NBER WORKING PAPER SERIESMORTGAGES AND MONETARY POLICYCarlos GarrigaFinn E. KydlandRoman SustekWorking Paper 19744http://www.nber.org/papers/w19744NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts AvenueCambridge, MA 02138December 2013We thank Morris Davis and Wouter den Haan for helpful suggestions. We are also grateful for commentsto seminar participants at St. Louis Fed and IHS Vienna and conference participants at SAET in Parisand London Macro Workshop. The views expressed are those of the authors and not necessarily ofthe Federal Reserve Bank of St. Louis, the Federal Reserve System, or the National Bureau of EconomicResearch.NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications. 2013 by Carlos Garriga, Finn E. Kydland, and Roman Sustek. All rights reserved. Short sectionsof text, not to exceed two paragraphs, may be quoted without explicit permission provided that fullcredit, including notice, is given to the source.Mortgages and Monetary PolicyCarlos Garriga, Finn E. Kydland, and Roman SustekNBER Working Paper No. 19744December 2013JEL No. E32,E52,G21,R21ABSTRACTMortgage loans are a striking example of a persistent nominal rigidity. As a result, under incompletemarkets, monetary policy affects decisions through the cost of new mortgage borrowing and the valueof payments on outstanding debt. Observed debt levels and payment to income ratios suggest the roleof such loans in monetary transmission may be important. A general equilibrium model is developedto address this question. The transmission is found to be stronger under adjustable- than fixed-ratecontracts. The source of impulse also matters: persistent inflation shocks have larger effects than cyclicalfluctuations in inflation and nominal interest ratesCarlos GarrigaFederal Reserve Bank of St. LouisP.O. Box 442St. Louis, MO 63166carlos.garriga@stls.frb.orgFinn E. KydlandDepartment of EconomicsUniversity of California, Santa BarbaraSanta Barbara, CA 93106-9210and NBERkydland@econ.ucsb.eduRoman SustekSocial Sciences, EconomicsUniversity of SouthamptonMurray Building (Bld. 58)Southampton, SO17 1BJUnited Kingdomrs1y10@soton.ac.uk1 IntroductionMost theories of how monetary policy aects the real economy rely on some form of nominalrigidity. Frequentlymadeassumptions, supportedbyempirical evidence, arethat pricesandwagesofindividualrmsorhouseholdsarepre-setinnominaltermsforagivenperiodof time, withtheresultthatnominal variablesunderthecontrol of amonetaryauthorityaectrelativepricesandreal incomes. Aspecicformof nominal rigidity, butsomewhatoverlooked in the literature, characterizes also standard mortgage loans. In particular, fully-amortizing mortgages require the homeowner to make nominal instalmentsregular interestand amortization paymentsfor the duration of the loan. The installments are calculated soas to guarantee that the principal is repaid in full by the end of the loans life. A conventionalxed-rate mortgage (FRM) in the United States, for instance, carries a xed nominal interestrate and prescribes constant nominal installments for the entire life of the loan,typically 30years. Anadjustable-ratemortgage(ARM), typical fortheUnitedKingdomorAustralia,also prescribes nominal installments, calculated each period so that, given the current short-termnominalinterestrate,theloanisexpectedtoberepaidinfullbytheendofitslife.1Thispaperstudiesthemacroeconomicconsequencesofthenominalrigidityinherentinstandardmortgageloans. Inparticular, our aimis tocharacterizethechannels throughwhichthe rigidityfacilitates the transmissionof monetarypolicyintothe real economy,especiallyintohousinginvestment, andtoinvestigatethestrengthof thetransmissioningeneralequilibrium. Inordertoisolatetheeectsoftherigidity, thepaperabstractsfromothernominalfrictions.1Themajorityof mortgageloansinadvancedeconomiesarefully-amortizingmortgageswithatermof15to30years,eitherFRMsorARMs. Onaverage,overtheperiod1982-2006,FRMsaccountedfor70%ofmortgage originations in the United States (Federal Housing Finance Agency, Monthly Interest Rate Survey,Table10); before1982, theywereessentiallytheonlymortgagetypeavailable. OthercountriesinwhichFRMswithinterestratesxedforatleast10yearshavetraditionallydominatedthemortgagemarketincludeBelgium,Denmark,andFrance(inaddition,thetypicalmortgageinGermanyandtheNetherlandshas rates xed for 5 to 10 years); in other advanced economies, ARMs (with an interest rate linked to a short-term market rate) or FRMs with interest rates xed for less than 5 years prevail; see ScanlonandWhitehead(2004)andEuropeanMortgageFederation(2012a). Suchcross-countryheterogeneityinmortgagemarketsappears to be due to dierent government regulations (e.g., GreenandWachter, 2005; Campbell, 2012). Thestructureofmortgagemarketsandmortgagecontractsistakenhereasgivenandweconsideronlythetwoextremes: FRMswithaninterestratexedfortheentiretermandARMs.Recent monetary policies in a number of advanced economies have aimed at reducing long-term interest rates or have committed to low short-term interest rates for long periods of time.One of the goals of such policies is to encourage housing investment (e.g., BoardofGovernors,2012). Concernshavealsobeenexpressedabouttheconsequencesofpotential futurerisesinshort-terminterestratesforexistinghomeownerswithARMs(BankofEngland, 2013).Themodel developedinthispaperprovidesasteptowardsaframeworkallowingformal,generalequilibrium,analysisoftheeectsofsuchpoliciesonaggregatehousinginvestmentandincomeredistribution.Mortgagepayments(interestandamortization)asafractionof incomethesocalleddebt-servicingcostsarenontrivial. Ourestimatessuggestthat,onaverageoverthepast30-40 years, they were equivalent to 15-22% of the pre-tax income of the 3rd and 4th quintilesof the U.S. wealth distribution, representing the typical homeowner (CampbellandCocco,2003). Asimilar picture emerges alsofromscatteredinformationfor some other coun-tries. HancockandWood(2004)reportthatintheUnitedKingdommortgagedebtservic-ingcosts(forpre-taxincome)uctuatedbetween15%and20%overtheperiod1991-2001.AndinGermany, mortgage debt servicingcosts are reportedtobe around27%of dis-posableincome(EuropeanMortgageFederation, 2012b). Mortgagedebtto(annual)GDPratios in advanced economies are also considerable, reaching on average around 70% in 2009(InternationalMonetaryFund,2011,Chapter3).The nominal rigidity in mortgages leads to two channels of monetary policy transmission.Onechannel worksthroughnewborrowing(apriceeect), theotherthroughoutstandingmortgagedebt (current andexpectedfuturewealtheects). As apreliminarystepillus-tratingtherealeectsthroughtherstchannel,Figure1showsthequantitativeimpactofalternative paths of the short-term (i.e., one-period) nominal interest rate on a typical mort-gage holders expected debt-servicing costs over the life of a typical 30-year mortgage, eitherFRM or ARM (one period here, equals one quarter). In this example, a household considersbuyingahousebytakingoutamortgageinperiod1worthfourtimesitsannual post-tax2income.2Let us assume no uncertainty (for easier exposition) and that short- and long-termnominalinterestrates,aswellasmortgagerates,satisfystandardno-arbitrageconditions3.Inaddition,assumethattherealinterestrateandthehouseholdsrealincomeareconstant(thereal rateis1%perannum)andthatthehouseholdsnominal incomechangesinlinewithination. Thetworealvariablesarepurposefullyheldconstantsothatanyrealeectsonthehouseholdsbudgetoccuronlyduetonominalfactors.Panel Aof Figure1considerstheeectof amean-revertingdeclineof theshortrate.Itssteady-statelevelis4%,whichisroughlytheaveragefortheperiodfrom1990onwards.As the right-hand side chart shows (lines labeled steady state), at the steady-state interestrate, debt-servicingcosts are front-loadedanddecline monotonicallyover the life of themortgage,herefrom29%to6.5%. Thisisthewell-knowntiltingeect,whichoccursduetoapositiveinationrate(3%).4Nowinsteadsupposethatinperiod1theshortrateisequal to 1% (monetary policy easing) and reverts back to the steady state with persistenceof 0.95, theaverageautocorrelationinthedata. Underthispath, thetiltingisweakened:atthefrontendof themortgagedebt-servicingcostsdecline, whileatthebackendtheysomewhatincrease. Forexample, inperiod1, theydeclineby9percentagepointsunderARM and by 4 percentage points under FRM. The decline under FRM is smaller than underARMbecausetheFRMinterestrate, duetothemean-revertingnatureof theshortrate,2Thisisbasedontheaverageratio, 1975-2010, of themedianpriceof anewhome(assumingaloan-to-valueratioof76%)tothemedianhouseholdincome(assuminganincometaxrateof23.5%). ThedataonbothhousepricesandincomesarefromtheU.S.CensusBureau. Theloan-to-valueratioistheaverageratio for single family newly-built home mortgages (Federal Housing Finance Agency, Monthly Interest RateSurvey,Table10). ThetaxrateisanestimatediscussedinSection4.3Specically,(i)theexpectationshypothesisi.e.,theinterestrateonann-periodnominalzero-couponbondisequal totheaverageof one-periodnominal interestratesbetweenperiods1andn; (ii)theFishereecti.e.,theone-periodnominalinterestrateattimetisequaltotherealinterestrateplustheinationrate between periods t and t+1; and (iii) mortgages are priced by arbitrage with the zero-coupon bonds (i.e.,in the case of FRM, the mortgage interest rate is such that when the installments are evaluated at the pricesof zero-coupon bondswhich are determined by the expectations hypothesisthe present value of a $1 loanis $1; in the case of ARM, the mortgage rate is equal to the one-period interest rate, implying again that thepresentvalueofa$1loanisequal to$1). Theprinciplesofmortgagepricingandinstallmentcalculationsarediscussedby,e.g.,Fabozzi,Modigliani,andJones(2010);inthecontextofatwo-periodmortgage,theyareexplainedinSection3.4Positiveinationdeatesthereal valueof mortgagepaymentsinlaterperiodsof thelifeof theloan,whichhastobecompensatedbyhigherreal paymentsatthebeginning, forthepresentvalueofa$1loantoequalto$1.3declinesbylessthantheshortrateitself. However, theimpactof adeclineof theshortrateondebt-servicingcostsisnotalwayslargerunderARM. Panel Bof Figure1depictsasituationahump-shapeddeclineof theshortratecharacterizedbyastrongerimpactunderFRM.ThisisbecausetheFRMrateanticipatesthefuturedeclineintheshortrate,thusdecliningimmediatelyinperiod1.5The two cases illustrate that changes in the path of the short rate, occurring due to purelynominal factors (the expected path of ination), redistribute the expected debt burden overthelifeoftheloan. Here, reducingreal mortgagepaymentsclosertothefrontend, wheredebt-servicingcosts are the highest. This lowers the eective cost of the loanunder aconcaveutilityfunctionandincreaseshousingdemand. Amonetarypolicytighteninghasthe opposite eect. An implicit assumption in this discussion,and a necessary condition forthiseecttomattertothehousehold, isthatthehouseholdcannotfullyosettheimpactoftheshortrateondebt-servicingcoststhroughothernancialinstruments.Inadditiontotheabove(price)eect, whichrelatestonewmortgageloans, inaworldwithuncertaintymonetarypolicyalsoaectshouseholddecisionsex-post, throughcurrentandfuturedebt-servicingcostsonoutstandingmortgagedebt(wealtheects). Inthecaseof FRM, onlytheinationratematters: ahigherinationratereducesthereal valueofoutstandingdebtandthusthereal valueof thepaymentshouseholdshavetomake. Thestrengthof thiseectincreaseswithinationpersistence. Inthecaseof ARM, boththeshort-termnominal interestrateandtheinationratearerelevant. Anequiproportionate(persistent)increaseinthetworates, forinstance, initiallyincreasesthereal payments, astheimpactofahighernominal interestratedominatestheeectofhigherination. Overtime, however, the eect of persistently high ination gains strength, reducing the real value5If thesteady-stateshortratewasequal to1%(i.e., zeroinationrate), debt-servicingcostswouldbeconstantat15%. Themorepersistentthe3percentagepointdeclineof theshortrateis, thecloserdebt-servicing costs get to 15% and the smaller is the dierence between the above eects under ARM and FRM.In the case of 8% steady-state short ratethe average for the period 1970-1989a mean-reverting decline intheshortrateby3percentagepoints(0.95persistence)resultsindeclinesindebt-servicingcostsintherstperiodby11percentagepointsunderARMand6percentagepointsunderFRM. Atthe8%steady-stateshortrate,thetiltingis42%intherstperiodand3%inthenalperiod,makingagivenreductionmorevaluable(underaconcaveutilityfunction)thaninthecaseofthe4%steady-staterate.4ofthepayments.Thesechannelsarestudiednumericallyinageneralequilibriummodelwithincompleteasset markets andlong-termmortgage loans. As inthe above examples, mortgages arepricedbyarbitrage, butunlikeintheexamples, theshortrate, thereal interestrate, andthehouseholdsrealincomeareendogenous. Thesevariablesaredeterminedbyamonetarypolicyrule,themarginalproductofcapital(ownedbymortgagelenders),andlaborsupplydecisionsbyhomeownersincompetitivefactormarkets. Thetwotypesof theshortratedynamicsconsideredinthenumericalexampleaboveariseendogenouslyinresponsetodif-ferentshocks. Themonetarypolicyruleconsistsoftwoparts: systematicresponsesofthecentral banktomovementsinoutputandinationandexogenouschangesinanimplicitinationtarget. Inequilibrium, thelatterworkslikealevel factorinmodelsof theyieldcurveandallowsthemodeltoreplicatethepersistenceandvolatilityoflong-termnominalinterestrates;theformeraectsthecyclicalvolatilityofthelong-shortspread. Duetothelong-termnatureofthemortgageloan,thepersistenceofnominalinterestratesaectsthequantitativeimportanceofthenominalrigidity.Theresults canbesummarizedas follows. First, monetarypolicyhas alarger eectonhousinginvestmentunderARMthanunderFRM.Broadlyspeaking,thisisbecausetheprice and wealth eects reinforce each other under ARM, but tend to oset each other underFRM.Second,theeectsofthestochasticpartofthepolicyrulearelargerthantheeectsofthesystematicpart. Inthelattercase, generalequilibriumadjustmentsintheexpectedfuturepathof thereal interestratetendtoosetthereal eectsof thenominal rigidityinmortgages, whereasintheformercasesuchosettingforcesareweaker. Third, higherinationredistributesincomefromlenderstoborrowersunderFRM,but(atleastinitially)from borrowers to lenders under ARM.6An implication of our ndings for the current policydebateisthat,otherthingsbeingequal,lownominalinterestratesarelikelytohavelargerreal eects in ARM than FRM countries and the impact will be larger the longer is the time6The result that monetary policy transmission is stronger under ARM than under FRM is consistent withcross-countryempiricalndingsofCalza,Monacelli,andStracca(2013).5horizonforwhichtheratesareexpectedtostaylow.Thepaperproceedsasfollows. Section2relatesittotheliterature. Section3usesasimple three-period problem to explain the nature of the nominal rigidity and the two chan-nelsoftransmission. Section4describesthegeneralequilibriummodelanditsequilibrium.Section5discussesthemappingbetweenthemodelandthedataandcalibratesthemodel.Section6reportsthendingsandexplainsthegeneral equilibriumadjustments. Section7concludes and oers suggestions for future research. A supplemental material contains a listofthemodelsequilibriumconditions,thecomputationalmethod,adescriptionofthedatacounterpartstothevariablesinthemodel, andestimatesofmortgagedebtservicingcostsfortheUnitedStates.2 RelatedliteratureThepaperisrelatedtodistinctstrandsoftheliterature. First,anumberofearlierstudiesrecognize that ination/nominal interest rates may aect housing demand and construction.The role of the tilting eect has been investigated in the context of mortgage contract design(LessardandModigliani, 1975), a supply-demand econometric model of the housing market(Kearl, 1979), andaconsumers problemunder aconstant inationrate(Schwab, 1982;AlmandFollain,1984).7Second, followingtheseminal contributionof Iacoviello(2005), anumber of dynamicstochastic general equilibrium (DSGE) models study the role of housing and housing nancein the monetary transmission mechanism (Iacoviello, 2010, contains various references). Thisliterature, however, is concerned with a dierent channel than ours, focusing on the interac-tion between sticky prices, borrowing constraints, and the collateral value of housing. In ad-7Inaddition, Poterba(1984) notes that, as the U.S. income taxbrackets are set innominal terms,mortgagenanceandinationalsointeract duetothetaxdeductibilityof mortgageinterest payments.Thisfeatureaddsanadditional layerof nominal rigidityintoamortgagecontract, butisabstractedfrominthispaper. Morerecently,BrunnermeierandJulliard(2008)arguethatthemainchannelthroughwhichination and mortgages aect housing decisions is money illusion, which makes households ignore the eectsofinationontherealvalueoffuturemortgagepayments.6dition, housing nance in this literature takes the form of one-period loans, which (as shownin Section 3) eliminates from the transmission mechanism the nominal rigidity we focus on.8IacovielloandNeri(2010)estimateaversionofIacoviello(2005)andndthathousingde-mandshocksmodeledasshockstothemarginalutilityofhousingareimportantdriversof housing investment and house prices over the business cycle. Shocks to the marginal util-ityofhousingarealsokeyinthelandcollateralmechanismofLiu,Wang,andZha(2013).The price channel in our model may be viewed as a structural interpretation of such shocks,asitshowsupinasimilarwayintheoptimalityconditionforhousing.Housingandmonetarypolicyhavealsobeenstudiedinthecontextofhomeproductionmodels (Edge, 2000; Aruoba,Davis,andWright, 2012) andmodels withliquidityeects(LiandChang, 2004; DresslerandLi, 2009; Ghent, 2012). ExceptforGhent(2012), whoworkswithFRMsspeciedinreal terms,thesestudiesabstractfrommortgageloans.9Third,mortgages(orlong-termhousingdebtmoregenerally)areconsideredbyanum-ber of studies focusingonissues unrelatedtomonetarypolicy: optimal mortgagechoice(CampbellandCocco, 2003), consumption smoothing (HurstandStaord, 2004; LiandYao,2007), equilibriumhomeownershiprates (Chambers,Garriga,andSchlagenhauf, 2009a,b),andequilibriumforeclosures (GarrigaandSchlagenhauf, 2009; ChatterjeeandEyigungor,2011; CorbaeandQuintin, 2011). The objects of analysis of these studies are either a singlehouseholdsdecisionsorsteady-stateequilibriainmodelswithoutaggregateshocks. Thisallows the inclusionof various option-like features,such as renancingor default,which ourmodelwithaggregateshocksabstractsfrom.108Theinterestrateontheone-periodloanisinthisliteraturespeciedaseitherthecurrentshortrate(Iacoviello, 2005, andmanyothers), aweightedaverageof currentandpastshortandlongrates(Rubio,2011), or evolving in a Calvo-style sticky fashion (GrahamandWright, 2007). A staggered evolution of theinterestrateintroducesaformofnominal rigidityintothehousingloan, butduetotheone-periodnatureof theloan, householdscanundoitseects. Calzaetal. (2013)distinguishbetweenone- andtwo-periodcontracts, aimedatcapturingARMandFRMrespectively. TheirFRMthuscontainsthenominalrigiditystudied here, but it lasts for only two periods; as a one-period loan, their ARM does not contain the rigidity.9Inadditiontothesequantitative-theoretical studies, anumberof authorsinvestigatetherelationshipbetween monetary policy and housing empirically, in various regression models (see Kearl,Rosen,andSwan,1975; Kearl, 1979) andstructural VARs (e.g., BernankeandGertler, 1995; IacovielloandMinetti, 2008;Calzaetal., 2013). Usingdatafor anumber of developedeconomies, Calzaetal. (2013) ndstrongermonetarytransmissioninARMthanFRMcountries.10An exception in this regard is Koijen,VanHemert,andVanNieuwerburgh (2009), who study a mortgage7Fourth, the paper is related to studies investigating the redistributive eects of monetarypolicywhendebt contracts are speciedinnominal terms (DoepkeandSchneider, 2006;Meh,Rios-Rull,andTerajima,2010;Sheedy,2013). Weshowthatinthecaseofmortgagesthedistributionalconsequencesdepend,evenqualitatively,onwhethertheloanisARMorFRM.Finally,forournumericalanalysisweuseanapproximationofmortgageloansproposedbyKydland,Rupert,andSustek(2012), whichmakesmortgageseasytohandleinDSGEmodels. Thefocusof theirpaperisthelead-lagcyclical patternof residential investment,rather than monetary policy transmission. To that end, they take the mortgage and inationratesasexogenous, followinganestimatedVARprocesswithtotal factorproductivity. Assuch, the interest and ination rates process fed into their model reects shocks and frictionsourmodelabstractsfrom.3 ThenominalrigidityandchannelsoftransmissionIn a deterministic three-period problem of a single household, this section explains the natureof the nominal rigidityandthe resultingtwochannels of monetarypolicytransmission.Using,atthisstage,adeterministicthree-periodexampleallowsustodescribetherigidityin a transparent way. An extension to an innite horizon and uncertainty is straightforwardbut at thecost of extranotation(probabilities andhistories of events) andcumbersomeexpressions.11Timeisdenotedbyt = 1, 2, 3. Eachperiodthehouseholdisendowedwithconstantrealincome w and in period 1 has no outstanding mortgage debt (we introduce outstanding debtchoice problem with some option-like features in a model with aggregate shocks. Their agents and mortgages,however,liveforonlytwoperiods.11Theissues discussedhereapplyequallytoother long-termloans withnominal installments, suchascar loans. The focus of the paper is onmortgages as theyhave muchlonger termthancar loans andhousingmakesupabiggerchunkofhouseholdinvestmentthanautomobiles. Asshownbelow,aparticularnominal rigiditycharacterizesalsolong-termcouponbonds, typicallyissuedbycorporations. Thispaperabstractsfromcorporatedebtforthereasonthat,incontrasttosingle-familyhousing,long-termcorporateassetsarepredominantly(morethan75%)nancedthroughretainedearningsandotherformsof equity(RajanandZingales,1995).8later in this section). In period 1, the household makes a once-and-for-all housing investmentdecision, nancing a fraction of the investment with a loan and a fraction 1 with income.The loan can be used only for the housing investment and the house lasts for periods 2 and 3.The life-time utility function of the household is V=3t=1t1u(ct)+3t=2t1g(h), where is a discount factor, ct is consumption of a nonhousing good in period t, h is housing, and u(.)andg(.)havethestandardproperties. Thehouseholdmaximizestheutilityfunctionwithrespect to c1, c2, c3, and h, subject to three per-period budget constraints: c1+h = w+l/p1,c2=w m2/p2, andc3=w m3/p3, wherel =p1histhenominal valueof theloan,m2andm3arenominal loaninstallments(tobespeciedbelow), andptistheaggregatepricelevelinperiodt(thepriceofgoodsintermsofanabstractunitofaccount). Assumethereisanancial marketthatpricesassetsbyarbitragebutinwhichthehouseholddoesnotparticipatedueto, forinstance, highentrycosts(intheactual model thisassumptionwillbepartiallyrelaxed). Assumealsothatmonetarypolicycontrolsaone-periodnominalinterestrateit. Theabsenceofarbitragerestrictsittosatisfy1 + r=(1 + it)/(1 + t+1),where 1+r is a gross rate of return on real assets, assumed to be constant and given by somepricingkernel= (1 + r)1,andt+1 pt+1/pt 1istheinationratebetweenperiodstandt + 1.3.1 MortgagesMortgageinstallmentssatisfym2 (iM2+)landm3 (iM3+ 1)(1 )l. Here,iMtdenotesthemortgageinterest rate(henceforthreferredtoas themortgagerate). Under FRM,iM2= iM3= iF; under ARM, iM2and iM3may be dierent. Further, is the amortization ratein the rst period of the life of the mortgage, when the outstanding nominal debt is l. In thesecondperiod,theoutstandingnominaldebtis(1 )l andtheamortizationrateisequaltoone(i.e.,themortgageisrepaidinfull). FRMprescribesconstantnominalinstallments:m2=m3. TheamortizationratethereforesolvesiF+ =(iF+ 1)(1 ), whichyields=1/(2 + iF) (0, 0.5), foriF>0. Notethatd/diF= 1/(2 + iF)2(0.25, 0). For9agivenl,m2andm3thereforeincreasewheniFincreases. UnderARM,= 1/(2 + iM2) (0, 0.5), for iM2>0. If iM3>iM2thenm3>m2andviceversa. It isalsothecasethatd/diM2 (0.25, 0)andthereforethatm2increaseswheniM2increases.3.1.1 MortgagepricingandhousinginvestmentunderFRMIntheabsenceofarbitrage,iFhastosatisfy1 = Q(1)1(iF+ ) + Q(2)1(1 )(iF+ 1), (1)where Q(1)1= (1+i1)1and Q(2)1= [(1+i1)(1+i2)]1are the period-1 prices of one- and two-periodzero-couponbonds,determinedaccordingtotheexpectationshypothesis. Condition(1)statesthatthepresentvalueofinstallmentsforamortgageofsizeoneisequal toone.Noticethatif=1, themortgagebecomesaone-periodbondandif=0, themortgagebecomesacouponbond. Itisstraightforwardtoshowthat, for [0, 1), i10whenBt0ascapturingapremiumforunsecuredconsumercredit,whichisincreasinginaggregateborrowing19;(.) < 0ismeantto capture some intermediation costs on household savings, which reduce the interest rate onsavingsbelowthemarketinterestrateit. Inordertoavoidtheparticipationcostaectingthedenitionof aggregateoutput, itisrebatedtothehomeownerasalump-sumtransfert=Btt1/pt. Inanonstochasticsteadystate, B=0andtherst-orderconditionsforbt+1andbt+1imply = andhenceH= 0.18As inthe three-periodexample, is treatedas aparameter. Similar assumptionis made alsobyChambersetal. (2009a)andhasempirical support: overtheperiod1973-2006, therehasbeenverylittlevariation in the cross-sectional average of the loan-to-value ratio for single family newly-built homemortgages(FederalHousingFinanceAgency,MonthlyInterestRateSurvey,Table10).19Thiscanbethoughofascapturingthenotionthatasaggregateunsecuredcreditgrows,thecreditwor-thinessofborrowersdeclines.18Thehousingstockevolvesasht+1= (1 H)ht + xHt, (12)whereH (0, 1). Mortgagepaymentsareagaingivenasmt= (Rt + t)dt, (13)wheredt+1= (1 t)dt + lt, (14)t+1= (1 t) (t)+ t, (15)Rt+1=(1 t)Rt + tiFt, ifFRM,it, ifARM.(16)4.1.3 TechnologyAnaggregateproductionfunction, operatedbyperfectlycompetitiveproducers, is givenbyYt=Atf(Kt, Nt), whereKtistheaggregatecapital stock, Ntisaggregatelabor, andf(., .)hasthestandardneoclassical properties. Total factorproductivity(TFP)evolvesaslog At+1=(1 A) log A + A log At+ A,t+1, whereA (0, 1), Ais theunconditionalmean, andAt iidN(0, A). Thereal rateof returnoncapital, rt, andthereal wagerate, wt, aredeterminedbythemarginal productsof capital andlabor, respectively. Theresourceconstraintof theeconomyisCt+ XKt+ qtXSt+ G=Yt, whereCtisaggregateconsumption, XKtis aggregate investment in capital, XStis aggregate investment in housingstructures, andGis(constant)governmentexpenditures. Here, qtisthemarginal rateoftransformationbetweenhousingstructures andthe other uses of output, andhence therelative price of structures. It is given by a strictly increasing convex function q(XSt), whichmakes the economys production possibilities frontier concave in the space of (Ct+XKt+G)and(XSt)aspecicationakintothatofHumanandWynne(1999). Itssolepurposeis19toensurerealisticvolatilityofhousinginvestmentinresponsetoshocks; iftheproductionpossibilitiesfrontierwaslinear, giventhecalibrationoftheshocks, thevolatilitywouldbetoohigh.AsinDavisandHeathcote(2005), newhousesconsistof structuresandlandandareproducedbyperfectlycompetitivehomebuildersaccordingtoaproductionfunctionXHt=g(XSt, XLt). Here, XHtistheaggregatenumberof newhomesproducedinperiodt, XLtistheamountofnewresidentiallandused,andghasthestandardneoclassicalproperties.Homebuilders choose XHt, XSt, and XLt to maximize prots pHtXHtqtXStpLtXLt, subjecttotheaboveproductionfunction.4.1.4 MonetarypolicyandgovernmentMonetary policy is modeled as an interest rate feedback rule with a stochastic ination target(e.g.,Ireland,2007)it= (i + t) + (tt) + y(yty). (17)Here, >1, y 0, i isthenonstochasticsteady-statenominal interestrate, tistheinationtarget, yt log Yt log Yt1istheoutputgrowthrate, andyisitsnonstochasticsteady-statevalue(equal tozero). Theinationtarget followsanAR(1) processt+1=(1)+t+,t+1, where is less than but close to one, is the nonstochastic steady-stateinationrate, and,t+1 iidN(0, ). AsshowninSection4.2.2., inequilibrium,theinationtargetshockworkslikealevel factor, movingshortandlongratesequally,andallows themodel toreproducetheobservedvolatilityandpersistenceof the30-yearmortgage rate. A number of studies document that the level factor accounts for over 90% ofthe volatility of yields across maturities (see, e.g., Piazzesi, 2006) and shocks to the inationtarget are often invoked as its structural interpretation (e.g., AtkesonandKehoe, 2008). Themodel is closed by the government budget constraint: G+Tt= K(rtK)Kt +N(wtNt) +,whereTtisatransfertocapitalowners.204.2 EquilibriumThe equilibrium concept is the recursive competitive equilibrium (e.g., HansenandPrescott,1995). First, letzt [log At, t, pt1, Yt1] bethevectorof exogenousstatevariablesandlaggedendogenousvariablespt1andYt1, st [kt, bt, dt, t, Rt] thevectorof thecapitalownersstatevariables,st [ht, bt, dt, t, Rt]thevectorofthehomeownersstatevariables,andSt [Kt, Ht, Bt, Dt, t, t] thevectorof aggregateendogenousstatevariables, wheretheelementsare, respectively, theaggregatecapital, housing, bonds, mortgagedebt, anditseectiveamortizationandinterestrates. Next, writethecapital ownersoptimizationproblemasU(z, S, s) = max[xK,(b),l]{u(c) + E[U(z, S, (s))|z]}, (18)whereaprimedenotesavaluenextperiodandtheconstraints(4)-(9)arethoughttohavebeen substituted in the utility and value functions. Similarly, write the homeowners problemasV (z, S, s) = max[xH,b,n]{v(c, 1 n, h) + E[V (z, S, s)|z]}, (19)where the constraints (10)-(16) are thought tohave beensubstitutedinthe utilityandvaluefunctions. LetWt [XKt, pt, iMt, XHt, Bt+1, Nt] bethevectorof aggregatedecisionvariablesandprices,whereiMt= iFtunderFRMandiMt= itunderARM.DeneafunctionWt= W(zt, St).Arecursivecompetitiveequilibriumconsistsof thefunctionsU, V , andWsuchthat:(i) UandV solve(18) and(19), respectively; (ii) rtandwtaregivenbytherespectivemarginal productsof capital andlabor, pHtandpLtaregivenbythehomebuildersrst-orderconditionsforstructuresandland, andqt=q(XSt); (iii)itisgivenbythemonetarypolicyrule(17)andthegovernmentbudgetconstraintissatised;(iv)thebond,mortgage,housing, andlandmarketsclear: (1 )bt+1+ bt+1=0, (1 )(lt/pt) =pHtxHt,xHt=g(XSt, XLt), andXLt=1; (v)aggregateconsistencyisensured: Kt=(1 )kt,XKt=(1 )xKt, Tt=(1 )t , XHt=xHt, Nt=nt, Bt=bt, Ht=ht,21(1 )mt=mt, (1 )dt=dt=Dt, t=t=t, andRt=Rt= t; (vi) theexogenous statevariables followtheir respectivestochasticprocesses andtheendogenousaggregate state variables evolve according to aggregate counterparts to the laws of motion forthe respective individual state variables; and (vii) the individual optimal decision rules of thecapital owner (for xK, (b), and l) and the homeowner (for xH, b, and n) are consistent withW(z, S),oncethemarketclearingconditions(iv)andtheaggregateconsistencyconditions(v)areimposed.ItisstraightforwardtocheckthatthegoodsmarketclearsbyWalrasLaw: Ct +XKt +qtXSt + G = Yt,whereCt=(1 )ct+ ct. Equationscharacterizingtheequilibriumarecontained in Appendix A; a computational procedure resulting in log-linear approximation ofW(z, S) around the models non-stochastic steady state is described in Appendix B. The rst-orderconditionsofthecapital ownerforxKt, bt+1, andltresultinno-arbitrageconditionsforcapital,bonds,andnewmortgages. Asaresult,thecapitalownerisindierentbetweenthe three assets and the allocation of his period-t savings is determined by the homeownerssdemandforbondsandnewmortgages.204.2.1 CapitalownerandhomeownerblocksIt will be convenient to view the economy as consisting of two blocks. Given a set of decisionrulesforXHt,Bt+1,andNt,thecapitalownerblockdeterminesanaggregatedecisionrulefor XKt and pricing functions for pt and iMt. Similarly, given a set of decision rules and pricingfunctions for XKt,ptand iMt,the homeowner block determines aggregate decision rules forXHt,Bt+1,andNt. Inequilibrium,thetwosetsofdecisionrulesandpricingfunctionshavetobemutuallyconsistentateachpointinthestatespace(z, S). Workingwiththesetwoblocks in partial equilibriumi.e., taking the other blocks decision rules as givenfacilitatesunderstandingofthegeneralequilibriumresults.2120In the case of ARM, iMt= itmakes the capital owner indierent between new mortgages and bonds andtherst-orderconditionforltcanbedroppedfromthedescriptionoftheequilibrium. InthecaseofFRM,therst-orderconditiondeterminesiFt.21Intermsofequations, thehomeownerblockconsistsoftheoptimalityconditionsforthehomeownersBellman equation, while the capital owner block consists of the optimality conditions for the capital owners224.2.2 TheequilibriumshortrateThecapital ownersrst-orderconditionsforbt+1andxKtyieldtheFisherequation. Inalinearized form: it= Ett+1+Etrt+1, where (abusing notation) the variables are in percentagepointdeviationsfromsteadystate. Givenastochasticprocessforrt,bysuccessiveforwardsubstitutiontheFisherequationandthemonetarypolicyrule(17)determineit. Excludingexplosive paths for ination (a common assumption) and given close to one, the resultingexpressionforitisit j=0(1)jEtrt+1+j + t, (20)where, anticipating calibration described in the next section, y= 0 has been imposed. Dueto its high persistence, tgenerates highly persistent movements in itand thus moves itandiFtapproximatelyoneforone. Inthissense, itworkslikealevel factor, movingall yieldsapproximatelyequally. Incontrast, therstterminequation(20)ismuchlesspersistentthant, mainlyduetoalower persistenceof theAtshock. It producesonlytemporarymovementsinitandthussmallermovementsiniFtthaninit. Asaresult, itmovesthelong-short spread, iFt it. In this sense it works like a slope factor. The equilibrium inationrate is determined from the monetary policy rule as t= (i)/+t+(itt)/, whereitisgivenby(20). Noticethatahigherreducesthevolatilityofitandtinresponsetomovementsintheexpectedfuturepathofrt.Thereal interestratertispinneddownbythemarginal productof capital. Inalog-linearizedform, rt=At+( 1)Kt+(1 )Nt. The equilibriumitthus depends onthe stochastic paths of four variables: the exogenous state variables Atandtandtheendogenous variables KtandNt. Anygeneral equilibriumadjustments of itthus occurthrough expected future paths of Ktand Nt. Recall that in equilibrium the capital owner isindierentbetweensavinginmortgages, bonds, orcapital. Anincreaseinthedemandformortgages,otherthingsbeingequal,thusreducesKt+1. Thisincreasesrt+1andhenceit.Bellmanequation; bothblocksalsocontaintheproducers conditionsdeterminingrt, wt, qt, pHt, pLt, andXSt, the government budget constraint, andthe monetarypolicyrule, sothat the prices andtransfersrelevanttoeachblockcanbepinneddown(givendecisions/pricesoftheotherblock).235 CalibrationA closed-form solution to the models equilibrium conditions does not exist and the modelsproperties can be studied only numerically for specic functional forms and parameter values.Thechoiceof parametervaluesisbasedoncalibration. Themodel isquarterlyandmostparameter values are obtained by requiring the model to reproduce long-run averages of thedatainnonstochasticsteadystate. Somesecondmomentsarealsoused. Asmostof therequiredhistoricaldataarereadilyavailablefortheUnitedStates, thecalibrationisbasedonU.S.data.Anextralayerof complication, relativetomostDSGEmodels, arisesduetotheneedtomatchdebt-servicingcostsof homeowners. Forthisreasonthemodel isrequiredtobeconsistentwithboththecross-sectionaldistributionofincome,aswellasthekeyaggregateratios: XK/Y =0.156, XS/Y =0.054, G/Y =0.138, K/Y =7.06, H/Y =5.28, averagesfor1958-2006(seeAppendixCforthedescriptionofthedata), andN=0.255(AmericanTime-UseSurvey, 2003, population16+). Ocial dataformortgagedebtservicingcostsarenotavailablefortheUnitedStates. Estimates,however,canbeobtainedfromdierentdata sources (see Appendix D), resulting in long-run averages (1972-2006) in the ballpark of18.5% of homeowners pre-tax income. The models steady-state counterpart to this ratio isM/(wN ),where M= (R + )D/(1 +)and Disrealmortgagedebt.Consistency with the cross-sectional distribution of income is achieved through the trans-fer. Recallthathomeownersinthemodelareanabstractionforthe3rdand4thquintilesoftheU.S.wealthdistribution,whilecapitalownersareanabstractionforthe5thquintile.Inthedata,the5thquintilederives 40%ofincomefromcapitaland53%fromlabor;inthecaseof the3rdand4thquintiles, 81%comesfromlabor(SCF, 1998). Asaresult, if theonlysourceof incomeof capital ownersinthemodel wascapital, andgiventheobservedaverage capital share of output ,theywould account fortoo smallfraction of aggregate in-come (28.3% in the model vs 48% in the data), while homeowners share would be too large(71.7%vs34%). Asaresult, thesteady-statedebt-servicingcostsimpliedbytheobserved24andH/Y ratio(andsteady-stateamortizationandinterestrates)wouldbetoolow. Theparameter adjustsforthisdiscrepancybytransferring, inalump-sumway, someof thelaborincomefromhomeownerstocapitalowners.225.1 FunctionalformsThecapital owners per-periodutilityfunctionis u(c) =log c; thehomeowners utilityfunction is v(c, n) = log c +(1 ) log(1 n), where c is the composite consumption goodc(c, h) =ch1. Theadditiveseparabilityof thehomeownersutilityfunctionfacilitatesatransparentinterpretationoftheresultsasmarginalutilitiesareindependentofthecon-sumptionof other goods. Further, thegoods productionfunctionis f(K, N) =KN1andthehousingproductionfunctionisg(XS, XL) = X1SXL. AsinKydlandetal.(2012),q(XSt)=exp((XSt XS)), where>0andXSisthesteady-statestructurestooutputratio(Y is normalizedtobeequal tooneinsteadystate). Asimilar functional formisusedalsoforthebondmarketparticipationcost: (B)=exp(Bt) 1, where>0and Bt=0insteadystate. ItisstraightforwardtocheckthatthisfunctionsatisesthepropertiessetoutinSection4.1.2.5.2 ParametervaluesThemodelsparametersaresummarizedasfollows: (population); K, H, , A, A, A,,(technology);K,N,G,(scal);,,(mortgages);(bondmarket);,,y,,(monetarypolicy);and,,(preferences). TheparametervaluesarelistedinTable1and are discussed in detail in what follows. Most parameters can be assigned values withoutsolving a system of steady-state equations. Four parameters (, , K, ) have to be obtainedjointly. A third set of parameters (, , ) is assigned values by matching second momentsof the data. Table 2 lists the steady-state values of the models endogenous variables impliedby the calibration and, where possible, the values of their data counterparts. As can be seen,22Thelumpsumtransfercanbeinterpretedaslaborincomeofcapitalownersobtainedbyinelasticlaborsupplyandaconstantwagerate.25despitethehighlystylizednatureofthemodel, thesteadystateisbroadlyconsistentwithanumberofmomentsnottargetedincalibration.Inordertobeconsistentwiththenotionofhomeownersandcapitalownersinthedata, is set equal to 2/3. The parameter corresponds to the share of capital income in outputand is set equal to 0.283, an estimate obtained by GommeandRupert (2007) from NationalIncomeandProductAccounts(NIPA)foraggregateoutputclosetoourmeasureofoutput(seeAppendixC). Theshareof residential landinnewhousingissetequal to0.1, anestimate reported by DavisandHeathcote (2005). The depreciation rates Kand Hare setequal to 0.02225 and 0.01021, respectively, to be consistent with the average ow-stock ratiosfor capital and housing investment, respectively. The level of TFP, A, is set equal to 1.5321,so that steady-state output is equal to one. The stochastic process for TFP has A= 0.9641andA=0.0082, estimatesobtainedbyGommeandRupert(2007)fortheSolowresidualof a production function with the same and measurements of capital and labor inputs usedhere (see Appendix C). The labor income tax rate is derived from NIPA using a procedure ofMendoza,Razin,andTesar(1994), yieldingN=23.5%. Asnotedabove, G=0.138. Themortgage parameter is set equal to 0.76, the average (1973-2006) of the cross-sectional meanof theloan-to-valueratiofor singlefamilynewly-built homemortgages (Federal HousingFinanceAgency, MonthlyInterestRateSurvey, Table10). Usingtheaverage(1972-2006)30-year FRM interest rate of 9.31% per annum, Kydlandetal. (2012) show that = 0.00162and = 0.9946provideacloseapproximationtotheinstallmentsofaconventional30-yearmortgage. Inabaselinecase, theweight oninationinthemonetarypolicyrule, , issetequalto1.35,whichfallsinthemiddleoftherangeofestimatesreportedbyWoodford(2003), Chapter1. Thisparameterwill betreatedasafreeparameterinmonetarypolicyexperiments. Theweightonoutput, y, issetequal tozero.23Thesteady-stateinationrate, , issetequal to0.0113, theaverage(1972-2006)quarterlyinationrate. Insteadystate, the rst-order condition for ltconstrains iFto equal to i. Given the values of i and ,23Experimentationwithalternativevaluesof ydidnotsignicantlychangethedynamicpropertiesofthemodel. Thisisbecauseoutputinthemodel respondstoshocksinthetypical mean-revertingfashion,producingonlysmallgrowthratesaftertheimpactperiod.26therst-orderconditionforbtimplies= 0.9883. Fortheparticipationcostfunction(.),the choice of is guided by available studies on prices of unsecured credit. Namely, setting equal to 0.035 gives approximately the same premium at B= 0.5 as that predicted by theunsecuredcredit pricingfunctionof Chatterjee,Corbae,Nakajima,andRios-Rull (2007),Figure6,white-collarworkers.24Given the above parameter values, , , K, and are chosen jointly to match the valuesofK/Y , H/Y , debt-servicingcosts, andN. Therelationshipbetweentheparametersandthetargetsisgivenbythesteady-stateversionsof therst-orderconditionsforxKt, xHt,andnt,andtheexpressionforsteady-statedebt-servicingcostsnotedabove(therst-orderconditionsarecontainedinAppendixA). Theserestrictionsyield=0.2478, =0.6009,K= 0.3362,and= 0.5886.25Conditional onall of the above values, the parameters , , andare calibratedbysimulationunderFRM. Theparametersof theprocessfortarecalibratedbymatch-ingthestandarddeviation(2.4%)andtherst-orderautocorrelation(0.97)of the30-yearFRM mortgage rate (annualized rate, unltered data).26The resulting parameter values are=0.994and=0.0015. Theparameter controlsthevolatilityof theexpenditurecomponents of output and is used to match the volatility of aggregate consumption, relativetothevolatilityofoutput. TargetingthevolatilityofconsumptionhastheadvantagethatapproximatelythesameparametervalueisobtainedregardlessofwhethertheFRMortheARMeconomyisused. Theresultingvalueis= 0.35.24WethankEricYoungforthissuggestiononhowtocalibratethecostparameter.25In principle, Kcan be measured from NIPA in the same way as N. Such alternative parameterization,however,is inconsistent with the observed capital to output ratio. This is because is already pinned downbytherst-orderconditionforbonds. Nevertheless,KimpliedbythemodelisnotfarfromtheNIPAtaxrateobtainedbyGomme,Ravikumar,andRupert(2011): 33.62%inthemodelvs40.39%inNIPA.26The10-yeargovernmentbondyieldisactuallyusedasaproxyforthe30-yearmortgagerate. Thetworatesco-movecloselyfortheperiodforwhichbothseriesareavailable(from1972), butthedataforthe10-year yieldarelonger (1958-2007), thusprovidingabetter estimateof thestochasticpropertiesof theinationtargetshock.276 FindingsWestartbypresentingresultsforaversionoftheeconomyinwhichhomeownersarecom-pletelyexcludedfromthebondmarket(i.e., = and, inequilibrium, bt=0). ThisisdoneinSubsection6.1. Themainresultsofthepaperarequalitativelyunaectedbythissimplicationbutthegeneralequilibriummechanismiseasiertoexplain. Theexplanationis providedinSubsection6.2. Subsection6.3thenpresents theresults for thecasewithhomeowners access to the bond market. In light of how the simplied economy works, theseresultsarequitestraightforward.6.1 NoaccessofhomeownerstothebondmarketFigure2plotsthegeneral equilibriumresponsesofselectedaggregatevariablestoa1per-centagepoint(annualized)increaseintheinationtargetinperiod1. RecallthattheonlyrigiditythatallowsthetransmissionofthisshocktorealvariablesisthestructureofFRMandARMcontracts. Thersttwoleft-handsidechartsshowthat, inlinewithequation(20),theshort-termnominalinterestrate,theFRMinterestrate,andtheinationrateallincrease approximately by 1 percentage point in period 1 and revert back to the steady stateveryslowly, moreorlessreplicatingtheautocorrelationoftheshock, 0.994. (Theinationrate and the short rate under ARM increase by a little more than 1 percentage point due toan increase in labor supply, discussed below, which increases the marginal product of capitaland thus the rst term in equation (20).) Next, under both FRM and ARM, the increase inthe ination rate reduces in period 1 the real value of payments on outstanding debt. This isthestandardwealtheectpresentalsointhecaseofone-periodloans. However,itisquitesmallandisdwarfedbytheeectsofinationinthesubsequentperiods. UnderFRM,thecumulativeeectofpersistentlyhighinationgraduallyreducestherealvalueofmortgagepayments. Incontrast,underARM,mortgagepaymentsincreasesharplyinperiod2,thenstart to decline over time as the ination eect starts to slowly dominate the nominal interest28rateeect.27AsdiscussedinSection3, underbothFRMandARM, anequiproportional increaseinthenominal interestandinationratesincreasesthereal mortgageinstallmentsof anewloanatthefrontendof theloanslife. This, otherthingsbeingequal, increasesH. Butgiven the relative sizes of the outstanding and new debt, housing demand is mainly driven bythewealtheects, ratherthanthepriceeect. UnderFRM, housinginvestmentgraduallyincreases as the real value of mortgage payments on outstanding debt declines. The increase,however,ismodest,reachingapeakofonly1.6%. Incontrast,inthecaseofARM,housinginvestment drops sharplyinperiod2, by6.3%, as mortgage payments onexistingdebtincrease.28As for investment incapital, it increases under bothFRMandARM. UnderFRM,thisisduetoanincentiveofthecapitalownertosavemoreinordertomakeupfortheexpectedfuturedeclineinincomefromoutstandingmortgages(apartoftheincreasedsaving goes into the new mortgage borrowing by homeowners). Under ARM, this is due to anincentivetosmooththeeectofthetemporarywindfall ofhigherreal mortgagepaymentsfromperiod2on. Finally, under FRM, output graduallydeclinesashomeownersreducelaborsupplyinresponsetothepositivewealtheects. UnderARM, outputincreasesashomeownersincreaselaborsupplyinresponsetothenegativewealtheects.Figure3showsgeneral equilibriumresponsestoa1%increaseinTFP. Twocasesareconsidered: loosepolicy(=1.05)andtightpolicy(=2.5).29Thetopchartsshow27Under both FRM and ARM, the path of real mortgage payments from period 2 on reects both, paymentsondebtoutstandinginperiod1aswell aspaymentsonnewloanstakenoutfromperiod1(inclusive)on.Theoutstandingstockinperiod1,however,isalmost40timeslargerthanthequarterlyowofnewloans,dominatingthustheresponsesofmt,atleastintherst30-40periods.28Inperiod1, housinginvestmentunderARMdropsalittleduetothepriceeect. Thepriceeectissmall becauseof achangeinthevaluationof theinstallmentsonanewloan: asconsumptioninperiod2drops due to the increase in mortgage payments on outstanding debt (and is subsequently expected to returnbacktosteadystate), consumptiongrowthfromperiod2onispositive, reducingt,t+1fromperiod2on.Forasimilarreason,inperiod1housinginvestmentunderFRMincreases: futureconsumptionisexpectedtoincreasewiththeexpectedfuturedeclineinreal mortgagepaymentsonoutstandingdebt, reducingthevaluation of the mortgage payments on a new loan suciently enough to even reduce Hand increase housinginvestmentinperiod1.29As, toalargeextent, outputisdrivenbytheshock, forspaceconstraints, theresponsesofoutputarenotincludedinthegure. Forspaceconstraintsthegurealsodoesnotincludetheresponsesofnominalinterestrates. UnderbothFRMandARM,thepathoftheshortrateapproximatelycopiesthethepathoftheinationrate,beingabovetheinationrateduetoanincreaseinthemarginalproductofcapital.29that loose policy lets ination deviate from target much more than tight policy. Under bothFRMandARM,theinationrateincreasesinresponsetotheshock,asexpectedfromourdiscussioninSection4.2.2. Underloosepolicy, theinitial increaseisabout0.6percentagepoints (annualized) under both contracts but the ination rate is more persistent under FRMthan under ARM (we will come back to this in the subsection below); under tight policy, theination rate increases only a little under both contracts, thus eectively producing the sameallocations as under an index-linked mortgage. Any signicant dierences in the dynamics ofhousinginvestmentacrossthetwotypesofloanwillthusbevisibleonlyunderloosepolicyandthereal eectsof thenominal rigidityundereachcontractcanbejudgedagainsttheresponsesundertightpolicy.The key observation to make from Figure 3 is that, in contrast to the case of the inationtarget shock, theresponses of housinginvestment areverysimilar across bothmortgagetypesandpolicies. Especiallyinperiod1theresponsesarealmostidentical. Fromperiod2on, somedierencesexistbetweenFRMandARM, astheincreaseintheinationrateincreasesthereal paymentsonoutstandingdebtunderARM,whileitreducesthemunderFRM. Incontrasttohousinginvestment, theresponsesof capital investment(underloosepolicy)diersignicantlyacrosstheloantypes. Thenextsubsectionexplainstheseresults.6.2 ExplainingthegeneralequilibriummechanismWeusepartial equilibriumanalysisofthecapital ownerandhomeownerblockstoexplainthegeneral equilibriummechanisminthemodel. Figure4, panel A, showstheresponsesof thecapital owner block(i.e., treatingXHtandNtas exogenous andconstant) to1%increase in At. The rst chart shows a response of XKtfamiliar from the neoclassical growthmodel. Theresponseisalittlehigherafterperiod2underARMthanunderFRMbecauseofa(relativelysmall)additional increaseinincomeduetohigherreal mortgagepaymentsfromoutstandingdebt under ARM, occurringduetoanincreaseinthenominal interestand ination rates plotted in the next two charts. These two nominal variables increase due30toanincreaseintherstterminequation(20); thebaselinevalue=1.35isused. TheFRM mortgage rate iFtalso increases, but substantially less than it, as it is expected to meanrevertrelativelyfast; theimpliedautocorrelationisabout0.95. Noticeforfuturereferencethatabout2/3oftheincreaseinittransmitintot+1.Panel Bof Figure4shows theresponses of thehomeowner block(i.e., treatingXKt,t, andiMtasexogenous)to1percentagepoint(annualized)meanrevertingincreaseinit,assuming autocorrelation of 0.95. It is further assumed that 2/3 of ittransmit into t+1andthat iFtis related to itas in panel A. The rst chart in panel B shows that in response to theshock, XHtdeclinesbymoreunderARMthanunderFRM.Asthenexttwoguresinthepanel show, underARMthepriceandwealtheectsworkinthesamedirection(bothHtandrealmortgagepaymentsonoutstandingdebtincrease),whereasunderFRMtheyworkin opposite directions (Ht increases but real mortgage payments on outstanding debt declineover time).30In addition, Htincreases by less under FRM than under ARM. The rst chartin the panel complements the responses of XHtwith its response to1% increase in At. Thisresponseisthesameunderbothcontracts, asiMtandtare, inthiscase, heldconstant.TakentheresponsesofXHttotheinterestrateandTFPshockstogether,wewouldexpecta positive TFP shock, triggering movements of the nominal interest and ination rates as inpanel A, toincreaseXHtsubstantiallymoreunderFRMthanunderARM. This, however,doesnotoccuringeneralequilibrium,asFigure3showed.31Panel C completes the picture. It shows the responses of the capital owner block to 10%increaseinXHt(10%isusedsothattheorderofmagnitudeoftheshockisinlinewiththeresponsesofXHtinpanelB).Asthecapitalownersuppliesanyamountofnewmortgagesdemanded by the homeowner, such shock crowds out XKt. As a result, Kt starts to graduallydecline and rtto gradually increase, at least until the capital owner (induced by a higher rt)suciently increases his overall saving. rtthus follows a hump-shaped path, which produces30Of course, in period 1, only the price eect is present. The positive wealth eects from period 2 on underFRMshowupintheresponseofXhtintherelativelyfastrecoveryofXHt(fasterthanthedeclineofHt),whereasunderARMthenegativewealtheectshowsupasfurtherdeclineinXHtinperiod2.31Notethatitincreasesbyonly0.3percentagepointsinpanelA,whereaspanelBshowsresponsestoanincreaseby1percentagepoint(anormalization).31hump-shapedresponsesof itandt, plottedinthechartsinpanel C. Anticipatingfutureincreasesinit,iFtjumpsimmediately. Incombinationwiththesluggishincreaseint,thisimplies higher initial real mortgage installments on a new loan under FRM than under ARM.In sum, while the partial equilibrium eect of the nominal rigidity on housing investmentin the presence of a TFP shock is stronger under ARM than under FRM, the general equilib-rium eect is stronger under FRM than under ARM. In combination, these two eects resultin similar responses of XHtto the TFP shock regardless of the mortgage type. The workingofthismechanismisapparentinFigure3inthemorepersistentresponseofinationunderFRMthanunderARMreectingthehump-shapedcomponentandinthehump-shapedresponse of capital investment under FRM.32As shown below, in the full model, the generalequilibrium adjustment is weakened and the responses are closer to what would be expectedfrompartialequilibriumanalysis.Why,incontrasttotheTFPshock,inthecaseoftheinationtargetshockthegeneralequilibriumresponsesofhousinginvestmentdieracrosscontracts? (ReferbacktoFigure2.) Roughlyspeaking, thisisbecausetheresponsesof thetwoagenttypesaremutuallyconsistent at, moreor less, aconstant real interest rate. Inthecaseof FRM, whilethehomeownerdemandsmoremortgageborrowing, thecapital ownerwantssavemore. (Thereal interest ratedeclinesalittleduetoasmall increaseincapital accumulation, asnotall desiredsavinggets absorbedbynewmortgageborrowing.) Inthecaseof ARM, thehomeownerreducesdemandformortgages, whereasthecapitalownerincreaseshisdesiredsaving. However, thedownwardeect onthereal interest ratefromtheresultingfastercapital accumulation is neutralized by higher labor supply by the homeowner, which he usestosmooththeincreaseinreal mortgagepaymentsonoutstandingdebt. (Thereal interestrate increases a little as the eect of higher labor supply is somewhat stronger than the eectofhighercapitalstock,aswasnotedinourdiscussionofFigure2above.)32Thesamegeneral equilibriumadjustment alsoproduces thesimilar responses of XHtunder thetwoalternativevaluesof,foragivencontract.326.3 ThefullmodelWithaccess tothe bondmarket, homeowners have anadditional marginwithwhichtosmooththeimpact of thetwoshocks. Inparticular, theyusethebondtoborrowwheneitherincomedeclinesorrealmortgagepaymentsincreaseandtolendwheneitherincomeincreasesorreal mortgagepaymentsdecline. Theleft-handsidechartof Figure5, panelA,showstheresponsesofhousinginvestmenttotheinationtargetshockforthebaselineautocorrelationoftheshockof0.994. UnderbothFRMandARM, theresponsesarenowsmootherandsomewhatmuted, relativetoFigure2, butthemainresultthattheeectoftheshockislargerunderARMthanFRMstillholds(amaximumdeclineof 3.7%underARMvsamaximumincreaseof1.4%underFRM).Theright-handsidechart inpanel Ashows theresponses for alower autocorrelationof theshock, 0.75. Inthis case, whiletheresponses arequalitativelysimilar tothoseintheleft-handsidechart,quantitativelytheyaremuchsmaller,especiallyintheARMcase:0.4%vs 3.7%inperiod1. Highpersistenceof theshockisthuscrucial forthequan-titativeimportanceof thenominal rigidity. Historically, throughthelensesof themodel,thebaselineautocorrelationof0.994isthemorerelevantone,asitreproducestheobservedautocorrelationofthelong-termnominalinterestrate.33AnimplicationofthispropertyofthemodelforpolicyespeciallyforARMcountriesisthat,inordertohavesizableeecton housing investment, changes in the short-term nominal interest rate have to be persistent.Homeowners access to the one-period bond market also weakens the general equilibriumadjustmentsinresponsetoTFPshocksdescribedintheprevioussubsection. Nowhome-ownersrespondtoapositiveTFPshockbyincreasingbothhousinginvestmentandbondholdings. Theresultingowof funds fromhomeowners tocapital owners andthelowerdemandformortgages,relativetothecaseofnoaccesstothebondmarket,meanthatthecrowdingoutofcapital investment, anditsimplicationforinterestrates, doesnotneedtooccurasmuchasbeforeinorderforequilibriumtobereached. PanelBofFigure5shows33High persistence of long rates is historically observed across developed economies, not just in the UnitedStates.33that,inresponsetothe1%positiveTFPshock,housinginvestmentnowincreasesbymoreunderFRMthanunderARM.ThisisconsistentwithFigure4,panelB,whichshowsthat,in the extreme case of no crowding out, the increase in interest rates in response to the TFPshockdampenstheresponseofhousinginvestmentmoreunderARMthanunderFRM.Anal set of results is containedinTable 3, whichreports standarddeviations andcorrelationswithoutputof themodelsvariablesandtheircounterpartsinU.S. data(seeAppendix C for a description of the data counterparts to the variables in the model). Giventhat the model has only two shocks, and purposefully abstracts from a number of empiricallyrelevant frictions, these statistics serve the purpose of onlygaugingthe models generalplausibility, rather than as a formal test of the theory. As is customary in the business cycleliterature, the statistics are for HP-ltered series, both in the model and in the U.S. economy.Even though the ination target shock has real eects, the TFP shock is the dominant shockandthemodelsbusinesscyclemomentsaremainlydeterminedbythisshock. AsTable3shows, for most statistics, the model is broadly in line with the data, but some discrepanciesareworthnoting. First, thecorrelationinthemodelbetweenYtandNtisnegative, underbothFRMandARM. This is because, withlimitedmeans tosmoothconsumptionovertime, the intertemporal elasticity of laborresponsible for the positive comovement betweenhours and output in real business cycle modelsis relatively small here and is dominated bywealtheects. Second, becausetheonlyshockdrivingthecomovementbetweentheslopefactorandoutputinthemodelistheTFPshock,thenegativecorrelationofthelong-shortspreadwithoutputinthemodelisstrongerthaninthedata. Thelong-shortspreadisalsoonlyhalf asvolatile, relativetooutput, asinthedata. Thislikelyreectstheabsenceoftime-varyingtermpremiainourmodel. Third, themodel underpredictsthevolatilityofhouseprices, relativetothat of output, accountingfor about twothirds of theobservedrelativestandarddeviation, andproducestoohighcomovementbetweenhousepricesandthe business cycle. This is because qt, determined in the model by housing demand, is muchless volatile than in the data and too strongly positively correlated with output. In order to34reproducetheobservedvolatilityofhousepricesandtheircorrelationwithoutput, shockstoqtareneeded.347 ConcludingremarksAparsimoniousmodel containingeitherFRMorARMloanswasconstructedinordertoinvestigate the equilibrium eects on the real economy, and on aggregate housing investmentespecially,ofthenominalrigidityinherentinfully-amortizingmortgages. Themodelecon-omyhasapopulationof homeownersandcapital ownerswithselectedkeycharacteristicsofeachgroupobservedinthedata. Duetothenominalrigidityandincompleteassetmar-kets,monetarypolicytransmitsthroughtheeectivepriceofnewhousingandcurrentandexpectedfuturewealtheectsofpaymentsonoutstandingmortgagedebt. ThekeyndingisthatmonetarypolicyaectshousinginvestmentmoreunderARMthanunderFRM. Inaddition, shocks to long-run ination have larger eects than cyclical uctuations in inationandnominalinterestrates,occurringduetoTFPshocks. Finally,thedistributionalconse-quencesofmonetarypolicydependonthetypeofthemortgageloan. Apersistentincreaseintheinationrateredistributesreal incomefromlenderstoborrowersunderFRM, butfromborrowerstolendersunderARM, atleastintheinitial periodsaftertheshock. Animplication of our ndings for the current policy debate is that, other things being equal, lownominalinterestratesarelikelytohavealargereectonthehousingmarketinARMthanFRMcountriesandtheeectofsuchapolicywill belargerthelongeristhetimehorizonforwhichtheratesareexpectedtostaylow.Ouraimwastomakeasteptowardsabetterunderstandingoftheaggregateandredis-tributiveconsequencesof monetarypolicyinthepresenceof standardmortgageloans. Inorder to isolate the channels under investigation, and to describe their eects in a transparentway,themodelhasintentionallyabstractedfromothernominalfrictions. Shockswerealso34Inthemultisectoralmodel ofDavisandHeathcote(2005), TFPshocksintheconstructionsectorplaysucharole.35limitedtoonlytwotypes,traditionalbusinesscycleshockstoTFPandshockstolong-runination. The long-run ination shocks work like the level factor in models of the yield curve,movingshortandlongratesapproximatelyequally, whereasTFPshocksresembleaslopefactor,movingthelong-shortspreadoverthebusinesscycle.Anatural nextstepistoincorporateotherrelevantshocks, marginsof adjustment, orfrictionstoalignthemodel morecloselywiththedataandinvestigatehowthequantita-tively important elements of these richer environments impact on the basic conclusion of thepaper. Based on our partial equilibrium results, we conjecture that mechanisms that weakenthegeneral equilibriumadjustmentsinthepathof thereal interestratewill increasetheimportanceofthenominalrigidityinthetransmissionmechanism.Thefocusofthepaperwasonlyonconditionalrstmomentsinagentsdecisions. Thatis, we have abstracted from the role of risk. Indeed, ARMs have dierent risk characteristicsthenFRMs. Furthermore, long-terminterestratescontainriskpremiathatvarywiththestateof theeconomy. Incorporatingtheseelementswouldbeanotherfruitful extensionofthemodel.Athirdrelevantextension, conditional onsuccessfullyachievingthesecondone, wouldbetoincludeoptimal renancingand/orthechoicebetweenFRMandARM. Aswedis-cussed, introducingsuchmarginsisgoingtoweakentheeectsofthenominal rigidity. Inthatsense, ourresultsarebestinterpretedasprovidinganupperbound. Thechallengeofsuchextensions, however, istoavoidabang-bangsolution. Doingsonecessarilyinvolvesthecomplicationofhomeowners heterogeneity. Thesamedicultyalsoappliestothein-troductionof theoptiontodefault. Inourmodel homeownersarenotallowedtodefault.When real mortgage payments on outstanding debt increase, homeowners respond by cuttingconsumption and investment and increasing hours worked. These adjustments are especiallyrelevant in the case of ARM. With default, depending on its costs, homeowners may insteadchoosetodefaultonmortgagedebt, especiallyinresponsetoverylargeincreasesinrealmortgagepayments.36Finally, aninterestingnormativequestionregardsoptimal monetarypolicy. Underin-complete markets,thenominalrigidity inmortgage loansgenerates bothadistortionintheoptimalityconditionforhousingandex-postredistributionofincomebetweenhomeownersand capital owners. With our mapping between these two groups of agents in the model andinthedata, thelattergrouphavebettermeansof smoothingconsumptionovertimeandstatesof theworld. Inaddition, mortgageincomemakesuponlyasmall fractionof theirtotal income. An optimal monetary policy may therefore essentially face a trade o betweeneliminatingthedistortionintheoptimalityconditionforhousingandprovidinginsurancetohomeownersagainstex-postuctuationsinreal mortgagepayments. Asthereal eectsofmonetarypolicydierdependingonwhetherloansareFRMorARM,optimalmonetarypolicy is likely to depend on the type of the loan contract. Additional complexity in the de-sign of optimal monetary policy is likely to arise when default is allowed. All these questionsandissuesare,however,beyondthescopeofthispaperandareleftforfutureresearch.37ReferencesAlm, J., Follain, J.R., 1984.Alternativemortgageinstruments, thetiltproblem, andcon-sumerwelfare.JournalofFinancialandQuantitativeAnalysis19,11326.Aruoba, S. B., Davis, M. A., Wright, R., 2012. Homework in monetary economics: Ination,homeproduction,andtheproductionofhomes.NBERWorkingPaper18276.Atkeson, A., Kehoe, P. J., 2008. Ontheneedforanewapproachtoanalyzingmonetarypolicy.NBERWorkingPaperNo.14260.BankofEngland,2013.FinancialStabilityReport,June.London.Benigno, P., Woodford, M., 2006. Optimal taxationinanRBCmodel: Alinear-quadraticapproach.JournalofEconomicDynamicsandControl30,144589.Bernanke, B. S., Gertler, M., 1995. Insidetheblackbox: Thecreditchannel of monetarytransmission.JournalofEconomicPerspectives9,2748.Boardof Governors, 2012. Minutes of the Federal OpenMarket Committee, September12-13.Washington,D.C.Brunnermeier, M. K., Julliard, C., 2008. Moneyillusionandhousingfrenzies. ReviewofFinancialStudies21,135180.Calza, A., Monacelli, T., Stracca, L., 2013. Housing nance and monetary policy. Journal oftheEuropeanEconomicAssociation11,101122.Campbell,J.Y.,2012.Mortgagemarketdesign.NBERWorkingPaper18339.Campbell, J. Y., Cocco, J. F., 2003. Householdriskmanagement andoptimal mortgagechoice.QuarterlyJournalofEconomics118,144994.Chambers, M., Garriga, C., Schlagenhauf, D., 2009a. Accountingforchangesinthehome-ownershiprate.InternationalEconomicReview50,677726.Chambers, M., Garriga, C., Schlagenhauf, D., 2009b. The loan structure and housing tenuredecisionsinanequilibriummodelofmortgagechoice.ReviewofEconomicDynamics12,44468.Chatterjee, S., Corbae, D., Nakajima, M., Rios-Rull, J.-V., 2007. Aquantitativetheoryofunsecuredconsumercreditwithriskofdefault.Econometrica75,152589.Chatterjee, S., Eyigungor, B., 2011. Aquantitativeanalysisof theUShousingandmort-gagemarketsandtheforeclosurecrisis. WorkingPaper11-26, Federal ReserveBankofPhiladelphia.Corbae,D.,Quintin,E.,2011.Mortgageinnovationandtheforeclosureboom.Mimeo.Davis,M.A.,Heathcote,J.,2005.Housingandthebusinesscycle.InternationalEconomicReview46,75184.38Doepke, M., Schneider, M., 2006. Ination and the redistribution of nominal wealth. JournalofPoliticalEconomy114,106997.Dressler, S. J., Li, V. E., 2009. Insidemoney, credit, andinvestment. Journal ofEconomicDynamicsandControl33,97084.Edge, R. M., 2000. The eect of monetary policy on residential and structures investment un-derdierenctialprojectplanningandcompletiontimes.InternationalFinanceDiscussionPaper671,BoardofGovernorsoftheFederalReserveSystem.European Mortgage Federation, 2012a. Study on mortgage interest rates in the EU. Brussels.EuropeanMortgageFederation,2012b.EMFFactsheet2012Germany.Brussels.Fabozzi, F. J., Modigliani, F., Jones, F. J., 2010. Foundations of Financial Markets andInstitutions,4thEdition.Pearson.Garriga,C.,Schlagenhauf,D.E.,2009.Homeequity,foreclosures,andbail-outs.Mimeo.Ghent, A., 2012. Infrequent housing adjustment, limited participation, and monetary policy.JournalofMoney,Credit,andBanking44,93155.Gomme,P.,Ravikumar,B.,Rupert,P.,2011.Thereturntocapitalandthebusinesscycle.ReviewofEconomicDynamics14,26278.Gomme, P., Rupert, P., 2007. Theory, measurement and calibration of macroeconomic mod-els.JournalofMonetaryEconomics54,46097.Graham, L., Wright, S., 2007. Nominal debt dynamics, credit constraints andmonetarypolicy.TheB.E.JournalofMacroeconomics7,Article9.Green,R.K.,Wachter,S.M.,2005.TheAmericanmortgageinhistoricalandinternationalcontext.JournalofEconomicPerspectives19,93114.Hancock, M., Wood, R., 2004. Household secured debt. Bank of England Quarterly BulletinAutumn,291301.Hansen, G. D., Prescott, E. C., 1995. Recursive methods for computing equilibria of businesscyclemodels. In: Cooley, T. F. (Ed.), Frontiersof BusinessCycleResearch. PrincetonUniversityPress.Human, G. W., Wynne, M. A., 1999. Theroleof intratemporal adjustment costs inamultisectoreconomy.JournalofMonetaryEconomics43,31750.Hurst,E.,Staord,F.,2004.Homeiswheretheequityis: Mortgagerenancingandhouse-holdconsumption.JournalofMoney,Credit,andBanking36,9851014.Iacoviello, M., 2005. House prices, borrowing constraints, and monetary policy in the businesscycle.AmericanEconomicReview95,73964.39Iacoviello,M.,2010.HousinginDSGEmodels: Findingsandnewdirections.In: deBandt,O.,Knetsch,T.,Pe nalosa,J.,Zollino,F.(Eds.),HousingMarketsinEurope: AMacroe-conomicPerspective.Springer-VerlagBerlin.Iacoviello,M.,Minetti,R.,2008.Thecreditchannelofmonetarypolicy: Evidencefromthehousingmarket.JournalofMacroeconomics30,6996.Iacoviello, M., Neri, S., 2010. Housing market spillovers: Evidence from an estimated DSGEmodel.AmericanEconomicJournal: Macroeconomics2,12564.International Monetary Fund, 2011. Housing nance and nancial stabilityback to basics?In: Global Financial StabilityReport, April 2011: DurableFinancial StabilityGettingTherefromHere.InternationalMonetaryFund,Washington,DC.Ireland, P. N., 2007. ChangesintheFederal Reservesinationtarget: Causesandconce-quences.JournalofMoney,Credit,andBanking39,185182.Kearl, J. R., 1979. Ination, mortgage, and housing. Journal of Political Economy 87, 111538.Kearl, J. R., Rosen, K., Swan, C., 1975. Relationshipsbetweenthemortgageinstruments,the demandfor housing, andmortgage credit. In: NewMortgage Designes for StableHousinginanInationaryEnvironment.FederalReserveBankofBoston,Boston.Koijen,R.S.J.,VanHemert,O.,VanNieuwerburgh,S.,2009.Mortgagetiming.JournalofFinancialEconomics93,292324.Kydland, F. E., Rupert, P., Sustek, R., 2012. Housingdynamicsoverthebusinesscycle.NBERWorkingPaper18432.Lessard, D. R., Modigliani, F., 1975. Inationandthehousingmarket: Problemsandpo-tentialsolutions.In: NewMortgageDesignesforStableHousinginanInationaryEnvi-ronment.FederalReserveBankofBoston,Boston.Li, V. E., Chang,C. Y., 2004. The cyclial behavior of household and business investment inacash-in-advanceeconomy.JournalofEconomicDynamicsandControl28,691706.Li, W., Yao, R., 2007. The life-cycle eects of house price changes. Journal of Money, Credit,andBanking39,13751409.Liu, Z., Wang, P., Zha, T., 2013. Land-pricedynamics andmacroeconomicuctuations.Econometrica81,114784.Meh, C. A., Rios-Rull, J. V., Terajima, Y., 2010. Aggregateandwelfareeectsofredistri-bution of wealth under ination and price-level targeting. Journal of Monetary Economics57,63752.Mendoza, E. G., Razin, A., Tesar, L. L., 1994. Eective tax rates in macroeconomics: Cross-countryestimatesoftaxratesonfactorincomesandconsumption. Journal ofMonetaryEconomics34,297323.40Piazzesi,M.,2006.Anetermstructuremodels.In: Ait-Sahalia,Y.,Hansen,L.P.(Eds.),HandbookofFinancialEconometrics.Elsevier,Amsterdam.Poterba,J.M.,1984.Taxsubsidiestoowner-occupiedhousing: Anasset-marketapproach.QuarterlyJournalofEconomics99,72952.Rajan, R. G., Zingales, L., 1995. What do we know about capital structure?Some evidencefrominternationaldata.JournalofFinance50,142160.Rubio,M.,2011.Fixed-andvariable-ratemortgages,businesscycles,andmonetarypolicy.JournalofMoney,Credit,andBanking43,65788.Rupert, P., Sterk, V., Sustek, R., 2013.Exactandapproximatemortgages: Acomparison.Mimeo,UniversityofSouthampton.Scanlon, K., Whitehead, C., 2004. International trends inhousingtenure andmortgagenance. Special report for the Council of Mortgage Lenders, London School of Economics.Schwab, R. M., 1982. Ination expectations and the demand for housing. American EconomicReview72,14353.Sheedy, K., 2013. Debt and incomplete nancial markets: A case for nominal GDP targeting.CEPDiscussionPaper1209.Woodford, M., 2003. Interest andPrices: Foundations of aTheoryof MonetaryPolicy.PrincetonUniversityPress.41A.Mean-revertingdeclineintheshortrate0 20 40 60 80 100 120012345QuartersPercentShortterm interest rate 0 20 40 60 80 100 1200.050.10.150.20.250.3Quarters since mortgage advanceSteady state "Easing" under FRM "Easing" under ARM B.Hump-shapeddeclineintheshortrate0 20 40 60 80 100 120012345QuartersPercentShortterm interest rate 0 20 40 60 80 100 1200.050.10.150.20.250.3Quarters since mortgage advanceSteady state "Easing" under FRM "Easing" under ARM Figure 1: Monetary policy and debt-servicing costs. The left-hand side panelsshow alternative paths of the short-term nominal interest rate. The right-handsidepanelsshowthecorrespondingmortgagepaymentsasafractionofpost-tax income for FRM and ARM; the label steady-state refers to the case whenthe short rate is at its steady-state level of 4%. The mortgage loan is equal tofourtimesthehouseholdspost-taxincome. Inallcases,therealinterestrate(1%)andrealincomeareheldconstant;nominalincomechangesinlinewithination.42Table1: CalibrationSymbol Value DescriptionPopulation 2/3 ShareofhomeownersTechnologyA 1.5321 Steady-statelevelofTFP 0.283 CapitalshareofoutputK0.02225 DepreciationrateofcapitalH0.01021 Depreciationrateofhousing 0.35 CurvatureofPPF 0.1 LandshareofnewhousingFiscalG 0.138 GovernmentexpendituresN0.235 LaborincometaxrateK0.3362 Capitalincometaxrate 0.5886 LaborincometransferPreferences 0.9883 Discountfactor 0.2478 Cons. compositesshareinutility 0.6009 Shareofmarketcons. incompositeMortgages 0.76 Loan-to-valueratio 0.00162 Initialamortizationrate 0.9946 AmortizationadjustmentfactorBondmarket 0.035 ParticipationcostfunctionMonetarypolicy1.35 Weightoninationy0 Weightonoutputgrowth 0.0113 Steady-stateinationrateExogenousprocessesA0.9641 PersistenceofTFPshocksA0.0082 Std. ofTFPinnovations0.994 Persistenceofin. targetshocks0.0015 Std. ofin. targetinnovations43Table2: Nonstochasticsteadystateandlong-runaveragesofdataSymbol Model Data DescriptionNormalized:Y 1.0 N/A OutputTargetedincalibration:K 7.06 7.06 CapitalstockH 5.28 5.28 HousingstockXK0.156 0.156 CapitalinvestmentXS0.054 0.054 HousingstructuresN 0.255 0.255 Hoursworked m/(wn ) 0.185 0.185 Debt-servicingcosts(pre-tax)iM0.0233 0.0233 MortgagerateNottargeted:AggregatemortgagevariablesD 1.61 2.35Mortgagedebt 0.0144 0.0118AmortizationrateCapitalownersvariables(1 K)(r K) 0.012 0.013Netrateofreturnoncapital[(r )k +m]/[(r )k +m +] 0.31 0.39,Incomefromassetstototalincome m/[(1 K)(r )k +m +] 0.089 N/A Mortg. paymentstototal(net)incomeHomeownersvariablesH0 N/A Housingwedge m/[(1 N)(wn )] 0.24 N/A Debt-servicingcosts(post-tax)(wn )/(wn ) 1.00 0.81IncomefromlabortototalincomeDistributionofwealth(K + D)/(K +H) 0.71 0.82Capitalowners(H D)/(K +H) 0.29 0.18HomeownersNote: Rates of return and interest and amortization rates are expressed at quarterly rates;capitalowners = the 5th quintile of the SCF wealth distribution; homeowners = the 3rd and 4th quintilesoftheSCFwealthdistribution. Upper bound for the mortgage debt in the model due to the presence in the data of equity loans,secondmortgages,andmortgagesforpurchasesofexistinghomes.Foraconventional30-yearmortgage.NIPAestimatebyGommeetal.(2011).1998SCF;themodelcounterpartisdenedsoastobeconsistentwithSCF.Thesumofcapitalandbusinessincome.440 10 20 30 4000.20.40.60.81Percentage point, annualizedInflation rate (pi)ARM FRM 0 10 20 30 408642024PercentHousing investment (XH)FRM ARM0 10 20 30 4000.20.40.60.81Percentage point, annualizedShort rates (i) and FRM rate (iF)ARM FRM (i)FRM (iF)0 10 20 30 401012345PercentCapital investment (XK)ARM FRM 0 10 20 30 40202468PercentReal mortgage payments (m/p)FRM ARM0 10 20 30 400.20.100.10.20.30.40.5PercentOutput (Y)FRM ARMFigure2: General equilibriumresponsesto1percentagepoint(annualized)increaseintinperiod1; versionwithoutaccessofhomeownerstothebondmarket.45FRM ARM0 10 20 30 400.200.20.40.60.8Prcentage point, annualizedInflation rate (pi)Tight Loose 0 10 20 30 400.200.20.40.60.8Percentage point, annualizedInflation rate (pi) Loose Tight 0 10 20 30 401012345PercentReal mortgage payments (m/p)Tight Loose 0 10 20 30 401012345PercentReal mortgage payments (m/p)Loose Tight 0 10 20 30 402024681012PercentHousing investment (XH)Tight Loose 0 10 20 30 402024681012PercentHousing investment (XH)Loose Tight 0 10 20 30 4010123PercentCapital investment (XK)Tight Loose 0 10 20 30 4010123PercentCapital investment (XK)Tight Loose Figure 3: General equilibriumresponses to1%increase inAtinperiod1;version without access of homeowners to the bond market. Loose policy: =1.05;tightpolicy: = 2.5.46PanelA PanelB0 10 20 30 4000.511.522.533.5PercentCapital investment (XK)ARM FRM 0 10 20 30 402015105051015PercentHousing investment (XH)FRMARM Response to A, both mortgages 0 10 20 30 400.100.10.20.30.4Percentage point, annualizedShort rates (i) and FRM rate (iF)ARM FRM (i)FRM (iF)0 10 20 30 4000.20.40.60.811.2Percentage pointHousing wedge (tauH)FRMARM 0 10 20 30 400.100.10.20.30.4Percentage point, annualizedInflation rate (pi)ARM FRM 0 10 20 30 40202468PercentReal mortgage payments (m/p)FRMARM PanelC0 10 20 30 4000.020.040.060.080.10.12Percentage point, annualizedShort rates (i) and FRM rate (iF)FRM (i) ARM FRM (iF)0 10 20 30 4000.020.040.060.080.10.12Percentage point, annualizedInflation rate (pi)ARM FRM Figure 4: Explainingthe general equilibriumadjustments toapositive Atshock using partial equilibrium responses of the capital owner and homeownerblocks. PanelA:capitalownerblockresponsesto1%increaseinAt. PanelB: homeowner blockresponses to 1 percentage point (annualized) increase init(with2/3pass-throughtot+1); intherstchartcomplementedwiththeresponse of XHt to 1% increase in At. Panel C: capital owner blockresponsesto10%increaseinXHt.47A.1perc. point(annualized)increaseint= 0.994 = 0.750 10 20 30 404321012PercentHousing investment (XH)ARMFRM0 10 20 30 404321012PercentHousing investment (XH)ARMFRMB.1%increaseinAtFRM ARM0 10 20 30 4010123456789PercentHousing investment (XH)Tight Loose 0 10 20 30 4010123456789PercentHousing investment (XH)Loose Tight Figure5: Generalequilibriumresponsesofhousinginvestmentintheversionwithhomeownersaccesstothebondmarket. PanelA:eectsofvaryingthedegree of persistence of the ination target shock; panel B: responses to a TFPshockunderloosepolicy(= 1.05)andtightpolicy(= 2.5).48Table3: BusinesscyclepropertiesUSdata ModelFRM ARMStdY 1.92 0.94 1.04Rel. stdY 1.00 1.00 1.00C 0.42 0.42 0.35XS6.94 9.48 8.20XK2.45 1.76 3.01N 0.92 0.24 0.30 0.58 0.85 0.81i 0.58 0.85 0.85iF0.35 0.77 N/AiFi 0.42 0.21 N/Aq 0.58 0.18 0.15pH1.57 1.13 0.97Corr(Ct, Yt) 0.79 0.88 0.94(XSt, Yt) 0.60 0.99 0.85(XKt, Yt) 0.73 0.92 0.83(Nt, Yt) 0.84 -0.67 -0.05(t, Yt) 0.14 0.23 0.41(it, Yt) 0.36 0.32 0.48(iFt, Yt) 0.01 0.09 N/A(iFtit, Yt) -0.49 -0.98 N/A(qt, Yt) 0.41 0.99 0.85(pHt, Yt) 0.55 0.99 0.85Note: AllU.S.momentsareforHP-lteredseries,post-Koreanwardata. Interestandinationratesareannualized. The10-yeargovernmentbondyieldisusedasaproxyforiFtduetoitslongertimeavailability;theinationrateoftheGDPdeatorisusedfort;the3-monthT-billyield is used for it; the ratio of the residential investment deator to the GDP deator is used forqt; the ratio of the average price of new homes sold (Census Bureau) and the GDP deator is usedforpHt(1975-2006). Themodel momentsareaveragesof momentsfor150runsof themodel;thearticialseriesofeachrunhavethesamelengthasthedataseriesandareHPltered.49SupplementalmaterialappendicesAppendixA:EquilibriumconditionsThis appendixlists the conditions characterizingthe equilibriumdenedinSection4.2.Throughout, thenotationemployedis that, for instance, uctdenotes therst derivativeof thefunctionuwithrespecttoc, evaluatedinperiodt. Alternatively, v2t, forinstance,denotes the rst derivative of the function vwith respect to the second argument, evaluatedinperiodt.CapitalownersoptimalityTherst-orderconditionswithrespectto,respectively,xKt,bt+1,andlt:1 = Et{uc,t+1uct[1 + (1 K)(rt+1K)]},1 = Et[uc,t+1uct(1 + it1 + t+1)],1 = Et{Ud,t+1uct+ U,t+1uctDt [ (t)] + UR,t+1uctDt(iFtRt)}.Inthelastequation,whichasdiscussedinthetextappliesonlyintheFRMcase, Udt pt1Udtis anormalizationtoensure stationarityinthe presence of positive steady-stateinationandUdt, Ut, andURtare the derivatives of the capital owners value functionwithrespecttodt,t,andRt,respectively. ThesederivativesaregivenbytheBenveniste-Scheinkman(BS)conditions:Udt= uctRt+ t1 + t+ 1 t1 + tEt{Ud,t+1 + lt [(t)] U,t+1 + lt(Rt iFt)UR,t+1},Ut= uct( dt1 + t)( dt1 + t)EtUd,t+1+( dt1 + t){lt[ (t)] +(1 t)(t)11t1+tdt+lt}EtU,t+1+( dt1 + t)lt(iFtRt)EtUR,t+1,URt= uct( dt1 + t)+ (1t1+tdt1t1+tdt+lt)EtUR,t+1.50Intheseexpressions, dt dt/pt1,lt lt/pt,lt lt(1t1+tdt+lt)2 (0, 1),andDt 1t1+tdt(1t1+tdt+lt)2 (0, 1).Notice that for a once-and-for-all mortgage loan (lt= l in period t and lt= 0 thereafter) andnooutstandingmortgagedebt(dt=0inperiodt), Dt=0andl,t+j=0, forj=1, 2, ....Inthis case, therst-order conditionfor ltandtheBSconditionfor Udtsimplify. Oncecombined, the resulting equation is just an innite-horizon extension of the mortgage-pricingequation(1)inthetwo-periodmortgageexampleof Section3. Thecomplicationsinthegeneral casearisebecausethemortgagepaymentmtenteringthebudgetconstraintofthecapital ownerpertainstopaymentsontheentireoutstandingmortgagedebt, notjustthenewloan. ThesimpliedformalsoariseswhenRt=it1(i.e., ARM) and t=(i.e.,theamortizationrateisconstantthroughoutthelifeofthemortgage,whichisthecasefor = 1). Thisisbecauseinthatcasetheinterestandamortizationratesofmtarethesameasthoseofthenew(marginal)mortgagepayment.Thecapitalownersconstraints:ct+ kt+1 +bt+1 +lt= [1 + (1 K)(rtK)] kt + (1 +it1)bt1 + t+mt+ t+pLt1 , mt= (Rt+ t)dt1 + t,dt+1=1 t1 + tdt+lt,t+1= (1 t) (t)+ t,Rt+1={(1 t)Rt+ tiFt, ifFRM,it, ifARM,wheret lt/dt+1andbt bt/pt1.HomeownersoptimalityTherst-orderconditionswithrespectto,respectively,nt,xHt,andbt+1:vct(1 N)wt= v2t,vct(1 )pHt= Et{Vh,t+1 + pHt[Vd,t+1 + Dt( t )V,t+1 + Dt(iMtRt)VR,t+1]},511 = Et[vc,t+1vct(1 + it + t1 + t+1)],where Vdt pt1VdtandVht,Vdt,Vt,andVRtarethederivativesofthehomeownersvaluefunction. Further,DtisthehomeownersanalogtoDtandiMt= iFtintheFRMcaseandiMt= itintheARMcase. RearrangingthesecondequationyieldsvctpHt(1 +Ht) = EtVh,t+1,wherethewedgeHtisgivenbyHt Et[1 + Vd,t+1vct+ Dt( t )V,t+1vct+ Dt(iMtRt)VR,t+1vct].For the same reasons as in the case of the mortgage-pricing equation of the capital owner, thewedgeismorecomplicatedthaninthecaseofthetwo-periodmortgage. Again,itbecomesastraightforwardinnite-horizonextensionofeitherequation(2)or(3)whenthehousinginvestment decisionis once-and-for-all andthere is nooutstandingmortgage debt. Thederivativesofthevaluefunctionwithrespecttodt, t, andRtaregivenbyBSconditions,whichtakesimilarformstothoseofthecapitalowner:Vdt= vctRt + t1 + t+ 1 t1 + tEt[Vd,t+1 + lt (t ) V,t+1 + lt(RtiMt)VR,t+1],Vt= vct( dt1 + t)( dt1 + t)EtVd,t+1+( dt1 + t)[lt( t ) +(1 t)1t1t1+tdt +lt]EtV,t+1+( dt1 + t)lt(iMtRt)EtVR,t+1,VRt= vct( dt1 + t)+ (1t1+tdt1t1+tdt +lt)EtVR,t+1.Inaddition,thereisaBSconditionforthederivativewithrespecttoht:Vht= vht + (1 H)EtVh,t+1.Due to the aggregate consistency conditions (1 )dt= dt, t= t, and Rt= Rt, it isnotnecessarytoincludethehomeownerslawsofmotionforthemortgagevariablesamongtheequationscharacterizingtheequilibrium. Theconstraintspertainingtothehomeowner52are:ct + pHtxHtlt +bt+1= (1 N)(wtnt) + (1 +it1 + t1)bt1 + tmt + t,wheret= [bt/(1 + t)]t1,and mt= (Rt + t)dt1 + t,lt= pHtxHt,xHt= ht+1(1 H)ht.ProductionTheproducersrst-orderconditions:rt= Atf1 ((1 )kt, nt) ,wt= Atf2 ((1 )kt, nt) .Output:Yt= Atf ((1 )kt, nt) .Therelativepriceofstructures(i.e.,thecurvatureoftheproductionpossibilitiesfrontier):qt= q(xSt).HomebuildingUsingtheequilibriumconditionXLt= 1,the production function and the rst-order conditions of homebuilders (for the Cobb-Douglasproductionfunction):xSt=1(xHt)11,pHt= qt(xSt)1 ,pLt= pHt(xSt)1.For a given xHt, the rst equation determines xSt, the second pHt, and the third pLt. Noticethatwhen = 0,xHt= xStandpHt= qt.MonetarypolicyandthegovernmentThemonetarypolicyrule:it= (i + t) + (tt) + y(log Ytlog Yt1y).53Thegovernmentbudgetconstraint:G + (1 )t= K(rtK)(1 )kt + N(wtnt) +.MarketclearingThe land and structures market clearing conditions have already been imposed in the home-buildingsector. Theremainingmarketclearingconditionsareforthebondmarket:(1 )bt+ bt= 0;andmortgagemarket:(1 )lt= lt.ItisstraightforwardtoverifythattheWalraslawholds(i.e.,thegoodsmarketclearsandnationalaccountshold):(1 )ct+ ct + (1 )xKt + qtxSt + G = Yt= rt(1 )kt + wtnt.StochasticprocessesTFP:log At+1= (1 A) log A + A log At + A,t+1, where A,t+1 iidN(0, A).Inationtarget:t+1= (1 ) + t + ,t+1, where ,t+1 iidN(0, ).AppendixB:ComputationTherecursivecompetitiveequilibrium(RCE) is computedusingalinear-quadratic(LQ)approximationmethodfordistortedeconomieswithexogenouslyheterogenousagents(seeHansenandPrescott, 1995, fordetails). Inanutshell, theBellmanequationofeachagenttype (equations (18) and (19)) is LQ approximated. Following the split-up of the economy inSection 4.2 into the capital owner and homeowner blocks, the maximization problem of eachblock is solved in isolation,given a guess for the decision rules of the other block. The RCEof the entire economy is a xed point in which the guesses coincide with the outcomes of eachrespectiveblocksproblem. ThecenteringpointoftheapproximationisthenonstochasticsteadystateandtheapproximationoftheBellmanequationsiscomputedusingnumericalderivatives; all variables in the approximation are either in percentage deviations or percent-agepointdeviations(forrates)fromthesteadystate. Beforecomputingtheequilibrium,the model is made stationary by expressing all nominal variables in real terms and replacingratiosofpricelevelswiththeinationrate,asisdoneinAppendixA.Because the laws of motionfor the mortgage variables are nonlinear, andcannot besubstitutedoutintotheper-periodutilityfunctionasrequiredbythestandardLQapprox-imationmethod, themethodismodiedalongthelinesof BenignoandWoodford(2006).This involves forming a Lagrangian, consisting of the per-period utility function and the laws54of motionforthemortgagevariables. TheLagrangianisthenusedasthereturnfunctionintheBellmanequationbeingapproximated. Thisadjustmentisnecessarytoensurethatsecond-ordercross-derivativesoftheutilityfunctionandtheconstraintsaretakenintoac-count in the LQ approximation. This modication, as applied to the homeowner, is describedindetailbyKydlandetal.(2012). Thespecicationforthecapitalownerisanalogous. Wethereforereferthereadertothatpaper.An alternative procedureimplemented, for instance, by Dynarewould be to log-linearize the models equilibrium conditions in Appendix A and use a version of the Blanchard-Kahnmethodtoarriveattheequilibriumdecisionrulesandpricingfunctions. Asiswellknown, the two procedures yield the same linear equilibrium decision rules and pricing func-tions, approximationstothesetoffunctionsW(z, S). WehaveaslightpreferencefortheLQ method as, in the future, it can be easily adopted to a specication of the model with re-cursive preferences, which price in long-term risk, and are thus a natural choice for studyingtheimplicationsoftheriskcharacteristicsoflong-termmortgageloans.In computing the partial equilibrium results, we treat XKt, iMt, and tin the homeownercase, andXHt, Bt+1, andNtinthecapital ownercase, inthesamewayastheexogenousstatevariablesinthevectorzt. Specically, thevariablesareassumedtofollowadiagonalVAR(1)process, withtheparametervaluesspeciedinthetext, andareincludedinthevector ztof exogenousstatevariablesintherespectiveBellmanequations. TheBellmanequationof eachblockis thenLQapproximated. Thehomeowner blockgives aggregatedecisionrulesforXHt,Bt+1,andNt,whilethecapitalownerblockgivesaggregatedecisionrulesandpricingfunctionsforXKt, iMt,andt. Thesearelinearfunctionsofthevariablesineachblocks(modied)vectorzandineachblocksvectorofendogenousstatevariables:[Kt, Dt, t, t]inthecapitalownerscaseand[Ht, Bt, Dt, t, t]inthehomeownerscase.AppendixC:DatacounterpartstovariablesThisappendixexplainstheconstructionofthedatausedtocalculatetheaggregateratiosemployedincalibratingthemodel. Adjustmentstoocial dataaremadetoensurethatthedatacorrespondconceptuallymorecloselytothevariablesinthemodel. Tostart, forreasonsdiscussedbyGommeandRupert(2007), thefollowingexpenditurecategoriesaretaken out of GDP: gross housing value added,compensation of general government employ-ees,and net exports. In addition,we also exclude expenditures on consumer durable goods,as our homecapital includes onlyhousing, andmultifamilystructures, whichsincethemid-1980s rely much less on mortgage nance than single-family structures (Kydlandetal.,2012). Withtheseadjustments, thedatacounterpartstotheexpenditurecomponentsofoutputinthemodelareconstructedfromBEAsNIPAtablesasfollows: consumption(C)= the sum of expenditures on nondurable goods and services less gross housing value added;capital investment (XK) =the sumof nonresidential structures, equipment &software,andthechangeinprivateinventories;housingstructures(XS)=residentialgrossxedpri-vateinvestmentlessmultifamilystructures; andgovernmentexpenditures(G)=thesumof government consumption expenditures and gross investment less compensation of generalgovernment employees. Our measure of output (Y= C+XK+XS+G) accounts, on average(1958-2006),for74%ofGDP.BEAs FixedAssets TablesandCensus Bureaus M3dataprovidestockcounterparts55tocapital andhousinginvestment: capital stock(K)=thesumof privatenonresidentialxedassetsandbusinessinventories; housingstock(H)=residential assetsless5+unitproperties.35FederalReservesFlowofFundsAccountsprovidedataonmortgagesandweequalizemortgagedebtinthemodel (D)withthestockofhomemortgagesfor1-4familyproperties. TheFlowof Fundsdata, however, includemortgagedebtissuedforpurchasesofexistinghomes,secondmortgages,andhomeequityloans. Incontrast,themodelspeaksonly to mortgage debt on new housing. The data thus provide an upper bound for Din themodel.AppendixD:EstimationofmortgagedebtservicingcostsAsdiscussedinthemaintext, akeymeasurementforcalibratingthemodel concernsthemortgage debt servicing costs of homeowners. Unfortunately, such information for the UnitedStates is not readilyavailable. Four dierent procedures arethereforeusedtoarriveatits estimate. Toasmaller or larger extent, the four procedures exploit the notionthatthehomeownersinthemodel correspondtothe3rdand4thquintilesof theU.S. wealthdistribution. Someoftheseestimatesarguablyoverestimatethedebtservicingcosts,whileotherunderestimateit. Nevertheless, all fourproceduresyieldestimatesintheballparkof18.5%ofpre-taxincome,thevalueusedtocalibratethemodel. ThisballparkissimilartotheestimatesfortheUnitedKingdomreportedintheliterature,notedintheIntroduction.The rst procedure, for FRM(1972-2006) andARM(1984-2006), combines dataonincomefromtheSurveyofConsumerFinances(SCF)andthemodelsexpressionfordebtservicing costs. Suppose that all mortgage debt is FRM. The models expression for steady-state debt-servicing costs, (R+)[D/(pwNp)], can then be used to compute the averagedebt-servicing costs of homeowners. The various elements of this expression are mapped intodatainthefollowingway: D/(pwN p)correspondstotheaverageratioof mortgagedebt (for 1-4unit structures) tothe combinedpersonal income (annual, pre-tax) of the3rdand4thquintiles, whichisequal to1.56; RcorrespondstotheaverageFRMannualinterestrateforaconventional30-yearmortgage,equalto9.31%;andcorrespondstotheaverageamortizationrateover thelifeof themortgage, equal to4.7%per annum. Thisyieldsdebt-servicingcostsof 22%. Thisestimateislikelyanupper