House Price Bubbles and Debt Default in a DSGE model *

Transcription

1 House Price Bubbles and Deb Defaul in a DSGE model * Rachaar Nilavongse Job Marke Paper Deparmen of Economics Uppsala Universiy November 9 4 Absrac This paper develops a micro-founded model of morgage defaul in a dynamic sochasic general equilibrium (DSGE) framework. We model a housing bubble burs as a negaive shock o he expeced fuure housing prices. The collapse of curren housing prices induces borrowing-consrained households o defaul on heir morgages as heir home equiy becomes negaive. The morgage defauls impair bank balance shees and he banks resore heir balance shees by reducing heir lending o he household secor and he business secor. The model can explain he following key aspecs of he U.S. financial crisis: () he decline in acual housing prices; () he rise in he amoun of morgage defauls; (3) he rise in ineres rae spreads; (4) he decline in he yield on Treasury bills; and (5) he conracion of he real economy; and (6) he increase in he governmen deb levels. JE classificaion: E3 E44 G G. Keywords: Banking secor financial crisis house prices ineres rae spreads morgage defauls * rachaar.nilavongse@nek.uu.se Mailing Address: Deparmen of Economics Uppsala Universiy P.O. Box Uppsala Sweden.

2 Inroducion The bursing of he U.S. housing bubble in 7 led o he wors economic downurn since he Grea Depression. As he housing prices fell morgage defauls increased dramaically. The rise in morgage defauls and he decline in he housing prices negaively affeced bank balance shees. Banks repaired heir balance shees by shifing heir asses from household and business loans oward safer asses such as U.S. reasury bills. The increase in ineres rae spreads on household and business loans exacerbaed he economic downurn. There are six noable feaures of he recen U.S. financial crisis; () he decline in acual housing prices; () he rise in he amoun of morgage defauls; (3) he rise in ineres rae spreads; (4) he decline in he yield on Treasury bills; and (5) he conracion of he real economy; and (6) he increase in he governmen deb levels. This paper inroduces a micro-founded model of morgage defaul in a dynamic sochasic general equilibrium (DSGE) framework. We use he model o explain he feaures of he U.S. financial crisis. Bernanke and Gerler (999) model a bubble shock as an exogenous shock o asse prices which capures a non-fundamenal shock in heir model. We model a housing bubble burs as a negaive shock o he expeced fuure housing prices. A fall in he expeced fuure housing prices leads o a decline in curren housing prices. The collapse of he curren housing prices brings he morgage loan repaymen o be above he housing value. The negaive home equiy riggers borrowing-consrained households o defaul on heir morgages. In our model he morgage defauls impair he bank balance shees. The banks cu back on heir loans o he household secor and he business secor in order o resore he bank balance shees. As a consequence his leads o a credi crunch. The main conribuion of his paper is o inroduce a micro-founded model of morgage defaul wih a housing bubble shock in a DSGE model and we use he model o explain he key characerisics of he U.S. financial crisis. This paper is based on he lieraure on collaeral consrains such as Iacovoiello (5) Monacelli (9) and Jermann and Quadrini (). Borrowers face collaeral consrains ha are ied o heir value of physical capial or he value of house. Financial inermediaries are absen in hese papers. Our model has banks ha serve as financial inermediaries beween savers and borrowers. Kollmann e al. () inroduce ineres rae spreads in a perfec

3 banking compeiion framework where a represenaive bank faces a capial requiremen. Kollmann e al. () and Iacoviello (4) model loan defaul as an exogenous shock. In our paper household loan defaul is endogenous and is based on a micro-founded approach. Forlai and amberini () inroduce an endogenous household loan defaul in which heir model is based on he financial fricion model of Bernanke e al. (999) and Chrisiano e al (9). In heir seing an increase in he riskiness of housing invesmen riggers morgage defauls. In our seing he collapse of housing prices riggers morgage defauls which is in line wih he observaions from he recen U.S. financial crisis. This paper is organized as follows. Secion describes he model. Secion 3 presens he sochasic process. Secion 4 presens he calibraion of he model parameers. Secion 5 presens he quaniaive resuls from he model. Secion 6 discusses. Model The economy is composed of five ypes of agens. The agens in he economy are a paien household an impaien household (borrowing-consrained household) an enrepreneur a banker and he governmen. The bank akes deposis from he paien household and provides loans o he impaien household and he enrepreneur. The bank asses consis of household and commercial loans and governmen bonds. The bank faces a capial requiremen. The governmen finances is public expendiure hrough issuing bonds and axaion.. Paien household The expeced uiliy of a represenaive paien household is: ( NP ) h + E β P ln CP + nhln HP () = h + where β P is he discoun facor P C is curren consumpion H P denoes he holding of housing sock N P denoes labor hours ν h is he weigh on housing services and η is he inverse of he Frisch labor supply elasiciy. The household earns he deposi ineres rae R on he las period deposis D. The household receives he wage rae W P for supplying N P hours o he enrepreneur. The household uses his income for consumpion C P and for invesing in he houses 3

4 ( ) q HP HP and make deposis D wih a bank. The household pays he lump-sum ax T o he governmen. The paien household maximizes he expeced uiliy subjec o he following budge consrain: ( ) C + D + q H H = R D + W N () P P P P P where q is he real house price (he price of housing in unis of consumpion). The paien household chooses C P D H and N P o maximize () subjec o () and he firs-order condiions are: P = βpre C P C P + q ν h q + = + β PE CP H P C P + (3) (4) W = N C. (5) η P P P Equaion (3) is a sandard consumpion Euler equaion ha describes a rade-off beween he curren and he fuure consumpion and he curren consumpion depends negaively on he deposi ineres rae. Equaion (4) is he opimal holding of housing sock. The lef hand side is he marginal cos of acquiring a uni of housing sock q C P while he righ hand side is he marginal benefi of holding an exra uni of housing sock which comprises he marginal uiliy of housing services ν h H P q + and he expeced resale value of house β PE. The C P + opimal hours worked is deermined by equaion (5). The marginal benefi of working W S η equals he marginal cos of working N P CP.. Impaien Household The expeced uiliy of a represenaive impaien household (borrowing-consrained household) is given by: ( NM ) h + E β M ln CM+ nhln HM. (6) = h + 4

5 The discoun facor of he impaien household is βm < βp which will induce he impaien household o borrow raher han save in equilibrium. C M is he curren consumpion H denoes he holding of housing sock N M denoes labor hours ν h is a weigh on housing services and η is he inverse of he Frisch labor supply elasiciy. The budge consrain is: ( ) C + q H H + R Z = + W N. (7) M M M M M M M M M The household receives he wage rae W M for supplying N M hours o he enrepreneur. Thus he labor income is WM N M. The impaien household pays he lump-sum ax T o he governmen. The impaien household akes new loans M from a bank. The impaien household invess in he houses q( HM HM ) M. The amoun of he loans ha he impaien household agrees o pay back is RM where M RM is he gross real ineres rae on he las period loans. However he impaien household can defaul on heir loans by paying he loans back less han he conracual obligaions. Hence Z M is he amoun of morgage defauls. The household canno borrow more han a fracion The collaeral consrain is expressed as: ( + ) M M M M m M of he expeced value of he house. R m E q H. (8) Morgage defauls The impaien household defauls if he amoun of he household deb RM M exceeds he value of house ( ω) M + where ω is he qualiy of house wih a probabiliy densiy funcion of f ( ω ). A negaive value of ω decreases he value of he house. We assume ha R < M M M. The impaien household defauls if R M M M > ω where R M M M < and ω <. We le ω o have a negaive value i.e ω. Thus he defaul RM M region lies beween ω and ω =. M 5

6 The area of he defaul is R - ( ω) M M M We inegrae over he defaul area given he lower limi is ω and upper limi is R M M M. The amoun of morgage defauls Z M can be found by aking he following inegraion: ( RM - M -/ M -)- ZM = R M - M -- M -( + ω) f -ω ( ω) The above inegraion gives us he amoun of morgage defauls ha is: Z R M M M M = M ( ω) f ( ω) where f ( ω ) measures he sensiiviy of he morgage defauls o he house prices is a probabiliy densiy funcion of ω. We will show he simulaion of a sudden drop in housing prices in he secion 5. The collapse of he housing prices induces he impaien household o defaul as he value of he house is lower han he morgage loan repaymen. The impaien household chooses C M N M H M and M o maximize (6) subjec o (7) (8) and (9). The following firs order condiions are: (9) Z M + = βmrm E + λm RM βm E C M C M + C M + M where () Z M RM + M E = RM ( ω) f ( ω) > M E( q+ ) H M q ν h q Z + M + = + βme + λm mme( q+ ) + βm E CM H M C M + C M + H M where () Z M RM + M E = ( ω) ( RM M + ( ω) E( q+ ) HM ) f ( ω) < H M H M E( q+ ) H M W = N C. η M M M () Equaion () is an Euler condiion for consumpion wih he borrowing consrain and he endogenous loan defaul. λ M is he muliplier on he borrowing consrain (8). A decline in he house price has wo opposing forces on he impaien household s curren consumpion. A decline in he house price ighens he collaeral consrain; hence his leads o a drop in he The full derivaions of he endogenous morgage defaul are in he appendix. 6

7 curren consumpion. However a decline in he expeced fuure house price leads o an increase in he expeced defaul Z E M + M which in urn induces he impaien household o increase his curren consumpion. In general he former force is sronger han he laer force; herefore a decrease in he expeced housing prices will lead o a drop in he curren consumpion. The opimal housing sock is shown by equaion (). The lef hand side is he marginal cos of purchasing an exra uni of house. The righ hand side conains four componens which are he direc uiliy gain of an exra uni of holding housing sock ν h H M q + he expeced resale value of he house β ME C M increase in he value of collaeral m E ( q ) M M + he marginal benefi of an λ + and he marginal benefi from he expeced defaul β M C M + Z E H M + M. Given he level of household loans a decrease in he housing sock will increase he expeced defaul decision is given by equaion (). Z E H M + M. The impaien household's labor supply.3 Enrepreneur An enrepreneur has he following uiliy funcion: E βe = ( ln CE). (3) The discoun facor of he enrepreneur is βm < βp which will induce he enrepreneur o borrow raher han save in equilibrium. C E is he enrepreneur s consumpion. The enrepreneur produces oupu Y using a Cobb-Douglas echnology given by: α ( µ ν) ( α )( µ ν) Y = K H N N (4) µ ν E P M where is K capial (ha depreciaes a rae δ) H E is housing inpu N P and N M are he paien household and impaien household labor inpus respecively. α measures he labor income share of he paien household. The parameers µ and ν measure he capial share and housing share respecively. The flow of funds consrain is given by: 7

8 ( ) C + q H H + R + W N + W N + I + ξ E E E E E P P M M K = Y +. E (5) Wage paymens o he households are WU NU + WF NF and he capial invesmen is I. In addiional o he capial invesmen he enrepreneur faces capial adjusmen coss ξ ψ I = δ k k δ K K where ψ k is a parameer of he capial adjusmen coss. The enrepreneur invess in he housing marke q( HE HE ). is he new loans. The enrepreneur is able o fully repay back he loans RE E o he bank a he gross ineres rae of RE. The bank imposes a collaeral consrain ha is he value of he housing sock imes an exogenous loan-o-value raio ( + + ) E E E E m E. R me q H. (6) E The enrepreneur chooses C E K I E N P N M and H E o maximize (3) subjec o (4) (5) and (6). The firs order condiions are: = β R E + λ R C E E E E E CE + q β Y = E υ + q + m λ E q C C H ( ) E + + E E + E E + E (7) (8) ( ) Y W N α µ υ = (9) P P ( α )( µ υ) Y W N = M M () ψ k I µ = δ C + E δ K () ψ I I ψ I µ Y µ = γe δ δ + γ E + µ + ( δ). () C E + δ K K δ K CE + K Enrepreneur's Euler consumpion condiion is shown by equaion (7) where λ E is he muliplier on he enrepreneur s borrowing consrain (6). The inerpreaion of he enrepreneur s Euler consumpion condiion is similar o he impaien household s case. The opimal invesmen in he housing marke is presened by equaion (8). The benefi of an addiional invesmen in he housing marke comprises he fuure marginal produc of house 8

9 β Y E E υ C + E + H E housing collaeral m E ( q ) β E and he expeced resale value of house E q+ and he value of C E + λ +. Equaions (9) and () are he labor demand equaions. E E The decision of capial invesmen is given by () and ()..4 Governmen This secion describes he funcions of he governmen in his economy model. The governmen finances is public spending G hrough axaion and new deb issues. The governmen collecs he lump-sum ax T from he paien and he impaien households. The governmen sells he one-period bonds B G o he banker where q G is he price of he bonds. The value of new deb issues is qg B G. The governmen budge consrain can be expressed as: G + B = T + q B. (3) G G G The governmen spending follows a counercyclical fiscal policy ha is when oupu decreases he governmen increases he public spending o sabilize he economy. However he level of public expendiure decreases when he governmen deb levels rise. Thus he governmen expendiure rule is specified as: ( ρ ) G = G ρ B Y (4) G G G where ρ G reflecs he weigh on he public deb and < ρ G <. The ax funcion is: ( ρ ) T = T + ρ Y + B. (5) y y G The ax rae rises as he oupu and he governmen deb rise and oupu. ρ y is a weigh on he.5 Bank A banker has a preference defined as E βb = ( ln CB) (6) where he banker s discoun facor βb < βp C B is he banker s consumpion. The bank s asses consis of loans o he impaien household M and loans o he enrepreneur E. 9

10 The bank s asses also include he holding of governmen bonds qg BB where q G is he price of he governmen bonds and B G is he newly bough governmen bonds. The bank akes he deposi from he paien household o finance is loan operaion. Thus he bank capial is expressed as: K = + + q B D. (7) B M E G B I follow Kollmann e al. () in assuming ha he bank faces a capial requiremen. The bank capial o asse raio γ is. (see for insance he evidence in Van den Heuvel 8). This implies ha he bank capial K B should no be smaller han a fracion of he bank s holding of risky asses ( E M ) γ +. However he holding of governmen bonds is no subjec o he regulaory requiremen. The bank s excess capial beween he bank capial and he required capial: ( ) B E M X is defined as he gap X = K γ +. (8) The bank s excess capial can be rewrien as ( γ )( ) X = + + q B D. (9) E M G G I assume ha he bank incurs a posiive cos when he excess bank capial is negaive. Thus he cos funcion φ is a funcion of X and he cos funcion is a convex funcion ha is φ > φ ( ) =. The bank can choose o hold less capial han he argeed capial level bu his incurs a cos o he bank. The bank faces he loan losses when he impaien household defauls on he morgage loans. The amoun of loan losses Z M is defined as: Z R M M M M = M ( ω) f ( ω) The bank s flow of funds consrain is: ( ). (3) C + q B + R D φ X = D + R + R + B Z (3) B G B M E M M E E B M The expendiure side of he banker includes he curren consumpion C B buying new issued bonds from he governmen qg B B he ineres paymen R D o he paien household new loans o he impaien household M and he enrepreneur E he adjusmen cos of he bank capial φ ( X ). The banker s flow of income includes he saving household deposis he holding of governmen bonds BB he repaymen of loans by he enrepreneur RE E

11 and he repaymen of loans by he impaien household RM M. However he impaien household can defaul on he morgage loans and hus he amoun of morgage defaul is Z. An increase in Z M has a negaive impac on he bank capial and hus he banker s M flow of income. The bank chooses C B B B E F and D o maximize (6) subjec o (9) (3) and (3). The bank s firs order condiions are: β σ σ ( + ) ( + ) E C E C = + (3) ( X ) B B BR φ σ σ CB CB β R σ B ( CB + ) E = + ( γ) φ ( X ) (33) B E σ B CB σ B ( ) E CB + Z M ( ) ( ) + β BRM = γ φ X βbe σ B C + + B M (34) Z M RM + M where E = RM ( ω) f ( ω) > M E( q+ ) H M q G E = C σ B ( CB + ) σ B B + φ ( X ) (35) Equaion (3) deermines he opimal demand for paien household deposis. Equaion (33) deermines he opimal loan supply o he enrepreneur. Equaion (34) deermines he opimal loan supply o he impaien household. The expeced amoun of morgage defauls (loan losses) is included in he opimal loan supply o he household secor Z E M + M. A decline in house price may push he value of house o be lower han he loan repaymen. This leads o an increase in he expeced amoun of morgage defauls and his has a negaive impac on he bank capial. The bank is subjec o a capial requiremen ha is he bank capial should no be smaller han a fracion of he bank s risky asses which are loans o he impaien households and he enrepreneur. The bank is required o resore is capial o asse raio and he bank has a number of margins on which o adjus. The banker can raise he bank capial by reducing he demand for deposis from he paien household bu his decreases funds for making loans which in urn furher reduces he bank capial. The banker can fully absorb he losses from

12 morgage defauls by lowering his consumpion while mainaining he exising level of loans. However his is no opimal since he banker due o he curvaure of he banker s uiliy. Therefore he banker resores he capial o asse raio by reducing his consumpion aking fewer deposis and decreasing he supply of loans o he impaien household and he enrepreneur. The opimal demand of governmen bonds is described by equaion (35). The governmen bonds are considered o be safe asses; herefore hey are no subjec o he capial requiremen. Hence he deerioraion of he bank balance shee induces he bank o increase he holding of governmen bonds. 3 Sochasic process In his secion we discuss a sochasic process of he bubble process. We follow a bubble process as in Bernanke and Gerler (999). To capure he bursing of he U.S. housing bubble we inroduce a negaive shock o he expeced fuure housing prices. The sochasic process of he shock o he expeced fuure housing prices follows an AR() process; ε = ρε q + uq where < ρ q < and u q follows a whie noise process wih mean zero and variance σ. q The sochasic process of he shock o he expeced fuure housing prices is included in he paien household he impaien household and he enrepreneur s housing demand equaions. For example he impaien household s housing demand equaion () is rewrien as: q ν h εq Z + M + = + βme + λm mme( εq+ ) + βm E CM H M C M + C M + H M where Z M RM + M E = ( ω) RMM+ ( ω) E( εq ) HM f ω H M H M E( εq+ ) H M ( + ) ( ) An unanicipaed shock o he expeced fuure housing prices will have a direc impac on he value of he house. A sudden drop in he expeced fuure housing prices induces he impaien household o reduce he housing sock and o defaul on he morgage loan since he collapse of housing prices causes he value of he house o be below he amoun of loan repaymen.

13 4 Calibraion We calibrae he model o mach U.S. daa during he financial crisis. The model is solved by using a linear approximaion of he equilibrium condiions. The paien household s discoun facor β P is se a.995 in order o obain a seady sae ineres rae on deposis slighly abou 3 per cen on annual basis. We follow Iacoviello (4) and se he impaien household s discoun facor β M and enrepreneur s discoun facor β E a.94 o ensure ha he impaien household and he enrepreneur will borrow in equilibrium. The value of he weigh of housing in households uiliy funcion ν h is se a.5 which implies a raio of residenial housing o annual oupu around 4 percen. The housing share ν is se a.5 which implies a raio of commercial real esae o oupu of 6 percen. The household loano-value (TV) raio m M and he business TV raio m E are se a.75. The paien household s labor share α is.64. The values of labor supply elasiciy η share of capial µ in he producion funcion he depreciaion rae δ he capial adjusmen cos ψ k are chosen from he common values esablished in he DSGE lieraure. The following values of η µ δ and ψ k are.3.3 and respecively. The parameer on he qualiy of house ω is se o.4 o obain a seady sae ineres rae on household loan R F abou 4 percen on annual basis. f ( ω ) measures he sensiiviy of he morgage defaul o housing prices. f ( ω ) is se o. which implies he bank losses is abou 5 percen of GDP. This number is in line wih esimaes of loan losses by he Federal Financial Insiuions Examinaion Council. The values of banking parameers are drawn from Kollmann e al. () and Iacoviello (4). The curvaure of he cos of excess bank capial φ is.5. The capial sufficiency requiremen γ is se a.. The discoun facor of banker β is se a.945. Togeher wih choice of ω (describe above) hese numbers imply an annualized seady-sae lending rae on household loans enrepreneur R F o be of 4 percen and an annualized seady-sae lending rae on loans o RE o be abou 3.5 percen. B 3

14 5 Quaniaive resuls Impulse response o a negaive shock o expeced fuure house prices In his secion we show he simulaion of he bursing of he housing bubble. Figure shows he responses of he economy model o a percenage poin decrease in he expeced fuure housing prices. An unanicipaed negaive shock o he expeced fuure housing prices has a direc impac on he collaeral values of he borrowing consrained household (impaien household) and he enrepreneur. As he expeced fuure housing prices decreases his ighens he collaeral consrains. The ighening of he borrowing consrains induces he impaien household and he enrepreneur o inves less in he houses and hus leads o a fall in he curren housing prices. The fall in he collaeral values has a negaive impac on consumpion and invesmen. GDP decreases by abou.45 percenage poins invesmen by.7 percenage poins and consumpion by abou. percenage poins. The impaien household s home equiy is negaive since he decline in he curren house prices brings abou he household deb o be higher han he value of he house. As a resul he impaien household defauls on heir morgages. In his model he impaien household defaul immediaely upon reaching he negaive home equiy. Therefore here is a lile persisence in he impulse response of morgage defauls. An ineresing inerplay beween he household and he bank secor is hrough he morgage defaul channel. The morgage defauls have a negaive impac on he bank asses. The bank is subjec o a capial sufficiency requiremen ha is he bank capial should no be smaller han a fracion of he bank s risky asses which are loans o he impaien household and he enrepreneur. To resore he bank capial-asse raio he banker reduces loans o he household and enrepreneurs. The bank raises ineres rae margins on household and commercial loans. The surge in he ineres rae spreads cripple he economy which resuls in a decline in aggregae consumpion invesmen and employmen. The employmen is decreased by abou.3 percenage poins. The rise in he lending spreads and he plumme of he housing prices have negaive impacs on he demand for residenial houses and commercial real esae. Governmen bonds are considered o be safe asses; hus hey are no subjec o he capial requiremen. This induces he bank o increase he holding of governmen bonds o help resore is balance shee. An increase in he demand for governmen bonds decreases he yield 4

15 on he Treasury bills. As he GDP falls he governmen spending auomaically increases bu i does no sop he recession. The increase in he governmen spending brings abou a rise in he governmen debs. The model shows he inerconnecions beween he household he business he banking and he governmen secors. The model can capure he key aspecs of he U.S. financial crisis which are () he decline in housing prices; () he rise in he amoun of morgage defauls; (3) he rise in ineres rae spreads; (4) he decline in U.S. reasury yield; and (5) he conracion of he real economy; and (6) he increase in he governmen deb levels. Figure : Impulse responses o a negaive shock o he expeced house prices GDP.4 Employmen Consumpion Invesmen

16 .5 Morgage Defaul. Firm ending Spread Household ending Spread Bank Capial House Price Household oans Business oans Governmen Deb

17 5 Residenial Houses 4 Commercial Real Esae Treasury bill yield Governmen Spending Discussion This secion oulines he main key resuls from he model and discusses he value added of he paper. We develop a micro-founded model of morgage defaul in a DSGE framework. The main mechanism of he model is borrowing-consrained households defaul on heir morgages when he value of heir house is lower han he loan repaymen. We use his model o explain he key characerisics of he U.S. financial crisis which are () he decline in housing prices; () he rise in he amoun of morgage defauls; (3) he rise in ineres rae spreads; (4) he decline in U.S. reasury yield; and (5) he conracion of he real economy; and (6) he increase in he governmen deb levels. The bursing of he housing bubble leads o he collapse of he housing prices. The home equiy of borrowing-consrained households becomes negaive his induces he borrowing consrained households o defaul on heir morgages. In our model an increase in he morgage defauls has a negaive impac on bank asses. Banks are subjec o a capial requiremen. To resore he bank capial-asse raio banks cu loans o he credi-dependen secors of he economy which are borrowing-consrained households and enrepreneurs. The conracion of he credi supply causes credi-consrained households o reduce heir 7

18 consumpion housing invesmen and he credi-consrained enrepreneurs o reduce heir real esae holdings labor demand and capial invesmen. Treasury bills are considered o be safe asses so hey are no subjec o he capial requiremen. This induces he banks o increase he holding of reasury bills o help resore he bank balance shees. An increase in he demand for Treasury bills decreases he yield on he Treasury bills. As he economy is in he recession he governmen increases is spending o simulae he economic aciviy. The increase in he governmen spending brings abou a rise in he governmen debs. Jermann and Quadrini () explain he U.S. financial crisis by focusing on credi supply disrupion as a cause for he financial crisis. They show ha a ighening of business credi condiions generae a recession. Nilavongse (3) has a ighening of household credi condiions ha generae a deflaionary recession. Iacoviello (4) shows ha a loan defaul by borrowing consrained households riggers he financial crisis. The loan defaul is modeled as an exogenous sochasic process in his model. Forlai and amberini () have an endogenous household loan defaul in a DSGE framework. An increase in he riskiness of housing invesmen riggers morgage defauls. Their model is based on Bernanke e al. (999) which is derived from informaion asymmery heory. Our model is based on Kiyoaki and Moore (997) and Iacoviello (5) he borrowing consrain in heir model is derived from limied enforcemen heory. The limied enforcemen is he borrower canno be force o repay he deb. Hence he lending occurs if he borrower s physical asses (such as capials or real esae) can be used as collaeral which can be seized by he lender in he case of defaul. The maximum amoun ha he borrower can borrow is limied by he value of collaeral. There is no possibiliy of acual defaul by he borrower in Kiyoaki and Moore (997) and Iacoviello (5) model. In our model he borrowingconsrained household defauls on heir morgages when heir home equiy becomes negaive. Bernanke and Gerler (999) model a bubble shock as exogenous shock o asse prices. We model a housing bubble shock as an exogenous shock o he expeced fuure housing prices. In our model he bursing of he housing bubble which is a negaive shock o he expeced fuure housing prices riggers he collapse of acual housing prices. The collapse of he acual housing prices riggers morgage defauls which is in line wih he observaions 8

19 from he recen U.S. financial crisis. Our model incorporaes a banking secor which is absen in Forlai and amberini () model. There is inerplay beween he household and he bank secor in our model. The morgage defauls have a negaive impac on he bank asses. The main value added of his paper is o inroduce an endogenous morgage defaul wih a housing bubble shock in a DSGE model and we use he model o explain he key feaures of he U.S. financial crisis. References Bernanke B. and M. Gerler (999). Moneary policy and asse price volailiy. Federal Reserve Bank of Kansas Ciy Economic Review Bernanke B. S. Gerler M. & Gilchris S. (999). The financial acceleraor in a quaniaive business cycle framework. Handbook of macroeconomics Chrisiano. Moo R. & Rosagno M. (9). Financial facors in business cycles. Manuscrip. Forlai C. & amberini. (). Risky morgages in a DSGE model. Inernaional Journal of Cenral Banking 7() Iacoviello M. (5). House prices borrowing consrains and moneary policy in he business cycle. The American economic review 95(3) Iacoviello M. (4). Financial Business cycles. Review of Economic Dynamics forhcoming. Jermann U. & Quadrini V. (). Macroeconomic effecs of financial shocks. The American Economic Review () Kiyoaki N. & Moore J. (997). Credi Cycles. Journal of Poliical Economy 5() Kollmann R. Enders Z. & Müller G. J. (). Global banking and inernaional business cycles. European Economic Review 55(3) Monacelli T. (9). New Keynesian models durable goods and collaeral consrains. Journal of Moneary Economics 56() Nilavongse R. (3). Credi Disrupions and he Spillover Effecs beween he Household and Business Secors. Economics Discussion Papers No 3-48 Kiel Insiue for he World Economy. Van den Heuvel S. J. (8). The welfare cos of bank capial requiremens. Journal of Moneary Economics 55()

20 Appendix This secion presens he full derivaions of he morgage defaul. The impaien household defauls if he amoun of he household deb RM M is greaer han he value of house M ( ω) +. ω is he qualiy of house. We le ω o have a negaive value i.e ω. RM M Thus he defaul region lies beween ω and ω =. The area of he defaul is R - ( ω) M M M M We inegrae over he defaul area given he lower limi is ω and upper limi is R M M M. The amoun of morgage defauls Z M can be found by aking he following inegraion: ( R ) M - M -/ M - - Z = R - + f( ω) M -ω M - M - M - ( ω) where f ( ω ) which measures he sensiiviy of he morgage defauls o he house prices is a probabiliy densiy funcion of ω. ( RM - M - M -) -ω / - ( + ω) RM - M -- M - f ( ω) = ω RM M M ( RM M M ) ω M f( ω) M M M ( RM M M ) M f( ω) ω R = M + ω ( RM M M ) ω M f( ω) R M M R R q H R f M M M M M M M = M M M M M ( ω)

21 R M M + M ω+ M ω f ( ω) R R R ω M M M M M M = M + ω+ M M M f ( ω) R R ( ω) ω ω f( ω) M M M M = M + + M M R R ( ω) ( ω) f( ω) M M M M = M + M M This gives us he endogenous defaul on morgages: Z M R M M M = M ( ω) f ( ω). RMM We se he range of ω is beween and. Hence he amoun of morgage defauls in he seady sae posiive number. M Z F is a posiive number and he value of he house in he seady sae is a