Indebted Sellers, Liquidity and Mortgage Standards

Transcription

1 Indebed Sellers, Liquidiy and Morgage Sandards Allen Head Hongfei Sun Chenggang Zhou Queen s Universiy Queen s Universiy CMHC February 19, 218 Absrac The e ecs of households indebedness on heir house-selling decisions are sudied in a racable dynamic model wih search in he housing marke and defaulable longerm morgages. In equilibrium, boh sellers asking prices and ime-o-sell increase wih he relaive size of heir ousanding deb. For sellers in nancial disress, his behavior resuls also in a posiive relaionship beween deb and defaul risk a he individual level. Prices and ime-o-sell deermine boh he liquidiy of he housing marke and he overall risk of defaul. As such hey deermine he exen of pro able lending by compeiive morgagees. Calibraed o he U.S. economy he model generaes, as observed, posiive correlaions across sellers wih regard o asking prices and loan-o-value raios and over ime beween boh house prices and ime-o-sell and morgage loan-o-value raios a originaion. JEL Classi caion: E3, G21, R1, R31 Keywords: Housing; Morgages; Defaul; Search; Liquidiy; Block Recursiviy An earlier version of his paper was iled, Defaul, Morgage Sandards and Housing Liquidiy. Russell Cooper, Aaron Hedlund, Huw Lloyd-Ellis, Erwan Quinon, Shouyong Shi and paricipans a several seminars and conferences provided helpful commens. David Genesove kindly provided daa from his earlier work. The SSHRC and he Bank of Canada Governor s Award provided nancial suppor. The opinions expressed in he paper are our own and do no re ec he views of he Bank of Canada. All errors are our own. 1

2 1 Inroducion In his paper we sudy he in uence of a household s indebedness on is house selling decisions. We measure indebedness by he household s remaining morgage relaive o he marke value of is house and refer o i as he household s leverage, or is loan-o-value raio (LTV). To his end, we develop a dynamic model wih a fricional housing marke and defaulable long-erm morgage deb which gives rise in equilibrium o disribuions across households of house prices, deb, and defaul probabiliies. Morgage sandards (i.e. he maximum LTV a originaion) are deermined by he expeced defaul rae and he cos of defaul. These in urn depend on he exen of housing marke liquidiy; ha is, he speed wih which houses can be ransferred from one owner o anoher. In our heory, indebed households are never forced o defaul ourigh. Ye, in equilibrium, prospecive sellers wih larger ousanding deb ask higher prices, sell more slowly and when in nanical disress, defaul a higher raes. Such a relaionship beween deb levels and asking prices has been observed by Genesove and Mayer (1997, 21) and Anenberg (211). A he same ime, he more liquid he housing marke, he less likely sellers (of any deb level) are o defaul. House prices vary posiively over ime wih liquidiy, and so he heory also capures he observaion ha U.S. house prices move posiively wih morgage LTV s a originaion for rs-ime home buyers. 1 We model a growing populaion of ex ane idenical households ha ener a single ciy (on which we focus) when he value of doing so exceeds heir ouside opion. Once here, hey require housing and may eiher live as a rener or own one of a large number of idenical houses. These are produced and sold iniially by a compeiive developmen indusry. Residen households remain, eiher as reners or homeowners, unil hey leave he ciy as a resul of exogenous shocks. Wihin he ciy, houses are sold following he proocol of direced search as described by Moen (1997). Sellers o er houses in a range of sub-markes, each characerized uniquely by an announced asking price and disinc maching probabiliies for buyers and sellers. Search is direced in he sense ha sellers choose sub-markes opimally given he rade-o beween he posed price and he maching probabiliies implied by buyers opimal search behavior. House purchases are nanced by morgages issued by compeiive lenders (e.g. banks). We assume ha hese lenders issue loans of xed duraion a an exogenous ineres rae, 1 Beween 1987 and 212, he average LTV a originaion for rs-ime home buyers as measured by he American Housing Survey (27, 29, 211) exhibied a correlaion wih he Case-Shiller Index of.82 in levels and.181 in growh raes. 2

3 bu choose he size (and hus he LTV) of he morgage a originaion. Households ha do no own pay ren each period equal o a xed and exogenous fracion of income. Indebed homeowners are subjec randomly o nancial disress shocks which force hem o eiher sell heir homes hrough he search process or defaul and face foreclosure. A defauling household has is house seized by he lender, a foreclosure ag placed on is record, and is prohibied from paricipaing in he housing marke unil he ag is lifed, which also occurs randomly. Households in disress are no commied o sell, and based on heir speci c siuaions decide wheher and how o do so. Households hus e ecively choose opimally heir likelihood of defaul. In principle, a homeowner in nancial disress can avoid foreclosure by posing a su cienly low price and driving heir likelihood of selling o one. In equilbrium, however, hey do no do his. Raher, selling homeowners wih su cienly high LTV s pos prices ha are seeply increasing in heir ousanding morgage deb and are hus more likely o defaul han are less indebed ones. Moreover, all households choose prices associaed wih subsanial probabiliies of defaul. 2 As housing marke liquidiy deermines he probabiliy of sale a any given price, i a ecs no only he expeced defaul rae and homeowners expeced losses upon defaul, bu also he expeced reurns o foreclosure for lenders. Foreclosed houses sell more quickly and a higher prices in a liquid marke, lowering heir expeced carrying cos and hus he cos of defaul o lenders. Lenders hus relax lending sandards in he sense ha hey o er larger loans (i.e. higher LTV s a originaion), when he housing marke is expeced o be more liquid in he fuure. This occurs when income and house prices are high. We build on he model of Head e al. (214) which considers he dynamics of house prices and consrucion wih homogeneous households and complee nancial markes. 3 Here, we inroduce limied commimen and allow households o defaul a will, subjec o he foreclosure procedure. homes and heerogeneiy among households, ex pos. This generaes boh a role for morgage deb secured by Head e al. (214) focuses mainly on random search, while direced search, hrough sellers price seing decisions, is inegral o our analysis. 2 Some households may choose o defaul ourigh, making no aemp o sell. This occurs in cases of su cien negaive home equiy, a siuaion which may arise if house prices fall su cienly in response o an aggregae shock. As a resul, such shocks can cause paricularly severe waves of defaul and foreclosure if hey occur when he economy has a high proporion of highly levered homeowners. 3 As such, i conribues o he growing lieraure on search fricions in he housing marke (see, for example, Diaz and Jerez (213), Perosky-Nadeau and Rocheeau (216), He e al. (215), Head and Lloyd-Ellis (212), and Wheaon (199)). 3

4 We consider also he e ecs of emporary shocks o income in he ciy. The resuling aggregae (i.e. ciy level) dynamics wih regard o prices, populaion movemens and consrucion are much as in Head e al. (214). Here, however, we focus also on boh he aggregae dynamics of lending and foreclosure, and he di erenial e ecs of shocks on households wih di eren ousanding LTV s. Overall, we nd ha households wih di eren deb levels respond very di erenly as he ciy-wide housing marke evolves hrough he e ecs of aggregae shocks which resul in di erenial capial gains and losses depending on deb. Highly levered households experience relaively large ucuaions in home equiy in response o shocks, and heir pricing decisions re ec his. In paricular, he ciy will face a paricularly large wave of defauls if a negaive shock occurs when a relaively large number of homeowners already have high levels of ousanding deb. Oher papers ha sudy he relaionship beween housing marke liquidiy and lending aciviies are Ungerer (215), Hedlund (216b), Hedlund (216a) and Garriga and Hedlund (216). The las hree are more closely relaed o ours as hey consider models feauring direced search, long-erm morgages, and limied commimen. These papers all focus, however, on macroeconomic issues and feaure a di eren marke arrangemen in which heerogeneous buyers and sellers inerac only indirecly via inermediaries. Here, we focus on he di erenial decisions of sellers exclusively, and nd i useful o absrac from heerogeneiy on he par of buyers. We do his no for racabiliy per se, bu raher o focus speci cally on he sraegic decisions sellers and heir implicaions for he behavior of lenders, hereby simpli ng he analysis and inerpreaion of our ndings. 4 Along hese lines, in addiion o having a di eren maching environmen we also focus on housing markes a he ciy as opposed o naional level and sudy nie morgages a xed ineres raes raher han in nie-horizon morgage conracs. While his adds complexiy, we view our paricular resricions on conracs reasonable, as in he U.S. convenional morgages ypically have a 3-year erm and abou 7% of are a xed ineres raes. We also do no impose a shor-selling consrain on households as do boh Garriga and Hedlund (216) and Hedlund, Karahan, Miman and Ozkan (217). As noed above, in our model homeowners are never forced o defaul. decisions are driven enirely by sraegic consideraions. Raher, heir pricing and defaul 4 This seup renders he model block recursive and racable, in spie of he presence of heerogeneiy among boh buyers and sellers. In general, 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 Shi (21), Shimer and Smih (2), Smih (26), and Eeckhou and Kircher (21)). Shi (25) furher shows ha dynamics of soring wih wo-sided heerogeneiy can be racable in some seings. Neverheless, our model would be inracable for he purposes of he exercise we consruc if we allowed for wo-sided heerogeneiy wih endogenous saving alongside morgage choices. 4

5 In a New Keynesian model along he lines of Iacoviello (25) wih credi-consrained consumers and housing marke fricions, Ungerer (215) shows ha expansionary moneary policy leads o higher leverage among homeowners. A decrease in morgage 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 model di ers from ha sudied by Ungerer (215) in several respecs: Firs, in ha paper nancial fricions ake he form of a collaeral consrain along he lines of Kiyoaki and Moore (1997) and here is neiher defaul nor liquidaion of foreclosed houses in equilibrium. Second, deb is one-period, and so aggregae shocks may lead o forced deleveraging. Relaed o his, Ungerer (215) models houses as divisible so ha adjusmens can be made easily o he quaniy of housing owned. Finally, he housing sock is xed and here is no consrucion. Our paper is also relaed o hose of Ngai and Tenreyro (214) and Ngai and Sheedy (215) in ha we focus on selling behavior per se. 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 he paper is organized as follows. Secion 2 describes he environmen and Secion 3 formalizes a compeiive search equilibrium. Secion 4 describes our calibraion and Secion 5 characerizes he balanced growh pah of our search economy so parameerized. Secion 6 considers dynamics in response o aggregae shocks, and makes comparisons o an economy wihou search ha is described in an appendix. Secion 7 concludes. 2 The Environmen Time is in nie and discree, wih ime periods indexed by. The economy consiss of a single ciy, and he res of he world. The world is populaed by a measure Q() of ex ane idenical households, which grows exogenously a ne rae. Each household lives forever 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 5

6 follows a saionary sochasic process in logarihms. 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()] ; = 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: lim c!1 u (c) = and lim c! u (c) su cienly large. households have he common discoun facor, 2 (; 1). Boh consumpion goods and housing services are non-sorable and here is no saving by households across periods. 5 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 via he disribuion funcion G("), wih suppor [; "]. Households ha ener he ciy are separaed 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: All " c () = V b () + (1 )V p (); (1) where V b () and V p () are lifeime (from ime ) 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 which we refer o as developers. Each new house requires one uni of land, which can be purchased in a compeiive marke a price q() = Q(N()). developer also incurs consrucion cos k() = K(N()), where N() denoes he oal quaniy of new houses buil in period. The Houses require one period o build; hose consruced in period become available for sale a he beginning of period + 1. Home-ownership resuls immediaely in boh a uiliy bene o he owner-occupier and a per-period mainenance cos. or no hey are occupied. Houses depreciae over ime, regardless of wheher Depreciaion is, however, o se by he owner a mainenance 5 Because households experience idiosyncraic shocks ha are o some exen uninsurable, saving would complicae maers by rendering prospecive house buyers heerogeneous. 6

7 cos d 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; i.e. R() = &y(). The supply of renal accommodaion is oally elasic, and is no considered par of he ciy s housing sock. A he end of each period, all households in he ciy, regardless of heir ownership saus, may experience a shock which induces hem o leave he ciy permanenly. 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. exi he ciy receive coninuaion uiliy, V. All households which Exiing homeowners also have vacan houses ha hey may wan o sell, depending in par on heir ousanding morgage deb, if any. These households also have he opion of defauling. In he ciy, he housing marke is characerized by compeiive search. We imagine i as being characerized by a large variey of poenial sub-markes indexed by a price, p, and a pair of maching probabiliies; one for buyers and one for sellers. Wihin each sub-marke, 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 measures of buyers, B, and sellers, S, presen in he sub-marke) and p he posed ransacion price. ener. Boh buyers and sellers ake (; p) for all sub-markes as given and decide which o The maching probabiliies for buyers () and sellers () are given by () = () = M (B; S) = M 1; 1 B (2) M (B; S) = M (; 1) = (): S (3) Each buyer and seller can ener only a single sub-marke 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 sub-markes 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, developers sell newly buil homes. decide o sell. Second, homeowners who receive exi shocks as described above may Noe ha hese sellers are heerogeneous o he exen ha hey have 7

8 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, lenders sell foreclosed houses (see below). In our calibraion, prices ypically exceed per period income and as here is no saving, households mus borrow o nance house purchases. 6 Morgages are provided by a large number of perfecly compeiive lenders owned by risk-neural invesors who consume all pro s and losses ex pos. 7 To nance heir loans, hese lenders rade one-period risk-free bonds a an exogenous ineres rae, i, in an exernal bond marke. They also incur a proporional service cos, ; per period associaed wih he adminisraion of morgages. The deb conrac is a xed-rae morgage wih nie mauriy T. Le m and r represen he size and ineres rae on a morgage loan issued in period, respecively. Conrac, (m ; r ; T ), speci es a consan paymen per period: x(m ; r ) = r 1 (1 + r ) T m : (4) As he homeowner makes paymens, he principle balance on a period morgage, m n, evolves via m n+1 = 1 + r m n x m ; r (5) where n 2 f; T 1g and m = m. Since T is xed exogenously and r is consan over he life of he conrac, x() is unrelaed o afer originaion, and m n, for n 2 f; T 1g, represens he morgage balance a he beginning of he period in which he n+1s paymen will ake place. 8 A borrower can erminaeher morgage conrac a any ime by paying o he remaining balance. Morgages, however, are issued only on new home purchases. 9 A morgage erminaion is a defaul if he borrower does no repay all of he ousanding morgage balance. Defaul leads o foreclosure, whereby he lender akes conrol of he house, remiing o he borrower any surplus value of he house in excess of he ousanding loan 6 In he absence of saving, households would prefer o borrow o smooh consumpion even if he house price were less han period income. 7 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. 8 Tha is, a he beginning of period, m n represens he remaining balance on a morgage issued in period n 1, for n = ; : : : ; T 1. 9 In he absence of saving, a limiaion on re nancing is required o generae a disribuion of ousanding morgage debs on he balanced growh pah. In he absence of such a resricion homeowners would have incenive o re nance every period in order o smooh consumpion. 8

9 balance. Morgages are non-recourse, meaning (as here is no saving) ha lenders do no have access o eiher homeowners curren or fuure income in he even of a defaul. 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 illnesses 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 a he end of he curren period and eiher pay heir ousanding deb or defaul. In he even of defaul, a borrower s morgage balance is se o zero and a foreclosure ag is placed onher credi record. The lender repossesses he borrower s house, pus i in real-esae-owned (REO) invenory, and decides wheher and how o sell i saring 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 lender loses a fracion 2 (; 1) of he revenue o cover an exogenous cos, which we hink of as represening legal fees, coss of deferred mainenance, ec.. 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 = i++%, where i and are exogenously given as described above. The componen % is a risk premium, which compensaes for he risk of defaul. 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 1: Time Line 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. 9

10 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 on he submarkes, (; p), in which o search and lis heir houses for sale, respecively. Afer he housing marke closes, he morgage marke 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 (for mainenance, on new house purchases, morgages and/or rens), and consume he remainder. A he end of he period, moving shocks are revealed for all households and hose who receive hem leave he ciy immediaely. Figure 1 provides an illusraion of he iming of decisions. 3 Equilibrium We begin by describing in deail he behavior of agens and hen de ne an equilibrium for he environmen described above, which we 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. Throughou V () s will be used o denoe agen and house values a he beginning of period ; while W () s will be used o denoe agens values a he begining of he second sub-period of. Sub-scrips will be used o disinguish agen and house saes. In general, household values a he beginning of he second sub-period also depend on inra-period asse holdings, a. For agens who are home-owners, heir values, V o (m n ; ) and W o (m n ; ), depend also on heir ousanding morgage balance. Given ha house purchases ake place in he rs sub-period, new homeowners values depend also on he price a which hey purchased heir home, W o (p; m ; ). From his poin we will suppress he dependence of values on ime where possible. In so doing, primes (e.g., V o ) will be used o denoe fuure values The rs sub-period House rading and morgage defaul decisions are boh made in he rs sub-period. Le V b denoe he value funcion for a buyer. These households are eiher new enrans or 1

11 hose no owning a house and wihou a foreclosure ag, i.e. for whom f = : V b = max [()W o (p; m ) + (1 ())W b ()] : (6) (p;) In sub-period 1, a buyer will search for a house o buy, choosing opimally o ener submarke (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 W o (p; m ). The price paid and inial loan balance m (o ered by he lender and speci ed below) deermine he homeowner s down-paymen. 11 Wih probabiliy 1 (), he buyer fails o ge a mach, remains a buyer, and proceeds o sub-period 2 wih value W b (). W b () indicaes he inra-period asse balance, a. As noed above, he argumen of This will be non-zero only in he even ha he buyer has sold a house in he sub-period ha has jus ended. A buyer who eners he curren period wihou a house, necessarily eners he second sub-period wih a =. Wih buyers free o choose among hem, all acive sub-markes mus o er he same value, V b. Using (6) yields = 1 V b W b () (p) : (7) W b (p; m ) W b () Thus, free-enry of buyers deermines he relaionship beween he ransacion price and marke ighness across sub-markes. Le V o (m n ) denoe he value for a residen owner wih morgage m n 1 a originaion, who has made n 1 paymens and is no in nancial disress: 8 >< ((p s ))W b (max[; p s m n ])+ ( V o (m n ) = max (1 Dn )W o (m n ) p s >: [1 ((p s ))] max D n2f;1g +D n W f (max[; E V REO m n ]) This household decides wheher and in which sub-marke o sell her house. be represened by choice of asking price alone, given (7). ) 9 >= : >; (8) This can If he household eners submarke p s, hen wih probabiliy, ((p s )), he house is successfully sold. In his case, his individual repays as much ousanding deb as possible, and keeps he remaining pro, if any, o ha a = max[; p s m n ]. The household hen proceeds o he second subperiod as a buyer wihou he foreclosure ag and value W b (a). Noe ha he consrain 11 We measure loan-o-value (LTV) raios by he morgage balance relaive o he value of a house in REO invenory. Thus, while home buyers make di eren down-paymens depending on he sub-marke in which hey purchase, a each poin in ime he LTV a originaion is he same for all morgages. 11

12 p s m n is no imposed on on sellers. Tha is, he choice of selling price is no consrained o meeing/exceeding he ousanding deb. S hor sales are an opion in he sense ha morgage lenders allow indebed homeowners o clear heir deb (wihou he consequence of foreclosure) wih an amoun lower han he ousanding balance, as long as he owner makes he e or o lis he house and successfully sells a he lised price. 12 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, for n = ; : : : ; T 1, D n = 1 if a household who has made n 1 paymens on a morgage (issued in period n 1) chooses o defaul raher han making he nh paymen; and D n = oherwise. The value of a homeowner who has no defauled a he begining of he second sub-period is W o (m n ). A homeowner who has defauled has value W f (a), where a = max[; E [V REO ] m n ], a he beginning of he second sub-period. Such a homeowner e ecively sells heir house o he lender for he expeced discouned value of a vacan house in he lender s invenory a he beginning of he nex period, E [V REO ]. If his value is less han he household s ousanding morgage deb, 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. 13 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 b (max[; p sd m n ]). If he house is 12 For quaniaive exercises, we have compued he seller decision problems boh wih and wihou he consrain p s m n. The consrain does no make a di erence quaniaively. In paricular, we nd his consrain non-binding for all ypes of indebed sellers in boh he seady sae and he dynamics given he parameers and income shocks we adop in he paper. Theoreically, i is sraighforward o see why an indebed seller may nd i opimal o lis a a price p s > m n, even when shor sale is permied. In he seller s problem (8), for any choice of p s m n he problem amouns o choosing p s such ha ( (p s )) is maximized. Since ( (p s )) is decreasing in p s, his is equivalen o minimizing he choice of price. Say, p s is chosen such ha he resuling selling probabiliy is one, ignoring wheher such price exiss or no. For any p s m n, he maximized value of he righ-hand side of (8) can be no greaer han ha of he value given = 1. Given ha W b increases wih p s, he seller will opimally choose o lis he house a a price higher han he ousanding deb as long as here exiss some ~p s > m n such ha he maximized value is greaer han he seller s. We allow for shor sales of his ype o illusrae ha he posiive relaionship beween LTV and asking prices does no depend on shor sales being ruled ou. 13 In he even of nancial disress, i is always in an owner s bes ineres o aemp o sell if hey have posiive equiy. If he housing marke is liquid, disressed owners wih posiive equiy would never defaul because hey could sell immediaely pay heir morgage deb. According o he RealyTrac repor, however, less han 5% of homeowners who go ino foreclosure have negaive equiy. In our model, ime-consuming search and maching accoun for his feaure of he housing marke. 12

13 no 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 REO ] m n]). 14 Thus he value of a disressed residen owner wih deb m n is given by: V f (m n ) = max p sd ( ((psd )W b (max[; p sd m n ]) + [1 ((p sd ))] W f (max[; E [V REO ] m n]) ) : (9) 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 b (p nd ). Oherwise, she moves ono he nex sub-period as an owner wihou deb W nd. has value Such a homeowner V nd = max p nd f((p nd ))W b (p nd ) + [1 ((p nd ))] W nd g: (1) Nex, consider homeowners who have lef he ciy. Such households become irrelevan once hey are no longer homeowners. As long as hey are, however, hey sill make sales and defaul decisions. Such a homeowner has value: 8 ((p L )) u (max pl [; p L m n ] + y L R L ) + V + V L (m n ) = max p L >< >: [1 [(p L )]] 8 max D L n >< >: (1 D Ln ) +D Ln (u ( u(yl R L x n d) +E [V L (m n+1)] ) max [; [EV REO ] m n] +y L R L! + V ) 9 >= >; 9 >= : (11) Here, y L, R L, and d are income, ren and mainenance coss paid by he exiing household while lives ouside he ciy. 15 Also, x n = x(m ; r ) denoes households nh paymen on heir morgage issued n + 1 periods prior. Once he homeowner has eiher sold her house or defauled, she receives exogenous coninuaion value, V. The value of an owner who has lef he ciy wihou deb prior o moving is given by: V Lw = max p Lw ( ((plw )) u(p Lw + y L R L ) + V + [1 [(p Lw )]] fu(y L R L d) + E [V Lw ]g ) >; : (12) 14 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 wihou a ecing he resuls signi canly. 15 These quaniies are necesssary as long as he household remains a homeowner, because hey impinge on is defaul and pricing decisions. 13

14 Such a household s only decision is wih regard o wheher and a wha price o sell Vacan Houses A he beginning of he curren period, he values of vacan houses in consrucion rms invenories, V c, and of foreclosed houses, V REO are given, respecively, by V c = max p c f ((p c )) p c + [1 ((p c ))] [ d + E [V c ]]g (13) V REO = max p REO f ((p REO )) (1 ) p REO + [1 ((p REO ))][ d + E [V REO]]g :(14) Noe ha in (14), i can be seen ha he lender loses fracion of he proceeds of is sales as a cos of foreclosure Households in sub-period 2 As here is no saving, households behaviour in sub-period 2 e ecively is rivial: They consume heir income ne of ren and morgage paymens, as well as whaever asses wih which hey ener he sub-period. 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 she says in he ciy. Such a household s value is given by: V p = W p = u(y R) + p V + (1 p )E W p : (15) Wih probabiliy p, he perpeual rener receives a moving shock, leaves he ciy immediaely and receives he coninuaion value V. 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. As above, W f (a) is he value of such a buyer wih asse a a he beginning of sub-period 2. Thus, W f (a) = u(y + a R) + h V + (1 h ) f E W f() + (1 f )E [V b ] (16) Condiional on saying in he ciy, wih probabiliy f he foreclosure ag remains and he 14

15 household moves ono he following period wih expeced value Wf () (such households are inacive in he rs sub-period of nex period, as he foreclosure ag prevens hem from purchasing a house). Wih probabiliy 1 f, he foreclosure ag is lifed and his household will ener he nex period as a buyer wih value V b. 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 Vb 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 b (a) = u(y + a R) + h V + (1 h )E [V b ] : (17) A residen homeowner wih a morgage has he principle balance m n. The owner s periodic income is used o cover repaymen, mainenance cos and consumpion. W o (m n ) denoe he value of such an owner. I follows ha for n 2 [; T 2], Le W o (m n ) = u(y x n d) + z H + h E [V L(m n+1 )] + (1 h ) d E V f(m n+1 ) + (1 d )E [V o(m n+1 )] : (18) If he owner receives a moving shock, she exis he ciy immediaely and coninues wih value V L (m 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 Vf (m n+1). Oherwise, she eners he nex period as a non-disressed owner wih value Vo(m n+1 ). For n = T 1, a residen homeowner wih a morgage has value W o (m T 1 ) = u(y x T 1 d) + z H + h V Lw + (1 h )E [V nd] : (19) In his case, he curren morgage paymen is he homeowner s las. Thus, she will coninue on wih value VLw if hi by he moving shock (in which case she leaves he ciy owning a house bu having no deb) and wih value Vnd if she remains in he ciy. 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. Le W o (p; m ) denoe he value of a 15

16 new homeowner: W o (p; m ) = u(y (p m ) d) + z H + h V L(m ) + (1 h ) d E V f(m ) + (1 d )E [V o(m )] : (2) Finally, homeowners wihou morgage deb do no su er nancial disress shocks. They remain in he ciy unil hey experience a moving shock. The value of such an owner is given by W nd = u(y d) + z H + h V Lw + (1 h )E [V nd] : (21) 3.2 Developers As noed above, he consrucion indusry is comprised of a large number of compeiive rms. Free enry ino he indusry ensures ha in equilibrium he cos of building a house equals he expeced value of a vacan house for sale in period + 1: Q(N) + K(N) = E [V c ] : (22) 3.3 Lenders Because a lender has access o funds a a xed cos, i issues 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 exernal bonds plus he servicing cos. 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 lender a he poin of loan originaion. Therefore, lenders loan he same amoun, m, o all new borrowers in period, regardless of he price hey pay for heir house. Again suppressing ime super-scrips. Le P (m n ), for n 2 f; ; T 1g, be he presen a he beginning of he curren sub-period 2 of a morgage issued in n + 1 periods before and held by a residen homeowner. Tha is, he value of a morgage of original size m n 1 afer n 1 paymens have been made. Correspondingly, le P L (m n ), be he presen value 16

17 of such a morgage held by an owner ha has relocaed. 16 Then, for n 2 f; 1; ; T 1g, P (m n ) = x n I fn6=g + I fn6=t 1g 1 + i ((p L )) min 3 p L ; n+1 m + ( h 6 4 D [1 ((p L ))] Ln+1 min VREO ; ) m 7 n (1 D 2 Ln+1 )P L ( (m n+1)] ((p E sd )) min p sd ; ) m n+1 d + [1 ((p sd ))] min VREO ; + m n+1 8 (1 h ) ((p s)) min 9 p s; m n+1 >< + [1 ((p >= (1 6 6 d ) ( s))] 4 4 D n+1 min VREO >: ; ) m n+1 >; + (1 Dn+1)P (m n+1) (23) and for all n 2 f1; ; T 1g, 2 P L (m n ) = x n + I fn6=t 1g 1 + i + E 6 4 ((p L )) min [p L ; m n+1] + [1 ((p L ))] ( D Ln+1 min [V REO ; m n+1] +(1 D Ln+1 )P L (m n+1) ) (24) where p s; p sd ; p L ; D n+1; D Ln+1 are household policies (sales prices and defaul decisions) in period + 1 coningen on having morgage balance m n+1. Also, I fn6=g and I fn6=t 1g are index funcions, wih I fn6=g = I fn6=t 1g = ( ; if n = 1; oherwise ( ; if n = T 1 1; oherwise: (25) (26) Here, indicaes a morgage on which he borrower is making regular repaymens beginning wih he period afer originaion, while (26) indicaes a morgage ha maures afer he curren repaymen is made. The presen value, P (m n ), equals he curren period paymen x n on a morgage originaed n+1 periods ago, plus he discouned expeced value of he morgage in period in he nex. The laer is a eced by he probabiliies of he borrower receiving a moving and/or nancial disress shock, and he decisions he household 16 For such owners, we have n 1 as one repaymen has already been made by he beginning of he rs sub-period 2 following he household s relocaion. 17

18 will make regarding pricing and/or defaul in he even ha such shocks are realized. Noe again ha hese decisions do no depend on he price a which he homeowner originally purchased. To compue he presen value of a morgage conrac a originaion, we proceed recursively. Firs, we compue P (m T 1 ), and hen use backward inducion o obain P (m n ) for n 2 f; : : : ; T 2g. The value P L (m n ) is deermined in a similar way excep ha once relocaed borrowers experience neiher moving nor disress shocks. If a borrower sells her house in period +1, he amoun ha he lender will receive is he minimum of he sale proceeds and he ousanding deb m n. The equilibrium morgage loan size, m, is hen deermined using he morgage lender s zero-pro condiion: P (m ) m = : (27) 3.4 Laws of moion We now describe he evoluion of he disribuions of households and houses across saes. We express all in per capia erms (i.e. divided by he oal economy populaion, Q ) A he beginning of period, he households are divided ino reners; perpeual reners, F (); buyers wihou a foreclosure ag, B(); buyers wih a foreclosure ag, B f (), and homeowners. own a house. This laer group are eiher residens in he ciy or have relocaed and sill Wihin each ype, households are furher di ereniaed by heir morgage balance, if any. For n = ; : : : ; T 1, le H n () denoe he period measure of ciy residen homeowners who were issued a morgage in period n 1 and hus have made n 1 paymens prior o period. have relocaed by period holding such a morgage. Similarly, H Ln () denoes he measure of homeowners who Le H ; () and H L; (), respecively denoe residen and re-locaed homeowners who no longer have an ousanding morgage. Noe ha here is no need o keep rack of relocaed households once hey cease o be homeowners (because hey have already eiher sold or defauled). Houses for sale are eiher held vacan by relocaed homeowners, consrucion rms and morgage lenders or o ered by disressed homeowners. consrucion rms and morgage lenders by H c () and H REO () respecively. Denoe he invenories of Suppressing he ime indicaor, he oal sock of houses for sale in he curren period is hen given by: XT 1 XT 1 H s = H L; + H Ln + H c + H REO + d n=1 {z } vacancies 18 n= H n {z } disressed sellers homes : (28)

19 Similarly, he oal curren period measure of buyers searching o rade in he housing marke, B sum, can be wrien: B sum = XT 1 [(1 d ) (p s ) H n + d (p sd ) H n + (p L ) H Ln ] n= +(p nd )H ; + (p Lnd )H L; + (p c )H c + (p REO )H REO : (29) In paricular, he measure of buyers in an acive sub-marke equals he measure of sellers in ha sub-marke muliplied by he he corresponding marke ighness. For example, he measure of buyers searching for foreclosed houses sold by a lender equals he measure of REO houses, H REO, muliplied by he ighness of he lender s opimally chosen submarke, (p REO ). Coninuing o suppress he dependence of variables on ime, we now wrie ou he laws of moion for he socks households and houses in he various saes. To begin wih, he per capia measure of permanen reners nex period consiss of hose remaining from he curren period and hose who will have newly enered: (1 + )F = (1 p )F + (1 )G(" c): (3) Similarly, he measure of buyers wih foreclosure ags nex period includes hose remaining from he curren period who have neiher moved nor had heir ag removed randomly. To his is added he measure of residen homeowners who defaul his period. These homeowners may eiher have received a nancial disress shock and failed o sell or have defauled sraegically. Thus, we have 8 >< (1 + )Bf = (1 h ) >: f B f + d P T 1 n= (1 ( (p sd)))h n g+ (1 d ) P T 1 n= (1 ( (p s)))d n H n where as above, p sd, p s, and D represen opimal pricing and defaul decisions. Noe ha in general hese depend on homeowners ousanding morgages. The measure of buyers wihou foreclosure ags a he beginning of nex period consiss of newly-enering buyers, previously agged buyers whose ag has been removed, and non-relocaing buyers from he curren period who fail o buy a house. 9 >= >; (31) Noe ha he measure of buyers who successfully mach in he curren period equals he sum of he mea- 19

20 sures of he prospecive sellers of various ypes muliplied by heir corresponding maching probabiliies. Thus, we have (1 + )B = G(" c) + (1 f )B f + 8 >< B ((p Lw )) H L; (1 h ) ((p c )) H c ((p REO )) H REO >: P T 1 n=1 ((p L)) H Ln 9 >= : (32) >; The measure of indebed owners who have made n periodic paymens by he beginning of period + 1 on a morgage of size m n 1 a originaion evolves (for n > ) via: (1 + )H n = (1 h )(1 d )(1 ((p s )) (1 D n )H n 1 : (33) Tha is, he indebed owners wih an ongoing morgage going ino he nex period are he indebed owners from he curren period who do no move, experience nancial disress, successfully sell heir house, or defaul sraegically. For n =, H is he measure of residen homeowners who successfully purchase a house, remain in he ciy and do no experience nancial disress. This measure can be recovered from he number of sales in he curren period. Thus we have 8 >< (1 + )H = (1 h ) >: (1 d ) P T 1 n= ( (p s)) H n + d P T 1 n= ( (p sd)) H n + P T 1 n=1 ( (p L)) H Ln + ((p nd )) H ; + ((p Lw )) H L; + ((p c )) H c + ((p REO )) H T E"O 9 >= : (34) >; Finally, he measure of residen owners wihou a morgage evolves via (1 + )H ; = (1 h ) ( (1 d )(1 ( (p d )))(1 D T 1 )H T 1 +(1 ((p nd )))H ; ) : (35) This group is comprised of is previous members who have neiher move nor sell plus residen homeowners who make heir las morgage paymen in he curren period. Proceeding similarly for relocaed homeowners, HLn is he measure who made heir (n + 1)s paymens in he period. Again, he loan volume a originaion is m n 1 and 2

21 H L; is he curren period measure of relocaed owners wihou deb: (1 + )H Ln = (1 ((p L )))(1 D Ln 1 )H Ln 1 (36) + h (1 d )(1 ((p s )))(1 D n 1 )H n 1 ; (1 + )H L = h 8 >< >: (1 d ) P T 1 n= ( (p s))h n + d P T 1 n= ( (p sd))h n + P T 1 n= ( (p L))H Ln +((p nd )H ; + ((p Lw )H L; +((p c ))H c + ((p REO ))H REO 9 >= >; (37) (1 + )H L; = h ( (1 ((pnd ))H ; +(1 d )(1 ((p s )))(1 D T 1 )H T 1 +(1 ((p L )))(1 D LT 1 )H LT 1 +(1 ((p Lnd ))H L; : (38) ) As depreciaion is o se by mainenance, he per capia ciy housing sock evolves via (1 + )H = H + N; (39) where N is he measure houses buil in he curren period and available for sale in he nex. The per capia sock of houses in consrucion rms invenory a he beginning of he nex period includes houses held by consrucion rms ha go unsold in he curren period plus hose ha are newly buil: (1 + )H c = (1 ((p c )) H c + N. (4) Finally, he sock of houses in he REO invenory a he beginning of he nex period, 21

22 HREO, includes hose ha go unsold in he curren period plus he new foreclosures: (1 + )HREO = (1 ((p REO ))) H REO XT 1 + d (1 ((p sd ))H n n= XT 1 +(1 d ) (1 ((p s ))D n H n T 1 + n= X (1 ((p L )D Ln H Ln : (41) n=1 3.5 A Direced Search Equilibrium De niion. Given a morgage ineres rae, r; ren level, R; erminal coninuaion value, V ; and a sochasic process for ciy-level income, y, a direced search equilibrium is, for all periods, a collecion of 1. Household value funcions: V b ; W b ; V o ; W o ; V f ; W f ; V nd ; W nd ; V L ; V Lw ; V p ; W p (42) wih associaed policy funcions (choices of sub-marke o ener and wheher o defaul): p s ; p sd ; p nd ; p L ; p Lw ; D n ; D Ln ; n = ; : : : ; T 1; (43) 2. house values: V c ; V REO (44) wih associaed policies for developers and lenders: p c ; p REO ; (45) 3. an enry cu-o and morgage conrac: " c ; m ; (46) 22

23 4. and per capia measures of households and houses F; B; B f ; H n ; H Ln ; H ; ; H {z L; ; n = ; : : : T } 1; H; N; H c ; H REO : {z } (47) households houses Such ha: 1. New households ener he ciy opimally so ha (1) holds; 2. All agens opimize such ha he value and policy funcions lised in (42) - (45) saisfy (6), (8) - (21); 3. Free enry of consrucion rms: N sais es (22); 4. Free enry of lenders: m = m sais es (27); 5. The socks of households and invenories of houses evolve according o (3) - (39); 6. Marke clearing : B = B sum. Requiremens 1-5 in he above de niion are sandard and have been described in deail above. Requiremen 6 is an aggregae consisency condiion. In equilibrium he measure of buyers wihou foreclosure ags mus be consisen wih he oal measure of buyers acively paricipaing in housing search. Alernaively, i means ha all buyers ener some sub-marke. As has been menioned, sellers are heerogeneous, and each period heir disribuion is characerized by (H n ; H Ln ; H ; ; H L; ; H c ; H REO ). The decision problems faced by households, developers and lenders are, however, no a eced by his disribuion. In fac, as can be seen from (6), and (8) - (27), all of he value and policy funcions lised in (42) - (45), ogeher wih he morgage conrac m, are independen of he socks lised in (47). 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 Shi (29). As discussed here, block recursiviy arises because heerogeneous sellers selec hemselves opimally ino separae submarkes hrough he direced search mechanism. In doing so, hey ake 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 23

24 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 a seller cares only abou he price a which i can 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. Given De niion 3.5, we derive some analyical resuls ha will laer be helpful for us o undersand he opimal rading decisions of disressed sellers. Using (9), in he seady sae he gain from rade for a disressed seller as a funcion of her ousanding deb, m n, is given by: (m n ) = W b (p m n ) W f (max [; V REO m n ]) : Then we have he following resuls: Proposiion 1 Condiional on p > m n, we have: (i) If m n V REO, hen (m n ) < ; (ii) If m n < V REO, (m n ) > for any given p > V REO and (m n ) < for any given p < V REO. Lemma 1 Provided ha p > m n, (i) a disressed seller s rading probabiliy, ( (p)), sricly decreases in p; (ii) a disressed seller s rade surplus, (m n ; p), increases wih price p for any given deb level m n. 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 (9) is equivalen o max p; () (m n ; p) (48) where (p) = 1 V b W b () V o (p; m ) W b () 24 (49)

25 follows direcly from (7) evaluaed a he seady-sae. I is sraighforward o derive Lemma 1. Par (i) of 1 follows from he maching funcion lised in (2) and (3). Par (ii) is because 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. 4 Calibraion We now choose parameers o mach seleced characerisics of U.S. ciy-level housing markes along a balanced growh pah. In his seady-sae, he housing sock grows a he rae of populaion growh and all oher componens of he equilibrium, including he disribuion of agens across saes and real house values, remain consan. 17 A he ouse, we choose he following funcional forms: u(c) = ln(c) M(B; S) = $B S 1 k = 1 N 1 q = qn 1 (5) 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, while hose below he line are deermined joinly o mach he arges in he righ-mos column. A ime period is de ned as one year. 18 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 erminal coninuaion value, V, is equal o he seady-sae value of being a perpeual rener, V p. To deermine he morgage rae r, he annual yield on inernaional bonds i is se a 4%. The values of 17 We view his calibraion as permiing illusraive examples. Because of he complexiy of our model, we absrac from cerain aspecs of he economy, especially for clariy. For example, we absrac from house sales for reasons oher han nancial disress and inerciy relocaion; and we limi he ime period o one year. These decisions and oher preven us from argeing cerain aspecs of U.S. housing markes boh a he aggregae and ciy level. 18 While resricing he ime period o a year resuls in a longer average ime o sell han observed, noe ha we are concenraing only on disressed sales ha end in foreclosure, no, as noed above, ypical sales. Also, empirically, he average house price is roughly 3.2 imes annual income. In equilibrium, buyers can hus a ord o make he ypically observed 15% - 2% down-paymen using only heir periodic income. 25

26 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 &.16 Ren-price raio 5% Parameers deermined joinly.44 Loss severiy rae 27%.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 and % are deermined joinly in calibraion. The following parameers and arges are chosen following Head e al. (214) (HLS): The rae is chosen o mach he annual populaion growh during he 199 s. 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 Saiz (21). 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 Davis and Palumbo (28) and Saiz (21)). The 26

27 elasiciy of new consrucion wih respec o he price of housing,, is se equal o he median elasiciy for he 45 ciies sudied by Green e al. (25), = 5. The average house price is 3:2 imes annual income, and he mainenance cos d is chosen o be 2:5% of ha, following Harding e al. (27). Moreover, The value of is calibraed so ha he ownership rae in he ciy H=(H + B + F ) = 66:7%, as repored by he Census Bureau, where XT 1 H = H n + H ; n= denoes he oal measure of homeowners in he seady sae. We se he ren-o-income raio o & = :16. The Lincoln Insiue of Land Policy esimaes he average ren-o-price raio o be roughly 5% prior housing boom leading up o he 28 nancial crisis. We hus compue & as he produc of his ren-o-price raio and he average house price. 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. Foreclosed houses end o cause losses o lenders due o associaed ransacion and ime cos. Moreover, foreclosed houses are usually sold a a discoun relaive o oher houses wih similar properies. 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. Economiss have found he loss severiy raes range from as small as 2% during calm period over (Penningon-Cross, 23) o as much as over 75% during he Grea Recession (Andersson and Mayock, 214). In his paper, we choose parameers so ha in he even of a defaul, he value of minfv REO ; m n g m n = :73 (51) on average. This implies an average loss severiy rae of 27%. 27

28 Phillips and Vanderho (24) nd ha 3% of defauled convenional xed-rae loans and 5% of defauled convenional adjusable-rae loans ransiion o REO and Ambrose and Capone (1996, 1998) 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 =y = 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 ) (here g is he densiy of G) 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 HLS. 5 The Balanced Growh Pah For he baseline search economy, he seady-sae (i.e. balanced growh pah) conforms o he de niion of equilibrium in Secion 3.5, wih he addiional requiremens ha per capia income is consan and ha all funcions and values lised in (42) - (47) are ime invarian. In he seady-sae, all owners have sricly posiive home equiy 19. Residen owners who receive neiher moving nor nancial disress shocks do no aemp o sell heir houses and relocaed owners coninue o make repaymens unil a successful sale occurs or heir morgage is compleely paid o. Finally, all disressed owners aemp o sell. As such, here are no ourigh defauls, in he sense ha foreclosure occurs only as he resul of nancial disress followed by an unsuccessful aemp o sell. Figure 2 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. consisen wih he mobiliy and defaul rae arges from he calibraion. 2 Noe ha his is 19 Here home equiy represens he di erence beween he average house price and ousanding morgage deb 2 While his fracion is high overall, recall ha we have absraced from sales for reasons oher han nancial disress and inerciy relocaion. 28

29 Figure 3 presens he disribuions of residen homeowners (upper panel) and sellers (lower panel) by morgage saus. In he seady-sae, hese disribuions are deermined by ime and exogenous shocks alone. The measures of owners decrease wih he number of paymens 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 he sock of 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. 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 4 illusraes he disribuion of house prices in he seady sae. Indebed sellers se he lowes prices, followed by relocaed sellers and hen by developers and lenders, who sell only foreclosed houses. Tha developers and lenders ask he highes prices is unsurprising, as hey are risk neural and are concerned only wih he expeced reurn on a sale. Lenders ask slighly higher prices han developers, as hey require a reurn su - cien o recoup heir addiional holding cos (). For any level of ousanding morgage, disressed sellers have greaer incenive o sell han relocaed ones, as hey face he hrea of foreclosure and exclusion from he housing marke if hey fail o sell. Boh disressed and relocaed sellers generally ask higher prices he larger is heir ousanding deb. We urn o his relaionship now. 5.1 Leverage and seller behavior Figure 5 illusraes he relaionship beween a seller s opimal choice of sub-marke (which deermines boh her asking price and sales 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 is 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 29

30 13% 13% 48% 26% Disressed sellers Relocaed sellers Consrucion Firms REO invenory Figure 2: Composiion of house sellers period, while a relocaed seller does no, and moreover reains he choice of wheher o defaul in he nex period. Second, a relocaed seller receives coninuaion value V, 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 unil her foreclosure ag is lifed ( ve periods on average). Noe also he relaionship beween he posed asking price and LTV. For boh ypes of seller, he posed price is iniially weakly decreasing in LTV. A some poin, and his is more dramaic for disressed sellers, he relaionship becomes srongly increasing. For disressed sellers in paricular, he relaionship resembles closely ha repored by Genesove and Mayer (1997) (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). Figure 6 combines heir resuls wih ours in common unis. 21 The closeness of he relaionship is sriking, given ha none of he quaniies depiced for our economy are calibraion arges. The shape of he relaionship depiced in Figures 5 and 6 can be undersood using Proposiion 1. When a seller is su cienly indebed (m n V REO ), he gain from rade (m n ; p) is sricly decreasing in deb m n, for any p > m n. Essenially, heavily indebed 21 In our economy, he posed (asking) price is proporional o he mark-up, as all vacan (non-foreclosure) houses have a common value. 3

31 Number of paymens: n.15 Financially disressed seller Relocaed seller Number of paymens: n Figure 3: Seady-sae disribuions of morgage saus respecively among residen owners (upper panel) and household sellers (lower panel) sellers wih more are less concerned wih he likelihood ha hey successfully sell han wih he gain hey receive if hey do. The reason for his is ha hey receive residual pro only if hey sell a a su cienly high price. To hem, he foreclosure cos is xed; he marginal cos of defauling on a larger deb is borne enirely by he lender. A less indebed seller (wih m n < 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 m n. As such, a more indebed seller (bu wih m n < V REO ) will chose a lower price/higher sales probabiliy. Overall, for m n < V REO, he e ec of deb on he gain from rade (i.e., (m n )) is likely o be small in ha m n a ecs symmerically he reurns boh o selling and failing o do so (see (57)). Thus he relaionship is essenially a when m n < V REO, bu rapidly increasing for higher LTVs. Noe ha he condiion p > V REO is no resricive, a leas for in our baseline calibraion. Here V REO = 1:79, while he minimum selling price chosen by a seller is 3.4. In general, 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. While he proof of Proposiion 1 relies on he assumpion ha consumpion goods 31

32 Developers and Morgagees Relocaed sellers Disressed sellers Selling price Figure 4: Seady-sae disribuion of housing prices are non-sorable, he resul is in fac more general. In paricular, he derived properies of (d) require only ha Wb (p m n) > and W f (V REO m n ) Wb (p m n) > for m n < V REO. The former condiion requires ha he value of a buyer wihou he foreclosure ag is sricly increasing in her asse holdings. The laer requires ha he slope of he value of a buyer wih he ag a asse posiion V REO m n exceed ha of he value of an un agged buyer a p m n for lower levels of deb. Observing ha p > V REO in general, his requiremen is no resricive for value funcions such as W f () and W b (), which are sricly concave. 5.2 Maching and lending sandards We now consider he role of speci c assumpions regarding maching. We consider rs he e ecs of changing he maching coe cien $ and second he maching elasiciy, ceeris paribus. Our goal here is o examine how search fricions, which deermine in par he liquidiy of housing, a ec morgage lending sandards direcly. The op wo panels of Figure 7 illusrae he e ec of changes o $ on he average LTV (or alernaively, he down-paymen raio) a orginaion and he probabiliy of a morgage 32

33 .88 Maching probabiliy Financially disressed seller Relocaed seller Selling price Loan o value raio (%) Figure 5: 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. ending in foreclosure, respecively. 22 Consider a morgage issued in he curren period. The probabiliy of such a morgage ending in foreclosure, d, is given by d = TX (1 h ) n (1 d ) n 1 d [1 ((p sd ))]m n 1 n=1 + TX (1 h ) n 1 h [1 ((p Ld ))]m n 1 (52) n=1 where ((p sd )) and ((p Ld )) are he rading probabiliies in he opimal sub-markes chosen a period + i for residen and relocaed borrowers, respecively, who defaul afer having made n 1 paymens. The rs erm is he summaion of probabiliies of defaul over he enire duraion of morgage condiional on saying in he ciy. Similarly, he second erm is he summaion of probabiliies of defaul condiional on having relocaed elsewhere. 22 In equilibrium, households di er in LTV a originaion because hey purchase houses a di eren prices bu are advanced loans of he same size. Boh of he lef-hand panels of Figure 7 depic he e ecs of maching parameers on average, maximum and minimum LTVs a originaion, respecively. 33

34 Genesove and Mayer, 1997 (lef axis) Baseline Search (righ axis) Figure 6: Leverage and asking prices in he baseline search economy compared o he empirical ndings of Genesove and Mayer (1997). As is shown in Figure 7, LTVs a originaion are increasing and defaul probabiliies decreasing wih he value of $. The higher $, he more likely a seller is o mach, or equivalenly, he more liquid he housing marke. Thus, he expeced defaul rae is lower because more homeowners experiencing disress successfully sell. A he same ime, houses in REO invenory also sell more quickly. Overall, wih boh he likelihood and cos of defaul and foreclosure reduced, lenders are willing o advance larger loans, resuling in higher LTVs a originaion. The boom wo panels in Figure 7 demonsrae similar resuls for varying he value of, he elasiciy of he measure of maches wih respec o ha of buyers. Buyers surplus from housing ransacions and hus he value of being a buyer, increases wih. 23 This increases he value of living in he ciy, lowering he enry cuo, " c, and increasing he overall measure of buyers each period. Of course, a higher also implies a lower reurn o consrucion as developers receive less surplus from selling new houses. Overall, he housing marke is igher, and all houses sell wih higher probabiliy. Again, his lowers boh he expeced rae and cos of defaul, leading lenders o issue larger loans. Overall, hese exercises demonsrae ha morgage lending sandards are lower he 23 See Moen (1997) and Head e al. (214). 34