Spatial Asset Pricing: A First Step

Transcription

1 Spatia Asset Pricing: A First Step François Ortao-Magné University of Wisconsin Madison Andrea Prat Coumbia University Revised May 1, 2013 Abstract Peope choose where to ive and how much to invest in housing and financia assets, incuding whether to buy or to rent their home Traditionay, the first decision has been the domain of spatia economics, whie the second has been anayzed in finance Spatia asset pricing combines the two sets of decisions in a tractabe dynamic portfoio choice probem, which eads to an array of predictions on the geographica aocation of human capita, househods portfoio over space and time, housing rents, housing and asset prices Despite its simpicity, our framework provides a unified expanation for a number of known phenomena eg hedging roe of housing, home bias, housing investment over the ife cyce, ink between prices and oca productivity shocks and makes nove testabe predictions that span financia and spatia observabes eg cross-sectiona distribution of price-rent yieds, reation between rea estate returns and financia returns, deviations from production-maximizing aocation of human capita We are gratefu to Orazio Attanasio, Phiippe Bracke, Morris Davis, Christian Juiard, Robert Komann, Aex Michaeides, Sven Rady, Stijn Van Nieuwerburgh, Dimitri Vayanos, and Nancy Waace for hepfu discussions; and to seminar participants at Brown University, the University of Caifornia Berkeey, Coumbia University, ECARES, ESSET, the Federa Reserve Bank of St Louis, HEC Paris, Hong Kong University of Science and Technoogy, INCAE, the London Schoo of Economics, the University of Mannheim, the NBER Summer Institute, the University of North Caroina, Northwestern University, and the Tououse Schoo of Economics for usefu comments We are gratefu to CEPR, the Financia Markets Group at LSE, and the Tououse Schoo of Economics for their support fom@buswiscedu ap3116@coumbiaedu

2 1 Introduction The urban economics iterature provides insights into the aocation of househods over space and the cross-sectiona distribution of rents Given dividends for housing and other assets, the finance iterature provides insights into the demand and price of housing within standard asset pricing frameworks expanded to incude one uniform housing commodity So far, these two iteratures have evoved independenty The urban economics iterature abstracts from the stochastic nature of returns to housing, the investment demand for housing, and any risk premium buit into housing prices The finance iterature abstracts from the spatia heterogeneity of houses and the endogenous cross-sectiona distributions of househods investments and rents We expore the gains from merging the two iteratures We design a tractabe mode with cosed-form soutions that are comparabe to we-estabished resuts in both fieds The exercise reveas meaningfu interactions between spatia equiibrium and asset pricing, hence spatia asset pricing We demonstrate that choosing a ocation amounts to soving an expanded portfoio probem with the requirement to choose one unit of one of the free securities representing each ocation These securities deiver a stream of ocation specific net earnings, oca abor income minus housing rent As in standard spatia modes, rents are priced by agents who are indifferent across ocations The reevant attributes of each ocation incude not ony expected net earnings but aso their variance, or rather, the cost of hedging this variance: a function of the covariance of the agent s earnings with oca rents and asset prices The hedging motive for investment introduces home bias in portfoio choice Because of this home bias, the covariances of agents income within each ocation and the aocation of agents across ocations affect the aggregate demands and prices for a assets, incuding houses in every ocation As such, spatia asset pricing brings to the fore new testabe predictions spanning both financia and spatia variabes For exampe, we find the rater of homeownership in each ocation is reevant to the pricing of a assets Spatia asset pricing aso provides a unified conceptua framework to reconsider a variety of empirica questions concerning variations in househods portfoios over space and time, and the reationship between housing and financia asset prices The mode We assume four casses of assets: a risk-free bond, stocks, houses in a number of ocations, and non-transferrabe human capita As in standard asset pricing modes, agents may end and borrow at the risk-free bond rate without any constraint Agents may 2

3 aso invest in stocks, defined as caims over exogenous stochastic streams of dividends The dividend stream of houses, however, is determined endogenousy Houses provide access to a stochastic production technoogy that is specific to their ocation An agent s human capita determines the expected eve of his or her earnings at each ocation and the covariance of earnings with the ocation-specific production technoogy The distribution of individua characteristics across the popuation is expressed in a genera form 1 Houses differ ony in their ocation but one, the countryside, where the suppy is unimited The suppy of houses is fixed in every ocation Houses can be rented at the oca equiibrium market rate In our benchmark mode, houses can be purchased or sod even fractionay at the oca equiibrium price Agents may buy houses in every market, incuding in their city There are no frictions on any of the asset markets; eg, no credit constraints, no transaction costs for buying or renting, no imits to fractiona ownership We want to obtain cosed-form soutions and expressions that are comparabe to standard resuts obtained from mean-variance asset pricing modes, so we assume an overapping generations structure with finite ife and constant popuation size Agents have constantabsoute risk-aversion preferences with infinite easticity of intertempora substitution, and both city-productivity and stock-dividends stochastic shocks are normay distributed Agents choose where to ive at the beginning of their ife Living in a ocation requires consuming one unit of housing in this ocation Whie a investment decisions can be revisited in every period, the ocation choice is irreversibe moving costs are infinite 2 Location security Each ocation is represented as a free ocation security composed of two parts: 1 a stream of ocation-specific stochastic benefits that yied ocation and agent specific wages or enjoyment of oca amenities, and 2 a unit of oca housing, which requires a stream of endogenous stochastic rent payments, and satisfies the oca housing consumption requirement The ocation decision and the portfoio aocation probem of an agent can therefore be examined within the same dynamic optimization framework Besides choosing financia assets, each househod must pick one unit of one ocation security This characterization refects the discreteness of the choice of ocation 1 In most of this work, we interpret oca productivity as abor-reated and hence transated into abor earnings, but the mode has an equivaent interpretation in terms of eisure, where productivity is understood as the abiity of the agent to enjoy oca amenities We aso assume that there are no spiover effects across agents; that is, the productivity of an agent depends on its ocation but not on who ese ives in that ocation Later, we show that our characterization extends to a mode with generic economies of aggomeration 2 The assumption that peope cannot move is usefu for tractabiity of anaysis, but the assumption is not necessary for resuts about the roe of housing as a hedge Sinai and Souees 2009 provide evidence of the empirica reevance of this phenomenon, accounting expicity for US househod moving patterns 3

4 Spatia aocation and rents Rents are determined by the productivity of margina residents, househods that are indifferent between two or more ocations We demonstrate that our mode admits a set of hyper-margina residents: househods indifferent among a ocations, whether city or countryside These househods are a age one as ocation choice takes pace at age one, once and for a 3 Determining the set of hyper-margina househods is a key step to characterizing the equiibrium with stationary aocation of househods across ocations, inear housing rents and asset prices We conjecture a functiona form for rents and prices and we verify the vaidity of our guess We buid our guess by adapting standard resuts obtained in CARAnorma portfoio choice frameworks Loca rents respond to oca productivity shocks so as to keep the hyper-margina residents indifferent across a ocations The indifference condition of the hyper-margina resident pins down the reative eve of rents in different ocations The fact that one ocation the countryside has an unimited suppy of and determines the absoute eve of rents The ocation decision of any agent is determined by comparing that agent s ocationspecific set of productivity parameters with that of the hyper-margina residents By aggregating the investment demand functions of a agents, we obtain the asset pricing formuas both for rea estate in different ocations and for stocks We are then in a position to verify that the initia conjecture about the hyper-margina resident is correct and that it is indeed an equiibrium We thus prove the existence of an equiibrium where prices can be expressed as inear functions of the underying parameters, and the aocation of househods across ocation remains constant over time Uniqueness can be proven in specific cases Portfoio choice Because we assume househods choose their ocation at birth, margina residents are newborn househods As househods age, we do not put any restriction on the covariance of their current income with their income at birth A changing covariance exposes househods to the risk that shocks to their income may not provide fu insurance for the oca productivity shocks that affect their housing costs This risk can be hedged away by an appropriate hoding of oca rea estate The optima investment portfoio of every agent contains two components: 1 An investment in oca housing that depends on the agent s exposure to oca productivity shocks, and 2 a portfoio of stocks and houses, with identica weights across agents The first com- 3 An anciary contribution of our work is to show existence in a spatia genera equiibrium mode with an infinite number of types of agents Avaiabe existence resuts in the iterature use a different approach based on a finiteness Grimaud and Laffont,

5 ponent is a manifestation of home bias This hedging demand depends on the covariance between the agent s earnings and oca productivity shocks The second component is identica across agents because in our benchmark case, a agents fuy hedge their exposure to oca productivity shocks with a purchase of oca housing Asset pricing Conditiona on their purchases of oca housing for hedging purposes, a househods are fuy hedged against their idiosyncratic risk and thus identica with regard to aggregate risk Hence they a have the same investment demand for the remaining securities in the economy: the portfoio made up of a stocks and residentia properties in the economy minus the houses hed for hedging purposes Let us ca this portfoio the adjusted market portfoio Equiibrium requires that the price of a assets in the economy be such that tota investment demand beyond the hedging demand for houses equas the adjusted market portfoio A assets are therefore priced in this adjusted market portfoio Impications Our portfoio choice and asset pricing expressions specify ony objects such as prices and covariances that are in principe observabe We can thus deveop a number of empirica questions inking spatia and financia variabes Some of our predictions are aready known, but have so far been discussed mainy in isoation; others are, to the best of our knowedge, new The main strength of spatia asset pricing is that a of them can be anayzed within a consistent and tractabe conceptua framework Home bias With no friction on the housing market and identica homes within each ocation, oca homes are a perfect hedge for agents exposure to income minus rent risks This resuts in home bias that manifests itsef entirey in terms of oca housing purchases In a ess perfect word without fractiona ownership of housing and with oca consumption that extends beyond housing, we woud expect agents to use both oca housing and oca financia assets for hedging purposes- ie, stocks with strong covariance with the oca economy Our framework provides a compementary rationae for the preference for geographicay proximate investments pointed out in Cova and Moskowitz 1999 Homeownership over the ife-cyce Sinai and Souees 2005, and Davidoff 2006 aready pointed out the reevance of the hedging motive for househod s housing investment Our theory opens new empirica questions with regards to the measurement of the determinants of this hedging motive and provides a new rationae for the we-known hump-shaped ifecyce profie of homeownership 4 Suppose that, as agents, get oder, the covariance of their 4 For exampe, see Fernandez-Viaverde and Krueger 2011 and Attanasio et a

6 income with the income of newcomers to their city decreases Their income provides ess insurance against rent shocks This pushes them to purchase an increasing amount of oca housing for hedging purposes as they get oder Counter to this effect is the fact that as agents get oder, the time eft to ive is shortened and so their demand for insurance decines We show by exampe how this combination of effects can yied a hump-shaped homeownership profie Geographica aocation of agents The ceebrated ocation choice mode of Rosen 1979 and Roback 1982 determines jointy housing rents and the geographica distribution of agents, but it does so in a static, deterministic mode By moving to a dynamic, stochastic setting, we create a roe for investment When they choose where to ive, agents trade off expected net earnings opportunities expected wage minus expected rents against risk exposure voatiity of income minus rent Agents therefore do not necessariy choose the ocation that maximizes their net earnings They may prefer a ocation with ower expected income minus rents if their income in that ocation is ess correated with rents In such a ocation, the purchase of oca housing provides insurance benefits Nevertheess, they earn a risk premium on the oca housing because it is priced by outsiders to whom the voatiity of housing returns is a risk, not an insurance Inferences from the cross-sectiona dispersion of rents By the argument above, spatia asset pricing points out the imitations of any mode that impicity assumes ocation decisions based entirey on expected earnings and amenities and ignores variations in a househod s risk exposure The quantitative importance of the risk/hedging factor in ocation and investment decisions is of critica empirica importance Shoud this factor matter in any significant way, it woud ca into question standard approaches to the vauation of oca amenities that are based on a static frameworks Joint determination of housing and financia asset prices As a number of authors have noted, househods financia and housing investments and the prices of housing and financia assets can ony be determined jointy 5 Our paper offers a three-fod contribution First, we consider a arge number of assets in both asset casses and we aow a generic covariance matrix Second, rather than assuming an exogenous dividend process for houses we derive it endogenousy from geographica ocation equiibrium Third, we incorporate the hedging roe of oca housing 5 See for exampe Grossman and Laroque 1991, Favin and Nakagawa 2008, Henderson and Ioannides 1983, Goetzmann 1993, Brueckner 1997, Favin and Yamashita 2002, Engund, Hwang, and Quigey 2002, Iacovieo and Ortao-Magné 2003, and LeBanc and Lagarenne 2004, Cocco 2004, Yao and Zhang 2005, Piazzesi, Schneider, and Tuze

7 The fact that the quantity of homes in each ocation in the pricing-reevant adjusted market portfoio is determined endogenousy, adds a channe whereby the spatia aocation of househods affects the prices of a assets In other words, the spatia aocation of househods does more than determine the stochastic properties of the rents in each ocation; it aso determines what assets are part of the portfoio that is reevant to pricing systematic risk in equiibrium As a resut, our asset pricing formuas incude some spatia variabes such as the covariance between productivity shocks in different ocations and the proportion of owner-occupied housing in each ocation Whie the use of such variabes is new, they are a potentiay observabe The prices of stocks are therefore determined not ony by how their dividends co-vary with those of other financia assets but aso by how they co-vary with earnings of the hypermargina residents in each ocation Note that information reevant for asset pricing reated to the presence of housing in the economy cannot be represented by a singe aggregate housing good Cross-sectiona dispersion of rent-price ratios Our mode predicts the cross-sectiona distribution of rent-price ratios The oca yied depends on the average within-ocation covariance of the income of each resident with the income of the current and future margina residents For instance, in a one-company town, wages of a cohorts are highy correated with rents; residents there do not demand oca housing for hedging purposes, and prices are depressed with respect to rents Prices are higher in diversified towns where househods invest in oca housing for hedging purposes Extensions The anaytica resuts we obtain for the benchmark mode support two usefu extensions in the modeing of ocations and housing assets First, the benchmark mode can be extended to encompass economies of aggomeration and other forms of externaities among residents The equiibrium characterization of the benchmark mode remains vaid, but the presence of direct externaities ampifies the possibiity of equiibrium mutipicity If the aggomeration economies are strong enough, there wi be mutipe inear stationary equiibria corresponding to different aocations of taent across cities This means we may be abe to create inks between rea estate finance and the vast iterature on aggomeration effects Second, the benchmark mode assumes the ownership of rea estate is perfecty divisibe Househods are aowed to buy exacty the amount of oca housing they need to perfecty hedge their risk in income minus rent A number of frictions are ikey to ead househod away from the perfect hedge investment; eg, a preference for homeownership, preferentia tax treatment of homeownership, housing transaction costs, housing property indivisibiities 7

8 Any such impediment to obtaining a perfect hedge with oca housing eads househods to resort to expoiting the covariance between their oca risk and each of the financia assets Agents in different ocation therefore purchase a different portfoio of housing and stocks This hedging demand for stocks ends up affecting stock prices As a case in point, we propose expicit soutions for stock prices when a househods are required to own their home Whie in the baseine case the ony asset used for hedging was oca rea estate, the presence of indivisibiities induces agents to use stocks to hedge against their exposure or ack of exposure to oca rea estate This generates an additiona set of testabe portfoio impications 6 The paper is organized as foows: Section 2 reviews the reated iterature Section 3 sets out the mode Section 4 presents the main equiibrium characterization resut, through three propositions corresponding to: portfoio aocation Proposition 1, asset pricing Proposition 2, and ocation choice Proposition 4 Section 5 uses the main resut to discuss a number of reated issues Section 6 concudes A proofs are in the Appendix 2 Reated Literature This is to the best of our knowedge the first asset pricing mode where ocation choices, housing rents, and asset prices are endogenous Whie many of the phenomena that our mode describes have been anayzed esewhere, it is the first time that they can be expained within one mode, thus providing a unified conceptua framework to think about spatia asset pricing Our work is party inspired by an asset pricing mode outside the rea estate iterature: DeMarzo, Kanie, and Kremer 2004 They consider an economy with mutipe communities and oca goods as we as a goba good In this dynamic setting, some agents the aborers are endowed with human capita that wi be used to produce oca goods in future periods, but they are currenty subject to borrowing constraints Other agents the investors own shares in firms that produce the goba good This approach yieds a number of powerfu resuts Investors care about their reative weath in the community because they bid for scarce oca goods This generates an externaity in portfoio choice, which eads to the potentia presence of mutipe equiibria in the stabe equiibria, investors dispay a strong home bias And, if there is a behaviora bias, this externaity ampifies the bias through the portfoio decisions of rationa investors 6 Other extensions yied simiar predictions with regard to home bias Suppose househods enjoy more utiity from the same property if they own it than if they rent it Such an assumption impies that their investment in oca housing is not driven purey by hedging considerations Househods are wiing to distort their housing investment because of consumption benefits It then becomes optima to use stocks to dea with any residua risk in income minus rent not canceed with oca housing investment 8

9 Our mode differs from DeMarzo et a 2004 in a number of important dimensions: 1 Our oca good does not produce utiity directy, but it enabes agents to reaize their human capita potentia; 2 there are no credit constraints; and 3 our spatia aocation is endogenous We do share their goa of studying the properties of portfoio choice and asset pricing under uncertainty in the presence of community effects As in their mode, a home bias arises in equiibrium because of a hedging motive 7 Our contribution to the rea estate finance iterature, ies in endogenizing both housing prices and rents in a dynamic mode with mutipe ocations 8 Our approach to the modeing of housing as access to a ocation is in the tradition of urban economics Our ocation choice mode foows the standard muti-cities framework of Rosen 1979 and Roback 1982, where residentia properties provide access to the oca abor market, and ocations are differentiated by potentia surpus As in Rosen and Roback and the many more recent papers that buid on this framework eg, Gyourko and Tracy, 1991, Kahn, 1995, Gaeser and Gyourko, 2005, we assume househods face a unit housing consumption requirement and derive utiity from consumption of numeraire ony Lustig and Van Nieuwerburgh 2010 consider risk sharing across regions Empirica evidence indicates that the amount of housing weath in a region affects the sensitivity of oca consumption to oca income This paper is particuary cose to ours in that it considers mutipe ocations Lustig and Van Nieuwerburgh, however, assume exogenous ocation choice and that housing suppy is perfecty eastic in a ocations and hence rents depend ony on aggregate shocks Because we are concerned with portfoio choice in a dynamic environment, we assume househods are risk-averse Risk aversion in the face of stochastic streams of income and rent provides a motivation for ownership of oca residentia properties homeownership in our mode This approach buids on the work of Ortao-Magné and Rady 2002, Hiber 2005, Sinai and Souees 2005, 2009, Davidoff 2006 and others who provide evidence of the reevance of such motivation for housing investment We do not here review the vast iterature concerned with the determinants of housing prices Typicay in this iterature, rea estate prices are determined by a perfecty eastic suppy function Lustig and Van Nieuwerburgh, 2008 or by a perfecty eastic demand function Davis and Heathcote, 2005, Davis and Ortao-Magné, forthcoming, Gyourko, Mayer and Sinai, 2006, Kiyotaki, Michaeides and Nikoov, 2007, Van Nieuwerburgh and 7 Our resuts on home bias are aso reated to the internationa finance iterature on the home bias puzze Stockman and Deas, 1989, but we differ in our focus on rea estate and in that ocation choice is endogenous in our mode 8 A review of the empirica iterature on the cross-sectiona dispersion of housing prices is beyond the scope of this paper For recent evidence emphasizing variations in housing price premiums see Campbe et a

10 Wei, Mode Consider an overapping generation economy where a mass 1 of agents is born in every period Each agent in the t-cohort is born at the beginning of period t, ives for S periods, and dies at the beginning of period t + S Hence, at every time t, there are a mass S of agents aive in the economy 31 Geography There are L cities, denoted by index 1,, L and a countryside denoted by index 0 City has an exogenousy given mass of houses Let n be the mass of houses per cohort that wi be active on the housing market so that tota suppy of housing in city equas S n We assume that housing suppy is scarce in cities: L n < 1 1 but it is abundant once we incude the countryside: L n > 1 0 Each house accommodates exacty one agent 32 Production The income of a person who ives in the countryside is normaized to zero Productivity in city foows the process y t y t 1 + τ t where τ t is a random variabe, independenty and identicay distributed across time At birth, each agent draws: A vector of city-specific endowment surpus, ε [ε ] 1,,L, with ε, A matrix of city- and age-specific insuation parameters: ρ [ρ s] 1,,L,,S, with ρ s [0, 1] Assume ρ 0 0 for a The parameters ε, ρ are iid across generations Their joint distribution within a generation takes the genera form φ ε, ρ, with the ony requirement that it shoud be continuous and have fu support 10

11 At time t + s, the income of an agent iving in city, born at time t, with parameters ε, ρ is y t,t+s ε, ρ y t 1 + ε + s m0 1 ρ m τ t+m for s 0,, S 1 note the difference between yt, a city-wide variabe, and yt,t+s ε, ρ an individua specific variabe Hence, the income of each agent can be decomposed into a permanent part, which captures the initia productivity of the agent in a ocation and a time-dependent part, which is determined by the oca productivity shocks in the city and that agent s sensitivity to the city s shocks We ca ε the city-agent effect and ρ s the shock insuation effect We represent beow the income earned by an agent born at time t, iving in city, for each of the first three years of ife: y t,t y t,t+1 y t,t+2 ε, ρ ε, ρ ε, ρ city-agent effect {}}{ city-cohort effect {}}{ ε + yt ; year 1 innovation city-agent effect city-cohort effect {}}{{}}{{ }} { ε + yt + 1 ρ 1 τ t+1; city-agent effect {}}{ city-cohort effect {}}{ ε + yt + year 1 innovation year 2 innovation { }} { { }} { 1 ρ 1 τ t ρ 2 τ t+2 Simiar formuations determine the agent s earnings unti reaching age S 1 9 At age S, the agent does not earn anything It is mathematicay convenient to set ρ S 0 for a agents even if it is irreevant to the agents earnings The city-agent effect, ɛ, is a standard object in muti-city modes with heterogeneous agents Depending on their human capita, agents face different earning opportunities in different ocations The shock-insuation effect, ρ, captures two economic phenomena First, agents may be exposed to a technoogica cohort-specific effect documented by Godin and Katz, 1998 The human capita of certain peope, especiay the young, may be more fexibe When a technoogica innovation appears, the income of certain agents wi be more affected than the income of others Second, certain agents ike senior workers and pubic sector workers may be part of an impicit abor insurance agreement Their wages are more insuated from productivity shocks 9 The structure of ε and ρ coud be more compex and sti be amenabe to anaysis in our mean-variance set-up For instance, we coud say that the city-agent effect is not constant over the ife of the agent but rather it foows a random wak Aso, we coud assume that the extent to which a shock that occurs at age s affect future incomes depends on the age of the agent 11

12 It is reasonabe but not necessary for the anaysis to assume that the insuation parameter, for a shock that occurs at a given age, increases with in the age of the agent: ρ s+1 > ρ s The two extreme cases are fu insuation ρ s 1 and fu exposure ρ s 0 10 For concreteness, we interpret y t,t+s as monetary income, but there is an aternative interpretation in terms of non-monetary benefits that is equivaent from a mathematica standpoint The term y t,t+s can be viewed as a money-equivaent of the utiity afforded by the amenities present in ocation The utiity can be decomposed in turn into an agentcity effect a preference for that particuar ocation and a shock component perhaps an environmenta or a socia risk mutipied by the agent s sensitivity to that type of shock Of course, the mode can aso be interpreted as a mix of monetary and non-monetary benefits An agent who ives and thus produces in city, must rent exacty one unit of housing in city 33 Housing market The housing market is frictioness There are no transaction costs associated with renting, buying, or seing property There is no difference between iving in an owned or a rented house At birth, every agent chooses in what city or the countryside to ive The agent cannot move afterward Living in city at time t entais paying the market rent, on a unit of housing, r t Rents are determined in equiibrium Agents may invest in divisibe shares of any city s housing stock and revise their decision at every period Let a t,t+s denote the amount of housing of city owned by an agent born at time t of age s The market price of a unit of housing in city at time t is p t The agent revises his or her housing investment at the beginning of every period For accounting purposes, imagine that the agent iquidates a housing assets and then buys the desired amount in each period At the beginning of period t + s, the agent acquires a t,t+s units in city at tota cost a t,t+sp t+s During period t, the agent coects rent on the housing investment for a tota of a t,t+sr t+s At the beginning of the next period, the agent iquidates the housing investment and receives a t,t+sp t+s+1 We denote a t,t+s the vector of the agent s housing investments, a t,t+s [ a ] t,t+s 1,,L We find it natura to restrict ρ s to be between zero and one, but our mathematica anaysis is vaid even if ρ s > 1 the agent s productivity is negativey correated with oca shocks and ρ s < 0 the agent is overexposed to oca shocks 11 Given the frictioness nature of the housing market, derivative securities woud be superfuous In particuar, Case-Shier home price indices for our cities a security bought at time t which pays a price p t+1 at time t + 1 woud be equivaent to purchasing housing for one period, net of the rent coupon Given the 12

13 34 Stock market Besides housing, there is another cass of securities caed stocks These are caims on productive assets, that as in reguar asset pricing modes produce an exogenous stochastic stream of income There are Sz k units of type-k asset, with k {1,, K} and z k > 0 A unit of stock k produces dividend d k t at time t The dividend foows the stochastic process: d k t d k t 1 + ν k t where ν is iid across time with probabiity distribution as beow As is the case for housing, every agent can buy units of every stock and revise portfoio aocations in every period The market price of stock k at a particuar time is q k t At the beginning of period t + s, the agent acquires b k t,t+s units of stock k at tota cost b k t,t+sq k t+s During period t + s, the agent receives dividends on investment in k for a tota of b k t,t+sd k t+s At the beginning of the next period, the agent iquidates the stock investment and receives b k t,t+sqt+s+1 k b t,t+s [ b k ] t,t+s k1,,k 35 Distribution of random shocks We denote b t,t+s the vector of the agent s stock investments, There are two sources of exogenous shocks in our economy: a vector τ of oca productivity shocks, and a vector ν of dividend shocks The shocks are independenty and identicay distributed over time, according to a norma distribution with mean 0 and covariance matrix Σ: τ t, ν t N 0, Σ We do not impose any restriction on the correation between oca productivity shocks and dividends One industry may be more affected by shocks in a certain market, and vice versa We aso do not impose any restriction on the correation of productivity shocks across cities 36 Consumption and savings As our goa is to deveop a cosed-form expression for asset prices, we assume that agents derive CARA utiity exp γw from weath at the end of their ife, w, where γ is the standard risk-aversion parameter Agents face no credit constraints and can borrow and end freey at discount rate 0, 1 For simpicity, we assume that agents are born with no weath this does not affect their decisions, given that they have CARA preferences random-wak nature of a our shocks, ong-term securities are aso redundant because they can be repicated by sequences of short-term investments This incudes ong-term rentas or futures on rea estate 13

14 37 Non-negativity constraints Asset pricing modes with normay distributed shocks suffer from a we-known technica probem As the vaue of dividends can become negative, agents may want to dispose of assets they own If they coud, the distribution of asset vaues woud no onger be norma, and the mode woud not be tractabe Hence, a modes in this cass assume, impicity or expicity, that agents cannot dispose of assets Typicay, this assumption is unreaistic because in practice both agents and firms are protected by imited iabiity Instead, in the mode stocks can have negative prices, and their owners must pay to get rid of them Our CARA-norma framework inherits this non-negativity probem That is, productivity in a city coud become negative, and house prices there may be negative 12 The usua response to this criticism, which appies here as we, is that the unconstrained mode shoud be viewed as an approximation of the mode with non-negativity constraints, as ong as the starting vaues are suffi cienty far from zero 38 Timing The order of moves for an agent born at time t is as foows: 1 At birth, the agent chooses in which ocation to spend the rest of his or her ife 2 At the beginning of each period t + 0,, t + S, the agent earns the vaues of the random shocks for that period, ν t+s and τ t+s 3 For s 0,, S 1, the agent revises housing and stock investments a t,t+s and b t,t+s, pays rent r t+s for one unit of housing in the chosen ocation and coects dividends and rents on the assets owned 4 At time t+s, the agent iquidates a investments a t,t+ and b t,t+ and consumes a weath before death 13 4 Anaysis An equiibrium is an aocation of househods across cities, a vector of optima portfoio hodings of housing and stocks for each agent, housing rents and prices for each city, and stock prices such that: 1 The ocation choice and portfoio hodings sove the agents probem; 2 the housing markets space and ownership in each city cear; and 3 the stock markets cear 12 We assume homeowners have an obigation to rent their property they pay a fine if it is vacant 13 The agent does not work or pay rent in the ast period of ife t + S but rather consumes a weath at the beginning of the period before death 14

15 A stationary equiibrium is an equiibrium where the mass of agents of a generation t who ive in a given city is the same across generations 14 A inear equiibrium is an equiibrium where stock prices, rents, and house prices can be expressed, respectivey, as: q k t 1 1 dk t q k 1 rt yt + r 2 p 1 t 1 r t p 3 where q [ q k] k1,,k and p [ p ] 1,,L are price discounts; and r [ r ] 1,,L is a rent premium to be determined in equiibrium The rent is equa to oca productivity pus a oca constant House and stock prices are equa to the discounted vaue of a perpetuity that pays the current rent or dividend minus an asset-specific discount Price discounts can aso be interpreted as expected returns of zero-cost portfoios 15 Throughout the anaysis we describe p and q k depending on the context as price discounts or expected returns, Our strategy for finding equiibria is as foows We start by conjecturing that we are in a stationary inear equiibrium We postuate a feasibe aocation of agents to cities, and we sove the portfoio probem of a generic agent iving in a given city As it turns out, soving this agent probem is enough to characterize stock prices and house prices up to a vector of city-specific constants With this information, we compute the expected utiity of every agent, conditiona on city choice We determine aggregate ocation demand, given any price vector by comparing expected utiities across cities Finay, we consider the margina residents We show that for every vector of cityspecific constants there are a set of agents who are indifferent among a ocations the hyper-margina residents, whie a others have strict preferences The characteristics of the hyper-margina residents are monotonic in the vector of city-specific constants, and we can identify the hyper-margina residents so that the mass of agents who move to each city equas the oca housing suppy in each city This proves that our initia conjecture on inear prices is correct A non-stationary equiibrium be structured as foows As agents cannot move after they ocate to city, the stock of rented accommodation used by the t-cohort wi not become avaiabe unti members of the t-cohort die at the end of t + S Hence, if the t-cohort is, say, overrepresented, the t + S + 1-cohort wi be equay overrepresented The non-stationary equiibria are characterized by cyces of ength S For instance, the expected return of a zero-cost one-unit portfoio invested in housing in city evauated in today s doars is E [ ] p t+1 p t rt 1 r t p 1 r t + p 1 p 16 It is tempting to consider the two first parts of the anaysis portfoio choice and asset pricing in 15

16 As agents have CARA preferences their ifetime utiity can be decomposed into: [ ] [ ] [ ] E u t E wt γv wt Proposition 1 re-writes the two components of the agents utiity and uses them to compute his optima portfoio choice and his expected utiity In what foows we focus on one agent and we drop the argument representing the agent-specific characteristics: ε, ρ A proofs are in appendix Proposition 1 Portfoio Aocation Suppose that prices and rents are given by equations 1, 2, and 3, with given r, q and p Consider any aocation of agents to cities Consider an agent born at period t characterized by a vector ε and a matrix ρ If this agent ives in and chooses investment profies [a t,t+s, b t,t+s ],,, the expectation and the variance of the agent s end-of-ife weath can be written, respectivey, as: E [w t ] s S ε r S s 1 L K ρ s+1 p + ã j t,t+s pj + V ar [w t ] 2 L K 1 2 2s S V ar ã j t,t+s τ j t+s+1 + j1 k1 j1 b k t,t+sν k t+s+1 k1 b k t,t+s q k where ã j t,t+s { a t,t+s 1 S s 1 ρ s+1 if j a j t,t+s otherwise The agent s optima investment profie is given by [ ] [ ãt,t+s 1 3 b t,t+s 2γ s+2 S Σ 1 p q for s 0,, S 1, and expected og-utiity is U 1 S s ε r S s 1 ρ s+1 p [ 1 4 p + S 4γ 2 q ], ] Σ 1 [ p q Proposition 1 says that the optima portfoio of any agent can be decomposed into: isoation, but they are vaid ony if the third part is present too If one assumes a different ocation mode or an exogenous aocation of agents to cities, the three price processes in equations 1, 2, and 3 woud be different, and Propositions 1 and 2 woud no onger hod For instance, if agents coud move between cities during their ifetime, it is not cear that the rent the price in city woud depend ony on productivity in city We see this as both a weakness and a strength of spatia asset pricing On the one hand, one cannot have a meaningfu discussion about rea estate prices in mutipe ocations without an underying spatia mode On the other hand, this opens the door to a weath of testabe impications reated to spatia and financia variabes ] 16

17 Demand for rea estate in the city where the agent ives, a t,t+s ã t,t+s, driven by a desire to hedge shocks to disposabe income due to rent fuctuations As the price of a house is inear in the rent, a house in a certain city is a perfect hedge against rent fuctuations in that city The hedging demand is given by 1 S s 1 ρ s+1 Hence, the hedging demand depends on how we the agent is insuated from oca productivity shocks at time t The hedging demand varies across agents and across time for a given agent, but it does not depend on the expected return of rea estate in that city if a city has a high return, that wi be refected in the mutua fund share ony 17 Investment in a mutua fund incudes a stocks and houses in a cities, with weights ã, b The mutua fund is the same for a agents A agents within a cohort buy the same amount of mutua fund shares but oder agents buy more shares, purey because of the discount rate Given a vector of expected returns which for now is sti exogenous, the weights ã, b that the mutua fund puts on various stocks and rea estate assets are given by a standard CAPM aocation The portfoio puts more weight on an asset if its returns are ess correated with other assets and have a higher expected vaue Now that we have soved the portfoio aocation probem for any given vector of premiums, we sove for the equiibrium expected returns Denote any measurabe aocation of agents to cities with the indicator function I ε,ρ, which takes a vaue of 1 if agents with persona characteristics ε and ρ ocate to city, and zero otherwise such that L 0 I ε,ρ 1 for a ε and ρ Proposition 2 Asset Pricing Suppose that rents are given by equation 2, with given r Consider any aocation of agents over space so that a cities are popuated Then, prices are given by equations 1 and 3 with discounts: [ p q where R [ R 1,, R,, R L] and R 1 S ] 2γS SΣ 1 S s 1 ε ρ [ n R z ], I ε,ρρ s+1φ ε, ρ dεdρ 17 Davis and Wien 2000 obtain a reated resut their Proposition 1 in decomposition of the optima portfoio of agents who face abor risk into a specuative component and a hedging component 17

18 Houses and stocks are priced according to their contribution to systematic risk by a cassic CAPM formua Proposition 2 finds the correct definition of systematic risk for this mode The weights of stocks in the market portfoio correspond to the quantity of stocks avaiabe, as in the reguar CAPM The weights of rea estate, however, are reduced by the tota hedging demand Namey, the weight of houses in city is equa to the mass of homes n minus the integra of the hedging demand by residents of : R To expore the pricing expressions in Proposition 2 further, define the adjusted market portfoio M as a portfoio aocation that incudes n R z k Q Q units of housing in city for every city, and units of stock k for every stock k where Q L 1 n R + K k1 zk The mutua fund that a agents buy is the adjusted market portfoio Denote the expectation and the variance of the adjusted market portfoio, respectivey, by p M and V ar M Define Cov, M as the covariance between the return of rea estate in city and the return of M For every stock k, define Cov k, M simiary Then: Coroary 3 The expected return of rea estate in city is given by and the expected return of stock k is p p k Cov, M V ar M pm Cov k, M V ar M pm The expression in the Coroary is akin to the cassic CAPM pricing formua where Cov,M V arm is a beta-factor for housing in city The main innovation ies in identification of the adjusted market portfoio, for which this formua is true 18 Propositions 1 and 2 are reay intermediate resuts They rest on a specific conjecture about the stochastic process that determines oca market rents, described in equation 2 But rents are not primitives, and we must now check that for the ocation mode used here the conjecture is in fact correct It is usefu to reiterate that the conjecture woud in genera not extend to other ocation modes, impying that Propositions 1 and 2 are vaid ony if accompanied by the specific spatia aocation mode that we have chosen Besides cosing the fixed-point argument, we aso need to determine the vector of rent premiums r, and to find the vector of hedging demands R 18 For instance, if one defines the market portfoio without the R correction, such a beta representation woud not be vaid 18

19 For an agent with persona characteristics ε, ρ, the og-utiity of ocating in city is given by U in Proposition 1, where now p and q are defined in terms of primitives through Proposition 2 For every ε, ρ, et ū ε, ρ ε S p S 1 S s 1 ρ s 4 s1 with the utiity of being in the countryside: ū 0 ε, ρ ū 0 19 Aso et [ 1 4 p Ū S 4γ 2 q ] Σ 1 [ p q Then, we can write the utiity of ocating in city as: 20 ] U Ū S S ū ε, ρ r 1 Namey, the agent s utiity can be decomposed into a component that is common to a agents and depends on investment in the mutua fund and an agent-specific component that depends on the city-agent effect ε and the shock-insuation vector ρ that the agent faces if the choice is to ocate in city A given agent ocates in city if and ony if U max m U m For every L-vector ˆr, we can write the aggregate demand for ocation as ν ˆr We obtain: ε,ρ:ū ε,ρ ˆr max mū m ε,ρ ˆr m φ ε, ρ d ε, ρ Proposition 4 Location Choice A inear stationary equiibrium exists In it, an agent with persona characteristics ε, ρ ocates in city if and ony if ū ε, ρ r max ūj ε, ρ r j j0,l and r is the unique vaue of the vector ˆr such that ν ˆr n in a cities The equiibrium rent in city is r t y j t + r 19 Assuming that ε 0 0 is without oss of generaity If it was not, one coud redefine a the ε as differences with ε 0 20 To see this, note that: s ε S s ρ s+1 p 1 S ε 1 2 S S p 1 S s+1 ρ s s1 19

20 Proposition 4 vaidates the conjectures that aowed us to obtain Propositions 1 and 2 The most important step in Proposition 4 is determination of the identity of hyper-margina residents the agents who are indifferent among a ocations incuding the countryside As we argue beow, a key property of our set-up is that the characteristics of the hyper-margina residents are constant over time The indifference conditions for these agents determine market rents This means that the oca rent processes are the same, but for a constant term, as the oca productivity processes This vaidates the inearity assumption for the rent process buit into equation 2 Let us retrace, at an intuitive eve, the steps that ead to Proposition 4 Despite the fact that the payoff of an agent in a given city is determined by S +1 parameters ε pus the vector ρ, the expected utiity U of the agent in that city can be condensed into a simpe expression incuding ū ε, ρ For any possibe vector of rents ˆr, the demand function ν ˆr estabishes how many agents wi ive in each ocation Hence, for every vector of rent constants ˆr, we identify a set of measure zero of hypermargina residents such that their expected utiity is the same in every city and in the countryside: ū ε, ρ ˆr ū 0 for a Note that this correspond to mutipe persona characteristic profies: a the vectors ε, ρ that yied the same ū ε, ρ One can show that the vector of expected utiities of the hyper-margina resident in different ocation is monotonic in the rent constant vector ˆr This means that the mapping can be inverted Given the identity of the hyper-margina resident, there is ony one vector of rents that guarantees that the hyper-margina residents are indeed indifferent across a ocations The assumption that the distribution of individua characteristics φ ε, ρ has fu support guarantees that the demand function is continuous As the hyper-margina residents determine the vector of ocation demands, one can find a set of hyper-margina residents that guarantees that demand equas suppy in every ocation This set is associated with the rent constant vector r In equiibrium, we have that housing demand equas housing suppy on the space market in every city: ν r n; and that the identity of the hyper-margina residents is given by the set of vaues ε, ρ so that, given the equiibrium rent vector, their expected utiity is the same in every city and in the countryside: ū ε, ρ r 0 for a 20

21 A key feature of our ocation equiibrium is that the characteristics of the hyper-margina resident are cohort-invariant It is this feature that guarantees that the rent process is inear and that our equiibrium characterization is vaid If, for instance, agents coud change city, the time-invariance property woud not hod, and the rent process woud not be inear As a resut, the properties of portfoio aocation and asset prices woud differ We view this as a strength of spatia asset pricing modes The underying geographic mode which we can potentiay observe through demographic and abor data affects equiibrium in the asset market The issue of uniqueness is compex Obviousy, there can be non-inear and/or nonstationary equiibria Given an aocation of residents to cities, there is ony one inear stationary equiibrium There coud be mutipe spatia equiibria, however The agent s expected utiity in equation 4 incudes a mutipicative term p ρ s As Proposition 2 shows, the rea estate return p depends on R and hence on who ives in city, which creates a nontrivia fixed-point probem The economicay interesting possibiity remains that there are mutipe aocations of residents to cities that give rise to inear stationary equiibrium Uniqueness can be achieved under certain functiona assumptions, as some exampes iustrate Whie we obtained cosed-form soutions for portfoio decisions and asset premiums, Proposition 4 does not express rents in cosed form This is natura as the probabiity distribution over individua characteristics, φ ε, ρ, is eft in a genera form By making specific assumptions over persona characteristics and geography, one can obtain cosed-form expressions for a variabes, as the foowing exampe iustrates Assume that: Agents in each cohort draw city-specific endowments ε from a uniform distribution defined over [0, 1] L At each age, a agents face the same city-specific insuation parameter [ ρ ] 1,,L s,,s A cities are of the same size: n 1 LN for every, with N 0, 1 Proposition 5 An agent with human capita ε ocates in city if: 1 ε max m ε m ; and 2 ε 1 N 1 L The equiibrium rent in city is r 1 N L + 1 S p S 1 S s 1 ρ s If there are ony two cities L 2, we can provide a two-dimensiona representation of the equiibrium aocation If, for instance, we assume that n 1 n and hence n0 1 3, s1 21