It is hard to miss the important role the housing sector plays in the macroeconomy.

Transcription

1 Maroeonomi Models wit Heterogeneous Agents and Housing KARSTEN JESKE Te autor is a resear eonomist and assistant poliy adviser in te maropoliy group of te Atlanta Fed s resear department. He tanks William Roberds and Ellis Tallman for elpful omments. It is ard to miss te important role te ousing setor plays in te maroeonomy. In 24 te value of real estate eld in ouseold portfolios amounted to over $17 trillion, or 143 perent of annual nominal gross domesti produt (Board of Governors 25). Tis amount is larger tan all orporate equity eld diretly, troug mutual funds and pension funds ombined. Closely linked to ousing is te mortgage market, wi now totals over $7.5 trillion. On te average ouseold s balane seet, a ome mortgage is te largest item on te liability side, easily dwarfing onsumer redit, wi inludes redit ard debt and ar loans, by a fator of almost four to one. Maroeonomists, terefore, ave spent onsiderable effort on inorporating ousing and ousing finane into maroeonomi models. In understanding te effets tat ousing and te various related government poliies ave on ouseolds in partiular and te eonomy in general, reognizing te eonomi and demograpi diversity among ouseolds is ruial. Net wort, ousing value (gross and net of mortgages), and inome vary substantially among ouseolds not only by age but witin age groups. Maroeonomists tus employ models alled eterogeneous agent models tat an aommodate tis diversity. Tis artile is a progress report on were tis line of resear urrently stands. At te Federal Reserve Bank of Atlanta onferene Housing, Mortgage Finane, and te Maroeonomy in May 25, several papers sared a ommon framework: namely, using maro models wit eterogeneous agents. 1 Tis artile disusses four of te onferene papers, te issues te researers explored, te progress tey made, and te allenges tat lie aead. Te paper by Fang Yang (25) deals wit te life-yle beavior of ousing versus nonousing onsumption. Her objetive is to build a model to aount for two peuliar features of onsumption over te life yle. Te paper by Wenli Li and Rui Yao (25) studies te effet of ouse prie anges on te maroeonomy. Tey point out tat wile ouse prie anges may ECONOMIC REVIEW Fourt Quarter 25 39

2 Table 1 Homeownersip Rates by Age Group Age group Homeownersip rates (perent) Note: Te unonditional mean aross all ages is 67.7 perent. Soure: Board of Governors of te Federal Reserve System, Survey of Consumer Finanes 21 not ave a uge effet on te eonomy as a wole, different ouseolds in partiular tose of different age and asset oldings will be affeted very differently by ouse prie anges. An eonomi model wit eterogeneous agents is terefore neessary to measure te effets a prie inrease migt ave on agents wit different inomes and asset oldings. Te paper by Mattew Cambers, Carlos Garriga, and Donald Slagenauf (25b) asks ow different ouseolds are affeted by te availability of different mortgage ontrats: most importantly, wi mortgage ontrat is most suessful in allowing younger ouseolds (tat tend to ave less savings and tus migt ave trouble oming up wit te neessary down payment) to purase a ouse. Te paper by Karsten Jeske and Dirk Krueger (25) addresses te important poliy question of weter it is desirable to subsidize mortgage interest rates. One important subsidy is te impliit federal guarantee for government-sponsored enterprises (GSEs). Jeske and Krueger examine te distributional effet of su a subsidy. Tey find tat mostly ig-inome and ig-net-wort ouseolds benefit wile, bot in terms of welfare and omeownersip, low-inome and low-net-wort ouseolds will not benefit at all. Mirolevel Data on Housing and Mortgage Finane Trying to answer te questions mentioned above requires models tat are detailed enoug to aommodate a ousing market and, most importantly, eterogeneity among ouseolds. Tis eterogeneity is te ruial ingredient of any model attempting to determine te distributional effets of ousing poliy. Before examining te models, I first introdue some empirial fats oming out of mirolevel data to demonstrate ow te aggregate numbers on inome, real estate, and mortgage debt are distributed aross te population. Aggregate real estate values are approximately 143 perent of aggregate annual inome. But inome, real estate wealt, mortgage debt, and net wort are very unevenly distributed bot aross and witin age groups. Te data set used in tis artile s omputations is te Survey of Consumer Finanes (from te Federal Reserve Board) for te year 21, wi, among oter tings, ollets data on onsumer inome and balane seets. Table 1 sows omeownersip rates by age group ompared to te unonditional omeownersip rate (tat is, over all age groups) of 67 perent. Te omeownersip rate is te lowest, at about 3 perent, for ouseolds eaded by persons below te age of twenty-nine. Te rate inreases steeply over te next two age oorts, peaking at more tan 8 perent for te sixty to sixty-nine age group and moderating sligtly after tat. Not all omeowners are equal, as Table 2 sows. Not only do fewer young ouseolds own a ome, but teir omes tend to be smaller tan tose of older ouseolds. Te average omeowner inreases real estate oldings from about $1, until reaing is or er fifties, wen e or se olds about $216, in real estate, and ten gradually downsizes te value of te primary residene. Mortgage debt also displays a ump-saped profile, toug te peak ours in te tirty to tirty- 4 ECONOMIC REVIEW Fourt Quarter 25

3 nine age group. Tis pattern seems to be onsistent wit te following senario: Houseolds upgrade te size and obviously te value of te ouse tey are living in. Early in life most of te upgrading is finaned wit larger mortgages. Homeowners aged forty and older ten tend to finane upgrades out of savings rater tan larger mortgages. In omeownersip rates, even witin age groups tere is substantial variation. Te first panel of Figure 1 plots rates by inome quintiles witin ea age group. Te art sows tat omeownersip is assoiated wit ig inome, wit ouseolds in te bottom quintile aving below-average ownersip rates aross all age groups. Te seond panel of Figure 1 Table 2 Home Values and Mortgage Debt by Homeowners Age Group Value of Mortgage on primary primary Age group residene residene Note: Values and debt sown in tousands of dollars. Soure: Board of Governors of te Federal Reserve System, Survey of Consumer Finanes 21 yields te same qualitative pattern for net wort, but te differenes between ouseolds wit low and ig net wort ouseolds are even more severe. Houseolds in te bottom quintile ave only a 3 perent omeownersip rate even in te sixty to sixty-nine age group. In ontrast, for all ages after forty, te seond troug fift quintiles display omeownersip rates above te national average of 67 perent. 2 Vast differenes also exist between omeowners and renters. Te first panel of Figure 2 plots median inome by age group for te wole population and for omeowners versus renters. For te population as a wole, ouseold inome inreases from just below $3, for te youngest age oort to almost $6, for fifty to fifty-nine age group, after wi it delines to about $25, for te seventy and over age group. Homeowners tend to ave mu larger median inomes tan renters; te median omeowner makes rougly twie as mu as te median renter. Moreover, te ump in te inome proess is more pronouned for omeowners. Te same general pattern olds for average inomes (see te seond panel of Figure 2). Te ontrasts between omeowners and renters are even more obvious in terms of net wort positions. Te tird and fourt panels of Figure 2 plot median and average net wort, respetively, by age group. Houseolds aumulate savings during teir working years, moreso tan does te population as a wole. Te median omeowner aumulates more tan $2, during is or er lifetime. Beause te net wort distribution is so skewed, average net wort is onsiderably larger: Te average omeowner aumulates more tan $8, over a lifetime. Te median renter, owever, as negligible net wort onsistently under $1, for all age groups. In fat, 5 perent of all renters ave essentially zero net wort. Even average net wort for renters is below $1, in all age groups. 1. Te four papers introdued ere as well as te program and te remaining papers are available on te Atlanta Fed s Web site at under News & Events, Conferenes, Te utoff for te lowest inome quintile is $15,4 in te youngest age group, inreases to $26,7 for te fifty to fifty-nine age group, and ten delines to $11,3 for ouseolds aged seventy and above. Te utoff for net wort is zero for te youngest ouseolds and inreases to $39,1 for ouseolds aged seventy and above. ECONOMIC REVIEW Fourt Quarter 25 41

4 Figure 1 Homeownersip Patterns by Age, Inome, and Net Wort Homeownersip ratios by inome quintile and age group 1. Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile Age Homeownersip ratios by net wort quintile and age group Age Soure: Board of Governors of te Federal Reserve System, Survey of Consumer Finanes 21 In summary, ouseolds differ substantially, bot between and witin age groups, in teir net wort positions and asset alloations. As one would expet, wen ouseolds are young and middle-aged tey aumulate savings for retirement, and one form of savings is real estate. Even witin age groups, tere is onsiderable eterogeneity of ouseolds. Inomes and espeially net wort vary substantially aross age groups and between renters and omeowners. Wen trying to answer questions su as Wat is te effet of inreasing ouse pries? or Sould we subsidize mortgage interest rates? one sould take into aount tat different ouseolds will be affeted very differently by anges in ouse pries or government poliies. An inrease in ouse pries migt be benefiial to existing omeowners, but renters may not be affeted at all or, even worse, migt suffer if rental rates inrease. Likewise, subsidizing mortgage interest for example, troug mortgage interest tax-dedutibility or 42 ECONOMIC REVIEW Fourt Quarter 25

5 Figure 2 Inome and Net Wort by Age Group 8, 6, Median inome by age group All Homeowners Renters 12, 1, 8, Average inome by age group 4, 6, 2, 4, 2, Age Age , 2, Median net wort by age group 8, Average net wort by age group 15, 6, 1, 4, 5, 2, Age Age Soure: Board of Governors of te Federal Reserve System, Survey of Consumer Finanes 21 government guarantees on mortgage lenders migt mostly benefit existing omeowners, wo are well off already beause tey tend to ave above-average inome and net wort. Tis onsideration reinfores te importane of studying models wit eterogeneous agents to aount for te effets tat poliy as on agents wit different demograpi and eonomi bakgrounds. A Generi Model wit Housing A key ingredient in te model is te life-yle pattern, wi implies two important features te model must inlude. First, it must inlude te ump-saped earnings profile tat te average ouseold experienes, as sown in Figure 2. Seond, te model must generate a realisti life span, wi means ouseolds of different age groups fae different mortality risk. Around te trend pat of life-yle earnings, a ouseold reeives an unertain stream of inome over its life yle and ten tries to maximize its lifetime disounted utility subjet to a budget onstraint. A ouseold reeives utility from onsumption of bot nonousing goods and ousing goods. Inome omes bot from labor and investment inome. Te maximization problem of a ouseold of age t in reursive form is ten defined in te following way: Te value of starting period t wit assets a and labor produtivity y is te maximum value of urrent utility plus expeted disounted period t + 1 value as a funtion of future assets and future labor produtivity. = + ( + ) V a, y max u, ρβev a, y, t t t 1 b,, ECONOMIC REVIEW Fourt Quarter 25 43

6 subjet to a number of onstraints: + + b = wy + a; a = (1 δ ) + (1 + r)b ; b B ;,. In te objetive funtion I use a disount fator β ommon to all age groups as well as additional disounting in te form of ρ t, wi is te onditional survival probability tat depends on age. Te first onstraint is te budget onstraint. Labor inome measured as wage rate w times labor produtivity y plus assets a an be spent on nonousing onsumption, ousing, and oter savings b. Te seond onstraint speifies te next period s assets a as te sum of ousing stok net of depreiation and maintenane ost (1 δ ) and oter savings times te gross interest rate (1 + r)b. Te tird ondition is a borrowing onstraint tat speifies a lower bound on asset oldings in oter words, an upper bound on borrowing. Finally, te fourt ondition states tat bot types of onsumption ave to be nonnegative. Tis model losely resembles tat of Aiyagari (1994), wit some distinguising features: two types of onsumption, a very partiular borrowing onstraint, and a life-yle omponent. 3 Denote u and u te derivatives of te utility funtion wit respet to nonousing onsumption and ousing, respetively. Assuming for now tat te borrowing onstraint does not bind, we an immediately derive two optimality onditions, also alled Euler equations: (1) 1= ρβ t E u, ( 1+ r ), and u, (2) 1= ρβe u, u, ( 1 δ t )+. u, u, Te first ondition is ompletely standard: Te bond return, weigted by te intertemporal marginal rate of substitution (also alled te priing kernel) and disounted by ρ t β, equals one. Te seond ondition as a similar struture but also inorporates today s marginal rate of substitution between ousing and nonousing onsumption. Bot onditions guarantee optimality by ensuring tat no utilityimproving substitution exists between eiter onsumption and savings or onsumption and ousing. Next, ombine te two Euler equations into u, u, r = δ. r Tis ondition states tat te marginal rate of substitution between onsumption and ousing equals te ost of one unit of ousing omputed as te depreiation rate plus te opportunity ost of ousing measured as te bond rate r. I also divide by 1 + r beause te payoff from ousing is in te urrent period wile te bond pays off in te next period. Te rigt-and side is also alled te user-ost formula for 44 ECONOMIC REVIEW Fourt Quarter 25

7 ousing, and if tere were a rental market in tis model, (r + δ )/(1 + r) would be te rental rate per unit. To put some struture into te model, assume tat te utility funtion takes te following form: η η η + u (, ) = θ 1 θ 1. 1 σ Tis is te form normally used in te literature. Te parameter σ is te risk aversion oeffiient, η determines te substitutability between ousing and nonousing onsumption, and θ is a weigting parameter for ow mu te ouseold values nonousing onsumption. How good is tis model in aounting for te observed data? From te model, one an dedue two impliations. First, te user ost formula for ousing an be written as r + δ = 1 + r tat is, ousing over nonousing onsumption stays onstant over te life yle. Next, from te first Euler ondition (equation [1]) and te fat tat te / ratio stays onstant over time one dedues tat σ 1 θ θ E 1 =. ρβ 1 + r t 1 η 1, 1 σ Normally it will be te ase tat β(1 + r) < 1. 4 Sine te survival probability ρ t is less tan one, te rigt-and side is greater tan one. Sine σ >, te seond impliation of te model as now been sown: Witout borrowing onstraints, onsumption dereases over te life yle, at least in expeted terms. Tis result makes perfet sense. Aording to te permanent inome ypotesis, te mean pat of earnings is irrelevant in te absene of borrowing onstraints; all tat ounts is te disounted expeted pat of earnings. If te interest rate is lower tan te rate of disounting, ouseolds will oose a dereasing onsumption pat toug tere may be flutuations around tis derease beause of anges in labor produtivity. Life-Cyle Patterns of Housing Consumption Te four onferene papers mentioned in te introdution are all extensions of tis same basi model. Yang (25) observes tat bot impliations of te model, 3. Note tat te timing onvention ere is sligtly different from tat in te Yang (25) paper. Speifially, I assume ere tat te ousing oie enters today s utility instead of tomorrow s. Tis assumption makes te problem sligtly easier to andle beause agents pik ousing goods for te same period wen teir inome unertainty is revealed. Terefore, ousing is a riskless asset, wi makes te optimality onditions sligtly more tratable. Jeske and Krueger (25), for instane, use tis timing onvention. In teir paper ousing investment will neverteless be risky beause of unertain depreiation. In te Yang paper, on te oter and, ouseolds pik for te following period wen te future produtivity sok y is not known yet, a oie tat makes ousing a risky asset. 4. Eonomists usually pik te disount fator β so as to mat te model s apital-to-output ratio to tat in te data. One an sow tat for te realisti range of te remaining parameters of te model te equilibrium interest rate r implied by tis apital-to-output ratio satisfies β(1 + r) < 1. ECONOMIC REVIEW Fourt Quarter 25 45

8 Figure 3 Consumption over te Life Cyle 35, and over te life yle 8, 3, 7, Consumption (nonousing) 25, 2, 15, 1, 5, Housing onsumption Consumption (nonousing) 6, 5, 4, 3, 2, 1, Housing onsumption Age / (normalized to 1 at age 45) 2 Ratio Age 9 8 Implied exess user ost ξ 6 4 Perent Age 9 Soure: Yang (25) (first panel); autor s alulations using data from Yang (25) (seond and tird panels) 46 ECONOMIC REVIEW Fourt Quarter 25

9 a dereasing onsumption profile and a onstant ousing versus nonousing onsumption ratio, are at odds wit te data. Te first panel of Figure 3 sows average ouseold onsumption on bot ousing and nonousing goods by age group (of te ead of ouseold). Notie tat nonousing onsumption inreases from about $23, per year for ouseolds aged twenty to an average of $32, per year for age forty-five and ten dereases after tat. 5 Contrast tis observation to te beavior of ousing onsumption, wi displays an even sarper inrease up to about age fifty-five but ten does not drop as sarply as nonousing onsumption. Not surprisingly, ten, te / ratio is not onstant but rater inreases over te life yle, as sown in te seond panel of Figure 3. Can te inrease in te / ratio be aounted for witout radial anges to te model? One ould imagine tat younger agents fae a iger depreiation rate of ousing or a iger interest rate. 6 Formally, assume tat te onsumption ratio at age t now takes te following sape: (3) t r + δ 1 θ = + t ξ + r t 1 θ 1 η 1, were ξ t generates differenes in ouseolds subjetive user ost beause of differenes in depreiation and interest rates during te life yle. Is it possible to quantitatively mat te observed ratio wit realisti values of ξ t? How positive would ξ t ave to be for young agents and ow negative for old agents in order to generate te / funtion observed in te data? To ompute tis ost, I normalize te impliit ost of a forty-five-year-old ouseold to zero and bak out te ξ t from equation (3) using te observed /ratios. Te tird panel of Figure 3 sows tis impliit ost ξ t over te life yle. To aount for te observed onsumption patterns, older ouseolds would ave to find ousing around 3 perentage points eaper tan tose age forty-five do, and, likewise, younger ouseolds would ave to find ousing 6 perentage points more expensive. It is diffiult to justify differenes of tis magnitude almost 1 perentage points differene between age eigty-five and age tirty entirely troug different borrowing osts or depreiation rates. Yang s paper an be tougt of as finding oter explanations to generate tis large differene in subjetive user osts. Speifially, Yang introdues two ruial ingredients into te model to aount for te features observed in te data binding borrowing onstraints and transation osts and sows tat tey an quantitatively aount for te inrease in te onsumption ratio. 7 One an easily add tese two features into te model above. First, assume tat B = (1 γ), tat is, te ouseold an borrow only up to a maximum of a ertain fration of te ouse value. Parameter γ an be viewed as te minimum down payment on a ouse. Tis onstraint ensures tat wit a ump-saped labor inome profile (see te first and seond panels of 5. Yang is not te first eonomist to point out tis feature. Earlier work on te onsumption profile over te life yle inludes Blundell, Browning, and Megir (1994); Attanasio and Browning (1995); Attanasio and Weber (1995); and, more reently, Gourinas and Parker (22) and Fernandez- Villaverde and Krueger (22, 24). 6. For example, one ould justify tis assumption wit te fat tat ouseolds wit a younger ead tend to ave a larger family size, and tus more persons per square foot ause a iger depreiation rate. 7. Yang introdues even more ingredients, su as a bequest motive and a soial seurity system; owever, to illustrate te key intuition, a borrowing onstraint and transation osts are suffiient. ECONOMIC REVIEW Fourt Quarter 25 47

10 Figure 2) agents will be onstrained in teir borrowing early in life. In oter words, ouseolds annot raise teir onsumption to te desired level derived from te permanent inome model witout violating teir borrowing onstraint. Tis restrition binds beause inome is low early in life. Seond, adjusting te level of ousing inurs transation osts. Assume tat if a ouseold ad a ousing level of old last period and wants to ange it to today, it as to pay a transation ost of ψ( old, ). Assume for now tat te transation ost satisfies te following onditions: ψ( old, ) ; ψ( old, old ) = ; Tat is, te transation ost funtion takes on nonnegative values, tere is no ost of leaving te size of te ouse unanged, 8 and te transation ost funtion is onvex. 9 In tis eonomy te optimality ondition between ousing and nonousing onsumption beomes (4) 2 ψ ( ), old. 2 u, r + 1 µ δ + µ u, u, 1 + r u, =,, E u γ + ψ (, old )+ ρβ ψ 2 t 1( ), u, were µ is te Lagrange multiplier on te borrowing onstraint and ψ 1 and ψ 2 are derivatives of te transation ost funtion wit respet to te first and seond argument, respetively. Tis formula is te same one as before, linking te marginal rate of substitution between ousing and nonousing onsumption wit te ost of purasing ousing. Instead of te user-ost formula of ousing, one now as a weigted average between te previous user ost (r + δ )/(1 + r) and te down payment ratio γ, were te weigt on γ is proportional to te Lagrange multiplier of te borrowing onstraint. 1 One an interpret te first two terms on te rigt-and side of equation (4) as te subjetive user ost of ousing (net of transation ost). Te more te borrowing onstraint binds, te iger te ousing ost for te ouseold. Tis result is intuitive: If te onstraint binds, te only way to afford more ousing is to redue urrent onsumption, wi omes at a ost of γ units of onsumption for every unit of ousing purased. Te remaining two terms ome from te transation ost funtion. Tey determine te marginal ost of ousing anges tat is, adjusting ousing onsumption from old to in te urrent period and adjusting from to in te following period. Using te same elastiity of substitution between te two onsumption types as in te Yang paper tat is, η = one ten obtains te following onsumption ratio: (5) 1 r = θ µ + δ µ 1 + θ u r u (, ) 1 +, γ + ψ E u, 2 (, old )+ ρβ ψ, t u, 1. 1 One way to interpret Yang s paper is tat te additional terms in te square brakets tend to derease in age; tus, inversely, te onsumption ratio inreases in age. For young agents tere are two annels: te borrowing onstraint and te transation ost. A binding borrowing onstraint as te same effet as a iger interest rate. 48 ECONOMIC REVIEW Fourt Quarter 25

11 Sine te subjetive user ost for young agents (before transation osts) is iger tan te ost (r + δ )/(1 + r) for unonstrained agents, te / ratio will inrease just as in te data beause te subjetive user ost dereases wen te borrowing onstraint is relaxed over te lifetime. So far, te larger impliit ousing ost auses young ouseolds to substitute away from ousing and into nonousing goods, even witout any transation osts. Transation osts add to tis effet. Sine te borrowing onstraint indues a umpsaped profile for onsumption (bot ousing and nonousing), young agents ave te tendeny to inrease teir. Consequently, te following will be true for any tree onseutive ousing onsumption values: old. In tat ase ψ 2 ( old, ) In understanding te effets tat ousing and related government poliies ave on ouseolds and te eonomy, reognizing te eonomi and demograpi diversity among ouseolds is ruial. ψ 1 (, ). Sine te seond term is disounted by ρ t βe[u (, )/u (, )], one would expet te total effet to be positive. Bot effets from borrowing onstraints and transations osts explain wy for young agents / an be lower tan te ratio derived in te fritionless version above. For older agents te borrowing onstraint will likely no longer bind. Instead, all of te ation omes from te transation ost terms. Te reverse of te argument for young agents applies. Older agents want to redue onsumption beause of disounting due to lower survival probabilities. But piking old implies ψ 2 ( old, ) ψ 1 (, ). Again, te last term in equation (5) is subjet to disounting, so te ψ 2 ( old, ) dominates, making te / ratio iger tan in a fritionless eonomy. Te bottom line of te Yang (25) paper is tat by adding te two modifiations te model mates te data qualitatively and quantitatively. Te Effet of House Prie Canges Li and Yao (25) study te effet of ouse prie anges on te maroeonomy. Te first panel of Figure 4 displays te U.S. ouse prie index, adjusted for inflation by deflating te nominal ouse prie index wit tree different ommonly used measures of inflation: namely, te onsumer prie index (CPI) and te deflators for bot gross domesti produt (GDP) and personal onsumption expenditures (PCE). Between 1975 and 1995 real ouse pries more or less stagnated if deflated by te CPI but moderately inreased by about 2 perent over te twenty years if deflated by GDP or PCE. 11 Of ourse, te pat of ouse pries was not monotoni. A deline in real ouse pries between 1979 and 1983 resulted from te ig inflation rates in te late 197s and te two reessions in te early 198s. Likewise, te 1991 reession 8. Reall tat in addition tere is maintenane ost, but tat enters te budget onstraint separately. 9. Te sape of transation osts is sligtly different from tat used in Yang s paper, or most of te literature for tat matter. Using te onvex transation ost, owever, it is easier to explain te intuition using te Euler onditions beause te neessary optimality onditions will also be suffiient. 1. Notie tat in equilibrium it is always true tat γ > (r + δ )/(1 + r). If tis were not te ase, one an sow tat ousing investment is stritly more profitable tan purasing bonds, and nobody would ever old positive amounts of bonds. Tis outome annot be an equilibrium outome sine te bond market as to lear tat is, bond savings ave to equal mortgage lending. 11. Tis result makes perfet sense: GDP and PCE deflators tend to be sligtly lower tan te CPI beause te CPI omputes prie anges of a fixed basket wile in te GDP and PCE onsumers obviously substitute into eaper goods, tus lowering tose two indexes. ECONOMIC REVIEW Fourt Quarter 25 49

12 Figure 4 Consumption over te Life Cyle Real ouse prie index 2 18 Index (1975 = 1) Wit PCE deflator Wit GDP deflator Wit CPI House prie index growt rates Perent (year-over-year) 12 Wit PCE deflator Wit GDP deflator Wit CPI Soure: Offie of Federal Housing Enterprise Oversigt, Bureau of Labor Statistis, and Bureau of Eonomi Analysis took a bite out of ouse pries, and even during te first four years of te reovery, ouse pries stagnated. After 1995, owever, ouse pries staged an astonising inrease of almost 5 perent if deflated by CPI and 6 perent if adjusted by eiter of te deflators. Moreover, no deline ourred in ouse pries around te 21 reession, and no stagnation ourred after te reession. On te ontrary, as te seond panel of Figure 4 sows, year-over-year growt rates in real ouse pries even aelerated during and after te reession to a reord pae of 8 to 1 perent in 24. Wat is te effet of tese ouse prie flutuations, espeially te large run-up in pries sine te mid 199s? In a model witout eterogeneity, a life-yle earnings profile, and borrowing onstraints, te effet on maroeonomi variables is exatly zero in te following sense: Imagine an eonomy as only one representative on- 5 ECONOMIC REVIEW Fourt Quarter 25

13 sumer, earning exatly te average inome and olding exatly te average ouse wort 143 perent of average inome. An inrease in ouse pries would ave a positive wealt effet and a negative effet beause ousing in te present and future beomes more expensive. One an sow, owever, tat te two effets preisely anel ea oter out, so relative prie anges between and are neutral. Utilizing te earlier model, one an formalize tis intuition. Imagine one unit of ousing now osts P units of te onsumption good. Abstrating from transation osts, te optimality ondition for ousing versus onsumption now beomes (6) u (, ) µ P 1 δ 1 1 u, u, P 1 + r = µ + u (, ) γ. Tis struture is familiar: Te marginal rate of substitution must equal te relative prie of te two goods. Relative prie is te ouseold s subjetive ost, wi in tis ase is a mixture of te user ost formula 1 (P /P )[(1 δ )/(1 + r)] and te ost of te borrowing onstraint. Notie tat for pries P = P tis user ost formula Heterogeneity among ouseolds is te redues to equation (4) witout te terms ruial ingredient of any model attempting to determine te distributional effets for te transation ost. Notie also tat permanently anging all future real estate pries to te same level will ave no effet of ousing poliy. on te / ratio beause ouse pries sow up in te optimality onditions only as ratios of two onseutive ouse pries. One an ten sow tat in te one-person eonomy te onsumption pat witout te prie ange is still affordable after te prie ange. Also, beause tis pat satisfies te new optimality onditions, te optimal beavior of te ouseold is to leave te onsumption pat unanged, meaning tat a permanent ouse prie ange is ompletely neutral. Te only way ouse prie anges an ave nontrivial effets is if ouseolds are eterogeneous. Li and Yao (25) use a model similar to Yang s, toug modified in several dimensions. Most importantly, Li and Yao introdue a rental market in wi ouseolds an attain ousing servies via two routes: eiter purasing a ome, wit te purase subjet to a transation ost, or renting it instead at a ost of αp. One an tink of te parameter α as te rental return per unit rented. Li and Yao ten simulate te eonomy after a permanent prie ange to ousing. Not surprisingly, renters suffer from a ouse prie appreiation. As opposed to te representative ouseold mentioned above, renters ave no positive wealt effet at all beause, by definition, tey do not own real estate. Renters are, owever, it wit an inrease in teir future ousing osts regardless of teir future tenure deision: If tey deide to buy a ouse, tey ave to pay more, and if tey ontinue to rent, teir ost per unit of rental goes up beause te rental prie is a fixed portion of te ouse prie P. Li and Yao an even quantify te loss renters suffer. Te loss in utility due to an inrease in P by two standard deviations (11.5 perent) is equivalent to permanently reduing all urrent and future onsumption by about 4.5 perent. Te effet on renters is obvious and unambiguous, wi is not neessarily te ase for omeowners. Li and Yao sow tat ouse prie appreiation an in fat be disadvantageous to some omeowners. Te intuition for teir result as to do wit te pattern of omeownersip displayed in Table 2 (on page 17). Te average value of te primary residene inreases for every age group until around age fifty. As mentioned before, tis inrease is most likely due to te binding borrowing onstraint ECONOMIC REVIEW Fourt Quarter 25 51

14 tat keeps young ouseolds from attaining onsumption levels (bot ousing and nonousing) implied by te permanent inome ypotesis. Suppose a young ouseold owns $1, wort of real estate and experienes a 2 perent inrease in ouse pries. Instead of a $2, gain, te ouseold may view tis as a $2, loss if it was planning to upgrade to a larger ouse wort $2,, wi now osts $24,, beause te upgrade would now ost $12, instead of $1,. In onlusion, Li and Yao argue tat a substantial portion of te population does not benefit from a ouse prie appreiation. Te break-even age for omeowners is at around age fifty, wi inidentally is rougly te age at wi ouseolds finis teir upgrading. Older ouseolds gain substantially from ouse prie appreiation, partly beause many of tem start moving into smaller ouses or even beome renters again and are tus able to as out some of teir apital gains. Even witout downgrading, older ouseolds experiene a welfare gain beause te value of te bequest tey leave to teir eirs inreases. Different Mortgage Contrats and te Tenure Deision of Young Houseolds One an interpret te paper by Cambers, Garriga, and Slagenauf (25b) as a follow-up to teir earlier study (25a), wi uses a life-yle model mu like te ones presented earlier to explore potential reasons for te pat of te omeownersip rate over te past forty years. Homeownersip stayed rougly onstant at 64.5 perent between 1965 and 1995 but ten inreased signifiantly to now almost 68 perent. Cambers, Garriga, and Slagenauf (25a) argue tat te main reason for tis inrease is te availability of mortgage ontrats requiring lower down payments. Tey simulate te model eonomy, one wit a tigter and one wit a looser down payment onstraint, and determine tat relaxing te down payment onstraint an indeed quantitatively aount for te inrease in omeownersip. Speifially, tey study te effet of going from a 2 perent down payment onstraint to an ombo loan tat is, an 8 perent first mortgage, a 15 perent seond mortgage, and a 5 perent down payment. How an te arrival of a new mortgage ontrat inrease omeownersip, espeially among young agents? Reall tat younger ouseolds, say, between te ages of twenty and forty, are araterized by tree features: Tey are far less likely to be omeowners and ave bot lower inome and lower net wort tan te average population. Equation (4) reveals te main ulprit for low omeownersip among young ouseolds: Beause of te binding borrowing onstraint, te subjetive ousing ost for young agents is iger tan te rental ost, disouraging young agents from buying. Reduing te tigtness of te borrowing onstraint (lowering parameter γ) is ten te most diret way to enourage omeownersip among tose agents wit a positive Lagrange multiplier µ (a binding borrowing onstraint). Cambers, Garriga, and Slagenauf (25a) sow tat lowering γ from 2 perent in te benmark to 5 perent under te ombo loan will indeed inrease omeownersip among young ouseolds and an quantitatively aount for te rise in omeownersip. Te great innovation in te two papers by Cambers, Garriga, and Slagenauf is tat tey model mortgages wit mu more are tan in te rest of te literature. Te oter tree papers disussed ere Yang (25), Li and Yao (25), and Jeske and Krueger (25) take one ruial sortut wen modeling mortgages: Houseolds roll over mortgage debt every period, tat is, tey oose watever amortization sedule suits tem, onstrained only by te borrowing onstraint (in Yang and Li and Yao) or te interest rate dependene on te leverage ratio (in Jeske and Krueger). Cambers, Garriga, and Slagenauf, owever, assume tat a mort- 52 ECONOMIC REVIEW Fourt Quarter 25

15 gage follows a fixed amortization sedule as it would for, say, a tirty-year fixed rate mortgage. One an tink of a mortgage ontrat ten as not only an interest rate r and a down payment ratio γ but rater a wole sequene of ratios over te lifetime of te loan. Suppose a tirty-year mortgage wit an 8 perent loan-to-value ratio stipulates a sequene of γ starting at 2 perent and inreasing very slowly over te first ouple of years (beause initially most of te mortgage payment goes toward interest rater tan prinipal). Toward te end of te loan more and more of te payment goes toward reduing te prinipal, and eventually γ reaes 1 perent wen te loan is paid off. Anoter example would be a balloon loan, in wi ase γ stays at 2 perent Relaxing te down payment onstraint an greatly inrease omeownersip among younger ouseolds beause it redues te subjetive user ost. for te duration of te loan and after wi te balloon payment omes due, wi requires te ouseold to eiter pay off te loan (γ = 1) wit one large payment or refinane into anoter mortgage. Naturally, different loan designs imply different payment sedules. Sine te model is detailed enoug to mimi a wole range of different mortgage ontrats, Cambers, Garriga, and Slagenauf (25b) an study ow different ontrat types affet te tenure deision, espeially of young agents. Te onlusion of teir paper is tat mortgage ontrats tat tend to ave low payments early on, su as a balloon mortgage, are most suessful in relaxing te borrowing onstraint. Tis onlusion is intuitive: Instead of relaxing just one borrowing onstraint, a balloon mortgage as te same effet as relaxing te onstraints over te wole life of te mortgage relative to a fixed rate mortgage, wit te exeption of te last period wen te entire prinipal is due. In general, te autors sow tat mortgage ontrats wit an inreasing payment sedule and tus lower γ over time tend to enourage more young ouseolds to beome omeowners but derease omeownersip among older ouseolds. Tis finding makes perfet sense: Young ouseolds experiene an inreasing earnings profile (see te first two panels of Figure 2), wi makes a mortgage wit an inreasing payment sedule a good mat for tem. Te opposite is true for middle-aged and older ouseolds wose inome, at least on average, is stagnating or even dereasing, making a mortgage wit an inreasing payment sedule a mismat to teir earnings profile. 12 Cambers, Garriga, and Slagenauf ave terefore pointed to te great importane of te borrowing onstraint in determining tenure deisions over te life yle. Relaxing tis onstraint an greatly inrease omeownersip among younger ouseolds beause it redues te subjetive user ost in equation (4) if µ >. Tis effet ours bot for te borrowing onstraint in te initial period wen a ouseold purases a ouse as well as over te duration of te mortgage. Te Effet of Subsidies Te papers disussed so far learly speify a dual role of ousing namely, a onsumption role ( enters te utility funtion) and an investment role beause ousing is also a form of saving as long as te depreiation rate is below 1 perent. One 12. Notie tat in te model te autors assume tat only one single mortgage ontrat exists beause adding multiple ontrats into te same model involves too big a omputational burden. Tis result sould not interpreted as an argument against, say, balloon mortgages beause tey redue ownersip among older ouseolds. In reality, several types of mortgages are available, so older ouseolds an oose mortgages tat suit tem better tan a balloon mortgage. ECONOMIC REVIEW Fourt Quarter 25 53

16 ingredient missing in te literature so far as been te fat tat ousing investment is risky. Jeske and Krueger (25) set up a model in wi, in addition to idiosynrati labor inome risk, ouseolds fae an idiosynrati ouse depreiation sok. Tis setup generates some realisti features: Most importantly, ouseolds may end up wit negative equity on teir ome if A main finding of Jeske and Krueger is tat te depreiation sok is large enoug. Tis interest subsidies, if tey are indeed passed feature is one way to generate mortgage forelosure, wi as not been done before on to ouseolds, an ause overinvestment in te general equilibrium literature. Jeske in ousing and larger mortgages. and Krueger also assume tat tere is a deadweigt loss from forelosure, taking te following sape: If a property goes into forelosure, te bank reeives proportionally less tan te value of te property. In reality, tis forelosure loss is substantial. Pennington-Cross (24) estimates te deadweigt loss of forelosure to be 22 perent of te value of te property. Tis loss beomes ruial in te poliy experiment later beause a subsidy on mortgage rates will likely inrease mortgage default as well as te deadweigt loss. In addition, Jeske and Krueger allow ouseolds to oose any leverage ratio tey desire. Tis feature differs from te papers studied so far, in wi ouseolds fae a sarp borrowing onstraint tat requires a minimum or even fixed down payment ratio. Jeske and Krueger assume tat te interest rate ouseolds pay is a funtion tat inreases in te leverage ratio tey are taking on. Speifially, a bank will prie te forelosure risk into te mortgage, and, if a ouseold ooses iger leverage, a smaller depreiation sok will be required to trigger a default tat is, default is more likely implying a iger interest rate. Jeske and Krueger must also make some simplifying assumptions. As opposed to te previously disussed papers, tey do not onsider life-yle effets tat is, tey assume tat ouseolds are infinitely lived, and tey abstrat from transation osts. Bot assumptions are neessary to make te omputational burden tratable. Suppose tat a government were to provide a subsidy tat lowers mortgage interest. Jeske and Krueger try to study te maroeonomi effets of su a subsidy. Te subsidy te autors ave in mind as a ounterpart in reality government-sponsored enterprises (GSEs) like Fannie Mae and Freddie Ma, wi are in te business of borrowing in te bond market to ten purase large portfolios of ome mortgages. Te GSEs reeive a benefit from te federal government in te form of an impliit bailout guarantee. 13 Te Congressional Budget Offie (CBO) estimates tat te subsidy is wort about 42 basis points; tat is, GSEs an borrow at rates.42 perentage points below wat oter entities wit a similar redit rating would ave to pay in te bond market. Passmore, Serlund, and Burgess (24) estimate tat only a small fration 7 basis points atually makes it to te omeowner wile te GSEs keep te lion s sare of te subsidy to pass on to teir sareolders. Blinder (24), on te oter and, defends te GSEs by estimating tat tey indeed redue mortgage rates by almost te entire 42 basis points. Jeske and Krueger (25) do not attempt to judge wi of te two studies is orret but rater do te following tougt experiment: Supposing tat te GSEs indeed pass on te entire 42 basis points to omeowners as tey laim, ten wat is te effet on aggregate maroeonomi variables as well as te distribution of wealt and inome? A main finding of Jeske and Krueger is tat interest subsidies, if tey are indeed passed on to ouseolds, an ause overinvestment in ousing and larger mortgages. 54 ECONOMIC REVIEW Fourt Quarter 25

17 In fat, mortgage debt inreases proportionally more tan te ousing stok, implying more leverage on ousing. Tis finding, of ourse, is less tan surprising: Te subsidy makes ousing eaper and terefore auses more ousing investment, and lower mortgage rates enourage more leverage. Tanks to te endogenous borrowing onstraint, Jeske and Krueger are able to study one new aspet of ousing poliy, namely, ow mortgage subsidies affet te aggregate level of mortgage default. Tey find tat te subsidy inreases te proportion of mortgages in default by one alf, wi in turn makes te deadweigt loss 5 perent larger. Te next provoative question is, Wat are te distributional onsequenes of te subsidy? Te subsidy as little effet on omeownersip rates. In oter words, te assistane goes mainly to existing omeowners tose ouseolds already well off due to ig inomes and ig net wort wi are able to afford even larger ouses wile poor ouseolds are almost unaffeted in teir tenure oie. Tis result is onsistent wit te findings of te generi ousing model outlined earlier, espeially equation (4): Lowering interest rate r and tus te user ost for ousing (r + δ )/(1 + r) may ave only a small effet on ouseolds tat are onstrained by te down payment ondition. 14 Te main part of teir subjetive user ost, after all, omes from te borrowing onstraint (te seond term on te rigt-and side of equation [4]) lowering urrent onsumption. In te Jeske and Krueger model, anoter reason wy poor ouseolds will not neessarily invest in real estate and instead keep renting is tat te rental rate is determined endogenously; tat is, te rental rate is a market-learing prie ensuring tat rental demand equals rental supply. Wit a mortgage subsidy tat auses more investment in ousing, te rental prie will neessarily drop, making renting more attrative relative to ousing. Tese results may disappoint te proponents of government assistane for mortgage finaning. Eiter te subsidy in te form of impliit guarantees and lower interest rates goes to sareolders of GSEs or te interest rate redution is passed on to omeowners but ends up in te ands of already well-off ouseolds wile doing noting to promote omeownersip for poorer ouseolds. Te findings, by te way, are onsistent wit empirial work tat suggests tat mortgage rate redutions ave only a marginal impat on omeownersip rates wile te borrowing onstraint seems to play a mu larger role. 15 Jeske and Krueger onfirm tese findings but are able to do so in a logial and oerent framework using general equilibrium instead of mirosimulation teniques. Conlusion Houseolds in different age oorts and different positions in te inome and wealt distribution ave very different omeownersip ratios, real estate wealt, and mortgage debt. Likewise, renters and omeowners look very different in teir pat of inome and asset oldings over te life yle. Tis eterogeneity poses a allenge to researers trying to answer questions regarding ousing poliy beause one annot study te effet of a poliy merely on te average ouseold. Studying ousing in te framework of maroeonomi models terefore requires inorporating eterogeneity of ouseolds. Likewise, te size of te ousing stok is too large to ignore general equilibrium effets. 13. See Frame and Wall (22) for a survey on tis topi. 14. One must use aution ere: Te interest rate itself depends on te leverage ratio, so ouseolds will not view it as exogenous as tey did in te oter tree papers. Te intuition, toug, would be similar in a model wit an interest rate as a funtion of te leverage ratio. 15. See Painter and Redfearn (22) or Feldman (21) for a summary of a variety of empirial studies. ECONOMIC REVIEW Fourt Quarter 25 55

18 Tanks to advanes in omputational teniques and faster omputer ardware, eonomists are now able to write down and solve models detailed enoug to aommodate ousing. Te onferene osted by te Federal Reserve Bank of Atlanta featured (among oters) four papers, all of wi are variations of te generi ousing model desribed ere even toug tey tou on a large variety of issues. Yang (25) points out te importane of transation osts and borrowing onstraints in life-yle models wit ousing. Li and Yao (25) study te effet of ouse prie anges. Cambers, Garriga, and Slagenauf (25b) inorporate finanial innovations in te form of different mortgage ontrats. And Jeske and Krueger (25) study te effet of a subsidy on mortgage finane. In ea paper, one an dedue te main intuition from te simple equilibrium onditions. REFERENCES Aiyagari, S. Rao Uninsured idiosynrati risk and aggregate saving. Quarterly Journal of Eonomis 19, no. 3: Attanasio, Orazio, and Martin Browning Consumption over te life yle and over te business yle. Amerian Eonomi Review 85, no. 5: Attanasio, Orazio, and Guglielmo Weber Is onsumption growt onsistent wit intertemporal optimization? Evidene from te Consumer Expenditure Survey. Journal of Politial Eonomy 13, no. 6: Blinder, Alan S. 24. Letter on Federal Reserve working paper to CEO Raines from Dr. Alan S. Blinder, February 23. < pdf> (September 13, 25). Blundell, Riard, Martin Browning, and Costas Megir Consumer demand and te life-yle alloation of ouseold expenditures. Review of Eonomi Studies 61, no. 26:57 8. Board of Governors of te Federal Reserve System. 25. Flow of funds statistis. June 9 release. <www. federalreserve.gov/releases/z1/> (September 13, 25). Cambers, Mattew, Carlos Garriga, and Donald Slagenauf. 25a. Aounting for anges in te omeownersip rate. Florida State University working paper.. 25b. Mortgage ontrats and ousing tenureduration deision. Paper presented at te Federal Reserve Bank of Atlanta onferene Housing, Mortgage Finane, and te Maroeonomy, May Feldman, Ron J. 21. Mortgage rates, omeownersip rates, and government-sponsored enterprises. Federal Reserve Bank of Minneapolis annual report, Fernandez-Villaverde, Jesus, and Dirk Krueger. 22. Consumption and saving over te life yle: How important are onsumer durables? Proeedings of te 22 Nort Amerian summer meetings of te Eonometri Soiety Consumption over te life yle: Some fats from Consumer Expenditure Survey data. University of Pennsylvania working paper. Frame, W. Sott, and Larry D. Wall. 22. Finaning ousing troug government-sponsored enterprises. Federal Reserve Bank of Atlanta Eonomi Review 87, no. 1: Gourinas, Pierre-Olivier, and Jonatan Parker. 22. Consumption over te life yle. Eonometria 7, no. 1: Jeske, Karsten, and Dirk Krueger. 25. Housing and te maroeonomy: Te role of impliit guarantees for government sponsored enterprises. Paper presented at te Federal Reserve Bank of Atlanta onferene Housing, Mortgage Finane, and te Maroeonomy, May Li, Wenli, and Rui Yao. 25. Te life-yle effets of ouse prie anges. Paper presented at te Federal Reserve Bank of Atlanta onferene Housing, Mortgage Finane, and te Maroeonomy, May Painter, Gary, and Cristian L. Redfearn. 22. Te role of interest rates in influening long-run omeownersip rates. Journal of Real Estate Finane and Eonomis 25, nos. 2 3: Passmore, Wayne, Sane Serlund, and Gillian Burgess. 24. Te effet of ousing government-sponsored enterprises on mortgage rates. Federal Reserve Board Finane and Disussion Series No Pennington-Cross, Antony. 24. Te value of forelosed property: House pries, forelosure laws, and appraisals. Offie of Federal Housing Enterprise Oversigt Working Paper 4-1, January. Yang, Fang. 25. Consumption along te life-yle: How different is ousing? Paper presented at te Federal Reserve Bank of Atlanta onferene Housing, Mortgage Finane, and te Maroeonomy, May ECONOMIC REVIEW Fourt Quarter 25