Do Mortgage Subsidies Help or Hurt Borrowers?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Do Mortgage Subsidies Help or Hurt Borrowers? David E. Rappoport 2016-081 Please cite this paper as: Rappoport, David E. (2016). “Do Mortgage Subsidies Help or Hurt Borrowers?,” Finance and Economics Discussion Series 2016-081. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2016.081. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Do Mortgage Subsidies Help or Hurt Borrowers?* David E. Rappoport† Federal Reserve Board October5,2016 Abstract Mortgage subsidies affect homeownership costs by reducing effective mortgage ratesandincreasinghouseprices. Ishowanalyticallytheroleofmortgagesubsidies indetermininghousepricechanges,economicincidence,andefficiencycostsusing a theoretical framework for applied welfare analysis. I derive simple expressions for these effects, as functions of reduced-form sufficient statistics, which I use to measuretheeffectsfromeliminatingmortgagedeductions. Mymainresultscharacterizethedistributionalimpactofmortgagesubsidiesamongbuyersandownersand how house price responses attenuate efficiency losses. My results provide broader methodologicalinsightsintothewelfareanalysisofcredit policies. JELclassification : H22,R21,R28. Keywords: Public economics, mortgage subsidies, incidence, optimal taxation, houseprices,mortgageinterestdeductions,MID. *I specially thank my advisor John Geanakoplos for his advise and constant encouragement. I also benefited from helpful discussions with Mikhail Golosov, Andrew Metrick, Michael Peters, Guillermo Ordoñez, and Larry Samuelson; and comments from David Arseneau, Neil Bhutta, Amelia Biehl (discussant), Erica Blom, Don Brown, JuanEberhard,RebeccaEdwards,DrewFudenberg,ManolisGalenianos,ErikMaskin,AntonyMurphy(discussant), Daniel Ringo, Noam Tanner, Aleh Tsyvinski, Alexandros Vardoulakis and seminar participants at North American Summer Meeting of the Econometric Society, National Tax Association, Federal Reserve Bank of Dallas Applied MacroConference,Berkeley-Haas,Brown,Yale(Macro,TheoryandFinance),FederalReserveBoard,Universityof IllinoisatUrbana-Champaign,SUNYBuffalo,BancodeMexico,CentralBankofChile,MOOD,EconCon,Jerusalem School in Economic Theory, U Chile-CEA, PUC Chile, and Universidad Adolfo Ibañez. A previous version of this paper circulated under the title “Can Borrowers Benefit from Taxes on Borrowing?” (June, 2011). The opinions expresseddonotnecessarilyreflectthoseoftheFederalReserveBoardoritssta ff. †Email: david.e.rappoport@frb.gov. 1

1 Introduction Subsidiesthatreducethecostofdebtforhouseholdsorfirmsarecommonlyusedwiththeobjective to promote the capital expenditures financed with this debt. For example, mortgage subsidies are used to subsidize housing, reduced rates on student loans are used to subsidize higher education, and tax breaks on corporate debt are used to subsidize corporate investment. To evaluate the effect of these policies, it is necessary to consider the effect they will have both on the cost of debt and on the price of debt-financed capital. Using a su fficient statistics approach for welfare evaluation requires one to identify the necessary economic parameters for welfare evaluation. But the identification of the interest rate demand elasticity for capital in these settings is complicated by the interplay between debt and capital markets. In other words, any change in interest rates is expected to influence the demand for capital and thus its price, complicating the identification of onlyachangeininterestratesonthedemandforcapital. Thispaperevaluatesthewelfareeffectof mortgagesubsidiesaccountingfortheseconsiderations. Mortgage subsidies are provided in many countries and are economically large. For example, American mortgage borrowers can deduct mortgage interest payments and these deductions were the second largest federal tax expenditure in 2012, reaching $100 billion (Office of Management andBudget2012).1 In the first half of the paper, I investigate theoretically the e ffect of mortgage subsidies when these subsidies can affect both the mortgage interest rate and house prices. I characterize the welfare effects by simple formulas as functions of reduced-form sufficient statistics that can be empirically identified. In the second half of the paper, I use the simple formulas to estimate for 269U.S.metropolitanareastheeffectofeliminatingmortgageinterestdeductionsonhouseprices, households’welfare,andefficiencygains. To characterize theoretically the effect of mortgage subsidies, I extend the classic framework forappliedwelfareanalysistoanintertemporalsetting,wherehouseholdspurchasedurablehouses andfinancethemwithmortgagedebt. 2 Importantly,financialmarketsareimperfectsohouseholds facedifferentborrowingandsavingrates. Thisframeworkissimpleenoughtoallowforanintuitive analysisofwelfareasinthetextbookpublic-financemodel,butitisalsorichenoughtoincorporate the relationship between mortgage and housing markets. I derive welfare effects, as functions of reduced-form sufficient statistics, considering the effect of these subsidies on both mortgage rates andhouseprices. Thischaracterizationofwelfareeffectsyieldsinterestingresults. 1Inaddition,Americanborrowersreceivemortgageratesubsidiesthroughtheintermediationofmortgagecreditby government-sponsoredenterprises,likeFannieMaeandFreddieMac,whichguaranteemortgagessoldinsecondary markets. It has been estimated that this guarantee reduces effective mortgage rates by about 25 basis points (CBO (2001),Ambroseetal. (2004),Sherlund(2008),andDeFuscoandPaciorek(2014)). 2For a textbook version of this model, see Mas Colell, Whinston, and Green (2005). Kotlikoff and Summers (1987)andAuerbach(1985)surveytheincidenceandefficiencycostsresultsinthisliterature,respecteively. 2

A first result regards the characterization of the economic incidence of mortgage subsidies. I showthatinthepresenceoffinancialfrictions,whereborrowers’financingdecisionsareuniquely pinned down and can be summarized by the loan-to-value (LTV), the LTV is an additional key statistic to describe the incidence of mortgage subsidies on households, in addition to the demand and supply elasticities as in the canonical case. Intuitively, the LTV reflects that the benefit from lower mortgage interest payments only accrues to the fraction of housing expenditure financed withmortgagedebt. A surprising result I obtain is that mortgage subsidies can hurt homebuyers. This is surprising as it challenges the intuition from the classic analyses of taxes and subsidies, where subsidies always (weakly) benefit their recipients. As in the classic case, mortgage subsidies benefit buyers by reducing mortgage interest payments and create an offsetting effect by increasing the demand and price of housing. However, as buyers increase their LTV in response to these subsidies, they amplifythehousepriceincrease,whichcanmaketheincreaseinhousingexpenditurestrictlylarger thanthesavingsfromlowermortgageinterestpayments. Theconditionthatdeterminesthesignof the welfare effect on buyers is obtained from the sufficient statistics formula that I derive. On the one hand, the benefit of mortgage subsidies is accrued in the fraction of the house financed with mortgage debt, i.e., the LTV. On the other hand, the negative effect of the increase in house prices accrues to the total house expenditure. This adverse effect is proportional to the semielasticity of housepricestomortgagerates,whichismoreelasticwhenthesupplyofhousingismoreinelastic or when the demand for housing is more elastic.3 Thus, mortgage subsidies hurt borrowers when their initial LTV is low, resulting in small benefits from lower mortgage payments, or when the supply (demand) of housing is inelastic (elastic), resulting in large increases in house prices that hurtborrowers. Another result refers to the measurement of the efficiency costs generated by mortgage subsidies. Interestingly, only the distortions introduced in the mortgage market generate inefficiencies, but the size of the distortion does depend on the response of house prices. As in the classic case, the subsidy creates a deadweight loss in the mortgage market, as it distorts an optimal allocation so the welfare cost of collecting the required tax revenue to finance the subsidy is larger than its benefit perceived by households. In contrast, in the housing market the subsidy generates only a wealth redistribution through the house price, without contributing to additional efficiency losses. As house demand and house prices increase, the loss to buyers is offset one-for-one with the gain to sellers (homeowners and house producers). Moreover, the magnitude of the efficiency loss dependsontheelasticityofhousepricestomortgagerates,ashousepricesdeterminethedemandfor mortgagedebtboththroughtheeffectonthedemandforhousingandtheeffectonhousingexpenditure: higherhousepricesreducehousingdemandbutincreasehousingexpenditure. Considering 3ThroughoutthispaperIusetheconventionthatmoreelasticreferstoalargerelasticityinabsolutevalue. 3

thecompensatedresponseofmortgagedemand,Ishowthattheeffectofhousepricesattenuatesthe efficiencylossfrommortgagesubsidies,ashigherhousepricesreducethe(compensated)mortgage demandandthusthedistortiongeneratedbythesubsidy. In addition, the model allows me, under very general assumptions, to establish theoretically that the mortgage rate semielasticity of house demand equals the ratio between the price demand elasticityandtheindividualusercostofhomeownership(Lemma1). Thisrelationshipenablesme toovercometheinherentaforementionedchallengefortheempiricalidentificationofthemortgage ratedemandelasticity. Usingthisrelationshiptogetherwithexistingestimatesforthepricedemand elasticityofhousingandtheusercost,Iestimateanaveragemortgageratesemielasticityof 15.3 − (Section3). In the second part of the paper, I use the welfare formulas to provide new estimates of the effect of eliminating MID in the U.S. I consider LTV ratios and mortgage rates from a sample of 17.5 million mortgage originated in 2010-2015 together with my estimates for the mortgage rate house demand semielasticity and available estimates for the other key parameteres. I assume that housing markets are geographically segmented and apply the welfare formulas derived in this paper to produce estimates for 269 metropolitan areas of the effect of eliminating MID on house prices,householdswelfare,andefficiencygains. First, I estimate the effect of eliminating MID on house prices for the 269 metropolitan areas considered. Using the sufficient statistics formulas, I estimate a (house-value-weighted) average mortgage-rate semielasticity of house prices of 6.9, ranging from 9.6 in Miami, Florida, where − − house supply is the most inelastic, to 1.2 in Pine Bluff, Arizona, where the supply is the most − elastic. My estimates are broadly in line with other estimates in the literature that estimate these elasticities directly from the data but my estimates offer a higher level of regional granularity. These estimates imply a (house-value-weighted) average decline of house prices of 6.9% from eliminatingMID. Second,Iprovideestimatesfortheincidence,ordistributionalimpact,onhouseholds’welfare of eliminating MID in different metropolitan areas, depending on households’ LTV ratios and the estimated house price decline. As highlighted by the theory, the elimination of MID has a different effect on first-time homebuyers and homeowners. Both sets of households are hurt from the elimination of MID, which increases their effective mortgage interest rate. However, homebuyers benefit from the drop in house prices, whereas homeowners are additionally hurt by it. I estimate thatonaveragehomeownerswelfaredropsby11.5%ofthevalueofthehouse,whereashomebuyers welfare drops only by 8.5% of the value of the house. For an average house value of $320,000 these effects correspond to a present value loss of $36,800 for homeowners and a loss of $27,200 for homebuyers. In other words, current MID are more helpful for existing homeowners relative tofirst-timebuyers. Giventhedi fferentialresponseofhousepricesanddifferencesinhouseholds’ 4

characteristicstheaverageincidenceisalsoestimatedtovaryacrossregions. Iestimatethatonaveragehomebuyers’welfaredeclineby12.6%ofthehousevalueinAlexandria,Louisiana,whereas homebuyersinSanFrancisco,wherehousepricesareestimatestodropmore,areestimatedtolose only 3.8% of the value of the house (representing losses of $40,320 and $12,160 for a $320,000 house,respectively). Finally, I estimate the efficiency gains from the elimination of MID. As the theoretical framework highlights, these gains are attenuated by the effect of mortgage rates on house prices. I estimate that for my sample of 17.5 million households efficiency gains total $2.6 billion, or an averageefficiencygainof5basispointsofthehousevalue. Extrapolatingtothe49millionhouseholdsthatfinancetheirhomeswithmortgagedebttotale fficiencygainswouldincreasetoamodest $7.3 billion.4 My estimates imply that without the attenuating effect of house prices these losses wouldbe60%higher. Thepapermakesthreemaincontributions. First,mypapercontributestothepubliceconomics literature that utilizes the sufficient statistics approach to analyse imperfect financial markets. The sufficient statistics approach traces its origins to the work by Harberger (1964) and combines the advantage of the cleaner identification of reduced-form parameters with the ability of structural modelstodescribewelfareeffects(Chetty,2009). UsingthisapproachMatvos(2013)studieshow covenants create benefits for corporate borrowers by completing debt contracts. Dávila (2015) analyzes optimal bankruptcy exceptions for unsecured debt. Other applications of this approach includetheanalysisofmonetarypolicy(Alvarez,LeBihan,andLippi,2016;Auclert,2016),welfare effects of trade liberalization (Arkolakis, Costinot, and Rodríguez-Clare, 2012), and welfare analysiswithbehaviorallybiasedconsumers(Chetty,Looney,andKroft,2009;AllcotandTaubinsky, 2015). Although I focus on the case of mortgages and housing, the techniques and insights developedherearesuitabletoanalyzedebtsubsidiesinothercontextswheredebtisusedtofinance capitalexpenditures,likecorporateinvestmentinfixedassetsorcollegestudents’investmentinhumancapital. Myresultsopennewavenuesforappliedwelfareanalysisinthesesettings. There is a long tradition in public economics analyzing housing policy, and mortgage policy in particular (Laidler, 1969; Aaron, 1972; Rosen, 1979; Poterba, 1992; and Poterba and Sinai 2008). This literature considers how mortgage subsidies and other policies affect the house rental rate and evaluate the welfare effect of these policies using a rich description of the U.S. tax code. RelativetothisliteratureIfocusonthetaxprovisionsthataffectthecostofmortgagedebt,Irelax the assumption that house prices are fixed, and I consider individual heterogeneity in mortgage contract characteristics. My analysis shows that not considering the offsetting effect of house prices for the distortions introduced by mortgage subsidies will overstate the efficiency cost of 4HousingunitsinformationfromU.S.CensusBureau;2010-2014AmericanCommunitySurvey5-YearEstimates, TableB25096;generatedusingAmericanFactFinder;http://factfinder2.census.gov;(22September2016). 5

these subsidies by almost 60%. That is, the efficiency costs of MID are about 40% smaller than previouslyestimated. A strand of the literature has looked at the effect of MID on homeownership, or the extensive margin of housing demand. For example, Bourassa and Yin (2008) conclude that MID reduce homeownership rates of young households due to the effect on house prices. In addition, Glaeser andShapiro(2003),analyzingtimeandcross-statevariationinMID,findthatthee ffectofMIDin homeownership is small. Similarly, Hilber and Turner (2014) present evidence based on withinand across-state variation in MID over time that shows this subsidy is ineffective in promoting homeownership. HilberandTurner(2014)arguethatthecapitalizationintohousepricesoffsetsthe reductiononhomeowners’rentalratesbroughtaboutbyMID.Furthermore,SommerandSullivan (2016) study the impact of MID on a quantitative macroeconomic model with endogenous tenure choice,rents,andhouseprices. CounterfactualanalysisintheSommer-Sullivanmodelshowsthat eliminatingMIDwillincreasehomeownershiprates,insteadofreducingthem. Thesestudiesshare with my analysis the emphasis on the capitalization into house prices of mortgage subsidies, but unlikethesestudiesIfocusontheintensivemarginofhousingdemand. Second,mypapercontributestotheliteraturethatstudiestheeffectofmortgagecreditonhouse prices. UsingmysufficientstatisticsformulasIestimateanaveragemortgageratesemielasticityof housepricesof 6.9acrossthe269metropolitanareasinmysample,whichisbroadlyinlinewith − direct estimates from empirical studies (Glaeser, Gottlieb, and Gyourko, 2012; Adelino, Schoar, and Severino, 2014; Kung, 2015). The simple sufficient statistics formulas can also be used to estimate the effect of the quantity of credit on house prices: the elasticity of house prices to the volume of mortgage loans. I obtain an average estimate of 0.3 in line with the direct estimates of FavaraandImbs(2015)andDiMaggioandKermani(2015). Myestimatessupporttheconclusion ofGlaeser,Gottlieb,andGyourko(2012)thatthedeclineininterestratesintheearly2000scannot explain the increase in house prices in this period. Like these authors, I derive a formula for the semielasticity of house prices with respect to mortgage rates, which incorporates endogenous house supply. But instead of focussing on the extensive margin of house demand, I focus on the intensive margin. As in Glaeser, Gottlieb, and Gyourko (2012), when house supply is totally inelastic,Irecoverasemielasticityofhousepriceswithrespecttorealmortgageratescloseto 20, − as in Himmelberg, Mayer, and Sinai (2005) and as prescribed by the static asset market approach tohousevaluation(Poterba,1984). Finally, my paper relates to other studies of mortgage policies and the effect of MID using a structural approach. Hurst et al. (2015) document that interest rates of mortgages intermediated bygovernment-sponsoredenterprisesexhibitnoregionalvariationdespitecreditriskbeingheterogeneous across regions. The authors use a structural spacial model to quantify the redistributional impact of this pricing of mortgage credit across regions. Other studies have used quantitative 6

macroeconomic models to evaluates the effect of MID, other mortgage subsidies, and tax provisions that affect housing demand (see, among others, Gervais, 2002; Jeske, Krueger, and Mitman, 2013; Sommer and Sullivan, 2016). These studies find that MID, or other mortgage subsidies, reducewelfareandincreasehousepricesinmodelsinwhichthesepricesareendogenous. The rest of the paper is organized as follows. Section 2 presents the theoretical framework and the theoretical results. Section 3 describes the data used to quantify the welfare effect of eliminatingMID.Section4presentsmyestimatesoftheeffectofeliminatingMIDacrossdifferent metropolitan areas on house prices, households’ welfare, and efficiency gains. Section 5 provides someconcludingremarks. AndtheAppendicescontainadditionalmaterial. 2 Theoretical Framework In this section I present a simple model for applied welfare analysis to analyze the effect of mortgage subsidies. Importantly, the cost of mortgage debt affects households’ housing demand and financingdecisions—theLTVontheirhousepurchases. Moreover,housedemanda ffectstheprice ofhousing,whichinturninfluenceshousingandmortgagedemand. Ishowanalyticallytheroleof mortgage subsidies in determining house price changes, economic incidence, and efficiency costs. I derive simple formulas for these effects, as functions of reduced-form sufficient statistics, which generalizetheclassicformulasforincidenceandefficiencycoststothiscase. 2.1 Setup I consider an economy with two periods, t = 0,1. The economy is populated by households (homebuyersandhomeowners),houseproducers,lenders,andagovernment. Therearetwogoods, durable housing and perishable consumption, which is the numeraire. In addition, household can borrowfromlendersusingmortgages,whichmaybesubsidizedbythegovernment. Homebuyers. There is a continuum of mass 1 of identical homebuyers who derive utility from housing purchased in period 0, x, and period 1 consumption, c. I abstract away from non-housing consumptionthatcouldtakeplaceinperiod0forsimplicityandconsiderperiod1consumptionto capture the intertemporal nature of mortgage borrowing. Buyers’ preferences are represented by u(x,c), which is increasing and concave in each argument. This preference specification is very general as it does not impose separability between the utility derived from housing and period 1 consumption. Homebuyers receive income y in period 0 in units of the numeraire. They have no initial housing units, but can purchase them in period 0 at price p. Homebuyers can finance their house purchases with their income or mortgage debt, denoted by m. Each unit of mortgage borrowing 7

requires the homebuyer to pay a unit of the numeraire in period 1 in exchange for q units of the numeraire in period 0. A mortgage subsidy t adds to the amount received by borrowers in period 0 so after the subsidy borrowers receive q + t units in period 0, per unit promised.5 Using this notation the loan-to-value (LTV) ratio equals (q+t)m/px. Homebuyers also pay lump-sum taxes T inperiod0. In the model, house prices in period 1 are exogenous, as the model abstract away from the equilibrium of the housing market in that period. However, in order to account for the main determinants of the user cost of housing—expected capital gains and depreciation—I assume that the house price in period 1 is proportional to the endogenous house price in period 0. In particular, I assume that this price reflects (expected) house price appreciation π and the depreciation of the housing stock δ, thus the house price in period 1 equals (1 + π δ)p. Under these assumptions − the budget constraint in period 0 and 1 are given, respectively, by px + T y + (q + t)m and ≤ c (1+π δ)px m. Finally,Iassumethatneitherconsumptionnormortgagescanbenegative.6 ≤ − − Ingeneral,homebuyerswillchoosedifferentcombinationsofhousing,consumption,andmortgage debt depending on the price of housing and mortgages, and households’ preferences and income. Toillustratetheeffectofmortgagesubsidies,Ifocusonbuyersataninteriorsolutionwhere optimalityimplythat u x = [r(t)+δ π]p , (1) u − c wherer(t) = 1/(q+t) 1correspondstotheeffectivemortgageinterestrateafterthesubsidy. The − term in square brackets in equation (1) corresponds to the user cost of homeownership, which is increasingintheeffectivemortgagerateanddepreciation,anddecreasinginexpectedcapitalgains (cf. Poterba, 1984, or Himmelberg et al., 2005). Note that the mortgage subsidy fully distort the costoffundsintheusercostexpression,i.e.,theeffectivemarginalusercostisthesameregardless of what fraction of the house is financed with mortgage debt—the LTV—and what fraction is financed with a downpayment. This result holds under much general conditions than the model’s assumptions and will have important implications for the effect of mortgage subsidies on house pricesasIdiscussbelow. Lenders. Thereisacontinuumofmass1ofidenticallenders,whomaximizeprofits. Lendershave deep pockets and an opportunity cost of funds given by r . For each loan, lenders give borrowers f q = 1/(1 + r) funds in period 0 and are promised 1 unit in period 1. Lenders operate a constant return to scale technology, which reflects origination and servicing costs ρ per loan. Thus, lenders 5Usingthemortgagepriceq,insteadofthemortgageinterestrate,simplifiestheanalysisofe fficienycostsbelow. Butthisisequivalent,up-toafirstorderapproximation,toworkingwithasubsidyonthemortgageinterestrate:where mortgagesprovideaunitofthenumeraireinperiod0inexchangeforapaymentof1+r=1/qinperiod1. 6Non-negativeconsumptionimposesanaturalborrowinglimit,andthenon-negativityofmortgagespreventbuyersfromsavingatthemortgagerate,whichiswithoutlossofgeneralityasIfocusonborrowers. ForthisreasonIalso abstractawayfromothersavingalternativeshomebuyersmayhave. 8

maximization problem corresponds to max(r r ρ)l. Lenders optimal behavior will pin down l f − − thelendingmortgagerater = r +ρ. Thatis,mortgagesupplyiseffectivelytotallyelasticatr +ρ. f f This is a consequence of the simplifying assumptions on this part of the model: constant funding costandconstantreturntoscaletechnology. Homeowners and house producers. To highlight the distributional effects through house prices onexistinghomeownersandhouseproducers,Iconsidertheseagentsseparately. Thereisacontinuum of mass 1 of identical homeowners and house producers. Homeowners have a fixed endowmentofhouseshandhavelinearpreferencesfortheproceedsofhousesales, ph. Homeownersderiveutilityofhousesales,sotheywillalwayssellalltheirhouseendowment.7 Inaddition,houses areproducedbypricetakingfirms,whoproduce zhousingunitsatacostκ(z)thatisincreasingand quasi-convex. Firms optimal behavior imply p = κ (z), which implicitly define producers’ supply. 0 Therefore,totalhousesupplyisdenotedbyS = h+z. Government. Thegovernmentcollectslump-sumtaxesfromconsumersinperiod0,T,inorderto financemortgagesubsidies. Givenagovernmentpolicy, t,T ,thegovernmentneedstobalanceits budgetinperiod0,i.e.,tm = T. Itisassumedthatthegovernmentcollectsnon-distortionarytaxes (cid:8) (cid:9) in terms of period 0 income to simplify the efficiency analysis in the presence of income effects (seeAuerbach,1985). Inthisenvironmentacompetitiveequilibriumisdefinedasfollows. Definition 1 (Competitive Equilibrium) A competitive equilibrium consists of a house price, p, a mortgage rate, r, allocations for homebuyers, x, c, m , loan supply, l, house production, z, homeowners’ sales, h, and government policy, t,T , such that: homebuyers, homeowners, house (cid:8) (cid:9) producers,andlendersbehaveoptimallytakingpricesasgiven,thehousingandmortgagemarkets (cid:8) (cid:9) clear,andthegovernmentrunsabalancedbudget. InanefforttomaintainthesimplicityofthemodelIhaveabstractedawayfromseveralfeatures thatarerelevantinpractice. Thesefeaturesarenotrequiredtodescribetheresultsbutinfluencethe welfare estimates presented in section 4. First, the model abstract from uncertainty about income shocks and future house prices. The former will introduce a precautionary motive reducing the demandformortgagedebt. Thelatterincreasestheusercostofhomeownership. Infact,following Poterba (1992) and others, for the measurement exercise of section 4 I consider that the user cost comprisesatermthatcapturesariskpremiumforhousinginvestment.8 Second, the model abstract from other forms of borrowing and savings. This simplification is not instrumental for the results in this setting without uncertainty, as long as saving instruments 7Inkeepingwiththesimplicityofthemodel,inthissectionhomeownersareassumedtoseeltheirhousesinelastically. Nonetheless,intheanalysisofsection4homeownerswillbeidentifiedwithmortgagerefinancing,sotheywill beaffectedbothbythereductionofeffectivemortgageratesandthechangeinhouseprices. 8WhenIcalibratetheusercosttothedataIwilltakeintoaccountthepresenceofthisandadditionaltermsofthe usercostofhomeownershipthatIhaveabstractedawayinthemodel. 9

offer an interest rate lower than the mortgage rate, and other forms of borrowing have higher interest rates than mortgages. These conditions seem plausible: risk-adjusted saving rates are lower than borrowing rates, as reflected by positive bank interest rate spreads in practice and as required by no-arbitrage conditions in theory. On the other hand, mortgage (and other securitized borrowing)ratesare lowerthanratesonunsecured forms ofcredit, ascollateralenhanceslenders’ recoveryrates. Finally, I abstract away from homebuyers’ income in period 1. Future income affects the demandforhousing,asitaffectshomebuyers’lifetimeincome,butitdoesnotchangetheoptimality conditionforaninteriorequilibrium(equation(1)). Therefore,theanalysiswillremainunchanged when future income is considered, unless the homebuyer is constrained in the amount she can borrow using mortgage debt. This will be the case when there are minimum downpayment requirements,orequivalently,maximumLTVlimits. ThecasewithLTVlimitsisdiscussedbelow. 2.2 The Incidence of Mortgage Subsidies How are the costs and benefits of a mortgage subsidy shared between homebuyers, homeowners, house producers, and lenders in competitive equilibrium, when these subsidies house prices? To answer this question, in this section I derive formulas for the incidence of mortgage subsidies on theseagentsthatparallelthederivationsofKotlikoff andSummers(1987). Let D be the aggregate demand for houses, which from equation (1) depends on the house price, p, and after-subsidy mortgage rate, r(t). In addition, the uncompensated individual and aggregate demand functions will depend on households’ income, y. On the other hand, the total supplyofhouses,S,isonlyafunctionofhouseprices,ashomeownerswillalwaysselltheirhouse endowmentandhouseproducerswilladjusttheirproductionplansdependingonthelevelofhouse prices. Then,housemarketclearingrequiresthat D(p,r(t),y) = S(p) . (2) To describe the behavior of house prices and the incidence of mortgage subsidies it is useful to introduce the following notation. Let ε = (∂D/∂p) p/D and ε = (dS/dp) p/S denote D,p S,p the price elasticity of housing demand and supply, respectively, let ζ = (∂D/∂r)/D denote the D,r mortgage-rate semielasticity of house demand, and let ζ = (∂p/∂r)/p denote the mortgage-rate p,r semielasticity of house prices. In addition, for a given mortgage subsidy, t, let p(t) denote the housepricethatobtainsinequilibrium. Thefollowingresultfollows. Proposition1(IncidenceofMortgageSubsidies) Theincidenceofincreasingthemortgagesubsidy,t,equalszeroforlenders, (1+r(t))2zpζ forhouseproducers, (1+r(t))2hpζ forhomep,r p,r − − 10

owners,and u px(1+r(t))2 ζ [r(t)+δ π]+LTV (3) c p,r − (cid:16) (cid:17) forhomebuyers,wherethethemortgage-ratesemielasticityofhousepricesisgivenby ζ D,r ζ = 0 . (4) p,r ε ε ≤ S,p D,p − The formal proof is relegated to the appendix. Intuitively, the first order e ffect of the subsidy is brought about by price changes, which can are brought about by the adjustment in the demand for mortgage debt and housing depicted in Figure 1. With a totally elastic supply of mortgage debt the interest rate charged by lenders, r, remains fixed; thus, buyers see their borrowing cost drop from r to r t, as depicted by the arrow in panel (a). This is the most favorable outcome in − the mortgage market for buyers/borrowers. The mortgage subsidy, then, lowers the user cost for housingservices(equation(1))increasingthedemandforhousing. Theincreaseinthedemandfor housing, depicted by the first arrow in panel (b), depends on the mortgage rate semielasticity of house demand and the change in the mortgage rate. The second arrow in panel (b) shows how the equilibriuminthehousingmarketisrestoredviaanincreaseinhousepriceswiththecorresponding movements along the demand and supply for housing, which depend on the corresponding price elasticities. Higher housing consumption at higher house prices is financed with higher mortgage debt,sothedemandformortgagedebtincreases(panel(a)). As described above the upshot of the mortgage subsidy is a reduction—one-for-one in this case—of the effective mortgage rate and an increase in house prices. These two price changes have opposite effects on buyers’ welfare as shown in equation (3). This equation presents the two effects normalized by the house value and the marginal value of income (the term in front of the brackets).9 On the one hand, lower mortgage rates benefit home buyers by lowering their mortgageinterestpayments(orequivalentlyincreasingtheirmortgageborrowingforagivenfuture repayment). This effect is captured by the second term inside the brackets in equation (3) and it is proportional to the LTV on the house purchase, as the benefit from lower mortgage rates only accruedtothefractionofthehousefinancedwithmortgagedebt. Ontheotherhand,higherhouse prices hurt buyers as it increases the house rental rate, so households give up a higher fraction of their lifetime income for house consumption. This effect is captured by the first term inside the bracketsinequation(3),ζ [r(t)+δ π] 0. p,r − ≤ As the subsidy increase house prices, house producers and homeowners benefit proportionally to the value of the houses they sell, depending on the response of house prices given by the house pricesemielasticitytomortgagerates ζ . p,r 9The term (1+r(t))2 appears due to the assumption that mortgage subsidies increase the loaned amount, but it disappearsifthesubsidyisapplieddirectlytothemortgagerate. 11

Note that the incidence on lenders is zero because I assumed that lenders operate a constantreturn-to-scale technology and have constant opportunity cost of funds. Allowing banks’ operational or funding costs to increase as the supply of mortgage debt increases will attenuate the reduction of the effective mortgage rate from mortgage subsidies. Intuitively, banks origination and servicing costs may increase as the volume of mortgage lending increases, or alternatively, as banks increase their demand for funds to originate more mortgages they need to offer a higher compensationstotheirlenders,e.g.,depositors. Allowingthesegeneralequilibriumeffects,then,is expectedtoattenuatetheincidenceonhouseholdsandhaveanon-negativeincidenceonlenders.10 Proposition 1 is related to two previous results in the incidence literature. First, the result is related to the incidence of changes in interest rates on intertemporal consumption. A reduction in the interest rate makes current consumption cheaper incentivizing agents to increase current consumptionandincrease(decrease)borrowing(savings). Ontheotherhand,adeclineininterestrates generate a positive (negative) income effect for borrowers (savers). The total effect on intertemporal consumption and borrowing/savings decisions depends on both of these effects. The result in Proposition 1 can also be described in terms of substitution and income effects. A reduction in the mortgage rate, which is the relevant intertemporal price of consumption for households, reducestheusercostsofhousepurchasesandincreasesthecostoffutureconsumption. Households borrow more in order to substitute future consumption for additional housing. But the additional house demand pushes house prices up, generating a negative income effect for household, which is proportional to the entire house purchase. Lower mortgage rates, on the other hand, generate a positive income effect for borrowers proportional to the LTV of the house purchase. These two incomeeffectsdeterminetheincidenceofthesubsidy. Second, the result of Proposition 1 is related to the incidence of non-linear taxes. Reiss and White (2006) show that the incidence of nonlinear taxes equals the traditional expression for the compensatedvariationplusthechangeinthepremiumpaidoninframarginalunits. Whenahouse ispurchasedwithapositiveLTV,theremainderfraction,1 LTV,isfinancedwithadownpayment. − The user cost on the marginal units financed with debt depends on the after-subsidy mortgage rate; in contrast, the user cost for the inframarginal units financed with a down payment depends on the opportunity cost of funds used for the downpayment that I assume to be independent of the subsidy.11 A mortgage subsidy affects the house rental rate through both its effect on house prices and the user cost. On the one hand, the effect of house prices on the rental rate is given by ζ [r(t) + δ π], the first term in equation ( 3). On the other hand, the reduction of the effective p,r − 10Theincidenceonlenderswillremainzeroiftheconstant-return-to-scaletechnologyassumptionismaintained, butitwillbebecomepositiveifthetechnologyisassumedtohavediminishingreturnstoscale. 11The model abstract away from saving alternatives for households in period 0. But when these alternatives are consideredandthehouseholdatthemarginissubstitutingbetweensavingsandhouseexpenditure. Theinterestrate onsavingsdeterminetheopportunitycostoffunds. 12

mortgageratereducestheusercostone-for-oneforthemarginalunitsfinancedwithmortgagedebt and does not affect the user cost for the inframarginal units financed with a downpayment. Thus, the change in the user cost plus the change in the premium paid on the inframarginal units is just the change in the user cost for the units financed with mortgage debt. Since the change was onefor-onethechangeintheusercostequalstheLTVratio,whichcorrespondstothefractionofhouse expenditurefinancedwithmortgagedebt,thesecondterminequation( 3). Equation(4)isinterestingonitsownasitprovidesareducedformexpression,intermsofkey economic parameters, for the effect of mortgage rates on house prices, specifically, the mortgage rate semielasticy of house prices ζ . But as discussed above the identification of the mortgage p,r ratehousedemandelasticityζ iscomplicatedbytheinterplaybetweenmortgageratesandhouse D,r prices. The following Lemma establishes a relationship between the mortgage rate house demand semielasticity,thepricehousedemandelasticityandtheusercosts. Lemma 1 (Mortgage Rate House Demand Semielasticity) In an interior solution to the householdproblem ε D,p ζ = . (5) D,r r(t)+δ π − The proof consists of a simple application of the chain rule. In fact, let R = [r(t) + δ π]p − denotethehouserentalrate. Then ζ = 1/x(∂x/∂p)(∂p/∂R)(∂R/∂r) = ε /[r(t)+δ π]QED. D,r D,p − Lemma 1 establish a relationship between the house price demand elasticity of housing and themortgageratedemandsemielasticityofhousing: housingismoresensitivetoaonepercentage point reduction of the mortgage rate than a one percent reduction in house prices, as a one percentagepointreductioninmortgagerateshasagreatereffectonthehousingrentalrate. Moreover, Lemma 1 allows me, under very general conditions, to obtain an estimate of the mortgage rate house demand semielasticity based on the price demand elasticity and the user cost, which can be empirically identified. In this way, Lemma 1 allows me to overcome the inherent challenge for the empirical identificiation of the mortgage rate demand elasticity, as mortgage rates a ffect the demandandthepriceforhousing. Substitutingequation(5)inequation(4)Iget 1dp 1 ε D,p ζ = = . (6) p,r p dr r(t)+δ πε ε S,p D,p − − The ratio of price elasticities in the right-hand side of equation (6) corresponds to the effect of house prices from introducing a one percent house price subsidy.12 In addition, the user cost r(t)+δ π < 1 so its reciprocal is greater than 1. Thus, the reciprocal of the user cost equals the − housepriceresponseamplificationfrommortgagesubsidies,relativetohousepricesubsidies,due 12Theresultfollowsfromimplicitdifferentiationofequation(2),consideringnopre-existinghousepricesubsidies. 13

to the fact that the rental rate is more sensitive to changes of mortgage rates relative to changes of houseprices. Furthermore,equation(6)leadstothefollowingcorollary. Corollary1(MortgageSubsidiesCanHurtBorrowers) Ifthedemandforhousingisdownward sloping with respect to the house price, ε 0, and the supply for housing is upward sloping, D,p ≤ ε 0, then 1 ζ [r(t) + δ π] 0 and mortgage subsidies hurt borrowers if LTV < S,p p,r ≥ − ≤ − ≤ ζ [r(t)+δ π]. p,r − − ThecorollaryisadirectconsequenceofProposition1,Lemma1,andthefactthatε , ε S,p D,p − ≥ 0implythat 1 ε /(ε ε ) 0. Corollary1describesasurprisingresult,asitestablishes D,p S,p D,p − ≤ − ≤ a sufficient condition for borrowers to be hurt by mortgage subsidies. This condition is satisfied whenever the original LTV ratio is low enough, or the supply (demand) is very inelastic (elastic). Theresultissurprisingasitchallengestheintuitionfromtheclassicanalysesoftaxesandsubsidies on commodities, where subsidies always weakly benefit their recipients. The di fference with the classicresultisaconsequenceofthenon-lineareffectofthesubsidyintheusercostofhomeownership. Asdescribedabove,theusercostforthemarginalunitsfinancedwithmortgagedebtdepends on the effective mortgage rate, whereas the user cost for the inframarginal units financed with a downpayment depends on the opportunity cost of the funds used for the downpayment. Mortgage subsidies affect the user cost on the marginal units and thus distort the demand for housing as if housingwasfinancedentirelywithmortgagedebt. Incontrast,theimpactofmortgagesubsidieson borrowers’ welfare takes into account that only a fraction of the house is financed with mortgage subsidies(equation(3)). Infact,thebenefitfromlowere ffectivemortgageratesonlyaccruestothe fractionfinancedwithmortgagedebt,whereasthenegativee ffectfromhigherhousepricesaccrues totheentirehouse. House price responses described in equation (6) are also amplified by the adjustment in LTV incentivized by mortgage subsidies. In order to show how this amplification channel operates it is useful to consider the problem of a homebuyer who is constrained by an LTV limit. This is the case, for instance, when the marginal utility of period 1 consumption is bounded and period 0 income y is low enough such that the natural borrowing limit c 0 binds. In this case mortgage ≥ borrowing equals (1 + π δ)px and the LTV = (1 + π δ)/(1 + r(t)), which is fixed for any − − level of the mortgage subsidy t. Given the borrowing constraint, the demand for housing is given by (1 + r(t))y/[(r(t) + δ π)p], from where it follows that ζ = ε LTV/[r(t) + δ π]. Given D,r D,p − − that LTV < 1 this attenuates the response of house prices to interest rates (in absolute value). It is interesting to note that in this case of an LTV limit, the incidence on homebuyes is always non-negative—as in the classic case of subsidies on commodities. In this case, the marginal and average effects of mortgage subsidies are aligned. This case highlights that it is the LTV increase generated by the mortgage subsidy that opens the scope for the subsidy to hurt homebuyers, who aretheintendedbeneficiariesofthesubsidy. 14

The increase of LTVs caused by mortgage subsidies in the model could be thought of as an upperbound,asinthemodeltheonlyadditionalsourceoffundstofinancetheincreaseinhousing expenditure are mortgages. However, in practice home buyers might respond by adjusting their overall portfolio and liquidate some other assets to finance the additional house expenditure. In addition,asbuyersincreasetheirLTVlendersmightincreasetheinterestratetoprotectthemselves against higher expected losses, or risk averse home buyers might refrain from taking additional leverage when they face house price or income risk. These channels suggest that the increase in LTV will be attenuated, but the evidence points to increases (decreases) in LTV when mortgage subsidiesareincreased(decreased),lendingsupportforthisimplicationofthemodel.13 Insection4.2Iuseageneralizationofequation(6)toestimatethemortgage-ratesemielasticity ofhouseprices,ζ ,for269metropolitanareas. p,r 2.3 Efficiency Costs from Mortgage Subsidies Whatistheefficiencylossfromthedistortionsintroducedbymortgagesubsidies,whenthesesubsidies affect economic behavior in both mortgage and housing markets? To answer this question, hereIderivetheclassicexcessburdenformulaformortgagesubsidiesthatparallelthederivations in Auerbach (1985). The expression I obtain can be represented graphically as the area between thesupplyanddemandfunctionsandthewedgeintroducedbythesubsidyinthemortgagemarket: theHarbergertriangle. To calculate the excess burden generated by mortgage subsidies additional assumptions are needed. One necessary assumption is that mortgage subsidies only affect house prices in period 0, whereas house prices in period 1 are fixed. This assumption is needed because the house price in period 1 is not determined by the equilibrium of supply and demand. Similar results would obtain if future price effects are considered together with all the determinants of house demand and supply in future periods. Another set of assumptions are required to carry on the calculations and are drawn from the literature to facilitate comparison with the classic results (see Auerbach, 1985, for a discussion of the techniques and assumptions needed for these calculations). First, profits from house producers are rebated lump sum to households. This assumption together with accounting for the welfare change of homeowners effectively makes the excess burden measure independentfromtheredistributionofresourcesfromhouseholdstofirms(orfirmstohouseholds). Second, preferences do not exhibit income effects, i.e., preferences take the following quasilinear form,u(x)+c. Thisassumptionisnecessarytomakethetriangledelimitedbytheuncompensated demandfunctionanaccuratemeasureofwelfare,butitcanberelaxedobtainingsimilarresults,as 13FortheeffectofmortgagesubsidiesonLTVsseeFollainandDunsky(1997),LingandMcGill(1998),Dunsky andFollain(2000),andHendershot,PryceandWhite(2002). 15

I discuss below. This assumption also allows to aggregate the welfare effects across households, fixingthemarginalutilityofincome. 14 Let v denote the indirect utility function and e(p,r,v) denote the expenditure function given a house price p, a mortgage rate r, and an indirect utility v.15 I follow Davidoff, Brown, and Diamond (2005) and specify the expenditure minimization problem as the problem to minimize period 0 expenditure to achieve the level of indirect utility v and imposing the budget constraint in period 1 as a constraint. Under these assumptions the excess burden of introducing a mortgage subsidy, t, denoted by EB(t), corresponds to the loss in consumer surplus (which includes the loss infirms’surplus),plusthelossforhomeowners,minusthechangeingovernmentrevenues. EB(t) = e(p(t),r(t),v) π(t) e(p(0),r (0),v)+π(0) (p(t) p(0))h+G(p(t),r(t),t,y) , M − − − − whereG(p,r,t,y)denotethegovernmentexpenditureonmortgagesubsidies,equaltotm(p,r,y).16 AsecondorderTaylorapproximationof EB(t)yieldsthefollowingresult. Proposition 2 (Efficiency Cost from Mortgage Subsidies) The efficiency loss from mortgage subsidiesequals 1 EB(t) = t Δm (7) 2 This result has an intuitive explanation that can be better described using Figure 2. Mortgage subsidies reduce the effective mortgage rate faced by borrowers by t, increasing the demand for mortgage debt. As the model assumptions ensure that the interest rate offered by lenders remains fixedat r = r +ρ,theeffectivemortgageratefacedbyborrowersbecomesr t. Thus,thedemand f − for mortgage debt increases until the difference between the original mortgage demand M(0) and mortgage supply equals t, as depicted in Figure 2. This increases borrowers’ surplus by the area abde. The government needs to finance a subsidy t for every unit of mortgage credit taken by borrowers, for a total cost of tm(t) equal to the area acde. This creates a deadweight loss equal to theareaofthetrianglebcd,ortΔm/2. ThisistheHarbergerdeadweightlosstriangleofmortgage subsidiesfromthedistortionintroducedinthemortgagemarket. Ontheotherhand,inthehousingmarkettheincreaseindemandforhousingraisespricesfrom p(0) to p(t). This price increase creates a loss for homebuyers equal to the area abcd, which is exactly the gain for sellers, i.e., home producers and homeowners (Figure 2). That is, the effect 14Inaddition, recallthatthegovernmentfinancesitselfwithlumpsumtaxesinperiod0. Alternatively, itcanbe assumedthatithassomeothernon-distortionaryformsofincome. 15This notation allows to consider both variational measures of welfare change for consumers in the case with non-zeroincomeeffects. Infact,ifvcorrespondstotheindirectutilityintheequilibriumattheoriginal(subsidized) prices,thenhouseholdswelfarechangesaremeasuresbythecompensated(equivalent)variation. 16Notethatinthegeneralcasewithincomeeffects,theexcessburdencalculationsaredoneconsideringthecompensateddemandfunctions,sothegovernmentsubsidyexpenditurewilldependonthecompensatedmortgagedemand. 16

of the mortgage subsidy on the housing market is a zero-sum redistribution between buyers and sellersthatcreatesnoadditionaldeadweightloss.17 The deadweight loss depends on the sensitivity of mortgage borrowing to the subsidy, Δm, which is comprised of two parts. First, as effective mortgage rates fall homebuyers increase their demandforhousingandthustheyincreasetheirdemandformortgagedebt. Second,asIemphaize in this paper as effective mortgage rates fall house prices increase affecting mortgage demand in two ways. Higher house prices increase housing expenditure, which at the margin is financed with mortgage debt. Moreover, higher house prices reduce housing and mortgage demand. So the interplay of mortgage and housing markets attenuates the reaction of the demand for housing to mortgage subsidies. All in all, the first order e ffect of lower effectivemortgage rates is to increase the demand for housing by px ε ζ +ζ > 0.18 That is, the demand for housing increases D,p p,r D,r − astheeffectoflowermortgageratesdominatesthecounterbalancingeffectofhigherhouseprices. (cid:0) (cid:1) Inaddition,housingexpenditureincreasesby pxζ > 0. Thus, p,r − Δm px ε ζ +ζ +ζ Δt . (8) D,p p,r D,r p,r ≈ − (cid:0) (cid:1) Therefore,theefficiencylossislargerwhenhousedemandismoreelastictomortgagerates,i.e.,as ζ islargerinabsolutevalue. Incontrast,theefficiencylossislargerwhenhousedemandismore D,r inelastic to house prices. Intuitively, as the demand for housing becomes less sensitive to house prices, there is a smaller offset from the reduction of house demand as house prices increase. The effect of the mortgage rate semielasticity of house prices depends on the price elasticity of house demand. Whenthedemandforhousingismore(less)thanunitelastic,theefficiencylossislarger (smaller)whenthesemielasticityofhousepricestomortgageratesismoreinelastic. Thedifferencebetween(7)andexcessburdencalculationsthatassumethatthepriceofhousing is fixed, as in Poterba (1992) and Poterba and Sinai (2008) among others, arises from the terms involving the semielasticity of house prices with respect to mortgage rates ζ . In fact, if these p,r termsarecanceledIrecovertheexdcessburdenformulausedinthesecalculations: EB = 1/2tΔx, whereusingmynotationΔx pxζ Δt. D,r ≈ − ToderiveProposition2Iassumedthatthedemandforhousesdoesnotdisplayincomeeffects. Thisassumptionissometimesjustifiedonthegroundsthatthemarketbeingstudiedissmall,making income effects negligible (Vives, 1987). In contrast, for most households housing is an importantexpenditurecategoryandhousingconstituteanimportantfractionoffinancialwealth,making income effects relevant. As we know from the classic results in public finance, income e ffects can be considered in the analysis by considering the compensated demand functions for housing and 17Notethatthechangeinbuyers’welfareistheareaunderthesupplyfunctionS,asopposedtotheareaunderthe demandfunction. 18Infact,fromequation(4)itfollowsthat px ε D,p ζ p,r +ζ D,r = pxζ D,r ε S,p ε S,p ε D,p − 1 >0. − − − (cid:0) (cid:1) (cid:0) (cid:1) 17

mortgagedebtandbyconsideringthattheformofcompensationtakeaparticularform(Auerbach, 1985). In fact, it is possible to extend the result of Proposition 2 assuming that compensation takes the form of period 0 income. In this case, Δm, the uncompensated response of mortgage borrowing,needstobereplacedby, Δmˆ,thecompensatedresponseofmortgageborrowing. To calculate the response of the compensated demand for mortgage debt, let ε , ζ , εˆ , M,p M,r M,p and ζˆ denote the uncompensated and compensated house price and mortgage rate elasticities of M,r mortgage demand following the notational conventions used previously. Then, the compensated response of mortgage demand can be approximated by Δmˆ m εˆ ζˆ + ζˆ Δt. Where ζˆ M,p p,r M,r p,r ≈ − correspondstothemortgageratesemielasticityofhouseprices,whenhomebuyersarebeingcom- (cid:0) (cid:1) pensated by the income effect of price changes, i.e., ζˆ = ζˆ / ε εˆ . The formula for the p,r D,r S,p D,p − response of the compensated demand for mortgage debt can be expressed in terms of the demand (cid:0) (cid:1) elasticities for housing using the Slutzky equations for mortgage demand and the relationship imposedonthedemandelasticitiesbytheperiod t = 0budgetconstraintforhomebuyers.19 Finally, suppose there is a preexisting subsidy t and the subsidy is changed to t . Let Δm = 0 1 1 m(t ) m(t )andusethesamenotationforothervariables,forinstance, Δt = t t . Then,itcan 1 0 1 1 0 − − beshownthattheefficiencylossisgivenbytheHarbergertrapezoidformula 1 EB(t) = t Δm + Δt Δm . 0 1 1 1 2 3 Data In this section I describe the data used to measure the effect of eliminating MID. This description precedes the generalization of the results of section 2, as the data availability will inform the modelingchoicestogeneralizetheseresults. Mortgagelevelinformation. IusemortgagelevelinformationfromMcDashAnalytics(formerly LPS). From this data source I consider individual mortgages originated between 2010 and 2015, which corresponds to the largest sample leaving out the financial crisis of 2007-2009. Granted, thisisaspecialperiodfollowingalargefinancialcrisis, 20 butitseemsthemostadequateperiodto characterizetheeffectoftheeliminationofMIDifitwhereimplementedtoday. From the McDash data I consider fixed mortgages—i.e., fixed monthly payment and fixed term—which have a first-lien on the property. I focus my analysis on the most common loan terms: 10,15,20,25,and30years. Thesemortgagesarethemostcommoncomprisingabout90% 19Infact,fromtheSlutzkyequtionsεˆ =ε +pxy 1ε andζˆ =ζ +pxy 1LTV(1+r(t)) 1ε ,whereε M,p M,p − M,y M,r M,r − − M,y M,y denotestheincomeelasticityof(uncompensated)mortgagedemand. Inaddition, differentiatingtheperiod0budget constraintε = LTV 1+LTV 1ε ,ζ =(1+r(t)) 1LTV 1ζ +(1+r(t)) 2 andε = (1+r(t))ym 1. M,p − − D,p M,r − − D,r − M,y − − 20The S&P/Case-Shiller U.S. National Home Price Index of house prices fell between July 2006 and February 2012by27%,andwas5%belowitsJuly2006peakinDecember2015. 18

of the mortgages originated in 2010-2015 (see Appendix B).21 The McDash data also provides the mortgage rate and the LTV ratio at origination. The former varying due to borrower specific characteristicslikecreditscoreandincome,amongothers,evenwhenthemortgagetermandLTV ratio are held constant. Finally, the information contains the ZIP code of the property, which I use tomaptheobservationstothecorrespondingmetropolitanarea. Additionally,Iconstructanindicatorforfirst-timehomebuyersusingdatafromEquifaxCredit Risk Insight Servicing (CRISM). This data source matches credit bureau data from Equifax with mortgage records in McDash. A mortgage is identified as a first time homebuyers if the mortgage was reportedly used to purchase a property (as opposed to refinance it) and the borrower did not have any opened mortgage account in his credit history over the last six months. Using this definition I identify 18.5% of mortgages that correspond to first-time buyers (Table 1). The details of thesedatasourcesandcalculationsareprovidedinAppendix B. Table 1 provides descriptive statistics for mortgage rates and LTV ratios for all borrowers in my sample. The average mortgage rate and LTV ratio in the sample are 4.2 and 77 percent, respectively. Table 2 presents similar statistics separately for first-time buyers and home owners by the term of the mortgage. As expected, mortgage rates and LTV ratios are higher for first-time buyers, with average mortgage rates and LTV ratios of 4.3 and 90 percent, respectively. Also as expected, the mortgage rate and the LTV increase with the term of the mortgage. The latter probablyreflectingthepresenceofincome-to-debt-servicelimitsimposedbylenders. To handle the heterogeneity in mortgage rates and LTV ratios available in the McDash data it willbeusefultointroducethefollowingnotation. Leti I indexthesetofmortgageborrowersand ∈ consider that each borrower i is offered a different after-subsidy mortgage rate r(t) and borrows i i usinganinitialLTVratio LTV.22 i Elasticities. IdrawfromavailablestudiesanduseLemma 1tocalibratetherelevantelasticities. Saiz (2010) uses land topology-based estimates of land availability to provide estimates of the price elasticity of house supply for 269 Metropolitan Statistical Areas (MSAs), over 10-year periods. I denote with ε the price elasticity of house supply in metropolitan area j. The estimated S,p,j valuesrangefromaslowas0.6toashighas12.1,andhaveapopulation-weightedaverageof1.8. Theempiricalliteraturesuggeststhatthepriceelasticityofhousingdemandiscloseto 1,e.g., − Rosen(1985)orDavisandOrtalo-Magne(2011). SoIset ε = 1. D,p − To calibrate the mortgage rate semielasticity of demand, ζ , I use the relationship of this D,r elasticity and the price elasticity of demand and the user cost established in Lemma 1. The price elasticity of demand was set to 1 and to estimate borrowers’ user cost I proceed as follows. The − 21Second-lienmortgageshavenotbeencommonafterthefinancialcrisis. 22Inthemodelofsection2borrowerswithidenticalpreferenceswillchoosedifferentLTVratios,iftheyborrowat differentmortgageratesr(t)orhavedifferentincomelevelsy. i i i 19

mortgage data from McDash provides the mortgage rate for each borrower. The other terms of the user cost, namely δ π in the model of section 2, are calibrated assuming that they represent − all the non-mortgage rate components of the user cost, some of which I have abstracted away in the model for simplicity. I follow Poterba (1992) and Himmelberg et al. (2005) and consider the following additional component of the user cost: τ , the marginal income tax, τ , property taxes, y p and φ the risk premium. Let i denote the nominal mortgage rate, which equals the real mortgage rate r plus the (expected) rate of inflation Π. Then I can express the real user cost, accounting for the deductibility of mortgage interest and property taxes, as r τ i+(1 τ )τ +δ π+φ. The y y p − − − values for these parameters are set following Himmelberg et al. (2005): τ = 25%,Π = 2%,τ = y p 1.5%,δ = 2.5%,π = 1.8%, and φ = 2%. With a slight abuse of notation I denote the real user cost by r τ i + δ π, and I set δ π = (1 τ )τ + δ π + φ = 3.8%. Considering my sample i y i y p − − − − − average nominal mortgage rate of 4.2% and a 2% inflation, I obtain a real mortgage rate of 2 .2% andasubsidyfromMIDofabout100basispoints. Theseparametervaluesgivearealusercostof housingof5%. Lemma 1 can be restated considering the deductibility of mortgage interest in the user cost of homeownership. In this case, the relationship between the mortgage rate semielasticity and the priceofhousedemandelasticityisgivenby ε (1 τ ) D,p y ζ = − . (9) D,r,i r τ i +δ π i y i − − This relationship can be used to compute the mortgage rate semielasticity of house demand at the individual level. Table 1 present descriptive statistics of my estimates for this elasticity. The sample average equals 15.3, with individual estimates displaying significant heterogeneity − rangingfrom 41to 5. − − Finally, let γ be the housing expenditure share, which in the model corresponds to the rental rateofhousingoverincome,andletε denotetheincomeelasticityofhousedemand. Following D,y Poterba(1992)andDavisandOrtalo-Magne(2011)Isetγ = 0.25,andfollowingPoterba(1992)I setε = 0.75. D,y 4 Estimates of the Effects of Eliminating MID In this section, I present my estimates of the effects of eliminating mortgage interest deductions (MID) for 269 American metropolitan areas. I begin in section 4.1 with a description of how I measure the welfare effects (incidence and efficiency loss) of MID. As illustrated by the results in section 2, the welfare effects will depend on the effect of the MID on house prices, so first I present estimates of the e ffect of eliminating MID on house prices for these 20

regions (section 4.2). Finally, I present the estimates of the incidence and efficiency losses for these regions (section 4.3). The estimates for the 269 metropolitan areas are available at http://www.federalreserve.gov/econresdata/feds/2016/files /feds2016081data.csv. 4.1 Measurement of Welfare Effects The characterization of the incidence and the efficiency loss of mortgage subsidies presented in section 2 was done in the simplest framework to highlight the economic mechanisms and the economic intuition. In contrast, in this section I generalize these results to measure the effects of eliminatingMIDincorporatingtherelevantfeaturesofthedatadescribedinsection3. Undertheassumptionthatthemarginaltaxrateis τ andthatthehouseholditemize,theeffecy tive MID can be expressed as τ i. Arguably this is a strong simplifying assumption as marginal y i tax rates vary substantially by household depending on their income. Nonetheless, given that the McDash and CRISM data do not contain income or other relevant tax-related information, I assume that the marginal tax rate τ = 25% for all households and that all households itemize their y deductions. Theseassumptionswillaffecttheestimatedwelfareeffects. ForthedistributionaleffectofeliminatingMID,Iwillattributethenegativeeffectofhighereffectivemortgageratesforallhouseholds although non-itemizing buyers (homeowners) will only benefit (su ffer) from lower house prices. Fortheefficiencyloss,Iwilloverestimatetheaggregateeffectasforsomehouseholdstheelinination of MID will have no effect on effective mortgage rates. This bias goes against my result that the aggregate efficiency gains from eliminating MID are small given the offset in the distortion of mortgagedemandgeneratedbythedeclineinhouseprices. Despitethesimplifyingassumptionthatmarginaltaxratesarethesameacrosshouseholds,note thattheactualsubsidyfromMIDwillstillvaryacrosshouseholdsreflectingthedi fferencesinnominal mortgage rates. To handle this variation analytically I introduce the following notation. Let ϕ [0,1]denotethefractionofmortgageintereststhatcanbededucted,sor(ϕ) = r ϕτ i denote i i y i ∈ − the effective real mortgage rate when a fraction ϕ of mortgage interests can be deducted. Then, ϕ = 1 represents the current condition, where all mortgage interests can be deducted, whereas ϕ = 0 represents the elimination of MID. In addition, I denote by Δr = r(0) r(1) the change i i i − in the mortgage rate (or any other variable) from the elimination of MID. Now I can present the simpleformulasfortheeffectofeliminatingMIDonhouseprices,incidence,andefficiencycost. House price effects. In the data different regions display a different price elasticity of supply and households (first-time buyers and owners) borrow using mortgage contracts with di fferent characteristics. Iassumethateachmetropolitanareacorrespondstoasegmentedhousingmarketwithno householdmobilityinresponsetoMID,soeachmetropolitanregioncanbeconsideredseparately. 21

Let I be the set of households in metropolitan region j, and ω be household’s i share of housj i ing consumption in the region. The aggregate demand for housing is given by x(p,r(t),y). i Ij i i i i ∈ Then, fully differentiating the house market clearing condition for region j, with respect to the P mortgage rate, I obtain the following expression for the mortgage rate semielasticity of house pricesinregion j ωζ ζ = i ∈ Ij i D,r,i . (10) p,r,j ε ε PS,p,j − D,p Similarly, I obtain that the effect on house prices in region j from removing MID can be approximatedby Δp ωζ τ i j i ∈ Ij i D,r,i y i . (11) p ≈ ε ε j P S,p,j − D,p In section 4.2, I provide metropolitan level estimates of the mortgage rate semielasticity of housepricesusingequation(10)andofthehousepricedeclineimpliedbytheeliminationofMID usingequation(11). Incidence. One feature of the data is that house investments and mortgage borrowing extend over manyyears. HereIdescribetheassumptionsmadetoextendProposition1toamultiperiodsetting. First, I assume no transaction costs. Then, the incidence of mortgage subsidies will only depend on the future trajectory of housing and mortgage demand, and will be independent of the moves a households makes in the period. If a household moves from one house to an identical house (as measured by their effective housing units represented by x in the model) and maintain the same pathformortgagebalancesthismovewillhavenoeffectontheincidenceofmortgagesubsidies. Second, I assume that the household does not adjust its housing or mortgage demand after the originationofhermortgage. Thatis,thehouseholdletitshousingstocktodepreciateandpaysoff hermortgageaccordingtothescheduleimpliedbytheoriginalfixedmortgage. 23 Third,Iassumethatafterthehouseholdpaysoffhermortgageshesellsthehouseandconsume the proceeds. Buying another house will make the incidence to depend on the adjustment to the housingstock,whichIcannotobserveinthedata. Thisassumptionaffectstheestimatesdepending on the future path of the housing stock and mortgage debt. For households that do not purchase anotherhousetheeffectofthisassumptiondependsonthedifferencebetweentheactualfuturesale date versus the date when the mortgage is paid off. For households that remain in the house after themortgagedebtwasscheduledtobepaidoff,theharmfromsellingthehouseatalowerpriceis front loaded making the estimated welfare effect of the subsidy worse for borrowers. In contrast, if the house is sold before the mortgage termination date the adverse effect of lower future house prices is back loaded in my calculations and bias the estimates making them more beneficial for 23Theassumptionofafixedhousingstockissimilartoassumingthatthehouseholdpaystherequiredmaintenance coststokeepitshousingstockfixed,exceptforthetimingofmaintenancecosts. 22

households. But, note that in the latter case the mortgage will be prepaid and the household will foregothereductionineffectivemortgageratesconsideredinmycalculations,makingthewelfare estimateslessbeneficialforhouseholds. I make the same assumptions to model the behavior of homeowners. The only distinction between homeowners and first-time buyers is that the former already own their optimal housing stock,sothechangeinhousepricesonlyaffectsthesehouseholdswhentheyselltheirhouse. LetT bethetermofthemortgageinyears,thenundermyassumptionsitispossibletoestablish i analyticallythefollowingresult. Proposition 3 (Incidence over multiple periods) The incidence of a permanent elimination of MID,equalszeroforlenders, p z (Δp /p )forhouseproducers, j j j j Δp j ΔV = u p x φ (r(1),T ) φ (r(1),T ) LTV (12) i c j i p i i m i i i − p − j ! for household i in metropolitan area j, where Δp /p < 0 is given by equation (11) and the price j j andLTVmultipliersare,respectively,givenby 1 (1 r(1) δ+π)Ti forfirst-timebuyers i − − − φ (r(ϕ),T ) = p i i  − (1 − r i (1) − δ+π)Ti forhomeowners (13) τ y i i  1 12T i φ (r(ϕ),T ) = . m i i 12(1+r i (ϕ))1 1 1 2  (1+r i (ϕ)) 1 1 2 1 − (1+r i (ϕ)) 1 1 2 (1+r i (ϕ))Ti 1  − − It follows that the welfare effec t of the elimination of MID can(cid:2)be expressed as(cid:3) the sum of two terms: one term representing the impact of the decline in house prices, and the other term representing the burden of higher effective mortgage rates, which depends on the LTV ratio at origination. The magnitude of these two effects depends on the mortgage rate, the other components of the user cost, and the term of the mortgage (assumed to equal the duration of the house investment). AlongermortgagetermincreasesthepresentvalueofMID,asitincreasesthepresent value of interest payments. On the other hand, longer house investments reduce the present value loss of selling the house at a lower price in the future. Note that the price multiplier is positive for first-time buyers ( φ > 0) and is negative for homeowners (φ < 0), reflecting the di fferential p p impact that the initial house price decline have on these two group of households. A reduction in house prices benefit first-time buyers as the benefit from purchasing their houses at lower prices outweighthepresentvaluelossfromsellingthesehouseunitsatlowerpricesinthefuture. Onthe contrary, under my assumptions, homeowners are only affected negatively from the lower house prices when they sell their houses in the future. Table 3 presents descriptive statistics of the price 23

andLTVmultiplierforownersandbuyersdependingontheirmortgageterm. Note that equation (12) makes the incidence on households comparable when they purchase houses of different values. In fact, the term in the RHS inside the parenthesis in this equation measures the incidence as a fraction of the house value. In section 4.3, I use this equation to provideestimatesoftheincidenceofMIDonhomeownersandfirst-timehomebuyers. Efficiency Costs. To measure the efficiency losses I need to maintain some of the assumptions made in section 2.3. In particular, I consider the same two period framework and maintain the assumption that house prices in period 1 are fixed. Given that the calculations of e fficiency losses abstract from distributional effects, I abstract from the distinction between owners and first-time buyers. In fact, I assume that all households have to buy their desire housing stock and at the sametimeareentitledtotheproceedsofthesaleoftheexistingstockofhousingandtheprofitsof house producing firms. In contrast, I relax other assumptions that are not needed to calculate the efficiency cost in practice. First, I consider income effects in the demand for housing. Second, I considerthatMIDarecurrentlyinplaceandcalculatethereductionintheefficiencylossaccruing from eliminating this deduction. Finally, I consider that in each metropolitan region there are heterogenous households who borrow using different mortgage rates and LTV ratios, as observed inthedata. Letπ (ϕ)betheprofitofhouseproducersinmetroarea jwhenthefractionofmortgageinterest j that can be deducted is ϕ, let h be the existing housing stock in metro area j, and let t(ϕ) be the j i period 0 mortage subsidy when a fraction ϕ of mortgage interest can be deducted.24 Using this notationtheexcessburdenofeliminatingtheMIDinmetropolitanarea jcanbeexpressedas EB (1,0) = e p (0),r(0),v π (0) p (0)h j i j i i j j j − − X i ∈ Ij (cid:0) (cid:1) e p (1),r(1),v +π (1)+ p (1)h +G p (0),r (0),0,e p (0),r(0),v , − i j i i j j j j Ij Ij j i i (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) where variables with subscript I denote the vector of values for all households i in metro area j j, for instance, r = r ; and G p ,r ,ϕ,e (p ,r,v) denote the government expenditure on Ij { i } i ∈ Ij j Ij Ij j i i mortgage subsidies, equal to t m p ,r,e p ,r,v = t mˆ p ,r,v , where the last i Ij (cid:0)i i j i i j i i(cid:1) i Ij i i j i i ∈ ∈ equalityusestheidentitybetweentheuncompensatedandcompensateddemandfunctions. P (cid:0) (cid:0) (cid:1)(cid:1) P (cid:0) (cid:1) AsecondorderTaylorapproximationaroundϕ = 1givestheHarbergertriangleformula25 1 1 EB (1,0) mˆ 0 mˆ 1 t(0)+t(1) = Δmˆ t(1) , (14) j i i i i i i ≈ 2 − 2 X i ∈ Ij (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)(cid:2) (cid:3) X i ∈ Ij 24Fromtheidentityq +t(ϕ)=(1+r ϕτ i) 1itfollowsthatt(ϕ)=ϕτ i(1+r) 1(1+r ϕτ i) 1. i i i y i − i y i i − i y i − − − 25NotethatHarberger’strapezoidformulawithpreexistingMIDturnsintoatriangleformulaonceIconsiderthe totaleliminationofMID. 24

wheremˆ ϕ = mˆ p (ϕ),r(ϕ),v isthecompensatedmortgagedemandfunction. i i j i i ToevaluatethisformulainthedataIusethatthechangeinthecompensatedmortgagedemand (cid:0) (cid:1) (cid:0) (cid:1) can be approximated by Δmˆ mˆ (1) εˆ Δpˆ /p + ζˆ Δr , where Δpˆ /p corresponds to i i M,p,i j j M,r,i i j j ≈ the decline in house prices in region j, when households are compensated for the price changes (cid:2) (cid:3) induced by the elimination of MID. The previous formula can be expressed in terms of the house demandelasticitiesusingtheSlutzkyequationsandtherelationshipimposedbytheperiod0budget constraint (see footnote 19). In fact, combining these equations I obtain that εˆ = ε /LTV , M,p,i D,p i astheincomecompensationformortgagedemandcancelswiththeeffectofhousepricesonhouse expenditure,andζˆ = ζ /LTV +r(ϕ)/(1+r(ϕ)) /(1+r(ϕ)),wherethelasttermcollectsthe M,r,i D,r,i i i i i effectoftheincomecompensationandtheeffectofmortgageratesonmortgageinterestexpenses. (cid:2) (cid:3) Itfollowsthat Δpˆ r(1)LTV j i i Δmˆ p x (1+r(1))ε + ζ τ i . (15) i j i i D,p D,r,i y i ≈ p − 1+r(1) " j i ! # housepriceeffect mortgagerateeffect That is, the distortion in the|(compe{nzsated) m}ortg|age dema{nzd can be e}xpressed as the sum of two effects. The house price effect, which reflects that as house prices decrease when the MID is eliminated (Δpˆ < 0) the demand for mortgage debt increases (ε < 0). On the other hand, the j D,p mortgagerateeffectcapturesthatastheeffectivemortgagerateincreases,thedemandformortgage debt declines (recall that ζ < 0). One important takeaway from decomposing the distortion of D,r,i mortgagedemandintothesetwotermsisthatthehousepriceeffect—whichispositive—attenuates thedistortionofmortgageratesinmortgagedemand,whichwillreducetheefficiencylossbrought about by mortgage subsidies. In section 4.3, I use equations (14) and (15) to provide estimates of efficiencygainsfromeliminatingMID. 4.2 Estimates of the Effect of Mortgage Rates on House Prices Usingthepreviouslyderivedformulastogetherwiththedatadescribedinsection3,Icanestimate the sensitivity of house prices to mortgage rates in the 269 metropolitan areas being analyzed. In fact, equations (10) and (11) provide, respectively, estimates for metropolitan area j of the mortgage rate semielasticity of house prices, ζ = 1/p (dp /dr), and of the decline in house p,r,j j j prices from the elimination of MID, Δp /p . These equations show that the effect of house prices j j differs across metropolitan areas given differences in the price elasticity of supply, ζ , and the S,p,j (house-value-weighted)averagemortgageratesemielasticityofdemand,ζ = ωζ . D,r,j i Ij i D,r,i ∈ Table4showsthatwhilethesupplyelasticitydisplayssignificantvariationacrossmetropolitan P areas,theaveragemortgageratesemielasticityisverystableacrossregions,despitetheindividual mortgage-rate semielasticities of house demand displaying considerable heterogeneity (Table 1). In fact, the former has a (house-value-weighted) mean of 1.5 and a standard deviation of 0.9. In 25

contrast,thelatterhasameanof 15.4andastandarddeviationofonly0.1.26 Thus,thesensitivity − of house prices to interest rates is determined primarily by the price house supply elasticity. Table 4 shows that the estimated (house-value-weighted) average decline in house prices from eliminating MID would be 6.9%. Similarly, the estimated (house-value-weighted) average mortgage rate semielasticity of house prices is 6.9. That is, the decline in house prices from eliminating MID − is about the same magnitude as predicted by a 1 percentage point increase in mortgage rates. This reflectsthatundertheassumptionthatthemarginaltaxrate τ equals25%andanaveragenominal y mortgage rate of 4.2% in my sample, the MID amount to a reduction in the effective mortgage rate of about 1 percentage point. This semielasticiy of house prices ranges from 9.6 in Miami, − Florida, where the price elasticity of supply is 0.60, to 1.2 in Pine Bluff, Arizona, where the − supplyelasticityis12.2. Figure 3 plots the estimated decline in house prices from eliminating the MID for the 269 metropolitan areas in my sample. The figure shows that my estimates for the decline in house prices depends primarily on the supply elasticity, and that my estimates are well approximated by 15.4/(ε + 1). This expression corresponds to equation (11) assuming an effective decline in S,p,j − mortgage rates of 1 percentage point, a mortgage-rate house demand semielasticity equal to the average of 15.4 (Table 4), and a price house demand elasticity equal to 1, as I have assumed. − − This approximation works well given that the mortgage-rate house demand semielasticity varies very little at the MSA level (Table 4). Note that equation (11) approximates the log-difference of houseprices,whichequals 15.4/(ε +1)Δr 15.4exp( ε )Δr,suggestingthatregressions S,p,j S,p,j − ≈ − − of(log)housepricechangesontheinteractionofmortgageratechangesandtheelasticityofhouseprice supply can identify the average mortgage-rate semielasticity of mortgage demand, when consideringatransformationofthepricehousesupplyelasticity. My estimates of the mortgage rate semielasticity of house prices ζ can be compared to p,r,j directestimatesfromempiricalstudies. Theempiricalevidenceisbroadlyinlinewithmyaverage estimate of 6.9. One strand of the literature studies the effect on house prices of changes in − mortgagerates,i.e.,thepriceofmortgagecredit. Glaeser,GottliebandGyourko(2012)regressan aggregatehousepriceindexofrepeatedsalesagainstthe10-yearTreasurybondrateandestimatea housepricesemielasticitytothisinterestrateof 6.8,butastheyacknowledge,thisestimatemight − bebiasedbytheendogeneityofinterestrates. Adelinoetal. (2014)usechangesintheconforming loan limit to measure the effect of lower mortgage rates on house prices. They estimate a house price semielasticity to mortgage rates between 9.1 and 1.2.27 In a related study, Kung (2015) − − 26The household-weighted average supply elasticity is 1.74 in my sample, in line with the household-weighted averagereportedbySaiz(2010)of1.75. 27It is interesting to note that the range of estimates provided by Adelino et al. (2014) is about the same as the rangeofvaluesforthemortgageratesemielasticityofhousepricesthatIestimate. However, thesetworangeshave different interpretations. Adelino et al. (2014) give a range of estimates for the average semielasticity in 10 MSAs 26

uses the variation in the conforming loan limit together with the original asking price to asses the likelihood that the change in this limit will affect a property and estimates a value of 6 for − this semielasticity. Himmelberg et al. (2005) and Hubbard and Mayer (2009) argue that the user cost model imply a much larger elasticity in absolute value. In fact, in order for the rental rate to remain constant, under plausible values for the key economic parameters of the user cost one obtains a value for this elasticity of about 19. But, in my model for the rental rate to remain − constant the supply of housing needs to be fixed, that is, the price elasticity of supply needs to equal zero. Taking a zero supply elasticity, formula 10 imply a value closer to the estimates of theseauthors. Another strand of the literature studies the effect on house prices of the quantity of credit supplied. Favara and Imbs (2015) and Di Maggio and Kermani (2015) use regulatory changes to instrument for changes in the supply of credit at the county level and find that the elasticity of housepricestothe(instrumented)volumeofmortgageloansisbetween0.2and0.33. Anenberget al. (2016)constructaninstrumentforthesupplyofcreditbasedonameasureofcreditavailability and estimate an elasticity of 0.9. Using my notation this elasticity, at a given metropolitan area j, can be expressed as ε , and it equals to the ratio of the mortgage rate semielasticity of house p,M,j prices, ζ , to the average mortgage rate semielasticity of mortgage demand, ζ . Using the p,r,j M,r,j data described above, I can compute this elasticity for each metropolitan area. In line with these empiricalstudies,the(house-value-weighted)averageofthiselasticityis0.3(Table4). The fact that the average estimated sensitivity of house prices is in line with the empirical studies lends indirect support to my simple framework for welfare analysis and provides external validitytothehousepriceeffectsusedinthewelfarecalculationsdescribednext. 4.3 Estimates of the Welfare Effects of Eliminating MID Iusetheinformationdescribedinsection3comprising17.5millionmortgagesoriginatedin2010- 2015tomeasurethedistributionaleffectofmortgagesubsidies—itseconomicincidence—andthe efficiencygainsfromeliminatingMID—thesizeofthenegativedeadweightloss. Incidence. As described in Proposition 3 the elimination of MID will hurt house producers proportionally to the decline in house prices. Estimates by metropolitan area of these price declines werepresentedinsection 4.2. Proposition 3 also describes how the elimination of MID will affect households through two effects: increasing mortgage interest payments and reducing house prices. The former hurt all households—homebuyers and homeowners, while the latter hurts homeowners but benefit homebuyers. Table 5 presents my estimates for the incidence of eliminating MID on households’ welconsideredintheiranalysis. Incontrast,Iprovidearangeofestimatesfor269MSAs,withthe(household-weighted) averageofmyestimatesforthesame10MSAsintheAdelinoetal. (2014))sampleequalto 8.1. − 27

fare. I consider the incidence measured as a percent of the house value, the term in parenthesis in equation(12), φ p /Δp φ LTV ,tomakethesemeasurescomparableacrosshouseholds. Tap,i j j m,i i − − ble5reportsseparatelyforhomeownersandhomebuyerstheaverageincidencebymortgageterm. Thelastthreecolumnsofthetablepresenttheestimatesoftheeffectofonlyhighermortgagerates causedbytheeliminationofMID,assumingnochangeinhouseprices. Highereffectivemortgage rates hurt both first-time homebuyers and homeowners, with households using longer mortgage terms being hurt more. Households using longer mortgage contracts are hurt more given that increasingthetermsofthemortgageeffectivelyincreasesleverage,ascapturedbytheincreaseinthe LTV multiplier (Table 3) and that LTV ratios and mortgage terms are positively related (Table 2). ComparingtheaveragewelfarereductionforthesamemortgagetermitisobservedthattheeliminationofMIDwillhurtbuyersmorethanowners,reflectingthatbuyersusemoreleverage—higher LTV—on average, conditional on the mortgage term (Table 2). Moreover, on average buyers are hurt more relative to owners reflecting that my sample of buyers uses relatively longer mortgage contracts. The first three columns of Table 5 presents the estimated incidence on households when both highereffectivemortgageratesandlowerhousepricesaretakenintoaccount. Lowerhouseprices increase the loss for homeowners. In contrast, lower house prices benefit first-time buyers who gainfrompurchasingtheirfirsthouseatlowerprices. Thee ffectofhousepricesismoreimportant in present value the shorter the mortgage term, given my assumption that the house is sold at the time the mortgage is repaid. All in all, on average homeowners loss from the elimination of MID corresponds to 11.5% of the value of their house, whereas on average first-time buyers only lose 8.5% of the value of their house. For an average house value of $320,000 these correspond to a lossof$36,800forhomeownersandalossof$27,200forhomebuyers. Given the differential response of house prices and households’ characteristics the average incidenceisestimatedtovaryacrossregions. Figure4depictstheaverageestimatedincidencefor thesetwogroupsofhouseholdsasafunctionofthehousepricesupplyelasticity. Inmoreinelastic regionsthisdiscrepancyislargerreflectingtheestimatedlargere ffectonhouseprices. My estimates display important variation across metropolitan areas for the incidence of the elimination of MID. Figure 5 presents a heat map for the average incidence on first-time buyers by metropolitan area. In the more inelastic coastal regions, the elimination of MID is estimated to cause a larger decline of house prices, thus it is estimated that homebuyers are hurt less in these areas, depicted by the (warmer) lighter pink colors. In contrast, in most of the midwest metropolitan areas are depicted in (colder) lighter blue colors, reflecting the higher elasticity of house supply in these regions that translates in an estimated smaller decline in house prices upon the elimination of MID. I estimate that on average homebuyers welfare decline by 12.6% of the house value in Alexandria, Louisiana, whereas homebuyers in San Francisco, where house prices 28

areestimatestodropmore,areestimatedtoloseonly3.8%ofthevalueofthehouse,representing dollarlossesof40,320and12,160fora320,000dollarhouse,respectively. AnotherwaytolookattheregionalvariationoftheeffectoftheeliminationofMIDisbytaking the average incidence for first-time buyers by state. Figure 6 presents a similar heat map as above considering the average incidence for buyers by state. The same pattern emerges with the coastal and more inelastic states displaying the smallest adverse effect for homebuyers of the elimination of MID, and the interior states displaying the largest adverse effect from the elimination of this subsidy. For first-time buyers, the benefit from lower house prices can more than o ffset the loss from higher effective mortgage rate, upon the elimination of MID, as reflected by the positive estimates reported in Table 5 for the maximum of the incidence on first-time buyers. My estimates imply that slightly more than 42,000 first-time buyers, in my sample, during 2010-2015 would have benefitedifMIDwerenotinplace. Thisestimateislikelytounderstatethenumberofhouseholds that would benefit, as all households are assumed to itemize. Non-itemizing buyers will only be benefitedfromlowerhouseprices. Efficiency Gains. As described above the efficiency loss generated by MID is proportional to the distortion generated on the (compensated) demand for mortgage debt (equation (14)). Table 6 presentsthecontributionofthechangeinhousepricesandeffectivemortgageratestothechangein this demand. The elimination of MID, on average, will increase effective mortgage rates by about one percentage point. This increase in effective mortgage rates is estimated to directly reduce the demand for mortgage debt by 15.7% of the (current) house value. Moreover, the increase in effectivemortgageratesindirectlyincreasethedemandformortgagedebtby5.8%ofthe(current) house value, as higher effective mortgage rates lower house prices. In fact, I estimate that the elimination of MID causes a decline in house prices of 5.7%, on average over households, when thecompensatedresponsesofhousedemandareconsidered.28 Table6presentsthetotaldistortion in the demand for mortgage debt as a percent of the house value. All in all, on average, the elimination of MID is estimated to reduce mortgage demand by 9.9%. The upshot is that the demand for mortgage debt is distorted about 40% less due to the offset coming from the decline in house prices, or in other words, absent the offseting effect of house prices the the distortion and efficiencygainwillbealmost60%largerthantheestimatesIobtain. 28The estimated price decline is lower when the income compensations are taken into account—compare 5.7% declinewithahousehold-weightedaveragedeclineinhousepricesof6.3%fortheestimatesreportedinsection4.2. FromtheSlutzkyequationswehavethatthecompensatedelasticitiesofhousedemandwithrespecttomortgagerates and house prices are less elastic than their uncompensated analogues. On the one hand, a less elastic mortgage rate semielasticity imply that house demand respond less to the same increase in effective mortgage rates. On the other hand, a less elastic price elasticity imply that house prices need to adjust more to re-equilibrate the housing market afteragivenincreaseinhousedemand. Myestimatesimplythattheformereffectdominatesandhousepricesdrop lesswhenincomecompensationsareaccountedfor. 29

Table 7 presents descriptive statistics for the efficiency costs for first-time buyers and homeowners by mortgage term. The first three columns present estimates of the e fficiency loss as basis points of the house value. The elimination of MID is estimated to create average efficiency gains of5.1basispointsofhousevaluesperhousehold. BytheHarbergertriangleformula,theestimated mortgagedemanddistortionisone-halfoftheproductofthechangeinthe(compensated)demand for mortgage debt and the size of the current mortgage subsidy. The former estimated to be 9.9% of the house value and the latter estimated to be roughly 100 basis points. These values imply an average efficiency gain of about 5 basis points of the house value. The last three columns of Table 7 present the estimated average dollar value of the efficiency losses from eliminating MID (negative values correspond to efficiency gains). The average efficiency gain is about $150 dollars per household, ranging from gains of $82,600 to losses of $10. Households who increase their (compensated)demandformortgagedebtinresponsetotheeliminationofMIDcontributetoefficiency losses. This is the case for households who are currently borrowing at very low mortgage rates so the effective increase in mortgage rates from eliminating MID is small in percentage points and forwhotheeffectoflowerhousepricesdeterminesthedirectionoftheresponseofmortgagedebt. Totalefficiencygains,formysampleof17.5millionmortgages,addupto$2.6billion. Assuming my sample is a random subsample of the 49 million households that finance their house with mortgage debt, according to the American Community Survey 2010-2014, an upper boundforthetotalefficiencygainsfromtheeliminationofMIDwouldbeamodest$7.3billion. 5 Conclusion Inthispaper,Iarguethatthewelfareevaluationofmortgagesubsidiesneedstoaccountfortheeffectsonboththeeffectivemortgagerateandhouseprices. Usingthesufficientstatisticsapproach, the welfare evaluation of mortgage subsidies requires a clean identification of the necessary economic parameters—in particular, the house demand semielasticity with respect to mortgage rates. But the empirical identification of this elasticity is complicated by the fact that changes in mortgage rates influence the demand for housing and thus its price, which feeds back into the demand forhousing. Lemma1allowsmetoexpressthemortgageratedemandsemielasticityastheratioof the price demand elasticity and the user cost, both of which can be cleanly empirically identified. In this way, I obtain individual level estimates of the mortgage rate house demand semielasticity, whichallowsmetocharacterizethehousedemandresponsetochangesineffectivemortgagerates. This characterization enables me to derive a sufficient statistic formula for the effect of effective mortgage rates on house prices. I use this formula to provide estimates of the mortgage rate semielasticity of house prices for 269 different metropolitan areas. The average of these estimates arebroadlyinlinewithdirectestimatesfromempiricalstudies(Section 4.2). 30

I describe the incidence, or distributional impact, of these subsidies as a function of reducedform sufficient statistics that capture the key economic parameters in these markets: the demand and supply elasticities for housing, the LTV, and the user cost of homeownership. My formulas traceouttheeffectofthesubsidythroughhouseprices,whichaffectdifferentiallyfirst-timehomebuyersandhomeowners. Bothtypeofhouseholdsarehelpedbylowereffectivemortgageratesbut buyers are hurt by higher house prices, whereas owners also benefit from higher prices. Using my formulas,Iprovidenewestimatesoftheeffectofeliminatingmortgage-interestdeductions(MID) across 269 U.S. metropolitan areas. I estimate that on average households will lose the equivalent of10.9%ofthevalueofthehouse,withhomeownerslosing11.5%andhomebuyersloosing8.5%. For an average home value of $320,000 this discrepancy in the effect between owners and buyers correspondsto$9,600inpresentvalue. Inaddition,thespilloverofmortgagesubsidiesintohouse prices introduces interesting regional variation through the differential response of house prices depending on the house supply price elasticity. My estimates imply that, on average, buyers in San Francisco, California, where the house supply is more inelastic lose only 3.8% of the house value, compared to buyers in Alexandria, Louisiana, where the supply is more elastic and buyers loose as much as 12.6% of the house value. Considering the average house value in my sample of $320,000, these estimates imply that homebuyers in regions with elastic house supply lose in present value $28,160 more than their peers in more supply inelastic areas. These differences are economicallysignificant. Moreover,Ishowthattheefficiencylossintroducedbymortgagesubsidiesisattenuatedbythe response of house prices. As house prices increase households reduce their demand for housing and thus, mortgage debt. The offsetting effect of house prices in the demand for mortgage debt reduces the size of the Harberger deadweight loss triangle, as it reduces the distortion of the compensated mortgage demand introduced by the subsidy (i.e., the base of the triangle). In practice, myestimatessuggestthatthismechanismreducestheefficiencylossduetoMIDbyabout40%. One important limitation of my analysis of mortgage subsidies is that it abstracts away from the rental market and the extensive margin of housing demand. Incorporating a rental market in the analysis is expected to influence the welfare evaluation of mortgage subsidies, in general, and MID,inparticular. Thesesubsidiesmayinducerenterstobecomehomeownersdirectlyimpacting households’ welfare. But, as my analysis and related literature emphasize, the capitalization into house prices of mortgage subsidies increases the rental rate of homeownership, which the subsidy aimedtodecrease. Theoveralleffectofmortgagesubsidiesontheincentivetoownversustorentis thusambiguous. TheavailableresearchontheeffectofMIDonhomeownershipratessuggeststhat the overall effect of these subsidies on homeownership rates is small (Glaeser and Shapiro, 2003; Bourassa and Yin, 2008; Hilber and Turner, 2014; Sommer and Sullivan, 2016). The conclusion of these studies suggest that most of the response to MID is expected to occur along the intensive 31

margin of house demand, which I considered in my framework. Future work should investigate howtheresultsoftheappliedframeworkforwelfareanalysispresentedinthispaperareinfluenced bytheadjustmentalongtheextensivemarginofhousedemand. My new evidence regarding the effect of eliminating MID helps to inform the public debate about the desirability of mortgage subsidies and the design of housing policy. If the government weretomaintaintaxsubsidiestoencouragehomeownershipandtheprogressivityofthetaxcode, myanalysissuggeststhatapreferredalternativewouldbetohaveafixed taxcreditforhomebuyers. This tax credit could be designed to span the duration of the house investment, in the same way that an interest deduction spans the duration of the mortgage.29 This would act like a lump sum subsidy,limitingthedistortionsintroducedbythispolicy. Infact,itshouldonlydistorttheowningversus-rentingdecision. Moreover,afixedtaxcreditwillbeprogressive,asitwillrepresentalarger fractionofhouseexpenditureforlowerincomehouseholdsthatpurchasemoreaffordablehouses. The techniques and insights developed in this paper can be used in other settings where debt andothermarketsinteract,likecorporateinvestmentinfixedassetsorcollegestudents’investment in human capital. In all these cases, a precise evaluation of debt policies should consider the spillovers of debt policies into the market for debt-financed capital. My analysis shows how to use the sufficient statistics approach to complete a key information requirement that arise in these settings: theinterestratedemandelasticityforthedebt-financedcapital. References Aaron, H. J. (1972), Shelter and subsidies: who benefits from Federal housing policies? , Studies insocialeconomics,BrookingsInstitution. Adelino, M., Schoar, A. & Severino, F. (2014), Credit Supply and House Prices: Evidence from Mortgage Market Segmentation. Available at SSRN: http://ssrn.com/abstract= 1787252. Allcott, H. & Taubinsky, D. (2015), ‘Evaluating behaviorally motivated policy: Experimental evidencefromthelightbulbmarket’, AmericanEconomicReview105(8),2501–38. Alvarez, F., Le Bihan, H. & Lippi, F. (2016), ‘The real effects of monetary shocks in sticky price models: Asufficientstatisticapproach’, AmericanEconomicReview106(10),2817–51. Ambrose,B.W.,LaCour-Little,M.&Sanders,A.B.(2004),‘Theeffectofconformingloanstatus onmortgageyieldspreads: Aloanlevelanalysis’,RealEstateEconomics 32(4),541–569. 29 In fact, one can consider that the households take an hypothetical mortgage, with fixed monthly payments, in ordertocalculatethedesiredtaxcreditthatwillbegrantedfortheremainingbalanceonthathypotheticalmortgage eachyear. 32

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Appendix A Proofs Proof of Proposition 1: Let V(t) = V(p(t),r(t),y) denote households’ indirect utility function. From the utilitymaximizationofhouseholdstheindirectutilityfunctionisgivenby V(t) = u(x,c) u (1+r(t))[px y (q+t)m] u [c (1 δ+π)px+m] , c c − − − − − − where I have substituted for the lagrange multipliers in an interior solution of the households’ problem. ApplyingtheEnvelopeTheoremIget dV dr(t) dp(t) dq(t) dp(t) = u [px y (q+t)m] u (1+r(t)) x +1 m +u (1 δ+π) x c c c dt − dt − − − dt − dt − dt " ! # 1dp(t) = u px [r(t)+δ π]+(1+r(t))2LTV , c −p dt − ! whereIusedthatdq/dt = 0ifr = r +ρ, px y (q+t)m = 0,1+r(t) = 1/(q+t),andLTV = (q+t)m/(px). f − − Moreover,sincedr/dt = (1+r(t))2,applyingtheChainRuleIget − dV 1∂p = u px(1+r(t))2 [r(t)+δ π]+LTV c dt p ∂r − ! Since home owners sell all their endowment of housing so the incidence on them equals (1 + − r(t))2h ∂p/∂r. Similarly, applying the Envelope Theorem to the profit maximization of house producers, Igetthattheincidenceontheseagentsequals (1+r(t))2z∂p/∂r. − Ontheotherhand,implicitdifferentiationofequation(2)yields dS(p)dp(t) ∂D(p(t),r(t))dp(t) ∂D(p(t),r(t))dr(t) = + . dp dt ∂p dt ∂r dt BytheChainRule(∂p/∂r)(dr/dt) = dp/dt. Substitutingfordp/dt intheexpressionabove,multiplyingby p/D,andrearrangingIget 1∂p ζ D,p = ζ = < 0 . p,r p ∂r ε ε S,p D,p − ProofofProposition2: Bydefinitiontheexcessburdenofamortgagesubsidy tisgivenby EB(t) = e(p(t),r(t),v) π(t) e(p(0),r (0),v)+π(0) (p(t) p(0))h+G(p(t),r(t),t,e(p(t),r(t),v)) . M − − − − In order to approximate the excess burden with a second order Taylor polynomial I calculate the following derivatives. First, using the Envelope Theorem in the households’ expenditure minimization problem 36

togetherwithafixmortgageinterestrate(price)andhousepriceinperiod1Iget de(t) dp(t) = x m , dt dt − where the derivative of the multiplier of the period-1-flow-budget constraint canceled given that that constraint is active. Second, using the Envelope Theorem in the producers’ maximization problem I get dπ(t)/dt = z dp(t)/dt, and noting the home owners sell all their endowment independent of prices, d(p(t)h)/dt = hdp(t)/dt. Finally,takingderivativesofthegovernmentsubsidyexpenditure,Iget dG(p(t),r(t),t,y) dm(p(t),r(t),y) = m(p(t),r(t),y)+t dt dt Therefore, dEB(t) dm(p(t),r(t),y) = t , dt dt whereIusedthat x = z+hinequilibrium. Takingsecondderivatives, d2EB(t) dm(p(t),r(t),y) d2m(p(t),r(t),y) = +t . dt2 dt dt2 IgnoringthecurvaturetermsandusingasecondorderTaylorapproximationIget dEB(0) 1d2EB(0) 1dm(p(0),r(0),y) 1 EB(t) t+ t2 = t2 = Δmt , ≈ dt 2 dt2 2 dt 2 where I used that dm/dtΔt = Δm, with Δt = t 0 and Δm = m(t) m(0). On the other hand, from the − − householdbudgetconstraintm(p(t),r(t),y) = p(t)x(p(t),r(t),y) y T so − − 1dm 1 EB(t) t2 = (1+r(t))2px ζ +ζ +ε ζ t2 p,r D,r D,p p,r ≈ 2 dt −2 (cid:16) (cid:17) ProofofProposition3: Let a tilde denote the monthly counterpart of variables. That is, let T˜ = 12T , let r˜(ϕ) be the monthly i i i mortgageratewith(1+r˜(ϕ))12 = 1+r(ϕ),andsoon. ThenIcanwrotetheproblemofafirst-timebuyeras i i maxu(x,c ,...,c ) i i0 iT˜ i s.t. p x +c +T y +m j0 i i0 0 i0 i0 ≤ c +(1+r˜(ϕ))m +T y +m i1 i i0 1 i1 i1 ≤ . . . c iT˜ i +(1+r˜ i (ϕ))m i,T˜ i − 1 +T T˜ i ≤ y iT˜ i + p jT˜ i (1 − δ˜+π˜)T˜ ix i Letλ betheLagrangemultiplierofthebudgetconstraintinperiodt,thenfromtheFOCwithrespectto t 37

mortgagedebtitfollowsthatλ = λ (1+r˜(ϕ)). Fromwhereitfollowsthat t t+1 i λ = (1+r˜(ϕ)) tλ . (A.1) t i − 0 Ontheotherhand,theEnvelopeTheoremimplythat dV d i ϕ (ϕ) = − λ 0 dp d j0 ϕ (ϕ) x i − T˜ i λ t m i,t − 1 dr˜ d i ( ϕ ϕ) +λ T˜ i (1 − δ˜+π˜)T˜ i dp d jT˜ ϕ i (ϕ) x i t=1 X In addition, by assumption a permanent increase in ϕ imply that dp /dϕ = dp /dϕ dp /dϕ. This j0 jT˜ i ≡ j relationship together with equation (A.1) and that the mortgage unpaid balance m for a fixed mortgage it withmonthlypaymentaandtermT˜ isgivenby i a 1 m = 1 , it r˜ i (ϕ) − (1+r˜ i (ϕ))T˜ i − t ! allowmetorewritetheincidenceas dV i (ϕ) (1 δ˜+π˜)T˜ i dp j (ϕ) = λ 1 − x dϕ − 0  − (1+r˜ i (ϕ))T˜ i  dϕ i   λ m 1 12T i dr˜ i (ϕ) 0 i0 − (1+r i (ϕ))1 1 2 1 − (1+r i (ϕ))1 1 2 (1+r i (ϕ))Ti 1  dϕ Usingthat(1 δ˜+π˜)T˜ i/(1+r˜ i (ϕ))T˜ i (1 r i (ϕ ) δ+π)Ti an − dthatdr˜ i (ϕ)/dϕ =(cid:2) τ y i i Iobtain − equ(cid:3)a tion(12). − ≈ − − − On the other hand, for homeowners everything is the same except for the period 0 budget constraint, whichwillbegivenby p x +c +T y +m + p h , j0 i i0 0 i0 i0 j0 i ≤ whereh isthehouseendowmentofhomeowners. Byassumption x = h sothetermrepresentingtheeffect i i i ofhousepricesinperiod0goesawayandIobtainequation(12). B Description of Mortgage Level Data The data corresponds to McDash Analytics (formerly LPS) and Equifax Credit Risk Insight Servicing (CRISM).TheformerisusedtocalculatetheLTVdistributionbyMSA,whereasthelatterisusedtoidentify first-timehomebuyers. I consider mortgages originated in 2010-2015, focusing on first-lien mortgages with LTV no greater than 150%, fixed rates, and terms of 10, 15, 20, 25, and 30 years. These mortgages are by far the most commonlyusedandrepresentmorethan90%ofthemortgagesoriginatedin2010-2015(TableB.1). ToidentifyfirsttimehomebuyersIuseCRISM,whichmatchescreditbureaudatawithmortgageinformation. Equifax uses a proprietary and confidential algorithm to match mortgage data from McDash /LPS using anonymous characteristics and payment histories. Each credit history is matched with a single borrowerintheLPSdata,includingfirst,second,andrefinancemortgages. Informationisincludedforthelife ofthemortgage,sixmonthsprecedingorigination,andsixmonthsfollowingtermination. BasedonmorethantwentyvariablesLPSandEquifaxrecordsarematchedandassignedamatchscore from 0 (no match) to 0.9 (close to perfect match). I restrict the sample to match scores of 0.8 and above, which according to Equifax corresponds to roughly 90% of mortgages. The data has a one year lag to 38

TableB.1: MortgagesOriginatedin2010-2015inLPS. Description Observations(millions) Mortgagesoriginatedin2010-2015 26.8 LTV>150% 0.7 Non-fixedratemortgages 1.6 Termsotherthan10,15,20,25,and30 0.3 Second-lienmortgages 0.02 Fixedratemortgages10,15,20,25,and30years 24.2 LostinmergewithCRISM 0.07 Inzipcodeswithelasticityinformation 19.5 Withoutinterestrateinformation 0.2 Non-owners 1.7 Finalsample 17.6 Source: McDashAnalyticsandEquifaxCreditRiskInsightServicing. ensurealltheinformationtoperformthematchispresentandavoidfalsepositives,soIrestrictthesample to2010:1-2015:4. First time home buyers are identified using two filters. The first filter is that the mortgage purpose is a purchase, as specified by variable ‘purpose_type’ in LPS. The second filter, using data from CRISM, is that neither the primary nor the secondary borrower associated to the mortgage record (‘loan_id’) has a mortgage open or a history of a previous mortgage over the six months previous to origination. This filter considers whether any of the following mortgage accounts was previously open: largest first mortgage (‘fm_lrg_opendt’), second largest first mortgage (‘fm_2lrg_opendt’), largest closed-end second (‘ces_lrg_opendt’),secondlargestclosed-endsecond(‘ces_2lrg_opendt’),largesthomeequitylineofcredit (‘heloc_lrg_opendt’),andsecondlargesthomeequitylineofcredit(‘heloc_2lrg_opendt’). Mortgages from LPS are assigned to MSA/NECMA divisions using ZIP codes. I map ZIP codes to counties in these divisions assuming that a ZIP code belongs to a county when the ratio of residential addresses in that county to the total number of residential addresses in the ZIP code is at least 50%. Since Saiz(2010)elasticitiesareforMSA/NECMAdivisionsusing1999codes,Iconsiderthecountycomposition fortheseregionsin1999. 39

Tables and Figures Table1: DescriptiveStatisticsofBorrower-levelCharacteristics. Description Mean Std.Dev. Min Max Nominalmortgagerate,i (percent) 4.16 0.63 0.001 18.00 i Effectivemortgagerate,r τ i (percent) 1.12 0.47 -2.00 11.50 i y i − Usercost(percent) 4.9 0.5 1.8 15.3 Mortgage-ratedemandsemielasticity,ζ -15.3 1.5 -41.1 -4.9 D,r,i LTVratio(percent) 77.2 21.3 0.0 150.0 Mortgageterm, T (years) 26.1 6.6 10.0 30.0 i Housevalue(dollars) 319,351 316,837 1,307 100,000,000 First-timebuyer 0.185 0.389 0.000 1.000 Notes: Author’scalculationsbasedonMcDashAnalyticsandEquifaxCreditRiskInsightServicing. Table2: MortgageRateandLTVRatioforBuyersandOwnersbyMortgageTerm. Term Numberof InterestRate(percent) LTVRatio (percent) (years) Mortgages Mean Min Max Mean Min Max Owners 14,331,587 4.13 1.00 18.00 74.3 0.0 150.0 10 541,909 3.59 1.00 18.00 46.9 0.2 150.0 15 3,035,368 3.66 1.00 13.38 63.7 0.0 150.0 20 930,503 4.18 1.50 13.55 71.2 0.6 150.0 25 144,844 4.41 2.00 12.19 79.9 0.4 150.0 30 9,678,963 4.29 1.00 18.00 79.4 0.0 150.0 Buyers 3,263,089 4.31 0.00 11.12 89.8 0.0 150.0 10 6,049 3.56 1.00 10.28 53.0 0.5 107.0 15 120,640 3.60 1.88 11.12 73.2 0.0 117.5 20 12,786 4.18 2.52 10.87 74.1 0.7 125.4 25 1,655 4.38 2.56 6.13 85.0 22.2 108.9 30 3,121,959 4.34 0.00 10.99 90.6 0.0 150.0 All 17,594,676 4.16 0.00 18.00 77.2 0.0 150.00 Notes: Author’scalculationsbasedonMcDashAnalyticsandEquifaxCreditRiskInsightServicing. 40

Table3: PriceandLTVMultipliersforBuyersandOwnersbyMortgageTerm. Term PriceMultiplier LTVMultiplier (years) Mean Min Max Mean Min Max Owners -0.30 -0.77 -0.01 11.9 3.7 16.2 10 -0.63 -0.77 -0.19 5.0 3.7 5.2 15 -0.50 -0.68 -0.15 7.3 5.7 7.9 20 -0.36 -0.55 -0.08 9.6 7.0 10.4 25 -0.27 -0.43 -0.05 11.7 8.5 12.9 30 -0.21 -0.46 -0.01 14.0 7.2 16.2 Buyers 0.78 0.23 0.96 13.7 4.3 16.9 10 0.37 0.23 0.63 5.0 4.3 5.2 15 0.50 0.39 0.80 7.4 6.0 7.7 20 0.64 0.53 0.88 9.6 7.6 10.1 25 0.73 0.61 0.81 11.7 10.9 12.6 30 0.79 0.42 0.96 13.9 10.1 16.9 All -0.10 -0.77 0.96 12.2 3.7 16.9 Notes:PriceandLTVmultiplierscorrespondstothecoefficientsthatmultiplythepriceeffectsandthemortgage rate effect (LTV) in the expression for the incidence on first-time buyers and homeowerns in Proposition 3. Author’scalculationsbasedonMcDashAnalyticsandEquifaxCreditRiskInsightServicing. Table 4: Descriptive Statistics of Key Economic Parameters and Effect of Mortgage Subsidies by MSA. Description Mean(1) Std.Dev.(1) Min Max Pricehousesupplyelasticity, ε (Saiz,2010) 1.49 0.90 0.60 12.15 S,p,j Mortgageratehousedemandsemielasticity, ζ -15.4 0.1 -16.1 -15.1 D,r,j Valueofthehousingstock(millions) 128,105 117,780 226 405,673 Mortgageratepricesemielasticity, ζ -6.85 1.94 -9.60 -1.18 p,r,j HousepricechangeeliminationMID,Δp /p (percent) -6.93 1.97 -9.83 -1.18 j j Comp. pricechangeeliminationMID,Δpˆ /p (percent) -6.33 2.04 -9.24 -0.96 j j Credithousepriceelasticity,ε 0.30 0.07 0.06 0.44 p,M,j Averageincidence(percentofhousevalue) -10.3 0.7 -12.0 -8.6 Totaldollarvalueofincidencetohouseholds(millions) -12,834 11,569 -39,358 -24 Averageefficiencyloss(basispointsofhousevalue) -4.6 1.0 -7.3 -3.2 Totaldollarvalueofefficiencyloss(millions) -52.1 44.3 -162.0 -0.1 Notes: (1)Totalmetropolitanareahouse-value-weightedmeanandstandarddeviations. Author’scalculationsbasedonMcDashAnalytics,EquifaxCreditRiskInsightServicingandSaiz(2010). 41

Table5: IncidenceofMortgageSubsidiesforBuyersandOwnersbyMortgageTerm. TotalIncidence Incidenceofhighermortgagerates Term (percentofhousevalue) (percentofhousevalue) (years) Mean Min Max Mean Min Max Owners -11.5 -36.4 -0.3 -9.6 -36.2 0.0 10 -6.0 -13.1 -1.1 -2.1 -10.5 0.0 15 -7.3 -21.3 -1.1 -4.3 -20.0 0.0 20 -9.4 -24.9 -1.7 -7.1 -23.9 -0.1 25 -11.9 -29.4 -1.0 -10.2 -28.9 -0.1 30 -13.3 -36.4 -0.3 -11.9 -36.2 0.0 Buyers -8.5 -23.6 8.2 -13.3 -27.2 0.0 10 -0.1 -6.2 4.0 -2.3 -8.9 0.0 15 -1.9 -9.1 6.5 -4.8 -12.4 0.0 20 -3.5 -12.1 4.8 -7.4 -16.0 -0.1 25 -6.6 -13.9 3.6 -10.9 -16.8 -2.6 30 -8.8 -23.6 8.2 -13.7 -27.2 0.0 All -10.9 -36.4 8.2 -10.2 -36.2 0.0 Notes:Totalincidenceconsidersbothhousepriceeffectsandlowermortgagerates.SeeProposition3fordetails. Author’scalculationsbasedonMcDashAnalytics,EquifaxCreditRiskInsightServicingandSaiz(2010). Table6: CompensatedMortgageDemandDistortionsofMortgageSubsidiesforBuyersandOwnersbyMortgageTerm. Housepriceeffect Mortgagerateeffect Totaleffect Term (percentofhousevalue) (percenthousevalue) (percenthousevalue) (years) Mean Min Max Mean Min Max Mean Min Max Owners 5.8 1.0 9.9 -15.7 -22.5 -7.3 -9.8 -19.0 1.9 10 5.6 1.0 9.8 -14.8 -22.3 -7.3 -9.2 -18.8 -1.8 15 5.6 1.0 9.9 -14.9 -21.2 -7.3 -9.4 -19.0 -1.3 20 5.7 1.0 9.7 -15.8 -21.4 -9.5 -10.1 -18.3 -3.5 25 5.7 1.0 9.8 -16.0 -21.0 -11.3 -10.3 -18.4 -4.3 30 5.9 1.0 9.9 -15.9 -22.5 -7.3 -10.0 -18.9 1.9 Buyers 5.6 1.0 9.7 -15.9 -20.6 0.0 -10.3 -17.1 3.4 10 5.4 1.0 9.5 -14.7 -20.3 -7.3 -9.3 -16.5 -2.4 15 5.3 1.0 9.7 -14.8 -20.5 -10.9 -9.5 -17.0 -2.1 20 5.4 1.0 9.4 -15.7 -20.5 -12.7 -10.3 -16.2 -4.5 25 5.3 1.0 9.4 -16.0 -17.9 -12.8 -10.7 -15.5 -5.4 30 5.6 1.0 9.7 -16.0 -20.6 0.0 -10.4 -17.1 3.4 All 5.8 1.0 9.9 -15.7 -22.5 0.0 -9.9 -19.0 3.4 Notes: See equation (15) for details. Author’s calculations based on McDash Analytics, Equifax Credit Risk InsightServicingandSaiz(2010). 42

Table7: EfficiencyLossfromMortgageSubsidiesforBuyersandOwnersbyMortgageTerm. Term Basispointsofhousevalue Dollarvalue (years) Mean Min Max Mean Min Max Owners -5.0 -32.6 0.2 -146 -75,790 13 10 -4.1 -32.6 -0.2 -116 -3,404 0 15 -4.2 -24.1 -0.2 -123 -39,175 -1 20 -5.1 -25.0 -1.1 -132 -7,800 -6 25 -5.5 -22.6 -1.1 -139 -36,727 -7 30 -5.2 -30.9 0.2 -157 -75,790 13 Buyers -5.4 -20.3 0.2 -103 -78,053 12 10 -4.1 -18.5 -0.4 -86 -1,245 -1 15 -4.2 -20.3 -0.5 -97 -41,317 -4 20 -5.3 -19.1 -1.7 -116 -13,597 -5 25 -5.7 -10.7 -1.9 -94 -740 -10 30 -5.5 -20.3 0.2 -103 -78,053 12 All -5.1 -32.6 0.2 -148 -82,599 10 Notes: Seeequation(14)fordetails. Author’scalculationsbasedonMcDashAnalytics,EquifaxCreditRiskInsightServicingandSaiz(2010). 43

Figure1: IncidenceofSubsidyinMortgageandHousingMarkets. (a)MortgageMarket (b)HousingMarket r p S p(t) dp = E 2 ∂S ∂D p(0) ∂p−∂p r S 1 t D(t) M(t) r t − D(0) M(0) 0 dt∂Mdr M 0 E = dt∂Ddr D ∂r dt − ∂r dt D ( τ ) Figure2: EfficiencyCostofMortgageSubsidiesinMortgageandHousingMarkets. (a)MortgageMarket (b)HousingMarket r p S a b p(t) p(0) d c a b c r S e f t D(t) r t e d M(t) − D(0) M(0) 0 m(0) m(t) M 0 x(0) x(t) D 44

Figure3: HousePriceEffectofEliminatingMIDbyMetropolitanArea. 0% -2% -4% estimate approximation -6% -8% -10% 0 2 4 6 8 10 12 Price House Supply Elasticity Notes: Estimates corresponds to the estimated values using equation (11). The approximation corresponds to 15.4/(ε +1),whereε isthehousepricesupplyelasticity. S,p,j S,p,j − Author’scalculationsbasedonMcDashAnalyticsandSaiz(2010). Figure4: AverageIncidencefromtheEliminationofMIDandPriceHouse-SupplyElasticity. -4% Buyers Owners -6% -8% -10% -12% -14% 0 2 4 6 8 10 12 Price House Supply Elasticiy Notes: Incidenceestimatesmeasuredaspercentofhousevalue, φ p /Δp φ LTV ,equation(12). p,i j j m,i i − − Author’scalculationsbasedonMcDashAnalytics,EquifaxCreditRiskInsightServicing,andSaiz(2010). 45

Figure5: AverageIncidenceforFirst-TimeBuyersofEliminatingMIDbyMSA Notes: Welfareismeasuredinpercentofhousevalue. Author’scalculationsbasedonMcDashAnalytics,EquifaxCreditRiskInsightServicing,andSaiz(2010). Figure6: AverageIncidenceforFirst-TimeBuyersofEliminatingMIDbyState Notes: Welfareismeasuredinpercentofhousevalue. Author’scalculationsbasedonMcDashAnalytics,EquifaxCreditRiskInsightServicing,andSaiz(2010). 46