Default, Mortgage Standards, and Housing Liquidity

Transcription

1 Defaul, Morgage Sandards, and Housing Liquidiy Allen Head Hongfei Sun Chenggang Zhou Queen s Universiy Queen s Universiy Universiy of Waerloo Sepember 18, 215 Absrac The in uence of households indebedness on heir house-selling decisions is sudied in a racable dynamic general equilibrium model wih housing marke search and defaulable long-erm morgages. In equilibrium, sellers behavior varies signi canly wih heir indebedness. Speci cally, boh asking prices and ime-o-sell increase wih he relaive size of sellers ousanding morgages. In urn, he liquidiy of he housing marke associaed wih equilibrium ime-o-sell deermines he morgage sandards o ered by compeiive banks. When calibraed o he U.S. economy he model generaes, as observed, negaive correlaions over ime beween boh house prices and ime-o-sell wih downpaymen raios. JEL Classi caion: E3, G21, R1, R31 Keywords: Housing; Morgages; Search; Liquidiy; Block Recursiviy Earlier versions of his paper were circulaed under he iles Housing Liquidiy and Lending Sandards and "Indebed Sellers, Housing Liquidiy and Morgage Sandards". We hank Huw Lloyd-Ellis for insighful commens and suggesions. Head hanks he SSHRC for nancial suppor. Sun graefully acknowledges suppor from he Bank of Canada Governor s Award. The opinions expressed in he paper are our own and do no re ec he views of he Bank of Canada. All errors are ours. Please direc correspondence o us a heada@econ.queensu.ca, hfsun@econ.queensu.ca, and cmzhou@uwaerloo.ca, respecively. 1

2 1 Inroducion In his paper we sudy he in uence of households morgage deb on heir house-selling decisions and he e ecs of he resuling housing marke liquidiy, by which we mean he ease wih which houses can be sold, on morgage lending sandards. To his end, we develop a dynamic model wih a fricional housing marke and long-erm morgage deb. The model gives rise in equilibrium o disribuions of boh house prices and deb, as well as o endogenous defaul probabiliies ha di er across households. Aggregae shocks drive morgage sandards hrough heir e ec on liquidiy, and his ogeher wih he morgage erms o ered o buyers a ec he selling decisions of households. Our heory demonsraes ha (i) house-selling decisions depend criically on sellers levels of home equiy, and (ii) liquidiy in he housing marke is an imporan facor in deermining he ease wih which households can borrow. A version of he economy calibraed o U.S. daa capures, qualiaively, boh he observed relaionship beween households morgage loan-o-value (LTV) raios and asking prices and he negaive correlaion beween house prices and morgage lending sandards over ime. Our paper is moivaed rs by he observaion ha morgage lending sandards are in some sense couner-cyclical. Speci cally, Figure 1 illusraes ha house prices are posiively correlaed wih LTV s a originaion (or negaively wih down-paymen raios) for rs-ime home buyers 1. This phenomenon has drawn paricular aenion for he period leading up o he U.S. sub-prime morgage crisis and subsequen house price collapse. Over his ime period, U.S. housing markes were booming in he sense ha prices were rising, sales volume increasing and ime-o-sell declining (Ngai and Sheedy, 215). he same ime, morgage lending sandards were relaxed, speci cally and signi canly in he sense of lowered down-paymen requiremens. 2 A I has been argued ha hese changes occured beyond hose associaed wih changes in regulaory consrains (see, e.g. Belsky 1 Acual down-paymens don necessarily re ec morgage sandards. Firs-ime buyers, however, are he mos likely o be a eced by down-paymen consrains. 2 Lending sandards may include morgage approval raes, down-paymen raios, documen requiremens, ineres raes, ec.. We focus on down-paymen raios, or LTV a originaion.? show ha many subprime loans in his period were characerized by such high LTV s.? consruc a series for loan-o-value raios (LTVs) faced by rs-ime home buyers and show ha he overall LTV raio increased from 2 o 25.? use daa compiled for over 2 U.S. ciies beween 2 and 28 o nd ha ineres-only (IO) morgages were used sparingly in ciies in which an elasic housing supply kep housing prices in check, bu were common in ciies wih an inelasic supply in which housing prices rose sharply and hen crashed.? documen and show ha lending sandards (denying raes) declined more in areas ha experienced larger credi booms (more applicans) and greaer price appreciaion.? nd ha regions wih high laen demand from 21 o 25 experienced large relaive decreases in denial raes, increases in morgages originaed, and increases in housing price appreciaion, despie he fac ha he same regions experienced signi canly negaive relaive income and employmen growh over his ime period. 2

3 and Richardson, 21 and Dow, 215). Here, we explore he incenive of pro maximizing lenders o relax lending sandards in response o a ho housing marke, by which we mean speci cally one in which prices are high and ime-o-sell is low by hisorical sandards. % of home purchase price 96% 94% 92% 9% 88% 86% Home Price Index (1 in 2Q1) % Year 55 2 % Percenage change (year ago) 2 % 1 % 1 % % % 1% 1% 2% Year Average LTV raio a originaion for 1s ime home buyers (lef axis) Case Shiller Home Price Index (righ axis) 2% Figure 1: Values and percenage changes (from one year earlier) in average rs-ime home buyers down-paymen raios and S&P/Case-Shiller U.S. Naional Home Price Index. Source: American Housing Survey (AHS) 27, 29, 211 naional daa. I has also been observed by?? and Anenberg (211), ha house sellers leverage a ecs boh heir asking prices and ime-o-sell. Speci cally, sellers wih high LTV s pos higher asking prices, wai longer o sell, and sell ulimaely a higher prices. This observaion suggess ha morgage deb a ecs no only prices, bu also households incenive o sell, as more heavily indebed sellers are evidenly willing o wai for a buyer who will pay a higher price. Our heory exends ha of? (HLS) which focuses on he dynamics of house prices, sales, consrucion, and populaion growh in an environmen wih homogeneous buyers and sellers and complee nancial markes. Here, we inroduce a form of limied commimen which allows households o defaul under cerain circumsances. This generaes a role for morgage deb secured by homes and generaes heerogeneiy among households, ex pos. Also, while HLS focuses mainly on random search, compeiive search is inegral o our 3

4 analysis and imporan for our resuls. 3 We consider ex ane idenical households which live eiher in a single ciy (on which we focus) or in a largely unmodeled res-of-he-world All residens of he ciy require housing, and may eiher ren or own one of a large number of idenical houses which are produced and sold iniially by a compeiive consrucion indusry. Households ener he ciy when he value of doing so, in par driven by ucuaions in income a he ciy level, exceeds heir ouside opion. Once here, households remain in he ciy eiher as reners or homeowners unil hey leave as a resul of exogenous shocks. Following he usual proocol of compeiive search, sellers o er houses for sale in a variey of sub-markes, inside each of which prospecive buyers and sellers are randomly mached. Each sub-marke is characerized by a unique combinaion of a posed price and pair of maching probabiliies (for buyers and sellers). These probabiliies deermine he expeced ime-o-buy and ime-o-sell for buyers and sellers, respecively. Search is direced in he sense ha buyers and sellers choose sub-markes opimally given he rade-o beween he posed price and he maching probabiliies. House purchases are nanced by morgages o ered by compeiive banks which conrol he erms o ered. Speci cally, banks decide he size of he morgage o o er, and his deermines he LTV a originaion. Morgages are of a xed lengh and a an exogenous ineres rae, which we model as deermined by aggregae condiions raher han hose wihin he ciy. Households who do no own pay ren each period equal o a xed and exogenous fracion of income. Morgage holders are subjec randomly o nancial disress shocks which force hem o eiher sell heir homes hrough he search process or defaul and face foreclosure. Households are no commied o sell, and based on heir speci c siuaions decide wheher and how o do so. Thus, households deermine opimally heir likelihood of defaul on morgage deb. If a household defauls, is house is seized by he morgage company, a foreclosure ag is placed on he is record, and i is prohibied from paricipaing in he housing marke unil he ag is lifed, which also occurs randomly. Thus, households face foreclosure in equilibrium when hey fail o sell heir houses in ime o avoid defaul. In principle, a homeowner could drive he probabiliy of defaul arbirarily low by posing a su cienly low price. In our calibraion, however, nancially disressed households do no do his. Raher, hey choose prices associaed wih subsanial probabiliies of defaul, and which generally rise wih he size of heir ousanding morgage deb. Moreover, some households choose o defaul ourigh, making no aemp 3 HLS does consider a case wih compeiive search, bu only as a robusness check. 4

5 o sell. This occurs for households wih negaive home equiy, which can occur if changing condiions drive home prices su cienly low. Because house sellers are heerogeneous here arises in inequilibrium a disribuion of house prices which evolves over ime owing o aggregae shocks. 4 Home buyers remain idenical, however, as we assume ha goods are non-sorable and rule ou household saving. Free-enry of hese homogeneous buyers ino he housing marke gives rise o he aforemenioned rade-o beween house prices and maching probabiliies. Heerogeneous sellers separae hemselves opimally ino various submarkes based on heir individual saes. As a resul, he individual decision problem is independen of he disribuion of sellers and he model is block recursive as in? and?. Regardless of he aggregae sae, above a cerain LTV, he prices posed in equilibrium by selling homeowners are seeply increasing in heir morgage size, a resul consisen wih he empirical ndings of (?, GenesoveMayer97). More highly levered sellers hus are more likely o defaul han are less levered ones. Negaive shocks (o ciy-wide income, for example) hus cause paricularly severe waves of defaul and foreclosure if hey occur when he economy has a high proporion of highly levered homeowners. Housing marke liquidiy a ecs morgage lending sandards hrough boh he expeced defaul rae and lenders expeced losses upon defaul. The more liquid he housing marke, he higher he probabiliy which which indebed households sell and hus he lower he rae of defaul and foreclosure. Similarly, morgage companies also sell foreclosed houses more quickly, lowering he expeced carrying cos of a foreclosed house and hus he cos of defaul. Also, houses ypically sell a higher prices in a hoer marke, regardless of who sells, furher lowering he cos of defaul. Thus, morgage companies are willing o o er larger morgages when he housing marke is more liquid. All ogeher, hese resuls imply a negaive correlaion over ime beween house prices and LTV s a originaion. The paper conribues o he growing lieraure on search fricions in he housing marke (see, for example,?,??,?, and?). None of hese papers, however, models morgage conracs and examines he long-erm housing-lending relaionship on which we focus. Two papers ha do sudy heoreically he relaionship beween housing marke liquidiy and lending sandards are? and?, wih he laer being he mos closely relaed.? also sudies a model ha feaures direced search, long-erm morgages, and limied commimen. Hedlund s model, however, feaures a di eren marke arrangmen in which buyers and sellers o no inerac direcly bu via he ineracion of real esae agens 4 Houses are sold by consrucion rms, morgage companies, and home-owners di ereniaed by boh heir reasons for selling and levels of morgage deb. 5

6 who buy houses from heerogenous sellers and hen sell hem, along wih newly consruced houses, o heerogeneous buyers. and racable. 5 Like ours, his seup renders he model block recursive Comparison of he wo models, however, sheds ligh on he role of he elasiciy of housing supply in generaing our main resuls. In Secion 5, we sudy a calibraed version of our model feauring a direced search framework based on Hedlund s, which we refer o as direced search wih middlemen. In summary, we nd ha our framework is beer able o accoun for a negaive correlaion beween house prices and lending sandards owing o is di eren implicaions for variaion in he elasiciy of housing supply han hose of his alernaive environmen. In addiion o he maching environmen, our model also di ers from ha of? in ha we sudy nie morgages a xed ineres raes raher han exible-rae in nie-horizon morgage conracs. Our moivaion here is principally realism, as in he U.S. convenional morgages ypically have a 3-year erm and abou 7% of hese morgages are a xed ineres raes. 6 While nie-horizon conracs add complexiy, hey enable us o sudy how opimal house-rading decisions vary across households a di eren sages of morgage repaymen. Anoher imporan di erence beween our paper and Hedlund s is ha we focus on housing and morgage markes a he ciy level, whereas he considers he business cycle dynamics of housing and morgage markes a he naional level. in a New Keynesian model along he lines of? wih credi-consrained consumers and housing marke fricions,? shows ha expansionary moneary policy leads o higher leverage among homeowners. In his model, a decrease in morgage ineres raes booss demand for housing. Wih more buyers in he fricional marke, lenders can liquidae foreclosed houses more quickly, e ecively reducing he expeced carrying cos of a foreclosed house and making lender more willing o nance larger fracions of house purchases. Our paper di ers from? in several respecs: raher han long-erm. Firs, in ha paper deb is one-period Thus, we are able o race ou he defaul decision of indebed households a every sage of heir morgage repaymen process. no defaul in equilibrium and hus no liquidaion of houses. Second, in? here is Financial fricions ake he form of Kiyoaki-Moore collaeral consrain wih lenders o ering deb o he exen ha defaul is prevened in equilibrium. In conras, in our heory lending sandards re ec 5 Direced search and soring wih wo-sided heerogeneiy is a challenging problem o ackle. There are a handful of papers ha characerize he seady sae of such an economy under cerain condiions (see?,?,?, and?).? furher shows ha dynamics of soring wih wo-sided heerogeneiy can be racable in some seings. Neverheless, our model would be inracable if allowing for endogenous saving decisions along wih endogenous morgage choices. 6 This percenage, however, has declined in recen years. 6

7 defaul probabiliies and he foreclosure invenory is a componen of housing supply. As a resul, marke liquidiy a ecs boh he expeced carrying cos of a foreclosed house and he expeced defaul rae. Boh facors conribue o he negaive correlaion beween house prices and LTV s a originaion. Finally, in? houses are divisible, he housing sock is xed and here is no consrucion secor. By focusing on he selling decisions of households, our paper is relaed o hose of Ngai and Sheedy (215) and Ngai and Tenreyo (214). Those papers focus, respecively, on he e ec of aggregae condiions and seasonal ucuaions in demand on he decisions of homeowners o pu heir houses on he marke. In conras, we focus speci cally on he selling decisions of heerogeneous homeowners disinguished by heir levels of morgage deb, wheher nanically disressed or no. As such, while boh he models and speci c issues sudied vary beween our paper and heirs, we view our work as complemenary. The remainder of his paper is organized as follows. Secion 2 inroduces he environmen of our baseline search economy. Secion 3 formalizes he equilibrium for he baseline. Secion 4 presens an alernaive model wih fricionless housing markes raher han compeiive search. Secion 5 inroduces a second alernaive model, using he direced search wih middlemen seup of?. Secion 6 provides jusi caions for our calibraion choices. Secion 7 presens he seady-sae. Secion 8 discusses he dynamic implicaions of our model in response o aggregae shocks. Secion 9 concludes. 2 The Environmen Time is in nie and discree, wih ime periods indexed by. The economy consiss of a single housing marke, which we refer o as he ciy, and he res of he world. The aggregae economy is populaed by a measure Q of ex ane idenical households, which grows exogenously a ne rae. Each household lives inde niely and supplies one uni of labor inelasically every period. In period, he uni of labor supplied earns income y, in unis of a single dae consumpion good. Income follows a saionary sochasic process in log-levels. Households in he ciy require housing, and may eiher ren or own one of a large number of symmeric housing unis. Households preferences are represened by " 1 # X U = E [u(c ) + z ] ; (1) = 7

8 where c denoes consumpion and z housing in period, respecively. We assume ha z = z H if he household owns he house in which hey live and z = oherwise. The funcion u() is sricly increasing, sricly concave and wice coninuously di ereniable, wih he boundary properies: u() =, lim c!1 u (c) = and lim c! u (c) su cienly large. All households have he common discoun facor, 2 (; 1). Boh consumpion goods and housing services are non-sorable and here is no echnology for households o save across periods. 7 A he beginning of each period, measure Q of new households arrive in he economy. Each of hese households has a bes alernaive value o enering he ciy, denoed ". These values are independenly and idenically disribued across new households according o he disribuion funcion G("), wih suppor [; "]. Households ha ener he ciy are, immediaely upon doing so, separed randomly and permanenly ino wo groups; hose ha value home-ownership and hose ha do no. The former we refer o as buyers and he laer as perpeual reners. Each period here exiss a criical alernaive value, " c, below which a new household sricly prefers o ener he ciy: " c = U + (1 )W p ; (2) where U and W p are of lifeime values of being a buyer and a perpeual rener, respecively, and 1 is he probabiliy of he enran becoming a perpeual rener. New houses are buil by a consrucion indusry comprised of a large number of idenical and compeiive rms. Each new house requires one uni of land, which can be purchased in a compeiive marke a price q = Q(N ). The builder also incurs consrucion cos k = K(N ), where N denoes he quaniy of new houses buil in period. Houses require one period o build; hose consruced in period become available for sale a he beginning of period + 1. Home-ownership boh provides a uiliy bene o he owner and requires cosly mainenance immediaely. occupied. Houses depreciae over ime, regardless of wheher or no hey are Depreciaion is, however, o se by he owner a mainenance cos m each period. Households in he ciy ha do no own houses ren. We absrac from mos aspecs of he renal marke, and assume ha ren is equal o a xed fracion of he ciy-level income, R = &y. The supply of renal accommodaion is oally elasic is no considered par of he ciy s housing sock. 7 This assumpion renders buyers in he housing marke homogenous boh heir labor incomes are idenical. This simpli es he analysis signi canly. 8

9 A he end of each period, all households in he ciy, regardless of heir ownership saus, experience shocks which induce hem o leave he ciy. For perpeual reners and buyers (regardless of wheher or no hey currenly own a house) hese shocks occur wih probabiliies p 2 (; 1) and h 2 (; 1), respecively. receive coninuaion uiliy, L. All households which exi he ciy Exiing homeowners also have vacan houses ha hey may wan o sell, depending in par on heir ousanding morgage deb, if any. households also have he opion of defauling. The housing marke is characerized by compeiive search. These We imagine i as being characerized by a large variey of poenial submarkes indexed by a price, p, and a pair of maching probabiliies; one for buyers and one for sellers. Wihin each submarke, maching akes place via a maching funcion, : M (B; S), which is increasing in boh argumens and has consan reurns o scale. Given his form we can index submarkes by (; p), where denoes he marke ighness (i.e., he raio of he measure of buyers, B, o ha of sellers, S) and p he posed ransacion price. Boh buyers and sellers ake (; p) of all submarkes as given and decide which o ener in search of a rade. The maching probabiliies for buyers () and sellers () are given by () = () = M (B; S) = M 1; 1 B (3) M (B; S) = M (; 1) = (): S (4) Each buyer and seller can ener only a single submarke in a given period, and here is no cos of enry. Free-enry generaes endogenously a rade-o beween he house price and he maching probabiliy across acive submarkes. Inuiively, higher-price submarkes have lower levels of ighness as buyers (who are all idenical) are willing o pay a higher price only if hey are compensaed wih higher probabiliy of maching wih a seller. The sock of searching buyers includes boh newly enered households and hose which have been searching unsuccessfully for some ime. As noed above, hese households are idenical. Sellers, however, are of a number of di eren ypes. Firs, consrucion rms sell newly buil homes. Second, homeowners who receive exi shocks as described above may decide o sell. Noe ha hese buyers are heerogeneous o he exen ha hey have di eren ousanding morgages. Home-owners may also sell as a resul of a foreclosure shock (described below), and again hey are di ereniaed by heir ousanding morgage. Finally, morgage companies sell foreclosed houses (see below). 9

10 In our calibraion, prices ypically exceed per period income and as here is no saving, households mus borrow o nance house purchases. 8 Morgages are provided by a large number of perfecly compeiive rms owned by risk-neural invesors who consume all pro s and losses ex pos. 9 To nance is loans, hese morgage companies rade one-period risk-free bonds a an exogenous ineres rae, i, in an inernaional bond marke. They also incur a proporional service cos, ; per period associaed wih he adminisraion of household morgages. The deb conrac is a sandard xed-rae morgage wih nie mauriy T. Le m and r m represen he size of a loan a originaion and he morgage rae, respecively. Conrac (m ; r m ; T ), speci es a consan paymen per period: x(m ) = r m 1 (1 + r m ) T m : (5) As he homeowner makes paymens, he principle balance, d, evolves via d(m ; n + 1) = (1 + r m ) d(m ; r m ; n) x (m ; r m ) (6) where n 2 f; T 1g and d(m ; ) = m. Since T and r m are xed exogenously, boh x() and d(; ) are unrelaed o afer originaion, and (m ; n) is su cien o represen he sae of an ongoing morgage. A borrower can erminae his/her morgage conrac a any ime by paying o he remaining balance, d(m ; n). he ousanding morgage balance. A erminaion is a defaul, if he borrower does no repay Defaul leads o foreclosure, whereby he morgage company akes conrol of he house, remiing o he borrower any surplus value of he house in excess of he ousanding loan balance. The lender does no have direc access o he homeowner s curren and/or fuure income, Homeowners wih ousanding morgage deb receive, wih probabiliy d each period a nancial disress shock. We inerpre hese shocks as represening circumsances such as accidens or unexpeced illness ha render he household unable o coninue morgage paymens. Recipiens of such shocks are referred o as disressed owners. They mus erminae heir curren morgage conrac wihin he same period and eiher pay heir ousanding deb or defaul. 8 In he absence of saving, households would prefer o borrow o smooh consumpion even if he house price were less han period income. 9 Alernaively, hese rms could be owned by households o whom hey would ransfer heir ex pos profis and losses lump-sum. This formulaion would, however, complicae he compuaion wihou changing our resuls signi canly. 1

11 In he even of defaul, a borrower s morgage balance is se o zero and a foreclosure ag is placed on his/her credi record. The morgage company repossesses he borrower s house, pus i in real-esae-owned (REO) invenory, and decides wheher and how o sell i beginning wih he following period. As noed above, he defauling homeowner receives he di erence beween he value of a house in REO invenory and he ousanding morgage balance, if posiive. Upon a successful sale, he morgage company loses a fracion 2 (; 1) of he revenue as coss, e.g., legal fees. As a penaly for defauling, buyers wih foreclosure ags lose access o he morgage marke and are hus excluded from he housing marke. Beginning wih he following period, he foreclosure ag eiher remains on a buyer s record (wih probabiliy f 2 (; 1)) or is removed. 1 In equilibrium, he morgage rae is given by r m = i + + %, where i and are exogenously given as described above. The componen %, however represens an endogenous risk premium, which compensaes for he risk of defaul, which occurs wih posiive probabiliy. Sub-period 1 Sub-period 2 +1 New enrans, Shocks on (y,d,f) Housing marke opens (buying/selling decisions) Credi marke opens (new morgages, defaul) Consumpion, morgage paymens Moving shock Figure 2: Time Line Each period consiss of wo sub-periods. A he beginning of sub-period 1, new households wih " " c ener he ciy. Income shocks, nancial disress shocks and shocks on he foreclosure ag are all revealed. Immediaely hereafer, he housing marke opens: Buyers and sellers decide he submarkes, (p; ), in which o search and lis heir houses for sale, respecively. Afer he housing marke closes, he morgage secor becomes acive: New owners ake ou morgages o nance heir purchases and curren morgage holders decide wheher or no o defaul. In sub-period 2, households receive income, make paymens (mainenance, down paymens, morgage paymens or rens), and consume he remainder. A he end of he 1 According o he policies of Fannie Mae and Freddie Mac, foreclosure lings say on a borrower s credi record for a nie number of years. 11

12 period, moving shocks are revealed for all households and hose who receive hem leave he ciy immediaely. Figure 2 provides a more deailed illusraion of he iming of decisions. 3 Equilibrium We begin by describing in deail he behavior of agens and de ning an equilibrium for he environmen described above, which we will refer o as our baseline search economy. 3.1 Households Consider households value funcions sequenially for he wo sub-periods of a ypical ime period : The rs sub-period House rading and morgage defaul decisions are boh made in he rs sub-period. Le U denoe he value funcion for a buyer. These households are eiher new enrans or hose no owning a house and wihou a foreclosure ag, i.e. for whom f = : U = max [()V o (p; m ) + (1 ())W ()] : (7) (p;) In sub-period 1, a buyer will search for a house o buy, choosing opimally o ener sub-marke (p; ). The buyer is mached wih a seller wih probabiliy (), in which case she proceeds o sub-period 2 as a new owner wih value V o (p; m ). The price paid and loan volume m ( deermined by he morgage company and speci ed below) deermine he homeowner s LTV. Wih probabiliy 1 and proceeds o sub-period 2 wih value W (f = ). (), he buyer fails o a mach, remains a buyer, Wih buyers free o ener any sub-marke, all hose which are acive mus o er he same level of U o all buyers. Rewriing (7) yields = 1 U W () (p) : (8) V o (p; m ) W () Thus, he free-enry of buyers deermines he relaionship beween he ransacion price and he marke ighness across sub-markes. Le S (m ; n) denoe he value for a residen owner wih deb m a originaion, who 12

13 has made n paymens and is no in nancial disress: S (m ; n) = max f( s )W (max[; p s d(m ; n)]) (p s ; s ) +(1 ( s )) max D 2f;1g f(1 D )V (m ; n) io +D W f (max[; E V+1 REO d(m ; n)]) subjec o s = (p s ) : (1) (9) This household decides wheher and in which sub-marke, (p s ; s ), o sell her house. Wih probabiliy, ( s ), he house is successfully sold. In his case, his individual repays as much ousanding deb as possible, and keeps he remaining pro, a = max[; p s if any. d(m ; n)], The household hen proceeds o he second sub-period as a buyer wihou he foreclosure ag who has value W (a). If he household chooses no o sell her house, or has failed o sell i, she hen decides wheher or no o defaul on her curren morgage conrac. Here D = 1 if he household chooses o defaul, and D = oherwise. The value of a homeowner who has no defauled a he begining of he second sub-period is V (m ; n). A homeowner who has defauled has value W f (max[; E V REO +1 d(m ; n)]) a he beginning of he second sub-period. Such a homeowner e ecively sells heir house o he morgage company for he expeced discouned value of a vacan house in he morgage company s invenory a he beginning of he nex period, E V REO +1. If his value is less han he household s ousanding morgage deb, d(m ; n), he household s asses are se o zero. The expecaion here is aken wih respec o aggregae shocks which a ec he value of vacan houses. If he value of he vacan house exceeds he deb, he defauling homeowner keeps he residual value. In eiher case, he home-owner acquires a foreclosure ag. Nex, consider a residen owner who receives a nancial disress shock a he beginning of period. 11 Such a homeowner mus erminae her morgage conrac wihin he same period. If he house is sold, he homeowner receives he residual value ne of deb and hen becomes a buyer wihou a foreclosure ag, W (max[; p sd d(m ; n)]). If he house is no 11 In he even of nancial disress, i is always in an owner s bes ineres o aemp o sell if hey have posiive equiy. In an economy wih perfecly liquid housing markes, disressed owners wih posiive equiy would never defaul because hey can immediaely sell heir houses and repay heir morgage. According o he RealyTrac repor, however, less han 5% of homeowners who go ino foreclosure have negaive equiy. Time-consuming search and maching is responsible for his observaion. The appraisal value of a house is no equal o is rue liquidaion value in decenralized housing markes. Appraisal value is ypically esimaed based on he mos recen sale prices of houses wih similar characerisics, whereas liquidaion value akes no only he price of a house bu also he probabiliy of sale ino consideraion. 13

14 sold, he owner defauls, he foreclosure ag is placed on her credi record. In his case, he homeowner receives he residual value of he house ne of he deb and eners he nex sub-period wih value W f (max[; E V+1 REO d(m ; n)]). 12 Thus he value of a disressed residen owner wih deb d(m ; n) is given by: S f (m ; n) = max ( sd )W (max[; p sd (p sd ; sd ) d(m ; n)]) + 1 ( sd ) W f (max[; E V REO +1 d(m ; n)]) subjec o sd = p sd : (12) o (11) Noe ha in (11) a disressed seller s choice of asking price is no consrained o exceed he ousanding morgage deb. A residen homeowner wihou a morgage decides wheher and how o sell her house. If he house is successfully sold, he owner moves on as a buyer wih value W (p sw ). Oherwise, she moves ono he nex sub-period as an owner wihou deb V. Such a homeowner has value S = max (p sw ; sw ) f( sw )W (p sw ) + (1 ( sw )) V g (13) subjec o sw = (p sw ) : (14) Nex, consider homeowners who have lef he ciy. Such households become essenially irrelevan once hey are no longer homeowners. A homeowner ha has lef he ciy wih an ousanding morgage deb has value: V L (m ; n) = max ( L ) u(max ; p L d(m ; n) + y (p L ; L L R L ) + L ) + (1 ( L )) max(1 D L D L ) u(y L R L x(m ) m) +E V+1(m L ; n + 1) (15) u(max ; E V+1 REO d(m ; n) + y L R L ) + L +D L subjec o L = p L : (16) Such a household makes decisions regarding wheher o defaul and in wha sub-marke, if any, o o er her house for sale. The possible scenarios are similar o hose described 12 Noe ha disressed residen owners can use proceeds from sales, bu no labor income, o pay o ousanding morgage deb. Relaxing his consrain would complicae he model and does no change he resuls signi canly. 14

15 above for households residen in he ciy. Here, y L, R L, and m are income received and ren and mainenance coss paid by he exiing household while i is in he alernaive locaion. 13 Once he homeowner has eiher sold her house or defauled, she receives exogenous coninuaion value, L. The value of an owner who has lef he ciy wihou deb prior o moving is given by: V L = max ( Lw ) u(p LW + y (p Lw ; Lw L ) R L ) + L + (1 ( Lw )) u(y L R L m) + E V L +1 (17) subjec o Lw = p Lw : (18) Such a household s only decision is wih regard o wheher and how o sell her house. Finally, he value of a vacan house in a consrucion rm s invenory V c, and he value of a foreclosed house V REO are given, respecively by V c = max ( c )p c + (1 ( c ))[ m + E V+1] c (19) (p c ; c ) subjec o c = (p c ) : (2) V REO = max ( REO )(1 )p REO + (1 ( REO ))[ m + E V (p REO ; REO +1 REO (21) ] ) subjec o REO = p REO : (22) Noe ha in (21), i can be seen ha he morgage company loses fracion of he proceeds of is sales as a cos of foreclosure Households in sub-period 2 Since here is no saving, households behaviour in sub-period 2 is essenially rivial: They simply consume heir income ne of ren and morgage paymens. Here, we esablish he value funcions for he various household saes a he beginning of his sub-period, which were used in he expressions above. A perpeual rener remains a rener (never seeking o purchase a house) he enire ime 13 These quaniies are necesssary as long as he household remains a homeowner, because hey impinge on is defaul and pricing decisions. 15

16 she says in he ciy. Such a household s value is given by: W p = u(c ) + p L + (1 p )E W p +1 (23) subjec o c = y R (24) Wih probabiliy p, he perpeual rener is hi by he moving shock, leaves he ciy immediaely and receives he coninuaion value L. Oherwise, she moves ono he nex period as a rener. Her consumpion is simply income ne of ren. A buyer wih he foreclosure ag on her credi record has access neiher o credi nor he housing marke. She will remain rening unil she moves ou of he ciy or he foreclosure ag is lifed from her record. Le W f (a) be he value of such a buyer wih asse a a he beginning of sub-period 2. The balance a represens his buyer s inra-period asse holding. This balance is sricly posiive if (i) he residen owner has defauled on her morgage in he preceding sub-period 1; and (ii) he value of a foreclosed house in he REO invenory is above he ousanding morgage deb. Oherwise, a =. Thus, h W f (a) = u(c ) + h L + (1 h ) f E W+1() f + (1 f )E U +1 i(25) subjec o c = y + a R : (26) Condiional on saying in he ciy, wih probabiliy f he foreclosure ag remains and he household moves ono he following period wih expeced value W+1() f (such households are inacive in he rs-subperiod of period + 1). Wih probabiliy 1 f, he foreclosure ag is lifed and his household will coninue on wih value U +1 as a buyer searching for a house in he sub-period 1 of period + 1. A buyer wihou he foreclosure ag a he beginning of sub-period 2 is eiher a residen owner who jus successfully sold her house or a buyer who has failed o purchase a house in sub-period 1. Such a buyer may have a posiive inra-period asse balance, a, coming from sale proceeds ne of he ousanding morgage deb in he previous sub-period. She will move on wih value U +1 and paricipae in he housing marke in he nex period if no hi by he moving shock a he end of he curren period. The value of such a buyer is given by W (a) = u(c ) + h L + (1 h )E U +1 (27) subjec o c = y + a R (28) 16

17 A residen homeowner wih a morgage has he principle balance d(m ; n), where n 2 [; T 1]. The owner s periodic income is used o cover repaymen, mainenance cos and consumpion. Le V (m ; n) denoe he value of such an owner. I follows ha for n 2 [; T 2], V (m ; n) = u(c ) + z H + h E V+1(m L ; n + 1) " # +(1 h ) d E S+1(m f ; n + 1) +(1 d )E S +1 (m ; n + 1) (29) subjec o c = y x(m ) m (3) If he owner receives a moving shock, she exis he ciy immediaely and coninues wih value V+1(m L ; n + 1). Noe ha her morgage deb does no vanish because she has relocaed. Condiional on no relocaing, in he nex period he owner receives a nancial disress shock wih probabiliy d. In his case, she coninues as a disressed residen owner wih deb S+1(m f ; n + 1). Oherwise, she eners he nex period as a non-disressed owner wih value S +1 (m ; n + 1). For n = T 1, a residen homeowner wih a morgage has value V (m ; T 1) = u(c ) + z H + h V L +1 + (1 h )E S+1 (31) subjec o c = y x(m ) m (32) In his case, he curren morgage paymen is he homeowner s las. Thus, she will coninue on wih value V L +1 if hi by he moving shock and wih value S +1 oherwise. A new owner who has purchased a house in he preceding sub-period pays he di erence beween he purchase price and oal deb m, ha is, he down-paymen. The periodic morgage paymen begins from he following period. The deb carried by his household ino he nex period is d(m ; ). Le V (p; m ) denoe he value of a new homeowner: V o (p ; m ) = u(c ) + z H + h V+1(m L ; ) " # d E S f +(1 h ) +1(m ; ) +(1 d )E S +1 (m ; ) (33) subjec o c = y (p m ) m (34) Finally, owners wihou morgage deb do no su er nancial disress shocks. They remain in he ciy unil experiencing a moving shock. The value of such an owner is given 17

18 by V = u(c ) + z H + h V L +1 + (1 h )E S+1 (35) subjec o c = y m (36) 3.2 Consrucion rms The consrucion indusry is comprised of a large number of compeiive rms. Building a new house requires one uni of land a he price q = Q(N ), where N is he measure of new houses buil in period and available a + 1. In addiion, he process of acually building he house resuls in consrucion coss k = K(N ). Free enry ino he indusry ensures ha in equilibrium he cos of building a house equasl he expeced value of a vacan house for sale in period + 1: Q(N ) + K(N ) = E V c +1: (37) 3.3 Morgage companies Because he morgage rm has access o funds a a xed cos, i will issue morgages unil i earns zero pro on each conrac. In paricular, he expeced reurn ne of expeced foreclosure coss on morgages will equal he opporuniy cos of funds, ha is, he ineres rae i of he inernaional bonds plus he servicing premium. Houses are idenical, households canno save over ime, and regular repaymens of all new morgages sar from he period following ha in which he house is purchased and he morgage iniiaed. As such, all new borrowers are idenical o he morgage company a he poin of loan originaion. Therefore, he morgage company loans of he same size, m ;, o all new borrowers in period, regardless of he price hey pay for heir house. Le P (m ; n) be he presen value of morgage = (m ; r m ) held by a residen homeowner a he beginning of sub-period 2 of period afer n paymens have been made, for n 2 f; ; T 1g. Correspondingly, le P L (m ; n), for n 2 f1; ; T 1g be he presen value of such a morgage held by an owner ha has relocaed (n 1 for such owners because one repaymen has been made by he beginning of he rs sub-period 2 afer he 18

19 household relocaed). Then, for n 2 f; 1; ; T 1g, P (m ; n) = x(m )I fn6=g + I fn6=t 1g 1 + i + (38) 2 2 ( L +1) min p L +1; d(m ; n + 1) 3 3 ( h 6 4 D +(1 ( L L +1(m ; n) min V+2 REO ; d(m ; n + 1) ) )) + (1 D +1(m L ; n))p+1(m L ; n + 1)] +(1 h ) 2 ( E ( sd +1) min p sd +1; d(m ; n + 1) ) 3 d + (1 ( sd +1)) min V+2 REO ; d(m ; n + 1) 8 ( s +1) min p s +1; d(m ; n + 1) 9 >< +(1 ( s >= 6 +(1 4 6 d ) ( +1)) 4 D >: +1 (m ; n) min V+2 REO ; d(m ; n + 1) ) >; + (1 D+1(m ; n))p+1(m ; n + 1) and for all n 2 f1; ; T 1g, P L (m ; n) = x(m ) + I fn6=t 1g 1 + i + 2 ( L 6 E ) min p L +1; d(m ; n + 1) +(1 ( L +1)) ( D L +1 min E V REO +2 ; d(m ; n + 1) +(1 D L +1)P L +1(m ; n + 1) ) (39) where p sd +1; sd +1; p s +1; s +1; p+1; L L +1; D+1; D+1 L are household policies in period + 1 coningen on having morgage balance (m ; n + 1). Moreover, I fn6=g and I fn6=t 1g are index funcions such ha I fn6=g = I fn6=t 1g = ( ; if n = 1; oherwise ( ; if n = T 1 1; oherwise (4) : (41) The rs index funcion I fn6=g re ecs ha a borrower sars making regular repaymens from he period afer originaion. The second, I fn6=t 1g, indicaes ha he morgage maures afer he curren repaymen is made. The presen value P (m ; n) is equal o he period- repaymen x(m ) plus he discouned expeced value of he morgage in period 19

20 + 1. The laer is a eced by he probabiliy of he borrower receiving eiher a moving nancial disress shock, and he decisions he household will make regarding pricing and/or defaul in even ha such shocks are realized. Noe ha hese decisions do no depend on he price which he homeowner originally paid for he house, only heir ousanding loan and expecaions of he fuure relaive reurns o sale and defaul. To compue he presen value of a morgage conrac a originaion, we proceed recursively. Firs, we compue he value for n = T 1, and hen use backward inducion o obain P (; ) for n 2 [; T 2]. The value P L (m ; n) is deermined in a similar way excep ha a relocaed borrower does no experience any moving or disress shock. If a borrower sells her house in period + 1, he amoun ha he morgage company will receive is he minimum of he sale proceeds and he ousanding deb d(m ; n + 1). The morgage company s zero-pro condiion is given by P (m ; ) m = : (42) Thus, (42) implicily deermines he equilibrium value of m ; in period. 3.4 Laws of moion We now describe he evoluion of he disribuion of households across saes. All measures are normalized by he oal populaion, Q. A he beginning of period, we have he per capia measures of perpeual reners F, buyers wihou a foreclosure ag, B, buyers wih a foreclosure ag, B f, indebed owners,, indebed and relocaed owners L, he consrucion rm s invenory c, and he sock foreclosed houses REO. The sock of perpeual reners in period consiss of hose remaining from he previous period and hose who have newly enered: (1 + )F = (1 f )F 1 + (1 )G(" c ): (43) The sock of buyers wih he foreclosure ag a he beginning of period includes hose remaining from previous period. These have neiher moved nor had heir ag removed randomly. To his are added residen homeowners who defauled in period 1. These homeowners may eiher have received a nancial disress shock and failed o sell or have 2

21 defauled sraegically. Thus, we have 8 >< (1+)B f = (1 h ) >: f B f 1 + (1 d ) P T 1 n= ( (1 ( s 1 (m ; 1 n ; n)))d 1 (m ; 1 n ; n) 1 (m ; 1 n ; n) P + T 1 d n= (1 (sd 1 (m ; 1 n ; n))) 1 (m ; 1 n ; n)g ) 9 >= >; (44) where 1 (m ; 1 n ; n) is he measure of morgage holders who had made n paymens by he beginning of 1 on a morgage of size m ; 1 n a originaion. The measure of buyers wihou foreclosure ags a he beginning of period consiss of newly-enering buyers, previously agged buyers whose ag has been removed, and non-relocaing buyers from he previous period who failed o buy a house. Noe ha he measure of buyers who successfully mached in he previous period equals he sum of he measures of he prospecive sellers of various ypes muliplied by heir corresponding maching probabiliies. Thus, we have (1 + )B = G(" c ) + (1 f )B R + (1 h ) 8 >< B 1 ( Lw 1) L 1 (;; ;) >: ( c 1) c ( REO 1 ) REO P T 1 n= L 1 (m ; 1 n ; n) L 1(m ; 1 n ; n) 9 >= >; (45) Le (m ; n) be he measure of indebed owners who have made n periodic paymens by he beginning of period on a morgage of size m ; n a originaion. For n >, his measure evolves via: (1 + ) (m ; n) = (1 h )(1 d )(1 ( s 1 (m ; n ; n 1))) (1 D 1 (m ; n ; n 1)) 1 (m ; n ; n 1) : (46) Tha is, he curren period owners wih an ongoing morgage are he indebed owners from he previous period who neiher moved nor experienced nancial disress, and neiher successfully sold heir house (wheher by choice or bad luck) nor defauled sraegically. For n =, (m ; ) is he measure of residen homeowners who purchased a house in he las period, have sayed in he ciy and have no experienced nancial disress. This 21

22 measure can be recovered from he number of sales in he previous period. Thus we have 8 >< (1 + ) (m ; ) = (1 h ) >: (1 d ) P T 1 n= (s 1 (m ; 1 n ; n)) 1 (m ; 1 n ; n) P + T 1 d n= (sd 1 (m ; 1 n ; n)) 1 (m ; 1 n ; n) + P T 1 n= (L 1 (m ; 1 n ; n)) L 1 (m ; 1 n ; n) +( sw 1) sw 1 + ( Lw 1) Lw 1 +( c 1) c + ( REO 1 ) REO 9 >= : (47) >; Finally, he measure of residen owners wihou a morgage (;; ;) is given by 8 >< (1 d )(1 ( s 1 (m ; T ; T 1))) (1 + ) (;; ;) = (1 h ) (1 D 1 (m ; T ; T 1)) 1 (m ; T ; T 1) >: +(1 ( sw 1)) 1 (;; ;) 9 >= : (48) >; This group is comprised of is previous members who have no moved plus residen homeowners who made heir las morgage paymen in period 1. Proceeding similarly for relocaed homeowners, L (m ; n) is he measure who have made n periodic paymens a he beginning of period. Again, he loan volume a originaion is m ; n and L (;; ;) he measure of relocaed owners wihou deb: (1 + ) L (m ; n) = (1 ( sl 1 (m ; n ; n 1)))(1 D L 1 (m ; n ; n 1)) (1 + ) L (m ; ) = h 8 >< >: L 1 (m ; n ; n 1) (49) + h (1 d )(1 ( s 1 (m ; n ; n))) (1 D 1 (m ; n ; n)) 1 (m ; n ; n) (1 d ) P T 1 n= (s 1 (m ; 1 n ; n)) 1 (m ; 1 n ; n) P + T 1 d n= (sd 1 (m ; 1 n ; n)) 1 (m ; 1 n ; n) + P T 1 n= (L 1 (m ; 1 n ; n)) L 1 (m ; 1 n ; n) +( sw 1) sw 1 + ( Lw 1) Lw 1 +( c 1) c + ( REO 1 ) REO 9 >= >; (5) 22

23 8 >< (1 ( sw 1)) 1 (;; ;) +(1 d )(1 (p s 1 (m ; T ; T 1))) (1 + ) L (;; ;) = h >: (1 D 1 (m ; T ; T 1)) 1 (m ; T ; T 1) >; +(1 ( sl 1 (m ; T ; T 1)))(1 D L 1()) L 1 (m ; T ; T 1) +(1 ( slw 1 )) L 1(;; ;): (51) As depreciaion is o se by mainenance, he oal ciy housing sock evolves via 9 >= H +1 = H + N ; (52) where N is newly buil houses available for sale in period. Le c be he sock of houses in consrucion rms invenory a he beginning of period. This includes unsold houses from he previous period and newly buil ones: (1 + ) c = 1 ( c 1) c 1 + N. (53) The sock of houses in he REO invenory a he beginning of period, REO, includes unsold houses from he previous period and new foreclosures: (1 + ) REO = 1 ( REO 1 ) REO 1 XT 1 + d (1 ( sd 1 (m ; 1 n ; n))) 1 (m ; 1 n ; n) n= XT 1 +(1 d ) (1 ( s 1 (m ; 1 n ; n)))d 1 (m ; 1 n ; n) 1 (m ; 1 n ; n) T 1 + n= X (1 ( sl 1 (m ; 1 n ; n))d L 1 (m ; 1 n ; n) L 1(m ; 1 n ; n): (54) n=1 Finally, for accouning purposes, he oal measure of buyers searching o rade in he 23

24 housing marke, B sum, can be derived as: XT 1 = (1 d ) s (m ; n ; n) (m ; n ; n) B sum X + d T 1 X T 1 + n= + sw n= n= sd (m ; n ; n) (m ; n ; n) sl (m ; n ; n) L (m ; n ; n) (;; ;) + slw L (;; ;) + c c + REO REO : (55) In paricular, he measure of buyers in an acive submarke equals he measure of sellers in ha submarke muliplied by he he corresponding marke ighness. For example, he measure of buyers searching for foreclosed house sold by he morgage company equals he measure of REO houses REO opimally chosen submarke, REO muliplied by he ighness of he morgage company s 3.5 De niion of equilibrium De niion. An equilibrium is a collecion of value funcions, associaed policy funcions, fu ; S (; ) ; S f (; ) ; S ; V L (; ) ; V ; V c ; V REO ; W p ; W (); W f (); V (; ) ; V o (; ) ; V L g (56) fp s (; ) ; s (; ) ; D (; ) ; p sd (; ) ; sd (; ) ; p sw ; sw ; p L (; ) ; L (; ) ; D L (; ) ; p Lw ; Lw ; p c ; c ; p REO ; REO g; (57) an enry cu-o value, morgage conrac, ren, f" c ; m ; ; R g ; (58) and per capia measures of households and houses ff ; B ; B R ; (; ) ; L (; ) ; c ; REO ; N ; H ; B sum g: (59) 24

25 Given he morgage ineres rae r m and he sochasic process for he general income y, he above funcions and values saisfy: 1. New households ener he ciy opimally so ha (2) holds; 2. All agens opimize such ha he value and policy funcions lised in (56) - (57) saisfy (7), (9) - (35); 3. Free enry of consrucion rms: N sais es (37); 4. Free enry of morgage companies: m ; sais es (42); 5. The socks of households and invenories F ; B ; B R ; (; ) ; L (; ) ; c ; REO ; H evolve according o (43) - (52); 6. Consisency: B = B sum. Requiremens 1-5 in he above de niion are sandard and have been described in deail above. Requiremen 6 saes ha in equilibrium he measure of buyers wihou foreclosure ags mus be consisen wih he oal measure of buyers acively paricipaing in housing search. As has been menioned, sellers are heerogeneous, and heir disribuion is characerized by (; ) ; L (; ) ; c ; REO. None, however, of he decision problems faced by households, consrucion rms and morgage companies are a eced by his disribuion. In fac, as can be seen from (7), and (9) - (42), all of he value and policy funcions lised in (56) - (57), ogeher wih he morgage conrac m ;, are independen of he sock variables lised in (59). This is rue despie he fac ha he socks hemselves depend on individual decisions, and ha he disribuion of sellers does a ec aggregae saisics. Thus, he model is block recursive in he sense of?. As discussed here, block recursiviy arises in our economy because hrough he compeiive search mechanism heerogeneous sellers selec hemselves opimally ino separae submarkes, aking he rade-o beween he price and he maching probabiliy as given. Given a paricular arge ransacion price, he only facor ha maers for a seller s rading decision is he probabiliy wih which i will be mached wih a buyer; he disribuion of sellers over oher price arges is irrelevan. Vice versa, for a given maching probabiliy ha he seller cares only abou he price a which i will sell. Block recursiviy grealy aids racabiliy by eliminaing he role of he disribuion of sellers in individual decisions. I is especially useful here as i enables us o examine he dynamics of he model in response o aggregae shocks. 25

26 4 An Economy wihou Search To accenuae he role of search fricions in our model, we also consider an economy in which he housing marke is perfecly compeiive. In his economy, houses are perfecly liquid in ha buyers (wihou a foreclosure ag) and sellers are able o rade immediaely. In his case, neiher consrucion rms nor morgage companies hold houses invenory. In his economy, nancial disress is exreme a he beginning of period, wih probabiliy f d an indebed residen owner may experience a defaul shock which forces her o defaul immediaely. A borrowers no hi by such a shock may choose o defaul only in he case in which heir housing equiy becomes negaive Value funcions. Household decisions in sub-period 2 are idenical o hose in he search economy. subperiod-1 household values in his non-search economy are disinguished by he superscrip n. A buyer wihou he foreclosure ag purchases a house a compeiive price p and has he value U n : U n The = V o (p ; m ): (6) An indebed residen owner who does no receive a defaul shock decides wheher and how o sell and wheher or no o defaul. Le H s = 1 and D n = 1 be indicaors of he selling and defaul decisions, respecively. If he homeowner sells, she repays as much of her ousanding deb as possible, keeps any remaining pro, and becomes a buyer wihou he foreclosure ag. If she decides no o sell, hen H s = and she decides wheher o defaul: S n (m ; n) = max H s ;Dn 2f;1g 8 >< >: (1 H s )W (max [; p d(m ; n)]) ( (1 D n )V (m ; n) +H s +D n W f (max ; E V nreo +1 d(m ; n) ) ) 9 >= : (61) >; An indebed owner who experiences a disress shock immediaely defauls. Such an owner has he value S nf (m ; n): S nf (m ; n) = W f (max[; E V+1 nreo d(m ; n)]): (62) 14 Noe, however, ha as defaul is cosly, no all owners wih negaive equiy will defaul. 26

27 A residen owner wihou deb decides wheher or no o sell and has value: S n = max H sw 2f;1g (1 H sw )W (p ) + H sw V : (63) Relocaed owners wih and wihou morgage deb make similar selling and defaul decisions and values V nl V nl (m ; n) = max H sl ;D nl 2f;1g 8 >< >: (m ; n) and V Ln, respecively: (1 H sl )(u(max [; p d(m ; n)] + y L R L ) + L) +H sl ((1 D nl )(u(y L R L x(m ) m) + E V+1(m nl ; n + 1)) +D nl V nl = max H slw 2f;1g (u(max ; E V+1 nreo d(m ; n) + y L R L ) + L) ( (1 H slw )(u(p + y L R L ) + L) +H slw (u(y L R L m) + E V nl +1) ) 9 >= >; (64) : (65) The values of vacan houses in a consrucion rms and morgage companies invenories are given, respecively by: V nc = p (66) V nreo = (1 )p : (67) For morgage conrac = (m ; r m ), he presen morgage values a he beginning of sub-period 2 afer n paymens are given, for relocaed and residen homeowners, respecively, are given by: P L (m ; n) = x(m ) + I fn6=t 1g 1 + i + 8 >< E >: (1 H+1) sl min [p +1 ; d(m ; n + 1)] +H sl +1 ( [D nl +1 min V nreo +2 ; d(m ; n + 1) +(1 D nl +1 )P L +1(m ; n + 1) ) 9 >= >; (68) 27

28 for n 2 f1; T 1g and P (m ; n) = x(m )I fn6=g + I fn6=t 1g 1 + i >< (1 H+1) sl min [p +1 ; d(m ; n + 1)] E >< >: h >: +H sl +1 8 >< +(1 h ) >: ( D nl +1 (m ; n) min V nreo +2 ; d(m ; n + 1) +(1 D+1 nl (m ; n))p+1(m L ; n + 1) n d min V+2 nreo ; d(m ; n + 1) +(1 8 n d ) (1 H+1) s min [p +1 ; d(m ; n + 1)] >< >: +H s +1 8 >< >: ) D n +1(m ; n) min V nreo +2 ; d(m ; n + 1) +(1 D n +1(m ; n)) P +1(m ; n + 1) 9 >= >; 9 >= >; 9 >= >; 9 >= >; (69) 9 >= >; for all n 2 f; ; T 1g, where, H+1, sl H+1, s D+1, n D+1 nl are policies ha households follow in period + 1 condiional on he aggregae shocks and having morgage balance (m ; n + 1). 4.2 Equilibrium. The de niion of equilibrium is similar o ha for he search economy excep ha he following housing marke clearing condiion replaces he consisency requiremen: B = S sum ; where B is he demand of houses and S sum is he oal supply of houses: S sum = TX (1 H s (m ; n ; n)) (m ; n ; n) n= + TX n= (1 H sl (m ; n ; n)) L (m ; n ; n) + c + REO : Finally, he corresponding per capia laws of moion are lised in Appendix A. 28

29 5 Direced Search wih Middlemen In his secion, we embed he mechanism of direced search wih middlemen inroduced by?. 15 In Secion 8 we conras he dynamics ha arise under his mechanism from hose displayed by our baseline economy. In paricular, his exercise is helpful for illusraing he imporance of he elasiciy of housing supply for our resuls. Here, we brie y describe he modi caions o our environmen required o embed his search process in our economy. 5.1 The environmen. The principal di erence beween his alernaive economy from our baseline perains o he search process. Le here be a coninuum of compeiive real esae rms. The housing marke is divided ino wo wo separae pars, which we refer o respecively as buying and selling markes. Real esae rms send agens (RE s henceforh) o boh markes o rade bi-laerally wih homeowners and prospecive buyers. In he buying marke, RE s sell homes o prospecive homeowners; In he selling marke, RE s buy houses from homeowners who desire o sell. A real esae rm incurs a cos b ( s ) for sending an agen o he buying (selling) marke. Each RE can sell/buy a mos one house a a ime. The wo markes open sequenially, wih he selling marke opening rs. Each of hese markes are characerized by compeiive search: There are a coninuum of poenial sub-markes disinguished by price and ighness. Boh real esae rms and households ake as given he combinaion of price and ighness across all submarkes. In addiion o hose which hey acquire in he selling marke, real esae rms also purchase new houses from consrucion rms a a compeiive price P c. We refer o his as he shadow price of housing, as does?. Thus, here exiss a compeiive marke for new housing o which only real esae consrucion rms have access. Real esae rms do no hold invenories of houses. Tha is, by conrolling he measures of RE s ha hey send o each marke, and by purchasing rms for re-sale immediaely from conrucion rms, real esae rms can mainain a zero ne ow of housing. E ecively, a law of large numbers assumpion renders real esae rms purchase and sale of houses deerminisic, alhough individual RE s mach randomly wihin sub-markes. In all oher aspecs, he environmen for his alernaive economy remains he same as ha for our baseline economy. In paricular, buyers remain idenical and sellers heero- 15 Hedlund uses his environmen o sudy racably an environmen wih wo-sided heerogeneiy. 29

30 geneous? 5.2 Value funcions All individual value funcions and morgage company decisions remain he same as in our baseline because nohing has changed from he households and he morgage company s perspecives. Individual choices of submarkes are, however, now denoed p b ; b and (p s ; s ), o disinguish paricipaion in he buying and selling markes, respecively. For i = b; s, le i p i ; i denoe he measure of REs ha a real esae rm sends o he buying and selling markes. Le N d denoe he measure of houses ha a real esae rm acquires from consrucion rms in he compeiive housing marke. A real esae rm solves he following saic problem: max f b (pb ;b );s (ps ;s );N d g Z Z (p b ; b )p b b b (p b ; b ) (7) [(p s ; s )p s s ] s (p s ; s ) P c N d Z subjec o N d + Pro maximizaion implies ha Z (p s ; s ) s (p s ; s ) (p b ; b ) b (p b ; b ) (71) (p b ; b ) p b P c b and b (p b ; b ) wih comp. slackness (72) (p s ; s ) (P c p s ) s and s (p s ; s ) wih comp. slackness. (73) As we can see, P c e ecively serves as he shadow price of houses. Tha is, i is he muliplier on (??) in a real esae rm s opimizaion problem. As in he baseline, consrucion rms sell newly consruced houses such ha supply of new houses N sais es This equaion e ecively replaces (37). k (N ) + q (N ) = P : (74) 5.3 Equilibrium The de niion of equilibrium corresponds o ha described in Secion 3.5, wih he following amendmens: The compeiive marke for new housing clears: N d = N ; 3

31 The condiion B = B sum is replaced by (??) holding wih equaliy. Noe ha because all buyers are homogenous, here will only be one acive buying submarke (p b ; b ) in equilibrium. 6 Calibraion We now choose parameers for boh he baseline search and non-search economies o mach seleced facs of he U.S. economy in a seady-sae equilibrium wih balanced growh. In his seady-sae, he housing sock grows a he rae of populaion growh and all oher componens of he equilibrium remain consan. For he baseline search economy, we choose he follwoing funcional forms: u(c ) = log(c ) M(B; S) = $B S 1 k = 1 N 1 q = qn 1 (75) where is he elasiciy of he measure of maches wih respec o he measure of buyers and represens he elasiciy of new land supply wih respec o land prices. Table 1 liss parameer values for he baseline search economy. Parameers above he separaing line are se o mach he corresponding arges direcly. Parameers below he line are deermined joinly o mach he se of parameers menioned in he able. A ime period is de ned as one year. 16 The discoun facor is se o re ec an annual ineres rae of 4%. Income in he seady sae is normalized o one. Thus, all presen values and prices are measured relaive o seady-sae per capia income. The coninuaion value upon leaving he ciy, L, is equal o he seady-sae value of being a perpeual rener, V p ss. To deermine he morgage rae r m, he annual yield on inernaional bonds i is se a 4%. The values of and % are deermined joinly in calibraion. 16 Seing a ime period as one year is due o speci cs of our model. In his seing, households canno save (so ha all buyers are homogeneous) and hus house buyers can only nance he down paymen wih heir periodic labor income. Empirically, he average housing price is abou 12.8 imes of quarerly income. If one period akes a quarer, hen a buyer s periodic income is oo small o a ord a ypical down paymen of 15% - 2%. In fac, borrowers could only a ord a down paymen less han 7:9% of he average housing price if he ime period was se as a quarer. 31

32 Table 1: Calibraion Parameer Values Parameer Value Targe Daa Parameers deermined independenly.96 Annual ineres rae 4.% p.12 Annual mobiliy of reners 12% h.32 Annual mobiliy of owners 3.2% 1.75 Median price-elasiciy of land supply 1.75 i.4 Inernaional bond annual yield 4.% T 3 Fixed-rae morgage mauriy (years) 3.12 Annual populaion growh rae 1.2% f.8 Average duraion (years) of foreclosure ag 5 q.96 Average land-price-o-income raio 3% m.8 Residenial housing gross depreciaion rae 2.5% 5 Median price elasiciy of new consrucion 5 &.5 Average ren-o-price raio 5% Parameers deermined joinly.44 Loss severiy rae 46%.246 Average down-paymen raio 2% %.74 Average annual FRM-yield 7.2%.57 Fracion of households ha ren 33.3% d.6 Annual foreclosure rae 1.6% z H.328 Average loan-o-income raio a originaion 2.72 $.56 Average fracion of delinquen loans repossessed 33.5%.137 Average housing price relaive o annual income Relaive volailiy of sales growh 1.32 p 6.2 Relaive volailiy of populaion growh.17 The following parameers and arges are chosen following?: The rae is chosen o mach he annual populaion growh during he 199s. The value of p is se o mach he annual fracion of reners ha move beween counies and h o mach he annual fracion of home owners who move beween counies according o he Census Bureau. The supply elasiciy parameer is se o = 1:75 following?. There, he supply elasiciy for 95 U.S. ciies is esimaed for he period beween 197 and 2. The esimaes vary from :6 o 5:45 wih a populaion-weighed average of 1:75 (2:5 unweighed). The seady-sae uni price of land q is se such ha he relaive share of land in he price of housing is 3% (see? and?). The elasiciy of new consrucion wih respec o he price of housing,, is se equal o he median elasiciy for he 45 ciies sudied by?, = 5. The mainenance cos m is chosen o be 2:5% of he seady-sae housing price according 32

33 o?. Moreover, he average house price is 3:2 imes annual income. The value of is calibraed so ha he ownership rae in he ciy P ss =( P ss + B ss + Bss R + F ss ) = 66:7%, where P ss denoes he oal measure of homeowners in he seady sae. The Census Bureau repors he ownership rae among households whose head is beween age 35 and 44 is roughly 66:7%. We se he average ren-o-price raio &, based on empirical ndings of he Lincoln Insiue of Land Policy. They esimae annual rens for owner-occupied unis based on daa from he Census Bureau. These esimaed rens are hen divided by he average self-repored value of owner-occupied unis o obain he ren-o-price raio. This number was fairly sable and hovered around 5% prior o he mos recen housing boom leading up o he 28 nancial crisis. The remainder of he parameers lised in Table 1 are deermined o mach joinly a number of arges based on he model. Firs, we se he average lengh of ime following a foreclosure unil a borrower is again allowed o access he morgage marke o ve years. This ime frame is consisen wih he policies of Fannie Mae and Freddie Mac, which guaranee mos U.S. morgages. Thus we se he probabiliy ha a foreclosure ag remains on a borrower s credi record o f = :8. According o he Federal Housing Finance Board, he average conrac rae on convenional, xed-rae morgages beween 1995 and 24 was 7:2%. We arge an average down-paymen raio of 2% and an annual defaul rae of 1:6%, which is close o he average annual foreclosure rae among all morgages during he 199s according o he Naional Delinquency Survey by he Morgage Bankers Associaion. The loss severiy rae is de ned as he presen value of all losses on a given loan as a fracion of he balance on he defaul dae. According o?, hese losses are caused by boh ransacion and ime coss associaed wih he foreclosure process, and by he fac ha foreclosed properies end o sell a a discoun relaive o oher houses wih similar properies. Using a daa se of 9; rs-lien liquidaed loans, he auhors esimae ha loss severiy raes range from around 35% among more recen morgages o as much as 6% among older loans. Based on hese numbers, we choose parameers so ha in he even of a defaul, he value of minfv REO ; dg d = :54 (76) on average. This implies an average loss severiy rae of 46%.? examine sub-prime morgage daa and nd ha 5% of delinquen loans wih loan-o- 33

34 value raios beween 8% and 9% end up being repossessed, compared o 55% of delinquen loans wih loan-o-value raios beween 9% and 1%, and 59% of delinquen loans wih loan-o-value raios above 1%. In addiion,? nd ha 3% of defauled convenional xed-rae loans and 5% of defauled convenional adjusable rae loans ransiion o REO and?? repor ha 32% o 38% of defauled FHA loans ransiion o foreclosure. Based on hese numbers, we choose parameers such ha in he even of nancial disress, he average probabiliy of a successful sale is 66:5%. Tha is, 33:5% of he homeowners who experience nancial disress ulimaely end up in foreclosure in he seady-sae. Evidence available from he American Housing Survey (AHS) suggess ha prior o 23 he raio of he original loan size o yearly income averages 2:72: Accordingly, we choose parameers such ha he seady-sae loan-o-income raio a originaion is given by m ;ss =y ss = 2:72. Finally, dynamics of our model depend crucially on wo elasiciies: he elasiciy of G(), evaluaed a " c, p = " c G (" c )=G(" c ) and he elasiciy of he maching funcion wih respec o he number of buyers,. These wo parameers are calibraed joinly by using esimaes of he relaive sandard deviaions of populaion growh and housing sales growh in response o income shocks as in?. For he non-search economy, all parameers remain a heir values in Table 1 excep for f d, zh and, which are adjused so ha he seady-sae saisics mach he relevan arges again. In he seady-sae of he non-search economy, he consrucion cos parameer is adjused so ha P = 3:2 given he rae of populaion growh. Appendix B provides liss of parameer values re-calibraed o he non-search and he direced-searchwih-middlemen economies respecively. 7 The Seady-Sae We now characerize he seady-sae of he baseline search economy. In he seadysae, per capia income (which is exogenous) remains consan over ime. The seadysae is de ned based on he de niion of equilibrium esablished in Secion 3.5, plus he requiremen ha all funcions and values lised in (56) - (59) are ime invarian. In he seady-sae, all owners have sricly posiive home equiy. Residen owners who receive neiher moving nor nancial disress shocks do no aemp o sell heir houses, regardless of heir ousanding morgage balances. All relocaed owners coninue o make repaymens unil a successful sale occurs or he morgage is compleely paid o. Finally, all disressed owners aemp o sell heir houses. As such, here no sraegic defauls in 34

35 he seady-sae, in he sense ha foreclosure occurs only as he resul of he a nancial disresse shock followed by an unsuccessful sale. Figure 3 depics he seady-sae disribuion of house sellers across ypes. Nearly half of he sellers in he marke are here as a resul of nancial disress. Noe ha his is consisen wih he mobiliy and defaul rae arges from he calibraion. Figure 4 presens he disribuions residen homeowners (upper panel) and sellers (lower panel) by morgage saus. The disribuion of homeowners is purely driven by exogenous shocks.the measures of owners decrease wih he number of paymens ful lled, for n = 1; ; 29, owing o he e ecs of boh moving and nancial disress shocks which a ec homeowners a consan raes over ime. The large bin a n = 3 represens all homeowners who have repayed heir enire morgage before experiencing eiher shock. While hese homeowners no longer face a risk of nancial disress, hey remain subjec o moving shocks and exi he ciy evenually wih probabiliy one. Similarly, he measure of disressed sellers decreases wih he number of paymens ful lled, for n = 1; ; 29, alhough here are no such sellers wih n = 3 by consrucion. The disribuion of relocaed sellers, in consras, is driven by households choice of selling probabiliy. These households are no required o sell, and hey are no longer hi by relocaion shocks. The fac ha some ener sub-markes wih high prices and low sales probabiliies accouns for he hump-shape of he disribuion. The spike a n = 3 arises from he fac ha residen homeowners who have paid o heir morgages are sill subjec o moving shocks, a which poin hey become relocaed sellers wihou a morgage. Figure 5 illusraes he disribuion of housing prices in he seady sae. 7.1 Leverage and seller behavior Figure 6 illusraes he relaionship beween a seller s opimal choice of sub-marke, which deermines boh her asking price and sales probabiliy on selling price and probabiliy, and her deb posiion. Overall, a disressed seller is more eager o sell han a relocaed seller and herefore, condiional on deb posiion (represened here by he LTV raio), poss a lower price and sells wih a higher probabiliy. The cos of failing o sell are higher for disressed sellers for wo reasons. Firs, a disressed seller has no choice bu defaul if she fails o sell her house wihin he period, while a relocaed seller reains he choice of wheher o defaul in he nex period. Second, a relocaed seller receives coninuaion value L, which is independen of her credi record, while a disressed seller who has defauled on her morgage and remains in he ciy is excluded from he housing marke for he nex ve 35

36 13% 13% 48% 26% Disressed sellers Relocaed sellers Consrucion Firms REO invenory Figure 3: Composiion of house sellers periods on average. Noe also he relaionship beween he posed asking price and LTV. For boh ypes of seller, he posed prices is iniially (very) weakly decreasing in LTV. A some poin, and his is mos dramaic for disressed sellers, he relaionship becomes srongly increasing. For disressed sellers in paricular, he relaionship resembles closely ha repored by? (see Figure 2, p. 267). in heir empirical sudy of condominium sales in Boson during he 199 s and is also consisen wih he ndings of Anenberg (211) and ohers. Figure 7 combines heir resuls wih ours in common unis. 17 The closeness of he relaionship is sriking, given ha none of he quaniies depiced for our economy are calibraion arges. 18 To undersand leverage-price relaionship illusraed in Figures 6 and 7, consider he case of a disressed seller. In he seady sae, he gain from rade for a such a seller as a 17 In our economy, he posed (asking) price is proporional o he mark-up, as all vacan (non-foreclosure) houses have a common value. 18 I is reasonable o believe ha he curvaure of he price choice of relocaed sellers would resemble more of he Genesove-Mayer resul if he coninuaion value, L, did depend on one s credi record in he ciy. 36

37 Number of paymens: n.15 Financially disressed seller Relocaed seller Number of paymens: n Figure 4: Seady-sae disribuions of morgage saus respecively among residen owners (upper panel) and household sellers (lower panel) funcion of her ousanding deb, d, is given be: (d) = W (p d) W f max ; V REO d (77) = ( W (p d) W f V REO d ; if d < V REO W (p d) W f () ; if d V REO (78) = f (1 h ) W f () U + ( u (y R + p d) u y R + V REO d ; if d < V REO u (y R + p d) u (y R) ; if d V REO ; : (79) This resul follows from (11) wih he dependence of he ousanding deb (d) on he speci cs of loan (m ; n) suppressed. Di ereniaing (79) wih respec o d, we have (d) = ( u y R + V REO d u (y R + p d) ; if d < V REO u (y R + p d) ; if d V REO ; (8) Given ha u > and u <, we have he following: 37

38 Selling price Figure 5: Seady-sae disribuion of housing prices Proposiion 1 (i) If d V REO, hen (d) < ; (ii) If d < V REO, (d) > for any given p > V REO and (d) < for any given p < V REO. In he seady-sae a disressed seller chooses a sub-marke o maximize her expeced gain from rade. Given he maching funcion and free-enry of buyers, he opimal submarke decision in (11) is equivalen o max p; () (d; p) (81) where = (p) = 1 U W () V o (p; m ) W () follows direcly from (8) evaluaed a he seady-sae. (82) I is sraighforward o show (i) ha ( (p)) is sricly decreasing in p given he properies of he maching funcion lised in (3) and (4); and (ii) ha he gain from rade (d; p) increases wih price p for any deb level, d; since u >. Thus, a higher selling price raises he gain from rade, bu reduces selling probabiliy. The opimal sub-marke choice re ecs his rade-o. The shape of he relaionship depiced in Figures 6 and 7 can be undersood using Proposiion 1. When a seller is su cienly indebed (d V REO ), he gain from rade 38

39 .9 Maching probabiliy.85.8 Financially disressed seller Relocaed seller Selling price Loan o value raio (%) Figure 6: Leverage and seller behavior. The op panel shows he choices of selling probabiliy by disressed and relocaed sellers. Correspondingly, he boom panel demonsraes he choices of selling price by he wo ypes of sellers. (d; p) is sricly decreasing in deb, d, for any given price, p. Essenially, heavily indebed sellers wih more deb worry less abou making sure ha rade happens han abou raising he gain from rade if i does. The reason for his is ha hey will receive no residual pro from a rade unless i is a a high price. The foreclosure cos is xed, and he cos of defauling on a larger deb is borne by he lender, raher han he seller herself. For a less indebed seller (i.e. one wih d < V REO ), has greaer incenive o sell, as failure o do so resuls in he loss of residual pro as well as he cos of he foreclosure ag. Moreover, for p > V REO, he gain from rade is sricly increasing in deb d. As such, a more indebed seller (bu wih d < V REO sill) will chose a lower price/higher sales probabiliy. Overall, for d < V REO, he e ec of deb on he gain from rade (i.e., (d)) is likely o be small in ha d a ecs he reurns boh o selling and failing o do so in a symmeric way (see (8)). Thus he relaionship is essenially a for lower LTVs bu rapidly increasing for higher LTVs. As a nal noe on he leverage-price relaionship, i is worhwhile clarifying ha he condiion p > V REO is no paricularly resricive. For example, in our baseline calibraion, he seady-sae value of V REO = 1:79, while he minimum selling price chosen by a seller 39

40 Figure 7: The red curve (lef axis) depics he raio of asking price o assessed value as measured by Genesove and Mayer (1997) ploed agains sellers LTV. The blue curve (righ) depics he same relaionship for he raio of he posed price o he value of a house in REO invenory for our baseline search economy. is 3.4. In general, he V REO ends o be much lower han he choice of selling price by any seller due o he foreclosure and carrying coss associaed wih houses in REO invenory. Noe ha while our proof of Proposiion 1 relies on he assumpion ha consumpion goods are no sorable, he resul is in fac more general. In paricular, he same properies for (d) arise as long as W (p d) > and W f V REO d W (p d) > for d < V REO. The former condiion requires ha he value of a buyer wihou he foreclosure ag is a sricly increasing funcion of her asse holdings. The laer requires ha he slope of he value of a buyer wih he ag a asse posiion V REO d be greaer han ha of he value of an un agged buyer a p d for lower levels of deb. Considering ha p > V REO in general, his requiremen does no seem overly resricive for value funcions such as W f () and W (), which are sricly concave. Indeed, we also gain suppor for his argumen given he mach beween he predicions of our calibraed economy and he evidence provided by?. 4