribuion a leas as high as some marke-deermined required rae of reurn ρ which is exogenous for any individual bank alhough endogenous in aggregae (his will be deermined laer). Therefore, ρ φ H b m bank. measures oal gross equiy reurns for a given The expression in he lef hand side of he inequaliy accouns for he oal equiy reurns associaed wih a porfolio of housing loans o he various members of he impaien dynasy. The erm µ m G ( ω m +1) reflecs he proporional verificaion coss µ H incurred in he repossession of he fracion G m ( ω m +1) of housing unis which defauling loans were borrowing agains, where G m ( ω m +1) = ω m +1 ω m +1f m ( ω m +1) dω m +1. The facor (1 Γ H (ω H +1)) plays a similar role o he facor (1 Γ m (ω m )) in (9) and accouns for bank leverage and he possibiliy ha an individual bank ha lend o households (idenified by he superscrip H) fails due o suffi cienly adverse idiosyncraic or aggregae shocks o he performance of is porfolio of housing loans. The full deails of he hreshold ω H +1 of he idiosyncraic shock below which he bank fails are presened in subsecion 3.4. Noe ha he assumpion of limied liabiliy and he fac ha bank liabiliies (deposis) are insured, a bank can mee he required reurn on equiy wih a lower lending rae. This suggess ha he limied liabiliy disorion acs in he direcion of expanding credi availabiliy for enrepreneurs and impaien households. I should also be emphasized ha he probabiliy of a defaul even hinges on R H, an aggregae variable which may be influenced by aggregae shocks (say, produciviy shocks). 1 In principle, he borrowing rae R m is par of he housing loan conrac and, hence, can be reaed as par of he decision variables of he impaien dynasy in period. However, reaing he inermediae variable x m as par of he conrac variables (ogeher wih b m and h m ) allows us o wrie he enire conrac problem wihou explici reference o R m. 2

21 Therefore, defaul is a funcion of boh idiosyncraic and aggregae shocks, unlike in BGG and mos of he lieraure on modelling financial fricions. Borrowing households opimizaion problem The decision problem of he borrowing households can be compacly wrien as a conracing problem beween he corresponding represenaive dynasy and is bank: max {c m +i,hm +i,lm +i,xm +i,bm +i } i= [ E (β m ) [log +i ( c+i) m + v m +i log ( ] ) h m ϕ m ( ) ] +i +i l m 1+η 1 + η +i (12) i= subjec o he budge consrain of he dynasy, ( ( )) x c m + q H h m b m w l m + 1 Γ m m R R +1q H H +1 H h m T m, (13) and he paricipaion consrain of he bank, ( ( ) ( )) ] x E [(1 Γ H (ω H +1)) Γ m m x µ m G m m R H R+1 H R+1 H +1 q H h m = ρ φ H b m, (14) which we impose wih equaliy wihou loss of generaliy. 3.2 Enrepreneurs We consider a sequence of overlapping generaions of wo-period lived risk-neural enrepreneurs. 11 Each generaion of enrepreneurs inheris wealh in he form of bequess n e from he previous generaion of enrepreneurs. Enrepreneurs are he only agens who can own and mainain he capial sock. They purchase new capial from capial goods producers and depreciaed capial from he previous generaion of enrepreneurs, and hen ren i o he conemporaneous producers of he consumpion good. Enrepreneurs finance heir capial holdings wih heir own iniial ne worh n e and wih loans b e received from he banks specialized in corporae loans. An enrepreneur born a ime values he ransfers made o he paien dynasy a ime 11 This assumpion makes enrepreneurs risk-neural in heir decision-making as assumed in BGG. 21

22 +1 ( dividends ), c e +1, and he bequess lef o he nex cohor of enrepreneurs ( reained earnings ), n e +1, according o he uiliy funcion (c e +1) χe (n e +1) 1 χe. Thus, once in period + 1, he enrepreneur will solve max (c e +1) χe (n e +1) 1 χe (15) c e +1,ne +1 subjec o: c e +1 + n e +1 W e +1. Opimizing behavior hen yields he dividend rule and he earnings reenion rule c e +1 = χ e W e +1 (16) n e +1 = (1 χ e )W e +1. (17) These rules guaranee easy aggregaion and generae he same ype of ne worh dynamics as in Bernanke, Gerler and Gilchris (1999). Assuming ha enrepreneurs derive uiliy from ransferring consumpion capaciy o he saving households raher han from heir own consumpion will allow us o focus our welfare analysis on households expeced lifeime uiliy bu wihou neglecing he consumpion capaciy associaed wih he profis of enrepreneurial firms. We reurn o his issue when describing bankers and in secion 6.1. The decision problem of he enrepreneur who sars up a can hen be wrien as: max k,b e,rf subjec o he period resource consrain E (W e +1) (18) q K k b e = n e, (19) 22

23 he definiion W e +1 = max [ ω e +1 ( r k +1 + (1 δ +1 ) q K +1 ) k R F b e T e +1, ], (2) and a bank paricipaion consrain which will be fully specified in he nex subsecion. In hese expressions, q K enrepreneur in period, b e is he price of capial a period, k is he capial held by he is he amoun borrowed from he bank in period, δ is he ime-varying depreciaion rae of each effi ciency uni of capial, r K effi ciency uni of capial, and R F he bank in period. Finally, T e o finance he deposi insurance. is he renal rae per is he conracual gross ineres rae of he loan aken from is he lump sum ax possibly imposed on he enrepreneurs Noe ha he depreciaion rae δ is ime-varying because we also consider a depreciaion shock ε δ, whereby and v is an i.i.d. shock. δ = δ + ε δ (21) ε δ = ρ δ ε δ + v δ (22) The facor ω e +1 ha muliplies he reurn from capial holdings is an idiosyncraic shock o he enrepreneur s effi ciency unis of capial. This shock realizes afer he period loan wih he bank is agreed and prior o rening he available capial o consumpion good producers in ha dae. Wih a role similar o he shock ω m +1 suffered by he housing held by borrowing households, he shock ω e +1 is a simple way o raionalize he exisence of idiosyncraic shocks o he enrepreneurs performance and o generae a non-rivial defaul rae on enrepreneurial loans. The shock is independenly and idenically disribued across enrepreneurs and follows a log-normal disribuion wih an expeced value of one, and densiy and cumulaive disribuion funcions denoed f e ( ) and F e ( ), respecively. Similar o all oher borrowers in our economy, an enrepreneur canno be held liable for any conraced repaymens due o banks (which amoun R F b e in period + 1) over and above he gross reurns ha she obains on he capial invesmen underaken in he 23

24 previous period, ( r K +1 + (1 δ +1 ) q K +1) ω e +1 k. This is why we have he max operaor in (2): i akes ino accoun limied liabiliy and he possibiliy ha enrepreneurs defaul on heir bank loans Enrepreneurial loans Le R K +1 = rk +1 + (1 δ +1 ) q K +1 q K denoe he gross reurn per effi ciency uni of capial obained in period + 1 ou of capial owned in period. Then he enrepreneur will repay her loan a + 1 whenever her idiosyncraic shock ω e +1 exceeds he following hreshold: ω e +1 RF b e q K k 1 R K +1 xe R K +1 (23) where x e RF be q K k denoes enrepreneurial leverage as measured by he raio of conracual deb repaymen obligaions a + 1, R F b e, o he value of he capial purchased a, q K k. Noice ha (23) implies (differenly from BGG where he conracual deb repaymens are made coningen on R K +1) ha flucuaions in R K +1 will (realisically) produce flucuaions in enrepreneurial defaul raes. When an enrepreneur defauls on her loan, he bank only recovers a fracion 1 µ e of he gross reurn of he capial available o he defauled enrepreneur, where µ e sands for verificaion coss incurred by he bank when aking possession of he reurns and selling he underlying capial o oher enrepreneurs. Hence a bank recovers R F b e from performing loans and (1 µ e ) R K +1q K ω e k from non-performing loans. Ex ane, lenders recognize ha under cerain realizaions of he idiosyncraic and he aggregae shocks, enrepreneurs will go bankrup, especially when heir ex ane leverage x e is high. The division beween enrepreneurs and heir bank of he oal gross reurns on a welldiversified porfolio of enrepreneurial invesmens a period can be compacly expressed 24

25 using noaion similar o he one already inroduced for borrowing households: Γ ( ) ω e e ω e = ω e +1f ( e ω+1) e dω e +1 + ω e +1 f ( e ω+1) e dω e +1, (24) ω e +1 which gives he share of he gross reurns (gross of verificaion coss) ha will accrue o he bank, and G ( ) ω e e ω e = ω e +1f ( e ω+1) e dω e +1, (25) which denoes he par of hose reurns ha comes from defauled loans. Then he verificaion coss incurred by he bank on is porfolio of loans o enrepreneurs will be µ e G e ( ω e +1), and he ne share of he oal gross reurns of he porfolio ha he bank appropriaes can be expressed as Γ e ( ω e +1) µ e G e ( ω e +1). We will use his expression below when inroducing he bank s paricipaion consrain ino he enrepreneur s opimizaion problem Enrepreneurs opimizaion problem The conracing problem beween he enrepreneur and her bank in period can be wrien as one of maximizing he enrepreneur s expeced wealh a + 1 [( ( )) x max E 1 Γ e e R K+1q ] K k x e,k R+1 K subjec o he paricipaion consrain of he bank: ( ( ) ( ))] x E [(1 Γ F (ω F +1)) Γ e e x µ e G e e R R+1 K R +1q K K +1 K k = ρ φ F (q K k n e ), (26) which we can wrie wih equaliy wihou loss of generaliy. Jus like in he case of he bank exending loans o impaien households in (1), equaion (26) saes ha he expeced payoffs appropriaed by he equiy holders of a bank which holds a porfolio of loans o enrepreneurs mus be suffi cien o guaranee he expeced rae of reurn ρ ha he bankers require on he wealh ha hey conribue o bank. Bankers equiy conribuion, φ F (q K k n e ), is deermined by he need o comply wih a capial requiremen φ F on each uni of lending. 25

26 The facor (1 Γ F (ω F +1)) ha muliplies he lef hand side of (26) accouns for bank leverage and he possibiliy ha an individual bank specialized in corporae loans (idenified by he superscrip F) fails due o suffi cienly adverse idiosyncraic or aggregae shocks o he performance of is porfolio of enrepreneurial loans. The full deails of he hreshold ω F +1 of he idiosyncraic shock below which he bank fails are presened below in subsecion 3.4. The final wealh of he enrepreneurs ha sar up in period can be wrien as: ( ( )) 1 Γ e W+1 e ω e +1 R K = { +1 (1 1 E ( ( Γ F ω+1)) ( F Γ e ω+1) e µe G ( e ω+1)) e R K }n e +1 ρ φ F and, since, a fracion (1 χ e ) of such wealh is lef as a beques o nex generaion of enrepreneurs, he law of moion of enrepreneurs aggregae iniial ne worh can be wrien as ( ( )) 1 Γ ω e n e +1 = (1 χ e ) +1 R K { +1 (1 1 E ( ( Γ F ω+1)) ( F Γ e ω+1) e µe G ( e ω+1)) e R K }n e +1. ρ φ F 3.3 Bankers We model bankers in a very similar way o enrepreneurs: here are overlapping generaions of risk-neural wo-period lived bankers. Bankers have exclusive access o he opporuniy of invesing heir wealh as banks inside equiy capial. Each generaion of bankers inheris wealh in he form of bequess n b from he previous generaion of bankers and leaves bequess n b +1 o subsequen one. Aggregae banker ne worh deermines, for given capial requiremen, he equilibrium required rae of reurn on bank equiy and hence he lending raes. A banker born a ime values he ransfers o he paien dynasy a + 1, c b +1, and he bequess lef o he nex cohor of bankers, n b +1, according o he uiliy funcion ( c b +1) χ b ( n b +1 ) 1 χ b. The banker who sars up a period receives a beques from he previous generaion of bankers and decides how o allocae his wealh as inside capial across he wo classes of exising banks: he banks specialized in housing loans (he H banks) and he banks specialized in enrepreneurial loans (he F banks). There is a coninuum of ex 26

27 ane idenical perfecly compeiive banks of each class. The ex pos gross reurn a + 1 on he inside equiy invesed in H and F banks a is denoed ρ H +1 and ρ F +1, respecively. If a banker saring up wih wealh n b invess an amoun e F in inside equiy of one or several F banks, and he res in one or several of he H banks, his ne worh afer one period will be: ( ) W+1 b = ρ F +1e F + ρ H +1 n b e F, (27) which he banker will devoe o ransfers c b +1 and bequess n b +1 by solving max ( ) c b χ b +1 (n b +1 ) 1 χb (28) c b +1,nb +1 subjec o: c b +1 + n b +1 W b +1. Opimizing behavior yields, he dividend rule and he earnings reenion rule c b +1 = χ b W b +1 (29) n b +1 = (1 χ b )W b +1. (3) Then he porfolio problem of he banker who sars up a can be wrien as follows: ( ) max E (W b +1) =E ( ρ F +1e F e F + ρ H +1 n b e F ). (31) So, inerior equilibria in which boh classes of banks receive sricly posiive inside equiy from bankers will require he following equaliy o hold where ρ E ρ F +1 = E ρ H +1 = ρ, (32) denoes bankers required expeced gross rae of reurn on equiy invesmens underaken a ime. This expeced reurn is endogenously deermined in equilibrium bu 27

28 boh individual banks and bankers ake i as given in heir decisions. Specifically, ρ plays an essenial role in wriing of he bank paricipaion consrains, (11) and (26), ha appear in he problems ha deermine he erms of he deb conracs esablished beween each class of banks and heir borrowers. Finally, he law of moion of he iniial wealh available o each generaion of bankers when hey sar up is: n b +1 = ( 1 χ b) ( ) [ ρ F +1e F + ρ H +1 n b e F ]. (33) This OLG se up makes bankers risk neuraliy operae as in a one-period model bu makes bank capial an imporan variable for aggregae dynamics (such as e.g. in Gerler and Kiyoaki, 211). As in he case of enrepreneurs, he ransfer of c b +1 o he savings households will allow us o focus he welfare analysis on households lifeime uiliy wihou neglecing he consumpion capaciy associaed wih bank profis. 3.4 Banks The banks which issue he equiy bough by bankers are insiuions specialized in exending eiher morgages or corporae loans. A bank lass for one period only: i is an invesmen projec creaed a and liquidaed a + 1. We assume a coninuum of banking insiuions of each class j = H, F. The equiy payoffs π j +1 generaed by a represenaive of class j afer is period of operaion is given by he posiive par of he difference beween he reurns from is loans and he repaymens due o is deposis: [ π j +1 = max ω j R ] j +1 +1b j R D d j,, (34) where b j and d j are he loans exended and he deposis aken by he bank a, respecively, R D is he gross ineres rae paid on he deposis aken a (which is uniform across all banks given he presence of deposi insurance), and R j +1 denoes he realized reurn on a well diversified porfolio of loans of class j. The max operaor reflecs he fac ha he 28

29 shareholders of he bank enjoy limied liabiliy, so heir payoffs canno be negaive. The bank s idiosyncraic failure risk comes from he exisence of an idiosyncraic porfolio reurn shock ω j +1 which is i.i.d. across he banks of class j and is assumed o follow a lognormal disribuion wih a mean of 1 and a disribuion funcion F j (ω j +1). Bank defaul will be driven by flucuaions in he aggregae loan reurn R j +1 (iself driven by firms or households defaul raes) and he bank-idiosyncraic shock ω j +1 which can be inerpreed as a shock o each individual bank s abiliy o exrac payoffs from is loans. A possible inerpreaion for his shock is a produciviy shock in he producion of banking services. When a bank fails, is equiy is wrien down o zero and is deposis are aken over by he deposi insurance fund which pays ou all deposis in full. The deposi insurance fund recoups his by aking over he failed bank s loan porfolio minus resoluion coss which are assumed o be a µ j fracion of oal bank asses. The bank also faces a regulaory capial consrain: e j φ j b j, (35) where φ j is he poenially ime-varying capial-o-asse raio of banks of class j. Thus, he bank is resriced by regulaion o hold equiy o a leas a fracion φ j of he asses i holds on is balance shee. I is possible o show ha in equilibrium he capial requiremen holds wih equaliy, so ha he bank s loans can be wrien as b j = ej and is deposis as d j φ j = (1 φj )ej. φ j Allowing he capial requiremen φ j o vary across differen classes of banks is consisen wih hinking of hem as risk-based (like under Basel III) or as secoral requiremens serving as ools of macroprudenial policy. Noe ha he role of he capial requiremen is he mirror image of he effec of he limied liabiliy ha was described earlier. A higher capial requiremen reduces leverage, forcing banks o ge funded wih a larger share of equiy, which, due o is scarciy, is more expensive. Moreover, i reduces he probabiliy of bank defaul and hence he subsidy implied by he deposi insurance, leading o a higher lending rae and more resriced access o credi for 29