Learning about the Neighborhood Zhenyu Gao Mihael Sokin Wei Xiong Otober 208 Abstrat We develop a model of neighborhood hoie to analyze information aggregation and learning in housing markets. In the presene of pervasive informational fritions, housing pries serve as important signals to households and apital produers about the eonomi strength of a neighborhood. Through this learning hannel, noise from housing market supply and demand shoks an propagate from housing pries to the loal eonomy, distorting not only migration into the neighborhood, but also the supply of apital. Our analysis provides testable, nuaned impliations on how the amplifiation of informational fritions varies aross neighborhoods. We are grateful to Itay Goldstein, Laura Veldkamp and seminar partiipants of 208 AEA Meetings, 208 NBER Asset Priing Meeting, Fordham University, and UC Berkeley for helpful omments. Chinese University of Hong Kong. Email: gaozhenyu@baf.uhk.edu.hk. University of Texas, Austin. Email: mihael.sokin@mombs.utexas.edu. Prineton University and NBER. Email: wxiong@prineton.edu.
Understanding the origins of the reent U.S. housing yle remains a paramount onern for maroprudential poliy. As explored in Cheng, Raina and Xiong 204 and Kaplan, Mitman, and Violante 207, widespread optimism has inreasingly beome reognized as an important driver of the housing boom. It is still unlear, however, how suh optimism developed on a national sale, how its expression differed in the ross-setion of U.S. metropolitan areas, and how it spread so pervasively from housing markets to loal eonomi ativity. In this paper, we address these questions by developing a heterogeneous agent model of information aggregation and learning in housing markets, and study its interation with population migration and loal investment deisions. We extend the problem of oordination with dispersed information to investigate its role in amplifying the agglomeration effets that underpin the formation of neighborhoods and ities. This allows us to provide novel ross-setional empirial preditions for testing the role of informational fritions in explaining the reent yle. Although informational fritions have long been appreiated as important in explaining housing market behavior, suh as in Garmaise and Moskowitz 2004, Kurlat and Stroebel 204, Favara and Song 204, and Bailey et al. 207, they have reeived relatively little attention as a potential soure of amplifiation during the reent U.S. housing yle. Leading auses for the yle range from redit expansion and fraudulent lending praties to speulation and optimisti, often extrapolative, expetations. By anhoring household expetations to loal eonomi onditions, our theory provides guidane as to where optimism and overreation had the most pronouned impat on housing and loal eonomi outomes during the boom. Consistent with the empirial observations of Glaeser 203, this mehanism an give rise to housing yles that are most extreme in areas of intermediate supply elastiity. By impating the demand urve for housing, informational fritions omplement the redit expansion and fraud hannels, and an also give rise to speulative demand. Speifially, we analyze how informational fritions affet the learning and beliefs of households and apital produers about a neighborhood, whih, in turn, drive both loal housing market dynamis and investment deisions. The model features a ontinuum of households in an open neighborhood, whih an be viewed as a ounty. Eah household has For redit expansion, see, for instane, Mian and Sufi 2009, 20 and Albanesi et al. 205. For fraudulent lending praties, see Keys et al. 2009 and Griffi n and Maturana 205. For speulation, see Nathanson and Zwik 207, DeFuso, Nathanson, and Zwik 207, and Gao, Sokin and Xiong 208. For extrapolative expetations, see Case and Shiller 2003, Glaeser, Gyourko, and Saiz 2008, Piazzesi and Shneider 2009, and Glaeser and Nathanson 207.
a hoie of whether to move into the neighborhood by purhasing a house, and has a Cobb- Douglas utility funtion over its onsumption of its own good and its aggregate onsumption of the goods produed by other households in the neighborhood. This omplementarity in households onsumption motivates eah household to learn about the unobservable eonomi strength of the neighborhood, whih determines the ommon produtivity of all households and leads to omplementarity in their housing demand. Eah household requires both labor, whih it supplies, and loal apital to produe its good aording to a Cobb-Douglas prodution funtion. Sine the prie of apital depends on its marginal produt aross households in the neighborhood, ompetitive apital produers must also form expetations about the neighborhood s eonomi strength when determining how muh loal apital to develop. It is intuitive that loal housing markets provide a useful platform for aggregating private information about the eonomi strength of a neighborhood. The traded housing prie reflets the net effet of demand and supply-side fators, in a similar spirit to the lassi models of Grossman and Stiglitz 980 and Hellwig 980 for information aggregation in asset markets. Different from the linear equilibrium in these models, our model features an important neighborhood seletion effet, through whih only households with private signals above a ertain equilibrium utoff hoose to live in the neighborhood. This seletion effet makes our model inherently nonlinear, whih poses a substantial hallenge to maintaining tratability in both household learning and information aggregation with dispersed information. Nevertheless, we are able to derive the equilibrium analytially, building on the utoff equilibrium framework developed by Goldstein, Ozdenoren, and Yuan 203 and Albagli, Hellwig, and Tsyvinski 205, 207 for asset markets. There are two key features that ontribute to this tratability. First, despite the equilibrium housing prie being a nonlinear funtion, its information ontent about the neighborhood strength an be summarized by a linear suffi ient statisti, whih keeps households learning from the housing prie tratable. Seond, despite eah household s housing demand being nonlinear, the Law of Large Numbers allows us to aggregate their housing demand, and to derive a utoff equilibrium for the housing market. In our setting, eah household possesses a private signal regarding the neighborhood s ommon produtivity. By aggregating the households housing demand, the housing prie aggregates their private signals. The presene of unobservable supply shoks, however, prevents the housing prie from perfetly revealing the neighborhood s strength, and ats as a soure of informational noise in the 2
housing prie. Our model allows us to analyze how informational fritions affet not only the housing prie, but also eah household s neighborhood, labor and prodution hoies. These, in turn, determine eah household s demand for housing and apital. Sine the housing prie serves as a signal about the neighborhood s strength, it plays a key role in determining agents expetations. Through this learning hannel, noise in the housing market, originating from either the demand or supply side, impats the housing prie and the loal eonomy. Noise that pushes up housing pries an indue more households to enter the neighborhood through learning, sine a higher prie signals a stronger neighborhood. This leads to a more pronouned housing yle, and, interestingly, to an oversupply of loal apital. Our analysis illustrates how the transmission of noise in housing markets to real estate and prodution outomes an vary aross different neighborhoods along two dimensions: the elastiity of housing supply in the neighborhood, and 2 the degree of households onsumption omplementarity. With respet to supply elastiity, the distortion to housing pries indued by noise through the learning hannel is hump-shaped. At one extreme, when housing supply is infinitely inelasti, the housing prie is fully determined by housing demand, and perfetly reveals the strength of the neighborhood; at the other extreme, when housing supply is perfetly elasti, housing pries are fully determined by housing supply, and are therefore not affeted by households expetations. At both extremes, learning does not distort the housing prie. As a result, the noise effet on housing pries from informational fritions is strongest at intermediate supply elastiities. With respet to the degree of households onsumption omplementarity, our analysis shows that distortionary effets in the housing market from learning tend to inrease with omplementarity, as greater omplementarity makes learning about the neighborhood s strength a more important part of eah household s deisions. In ontrast, omplementarity mitigates the distortions from learning to the market for apital, as greater oordination by households also lowers the average marginal produt of apital for a given strength of the neighborhood. Our insights into how harateristis of a neighborhood interat with informational fritions give rise to testable ross-setion hypotheses when sorting areas by supply elastiity or the degree of omplementarity of their industries. Our analysis shows that the distortionary effets indued by learning on population inflow and housing and real investment yles are most pronouned in areas with intermediate supply elastiities, rather than areas 3
with the most inelasti housing supply. This result helps explain why areas like Las Vegas and Phoenix with relatively more elasti housing supplies had more dramati housing yles than New York and San Franiso, as doumented, for instane, in Davidoff 203, Glaeser 203, and Nathanson and Zwik 207. Our analysis also highlights a learning externality in that, when making housing deisions, households do not internalize their impat on the expetations of apital produers through learning from housing pries. To the extent that any overbuilding in apital and ommerial infrastruture is diffi ult to reverse in the short or medium term, the exess supply an have prolonged, overhang effets on the loal eonomies. Gao, Sokin, and Xiong 207, for instane, find onsistent evidene that supply overhang in housing markets helped transmit the adverse impat of housing speulation to the real eonomy during the reent bust. By featuring a tratable utoff equilibrium framework, our work ontributes to a growing literature that analyzes information aggregation in nonlinear settings. Goldstein, Ozdenoren, and Yuan 203 investigate the feedbak to the investment deisions of a single firm when managers, but not investors, learn from pries. Albagli, Hellwig, and Tsyvinski 205, 207 fous on the role of asymmetry in seurity payoffs in distorting asset pries and firm investment inentives when future shareholders learn from pries to determine their valuations. These models ommonly employ risk-neutral agents, normally distributed asset fundamentals, and position limits to deliver tratable nonlinear equilibria. In ontrast, we fous on the feedbak indued by learning from housing pries to household neighborhood hoie and labor deisions in the presene of onsumption omplementarity and goods trading between households. We further analyze the spillover of this feedbak to investment deisions of apital produers. By showing that the utoff equilibrium framework an be adopted to analyze learning effets in this riher setting, our model substantially expands the sope of this framework. Our model differs from Burnside, Eihenbaum, and Rebelo 206, whih offers a model of housing market booms and busts based on the epidemi spreading of optimisti or pessimisti beliefs among home buyers through their soial interations. Our learning-based mehanism is also distint from Nathanson and Zwik 207, whih studies the hoarding of land by home builders with heterogeneous beliefs in ertain elasti areas as a mehanism to amplify prie volatility in the reent U.S. housing yle. It is also different from Piazzesi and Shneider 2009, whih investigates how a small population of optimists an inflate housing pries 4
by driving transation volume. Glaeser and Nathanson 207 presents a model of biased learning in housing markets, building on urrent buyers not adjusting for the expetations of past buyers, and instead assuming that past pries reflet only ontemporaneous demand. This inorret inferene by home buyers gives rise to orrelated errors in housing demand foreasts over time, whih, in turn, generate exess volatility, momentum, and mean-reversion in housing pries. Guren 206 develops a model of housing prie momentum, building on the inentive of individual sellers not to set a unilaterally high or low list prie beause the demand urve they fae is onave in relative prie. In ontrast to these models, informational fritions in our framework anhor on the interation between the demand and supply sides of the housing market, and feed bak to both housing pries and real outomes. This key feature is also distint from the amplifiation of prie volatility indued by dispersed information and short-sale onstraints featured in Favara and Song 204. By fousing on information aggregation and learning of symmetrially informed households with dispersed private information, our study differs in emphasis from those that analyze the presene of information asymmetry between home buyers and sellers, suh as Garmaise and Moskowitz 2004 and Kurlat and Stroebel 204. Neither does our model emphasize the potential asymmetry between in-town and out-of-town home buyers, whih is shown to be important by Chino and Mayer 205. In addition, there are extensive studies in the housing literature highlighting the roles played by both demand-side and supply-side fators in driving housing yles. On the demand side, Himmelberg, Mayer, and Sinai 2005 fous on interest rates, Poterba, Weil, and Shiller 99 on tax hanges, Mian and Sufi 2009 on redit expansion, and DeFuso, Nathanson, and Zwik 207 and Gao, Sokin and Xiong 208 on investment home purhases. On the supply side, Glaeser, Gyourko, and Saiz 2008 emphasize supply as a key fore in mitigating housing bubbles. Haughwout, Peah, Sporn and Tray 203 provide a detailed aount of the housing supply side during the U.S. housing yle in the 2000s, and Gyourko 2009b systematially reviews the literature on housing supply. By introduing informational fritions, our analysis shows that supply-side and demand-side fators are not mutually independent. Supply shoks an affet housing and investment demand by ating as informational noise in learning, and influene households and apital produers expetations of the strength of a neighborhood. 5
The Model The model has two periods t {, 2}. There are three types of agents in the eonomy: households looking to buy homes in a neighborhood or elsewhere, home builders, and apital produers. Suppose that the neighborhood is new and all households purhase houses from home builders in a entralized market at t = after hoosing whether to live in the neighborhood. Households hoose their labor supply and demand for apital, suh as mahines and offi e spae, to omplete prodution, and onsume onsumption goods at t = 2. Our intention is to apture the deision of a generation of home owners to move into a neighborhood, and we view the two periods as representing a long length of time in whih they live together and share amenities, as well as exhange their goods and servies.. Households We onsider a pool of households, indexed by i [0, ], eah of whih an hoose to live in a neighborhood or elsewhere. We an divide the unit interval into the partition {N, O}, with N O = and N O = [0, ]. Let H i = if household i hooses to live in the neighborhood, i.e., i N, and H i = 0 if it hooses to live elsewhere. 2 If household i at t = hooses to live in the neighborhood, it must purhase one house at prie P. This reflets, in part, that housing is an indivisible asset and a disrete purhase, onsistent with the insights of Piazzesi and Shneider 2009. To inorporate omplementarity in the households housing demands, we adopt a partiular struture for their goods onsumption and trading. Eah household in the neighborhood produes a distint good from other households. Household i has a Cobb-Douglas utility funtion over onsumption of its own good C i i and its onsumption of the goods produed by all other households in the neighborhood {C j i} j N : η Ci i U {C j i} j N ; N = C η N /i j i dj. η η The parameter η 0, measures the weights of different onsumption omponents in the utility funtion. A higher η indiates a stronger omplementarity between household i s onsumption of its own good and its onsumption of the omposite good produed by the other households in the neighborhood. One an view this omplementarity as apturing 2 See Van Nieuwerburgh and Weill 200 for a systemati treatment of moving deisions by households aross neighborhoods. 6
agglomeration and spillover effets from households and firms loating near eah other, or as refleting that households and firms require eah other s intermediate goods and servies as inputs to their own prodution. 3 As we will disuss later, this utility speifiation implies that eah household ares about the strength of the neighborhood, i.e., the produtivities of other households in the neighborhood. This assumption leads to strategi omplementarity in households housing demands, an important feature emphasized by the empirial literature, suh as in Ioannides and Zabel 2003. 4 The prodution funtion of household i is also Cobb-Douglas e A i Ki α l α i, where A i is its produtivity, l i is the household s labor hoie, and K i is its hoie of apital with a share of α 0, in the prodution funtion. We broadly interpret apital as both publi and private invesment in the neighborhood, whih an inlude offi e, mahines, omputers, and other equipment and infrastruture households an use for their produtive ativities. 5 As we desribe later, households buy apital from apital produers. When households are more produtive in the neighborhood, the marginal produtivity of apital is higher, and onsequently apital produers are able to sell more apital at higher pries. Introduing apital allows us to disuss how learning affets the prie and supply of not only residential housing, but also of loal investment in the neighborhood. Household i s produtivity A i is omprised of a omponent A, ommon to all households in the neighborhood, and an idiosynrati omponent i : where A N Ā, τ A A i = A + i, and i N 0, τ are both normally distributed and independent of eah other. Furthermore, we assume that i d i = 0 by the Strong Law of Large Numbers. The ommon produtivity, A, represents the strength of the neighborhood, as a higher A implies a more produtive neighborhood. As A determines the households aggregate demand for housing, it also represents the demand-side fundamental. 3 Similar speifiations of this utility funtion are employed, for instane, in Dixit and Stiglitz 977 and Long and Plosser 983 to give rise to input and output linkages in setoral prodution. One an view η η C i i η N /i C η j i dj as a final good produed by household i given intermediate goods {C j i} i N. 4 While our model builds on omplementarity in household onsumption, other types of soial interations between households in a neighborhood may also lead to omplementarity in their housing demand, as disussed in Durlauf 2004 and Glaeser, Saerdote, and Sheinkman 2003. 5 In the ase that K is a publi good, its prie an be interpreted as the tax a loal government that faes a balaned budget an raise to offset the ost of onstrution. Our model then has impliations for how housing markets impat the fisal poliy of loal governments. 7
As a result of realisti informational fritions, A is not observable to households at t = when they need to make the deision of whether to live in the neighborhood. Instead, eah household observes its own produtivity A i, after examining what it an do if it hooses to live in the neighborhood. Intuitively, A i ombines the strength of the neighborhood A and the household s own attribute i. Thus, A i also serves as a noisy private signal about A at t =, as the household annot fully separate its own attribute from the opportunity provided by the neighborhood. The parameter τ governs both the household diversity in the neighborhood and the preision of this private signal. As τ, the households signals beome infinitely preise and the informational fritions about A vanish. Households are about the strength of the neighborhood beause of omplementarity in their demand for onsumption. While a household may have a fairly good understanding of its own produtivity when moving into a neighborhood, omplementarity in onsumption demand motivates it to pay attention to housing pries to learn about the average level A for the neighborhood. We start with eah household s problem at t = 2 and then go bakward to desribe its problem at t =. At t = 2, we assume that A is revealed to all agents. Furthermore, we assume that eah household experienes a disutility for labor l i, and that a household in the neighborhood i.e., i N maximizes its utility at t = 2 by hoosing labor l i, apital K i, and its onsumption demand {C j i} j N : U i = max U {C j i} j N ; N l i {{C j i} j N,l i,k i} + ψ suh that p i C i i + N /i p j C j i dj + RK i = p i e A i K α i l α i, 2 where p i is the prie of the good it produes and R is the unit prie of apital. Households behave ompetitively and take the pries of their goods as given. At t =, eah household needs to deide whether to live in the neighborhood. In addition to their private signals, all households and apital produers observe a noisy publi signal Q about the strength of the neighborhood A: Q = A + Q Q, where Q N 0, is independent of all other shoks. As τ Q beomes arbitrarily large, A beomes ommon knowledge to all agents. This publi signal ould, for instane, be news reports or published statistis on loal eonomi onditions. 8
In addition to the utility flow U i at t = 2 from goods onsumption and labor disutility, we assume that households have quasi-linear expeted utility at t =, and, similar to Glaeser, Gyourko, and Saiz 2008, inur a linear utility penalty equal to the housing prie P if they hoose to live in the neighborhood and thus have to buy a house. Here we treat all housing units as homogenous with the same prie. Given that households have Cobb-Douglas preferenes over their onsumption, they are effetively risk-neutral at t =, and their utility flow is the value of their final onsumption bundle less the ost of housing. 67 Eah household makes its neighborhood hoie subjet to a partiipation onstraint that its expeted utility from moving into the neighborhood E [U i I i ] P must weakly exeed a reservation utility, whih, as in Glaeser, Gyourko, and Saiz 2008, we normalize to 0: max {E [U i I i ] P, 0}. 3 One an interpret the reservation utility as the expeted value of getting a draw of produtivity from another potential neighborhood less the ost of searh. The hoie of neighborhood is made at t = subjet to eah household s information set I i = {A i, P, Q}, whih inludes its private produtivity signal A i, the publi signal Q, and the housing prie P. 8.2 Capital Produers In addition to households, there is a ontinuum of risk-neutral apital produers that develop apital at t =, and sells this apital to households for their prodution at t = 2. Similar to many maroeonomi models, suh as Bernanke, Gertler, and Gilhrist 999, we model apital produers as a separate setor in the neighborhood. This introdues a market-wide supply urve for apital, and onsequently a market-wide prie, at t = 2, while avoiding introduing a speulative retrade motive into households apital aumulation deisions. 6 For simpliity, our model does not inorporate resale of housing after t = 2. As a result, we annot simply dedut the housing prie P as the housing ost from the household s budget onstraint at t = 2. Instead, we separately treat the housing ost as a utility ost proportional to the housing prie at t =. This utility ost is suffi ient to apture the notion that a higher housing prie implies a greater housing ost to the household without expliitly aounting for different omponents of the housing ost, suh as initial ost of purhase, ost of mortgage loan, and later resale value. 7 While we fous on a stati setting, introduing dynamis would reinfore our amplifiation mehanism stemming from learning. Sine future housing pries are related to aggregate produtivity growth in the neighborhood, households most optimisti about moving into the neighborhood beause of trading opportunities today would also be the most optimisti in speulating about the value of selling their house to other households in the future. 8 We do not inlude the volume of housing transations in the information set as a result of a realisti onsideration that, in pratie, people observe only delayed reports of total housing transations at highly aggregated levels, suh as national or metropolitan levels. 9
The representative produer ares about the prie of apital at t = 2, R, whih depends on apital s marginal produtivity. This, in turn, depends on the strength of the neighborhood, and whih households hoose to live in the neighborhood. The housing prie in the neighborhood serves as a useful signal to the produer when deiding how muh apital to develop at t =. We assume that eah apital produer an develop K units of apital by inurring a onvex effort ost λ Kλ, where λ >. While households buy apital from apital produers at t = 2, apital produers must foreast this demand when hoosing how muh apital K to develop at t =, in order to maximize its expeted profit: Π = sup E [RK λ ] Kλ I K where I = {P, Q} is the publi information set, whih inludes the housing prie P and the publi signal Q. It then follows that the optimal hoie of apital sets the marginal ost, K λ, equal to the expeted prie, E [R I ]: K = E [R I ] λ. The realized housing prie affets the expetation of apital produers about the neighborhood s strength A, whih, in turn, impats their hoie of how muh apital to develop. As a onsequene, in addition to altering the neighborhood hoie of potential household entrants, informational fritions in the housing market may also distort investment in the neighborhood..3 Home Builders There is a population of home builders, indexed on a ontinuum [0, ], in the neighborhood. Builder i [0, ] builds a single house subjet to a disutility from labor e +k ω i S i, where S i {0, } is the builder s deision to build and 4 ω i = ξ + e i is the builder s produtivity, whih is orrelated aross builders in the neighborhood through ξ. We assume that ξ = kζ, where k > 0 is a onstant parameter, and ζ represents an 0
unobserved, ommon shok to building ost in the neighborhood. From the perspetive of households and builders, ζ N ζ, τ ζ. Then, ξ = kζ an be interpreted as a supply shok with normal distribution ξ N ξ, k 2 τ ζ that e i d e i = 0 by the Strong Law of Large Numbers. At t =, eah builder maximizes his profit with ξ = k ζ. Furthermore, ei N 0, τ e suh Π s S i = max P e +k ω i S i. 5 S i Sine builders are risk-neutral, eah builder s optimal supply urve is S i = { if P e kζ+e i +k 0 if P < e kζ+e i +k. 6 The parameter k measures the supply elastiity of the neighborhood, whih an arise, for instane, from strutural limitations to building or zoning regulation. In the housing market equilibrium, the supply shok ξ not only affets the supply side of the housing market but also demand, as it ats as informational noise in the prie signal when households use the prie to learn about the ommon produtivity A. The elastiity parameter k determines the amount of this informational noise in the prie signal. Although onvenient for tratability and standard in the noisy rational expetations literature, our speifiation of the housing supply urve, S P = τ e + k log P + ξ, is not essential for our results. We ould instead have onsidered a more realisti model of housing supply with three neighborhoods: one with a perfetly inelasti housing supply, one with a perfetly elasti housing supply, and one in whih housing supply is prie-elasti and subjet to noisy supply shoks. As supply is fixed in the perfetly inelasti neighborhood, housing pries reflet only the housing demand fundamental, and are therefore fully revealing about the strength of the neighborhood A. In ontrast, sine additional houses an be built to absorb new housing demand in perfetly elasti areas, housing pries always equal the marginal ost of building, and ontain no information about A. It is in intermediate elastiity areas, where pries are driven by both demand and noisy supply-side fators, that households and apital produers fae a nontrivial filtering problem in inferring A from housing pries. Informational fritions are onsequently most severe in areas of intermediate supply elastiity, a key feature aptured in our more stylized model of housing supply.
.4 Noisy Rational Expetations Cutoff Equilibrium Our model features a noisy rational expetations utoff equilibrium, whih requires learing of the two real estate markets that is onsistent with the optimal behavior of households, home builders and apital produers: Household optimization: eah household hooses H i at t = to solve its maximization problem in 3, and then hooses { } {C j i} i N, l i, K i at t = 2 to solve its maximization problem in 2. Capital produer optimization: the representative produer hooses K at t = to solve its maximization problem in 4. Builder optimization: eah builder hooses S i problem in 5. at t = to solve his maximization At t =, the housing prie P lears the housing market: H i A i, P, Q d i = S i ω i, P, Q d e i, where eah household s housing demand H i A i, P, Q depends on its produtivity A i, the housing prie P, and the publi signal Q, and eah builder s housing supply S i ω i, P, Q depends on its produtivity ω i, the housing prie P, and the publi signal Q. The demand from households and supply from builders are integrated over the idiosynrati omponents of their produtivities { i } i [0,] and {e i } i [0,], respetively. At t = 2, the onsumption good prie lears market for eah household s good: C i i + C i j dj = e A i Ki α l α i, i N, N /i and the apital prie R lears market for apital: K i di = K di, 7 N where di represents the population of households that live in the neighborhood. N N 2
2 Equilibrium In this setion, we analyze the housing market equilibrium. We first analyze eah household s optimization problem given in 2, by onjeturing that only households with produtivities higher than a utoff A enter the neighborhood. We then derive a unique equilibrium utoff A that satisfies the learing ondition of the housing market. Finally, we verify at the end of the setion that the derived utoff equilibrium is the unique rational expetations equilibrium, in whih the hoie of eah household to live in the neighborhood is monotoni with respet to its own produtivity A i. 2. Choies of Households and Capital Produers We first analyze household hoies. At t = 2, households need to make their prodution and onsumption deisions, after the strength of the neighborhood A is revealed to the publi, and home builders and apital produers have also made their hoies at t =. Household i has e A i K α i l α i units of good i for onsumption and trading with other households. The following proposition desribes the household s onsumption, labor, and apital hoies. Its marginal utility of goods onsumption also gives the equilibrium goods prie. Proposition At t = 2, households i s optimal goods onsumption are C i i = η α e A i K α i l α i, C j i = and the prie of its produed good is p i = e η αψ++αψη A A i + 2 η 2τ η αψ++αψη Its optimal labor and apital hoies are τ A A η α e A j K α j l α j, τ αψ++αψη /2 + A A η τ A A. log l i = log K i = + ψ η α α ψ + + αψ η ψ A + η A i α α ψ + + αψ η α ψ log R + η α ψ log αψ++αψη + A A τ A A + l 0, + ψ + ψ α α ψ + + αψ η ψ η + ψ η A + A i α ψ + + αψ η ψ + α α ψ log R + α + ψ ψ η log αψ++αψη + A A τ A A + h 0, 3
with onstants l 0 and h 0 given in the Appendix. Furthermore, the expeted utility of household i at t = is given by [ ] E U {C j i} j N ; N l i + ψ I ψ i = α + ψ E [ p i e A i ] Ki α l α i Ii. Proposition shows that eah household spends a fration η of its wealth exluding housing wealth on onsuming its own good C i i and a fration η on goods produed by its neighbors N /i C j i dj. When η = /2, the household onsumes its own good and the goods of its neighbors equally. The prie of eah good is determined by its output relative to that of the rest of the neighborhood. One household s good is more valuable when the rest of the neighborhood is more produtive, as a result of the omplementarity in the household s utility funtion. Consequently, this proposition demonstrates that the labor hosen by a household is determined by not only its own produtivity e A i, but also the aggregate produtivities of other households in the neighborhood. Proposition also reveals that the optimal hoie of labor for eah household is loglinear with the strength of the neighborhood A, its own produtivity A i, and the logarithm of the apital prie is log R. The final nononstant term reflets seletion, in that only households with produtivity above A enter the neighborhood. Sine A is the mean of the distribution of household produtivity, it appears in this trunation. This proposition also demonstrates that household i s optimal hoie of apital has a similar funtional form. The household s optimal labor hoie and demand for apital are both inreasing in the strength of the neighborhood A, beause a higher A represents improved trading opportunities with its neighbors, while they are both dereasing in the prie of apital, log R. We now disuss eah household s deision on whether to live in the neighborhood at t = when it still faes unertainty about A. As a result of its Cobb-Douglas utility, the household is effetively risk-neutral over its aggregate onsumption, and its optimal hoie reflets the differene between its expeted utility from living in the neighborhood and the ost P of buying a house in the neighborhood. Then, household i s neighborhood deision is given by { if α ψ H i = E [ ] p i e A i Ki α l α i Ii P 0 if α ψ E [ ]. p i e A i K α i l α i Ii < P This deision rule supports our onjeture to searh for a utoff strategy for eah household, in whih only households with produtivities above a ritial level A enter the neighborhood. This utoff is eventually solved as a fixed point in the equilibrium. 4
Given eah household s equilibrium utoff A at t = and optimal hoies at t = 2, we an impose market-learing in the market for apital to arrive at its prie R at t = 2. Capital produers foreast this prie when hoosing how muh apital to develop at t =. These observations are summarized in the following proposition. Proposition 2 Given K units of apital developed by apital produers at t =, the prie of apital at t = 2 takes the log-linear form log R = + ψ ψ A α ψ + α ψ + α log K + + ψ ψ + α η log η αψ++αψη + α ψ ψ + α log + A A τ A A + r 0, αψ++αψη τ A A + A A τ /2 with onstant r 0 given in the Appendix. The optimal supply of apital by apital produers at t = is given by log K = λ α log E e A with onstant k 0 given in the Appendix. τ αψ++αψη /2 τ A A η τ αψ++αψη /2 τ A A + A A + A A ψ α η I + k 0, Proposition 2 reveals that the apital prie R at t = 2 is inreasing in the strength of the neighborhood A with the last two nononstant terms refleting seletion by households into the neighborhood, and is dereasing in the supply of apital K. It also demonstrates that the optimal supply of apital reflets expetations over not only the strength of the neighborhood A, but also the impat of trunation from the neighborhood hoie of households on the expeted prie of apital. The expetation term aptures not only the expeted produtivity from the terms-of-trade relative pries of household goods in the first ratio, but also the dispersion in labor produtivity in the seond ratio. 2.2 Perfet-Information Benhmark With perfet information, all households, home builders, and apital produers observe the strength of the neighborhood A when making their respetive deisions. Then, the optimal 5 8
hoie of apital K, given in Proposition 2, simplifies to log K = A + λ α λ α η log ψ + α + ψ log η τ αψ++αψη /2 τ A A τ αψ++αψη /2 τ A A + A A + A A + k 0, λ α where > 0, sine λ α > λ > 0. The supply of apital is log-linear with respet to the strength of the neighborhood A, with a orretion term for the trunation in the household population that ours beause of household seletion into the neighborhood. This trunation term reflets two fores. The first is that the prie at whih households harge eah other for their goods p i is also affeted by this trunation, while seond reflets that the smaller population has a higher average marginal produt of apital than the full population. We now haraterize the neighborhood hoie of households and the housing prie. Households will sort into the neighborhood aording to a utoff equilibrium determined by the net benefit of living in the neighborhood, whih trades off the opportunity of trading with other households in the neighborhood with the prie of housing. Despite the inherent nonlinearity of our framework, the following proposition summarizes a tratable, unique rational expetations utoff equilibrium that is haraterized by the solution to a fixed-point problem over the endogenous utoff of entry into the neighborhood, A. Proposition 3 In the absene of informational fritions, there exists a unique rational expetations utoff equilibrium, in whih the following hold:. Given any housing deision poliy for other households, household i follows a utoff strategy in its neighborhood hoie suh that { if Ai A H i = 0 if A i < A, where A A, ξ solves equation 22 in the Appendix. 2. The utoff produtivity A A, ξ is monotonially dereasing in ξ and inreasing humpshaped in A if η < > η, where η is given in 23 in the Appendix. 6
3. The population that enters the neighborhood is monotonially inreasing in both A and ξ. 4. The housing prie takes the following log-linear form: log P = + k τ τ e A A ξ. 9 5. The housing prie P, and onsequently the utility of the household with the utoff produtivity A, is inreasing and onvex in A. Proposition 3 haraterizes the unique rational expetations utoff equilibrium in the eonomy in the absene of informational fritions, and onfirms the optimality of a utoff strategy for eah household s neighborhood hoie when other households adopt a utoff strategy. Households sort based on their individual produtivity into the neighborhood, with the more produtive, who expet more gains from living in the neighborhood, entering and partiipating in prodution at t = 2. This determines the supply of labor at t = 2, and, through this hannel, the prie of apital at t = 2. The proposition also provides omparative statis of the equilibrium utoff household A A, ξ. This utoff is dereasing in ξ, sine a lower housing prie auses more households to enter the neighborhood for a given neighborhood strength A, and onsequently a higher population enters the neighborhood as ξ inreases. The utoff, in ontrast, is inreasing in neighborhood strength A, sine a higher A implies a higher housing prie and a higher prie of apital, but an be humped-shaped if there is suffi ient omplementarity beause the gains from trade for high realizations of A more than offset the inrease in pries. Though the utoff produtivity either inreases or is hump-shaped in A, more households ultimately enter the neighborhood beause a higher A shifts right in the sense of first order stohasti dominane the distribution of households more than it moves the utoff. Given a utoff produtivity A A, ξ, the housing prie P positively loads on the strength of the neighborhood A, sine a higher A implies stronger demand for housing, and loads negatively on the supply shok ξ, refleting that a disount is needed to ensure that a positive shift in housing supply is absorbed by a larger household population. As one would expet, the utoff A enters negatively into the prie sine only households above the utoff sort into the neighborhood. The higher the utoff, the fewer the households that enter the neighborhood, and the lower the housing prie that is needed to lear the market with the 7
lower housing demand. Despite its log-linear representation, the housing prie is atually a generalized linear funtion of τ τ e A ξ, sine A is an impliit funtion of A and log P. As a result of endogenous seletion into the neighborhood, the produtivity of the neighborhood is determined by whih households hoose to live there. The aggregate produtivity of the neighborhood A N is given by A N = log A e A j d j = A + 2 τ + log τ /2 + A A. τ /2 The first two terms are what one would expet without neighborhood hoie, while the third term reflets that the produtivity of the neighborhood is trunated by seletion. Importantly, sine A = A A, ξ, it follows that A depends on the housing prie in the neighborhood, introduing feedbak from housing pries to real deisions. Similar aggregation results exist for total inome N ea j p i K α l α j d j and labor supply l N jd j as well. 2.3 Equilibrium with Informational Fritions Having haraterized the perfet-information benhmark equilibrium, we now turn to the equilibrium at t = in the presene of informational fritions. With informational fritions, households and apital produers must now foreast the strength of the neighborhood A, and the prie of apital R at t = 2, when hoosing whether to live in the neighborhood, and when deiding the amount of apital to develop at t =. Eah household s type A i serves as a private signal about the strength of the neighborhood A. Sine types are positively orrelated with this ommon produtivity, higher types also have more optimisti expetations about A. This feature ensures that eah household will follow a utoff strategy when deiding whether to live in the neighborhood. As a result of the utoff strategy used by households, the equilibrium housing prie is a nonlinear funtion of A, whih poses a signifiant hallenge to our derivation of the learning of households and produers. It is the ase, however, that the equilibrium housing prie maintains the same funtional form as in 9 for the perfet-information benhmark. As a result, the information ontent of the publily observed housing prie an be summarized by a suffi ient statisti z P that is linear in A and the supply shok ξ: τ e z P = A ξ. 0 τ 8
In our analysis, we shall first onjeture this linear suffi ient statisti, and then verify that it indeed holds in the equilibrium. This onjetured linear statisti helps to ensure tratability of the equilibrium despite that the equilibrium housing prie is highly nonlinear. By solving for the learning of households and apital produers based on the onjetured suffi ient statisti from the housing prie, and by learing the aggregate housing demand from the households with the supply from home builders, we derive the housing market equilibrium. The following proposition summarizes the housing prie, eah household s housing demand, and the supply of apital in this equilibrium. Proposition 4 There exists a unique noisy rational expetations utoff equilibrium in the presene of informational fritions, in whih the following hold:. The housing prie takes the log-linear form: log P = τ A A ξ = τ z A + k τ e + k τ ξ. e 2. The posterior of household i after observing housing prie P, the publi signal Q, and its produtivity A i is Gaussian with the onditional mean Âi and variane ˆτ A given by  i = ˆτ A τ A Ā + τ Q Q + τ τ ξ z + τ A i, τ e ˆτ A = τ A + τ Q + τ τ e τ ξ + τ, and the posterior of apital produers, after observing housing prie P, the publi signal Q, is also Gaussian with the onditional mean  and variane ˆτ A given by  = ˆτ A τ A Ā + τ Q Q + τ τ ξ z, τ e ˆτ A = τ A + τ Q + τ τ e τ ξ. 3. Given any housing deision poliy for other households, household i follows a utoff strategy in its neighborhood hoie H i = { if Ai A 0 if A i < A, where A z, Q solves equation 25 in the Appendix. 9
4. The supply of apital takes the form: where F log K = λ α log F Â A, ˆτ A + A + k λ α 0, Â A, ˆτ A is given in the Appendix, and log K is inreasing in the onditional belief of apital produers Â. 5. The utoff produtivity A is dereasing, while the population entering the neighborhood and the housing prie P are inreasing, with respet to the noise in the publi signal Q. These properties also hold with respet to z under a suffi ient, although not neessary, ondition that 9 + k + τ τ e τ ζ τ A + τ Q k λ α α ψ + α + α η τ. α η + ψ τ e 6. The equilibrium onverges to the perfet-information benhmark in Proposition 3 as τ Q. Proposition 4 onfirms that, in the presene of informational fritions, eah household will optimally adopt a utoff strategy when other households adopt a utoff strategy. Informational fritions make the household s equilibrium utoff A z, Q a funtion of z P = + k τ e τ log P + A, whih is a summary statisti of the publily observed housing prie P, and the publi signal Q, rather than A and ξ as in the perfet-information benhmark. This equilibrium utoff is a key hannel for informational fritions to affet the housing prie, as well as eah apital produer s deision to develop apital. In the presene of informational fritions, the demand-side fundamental A and the supplyside shok ξ are not diretly observed by the publi and, as a result, do not diretly affet the housing prie and other equilibrium variables. Instead, their equilibrium effets are bundled together in the housing prie P through the speifi funtional form of z. Consequently, we an examine the impat of a shok to either A or ξ by analyzing a shok to z. The equilibrium housing prie in diretly implies that log P = τ z + k τ e A z 9 One may notie that a higher degree of omplementarity, η, tightens the suffi ient ondition, while muh of our analysis suggests that it amplifies the role of informational fritions. This is beause the ondition is not neessary, and is derived by omitting terms for whih η is relevant for amplifying the learning effet. 20.
That is, depending on the sign of A z, the equilibrium utoff A may amplify or dampen the housing prie effet of the fundamental shok z. Proposition 4 provides a suffi ient although not neessary ondition for A < 0. In this ase, there is an amplifiation effet. This z amplifiation effet makes housing pries more volatile, as highlighted by Albagli, Hellwig, and Tsyvinski 205 in their analysis of the utoff equilibrium in an asset market. 0 We shall analyze how different model parameters affet this amplifiation effet in the next setion. In the perfet-information benhmark, the publi signal Q has no impat on either the equilibrium utoff A or the housing prie beause both the demand-side fundamental A and the supply-side shok ξ are publily observable. In the presene of informational fritions, however, Q affets the equilibrium as it impats agents expetations. housing prie in demonstrates that log P Q = τ A + k τ e Q. The equilibrium By affeting the households expetations of A, and onsequently their utoff produtivity to enter the neighborhood, the noise in the publi signal Q affets the population that enters A the neighborhood and the equilibrium housing prie log P : P < 0 and > 0, as proved Q Q in Proposition 4. Furthermore, Q also affets the prie of apital, as well as eah apital produer s optimal hoie of how muh apital to develop. In the next setion, we fous our analysis on two key model parameters: the omplementarity in households onsumption and housing supply elastiity. Households onsumption omplementarity reinfores the effets of informational fritions. Without omplementarity, a stronger neighborhood, or a higher A, is bad news for households, beause a higher A raises not only the housing prie, but also the prie of apital. With omplementarity, however, a stronger neighborhood is also good news for households, beause it means that other households in the neighborhood are more produtive, and therefore represents a better opportunity for trade. In the presene of informational fritions, omplementarity gives eah household a stronger inentive to learn about A, and thus amplifies the potential distortionary effets from suh learning. Supply elastiity also plays an important and nuaned role in the distortionary effets of 0 This interesting feature also differentiates our utoff equilibrium from other type of nonlinear equilibrium with dispersed information, suh as the log-linear equilibrium developed by Sokin and Xiong 205 to analyze ommodity markets. In their equilibrium, pries beome less sensitive to their analogue of z in the presene of informational fritions. This ours beause households, on aggregate, underreat to the fundamental shok in their private signals beause of noise. 2
learning. It is instrutive to onsider two polar ases for supply elastiity. When supply is infinitely inelasti i.e., k 0, housing pries are only determined by the strength of the neighborhood A, and pries are fully revealing to households and apital produers. As a result, there is not any distortion from the learning when supply is infinitely inelasti. In ontrast, when supply is infinitely elasti i.e., k, pries onverge to log P = ζ, whih is driven only by the supply shok. In this ase, pries ontain no information about demand, and therefore no information about the strength of the neighborhood. Consequently, the learning from housing prie and the potential distortion of suh learning both dissipate as supply elastiity approahes infinity. These two polar ases demonstrate an important insight of our model that the distortion aused by learning on housing pries is humped-shaped with respet to supply elastiity. 2 We onlude this setion by establishing that the utoff equilibria we have haraterized, both with informational fritions and with perfet-information, is the unique rational expetations equilibria in the eonomy. Regardless of the pereived housing poliies of other households in the neighborhood, eah household with rational expetations will ontinue to follow a utoff strategy, whih establishes the uniqueness of the utoff equilibrium. This is summarized in the following proposition. Proposition 5 The unique rational expetations utoff equilibrium is the unique rational expetations equilibrium in the eonomy. Proposition 5 strengthens the preditions of our analysis, as they are now the unique preditions for behavior in the neighborhood. In the next setion, we explore the rosssetional impliations of our model for noise-driven housing yles. 3 Model Impliations We now investigate several impliations of our model regarding how informational fritions affet the housing market and loal real investment. We illustrate how several key aspets of the neighborhood and its real estate markets vary aross two dimensions: supply elastiity Note from equation 25 that A remains finite a.s. as k, allowing us to take the limit. 2 These two polar ases also larify that this insight is not dependent on the partiular form of housing supply speified in Setion.3. While our speifiation is standard and onsistent with existing models in the noisy rational expetations literature, we expet an alternative speifiation to deliver qualitatively similar results, as it would not alter these two polar ases. 22