Homeownership and the Scarcity of Rentals

Transcription

1 Homeownershi and the Scarcity of Rentals Jonathan Halket and Matteo Pignatti Aril 29, 2014 Abstract We model the rovision of owner-occuied versus rental houses as a cometitive search economy where households have rivate information over their exected duration. With ublic information, households with low vacancy hazard rates ay lower rents and search in thicker rental markets. With rivate information, rentals are under-rovided to long-duration households to discourage short-duration households from searching there. Ownershi is attractive in art because it cures the rivate information roblem. Using a novel data set of rental listings, the data show that homeownershi rates are high where rent-to-rice ratios are low but rentals are scarce and that long-duration households sort into scarce rental markets, consistent with our rivate information roblem. Keywords: adverse selection, cometitive search, housing tenure JEL Classification: C78, R13, R21, R23. This aer formerly circulated under Housing Tenure Choices With Private Information. We thank Melvyn Coles, Ricardo Lagos, Andreas Muller, Roland Rathelot, Morten Ravn, Thomas Sargent and esecially Gian Luca Violante for helful comments. We are also grateful to seminar articiants at the Atlanta, Boston and Cleveland Federal Reserves, Arizona State University, University of Toronto, Yeshiva University, Baruch College, New York University, Bank of Italy, Institute for Advanced Studies in Vienna, the Federal Reserve Board, the University of Essex Search and Matching Worksho, Paris SAM, HULM-Atlanta and the Nordic Macroeconomics Conference. Jonathan Halket thanks the ESRC for funding under grant number PTA and through the ESRC Centre for Microdata Methods and Practice (CeMMAP) grant number ES/I034021/1. University of Essex, Institute For Fiscal Studies and CeMMAP. jonathan@halket.com, Homeage: htt:// Confindustria Research Deartment and CeLEG. matteo.ignatti@nyu.edu, Homeage: htt://homeages.nyu.edu/~mm272. 1

2 1 Introduction There is a long list of lausible frictions that may create meaningful differences in the value of owning versus renting a home to a household. Many of the frictions that favor renting, such as the higher transactions costs of buying and selling a house and the downayment constraints in the mortgage market, aear in one form or another in nearly all life cycle models with a homeownershi choice 1. There is no such consensus on the frictions that favor owning, in art because they are under-modeled and under-measured. One oft mooted friction is that rentals are scarce in some arts of the market. Some studies that have inserted this friction in a reduced-form manner into their models (e.g. by imosing that large houses are only sulied on the owner-occuied market) have been otherwise successful at exlaining changes in homeownershi rates over the life cycle (Chambers et al. (2009a)), over time (Chambers et al. (2009b)), and across locations (Amior and Halket (Forthcoming)), among others 2. Another alternative is to assume a warm glow to owning (Iacoviello and Pavan (2009); Kiyotaki et al. (2008)). 3 In this aer we build a theory in which rental scarcity and (in an extension) warm glows are equilibrium outcomes rather than inuts. Our equilibrium yields the erhas counter-intuitive rediction that rentals are relatively chea where they are scarce. We then use a large, novel data set of for-rent and for-sale listings from Craigslist to show that, within a market (such as a city), the arts of the market (i.e. submarket ) where homeownershi rates are high are indeed also the arts where rentals are relatively chea but scarce. In other words, households are more likely to search for owner-occuied housing not when the relative rice of an equivalent rental is high, but rather when an equivalent rental is hard to find. Crucially, the data allow us to measure scarcity by measuring how quickly vacant homes are filled and not just the equilibrium suly of rental housing in a submarket. The data also show that households that have relatively long exected durations in their homes tend to live in the submarkets where rentals are scarce. Given the costs of vacancy for landlords, 1 e.g. Amior and Halket (Forthcoming); Cambell and Cocco (2007); Chambers, Garriga and Schlagenhauf (2009a,b); Cocco (2005); Diaz and Luengo-Prado (2008); Fisher and Gervais (2007); Gervais (2002); Halket and Vasudev (2014); Iacoviello and Pavan (2009); Kiyotaki, Michaelides and Nikolov (2008); Li and Yao (2007); Rios-Rull and Sanchez-Marcos (2008) 2 Further studies in this manner are Ortalo-Magne and Rady (2006); Rios-Rull and Sanchez-Marcos (2007). 3 One class of frictions that may work both ways is risk in the housing market, as in Sinai and Souleles (2005). Tax wedges may encourage ownershi (Diaz and Luengo-Prado (2008); Gervais (2002)) but would be welfare distorting absent some other reason to own. Some theories have a user cost remium of renting over owning, erhas due to excessive utilization of housing services on the art of renters (as in Henderson and Ioannides (1983)) but this should imly that rent-to-rice ratios are higher in the arts of the market where homeownershi rates are high. 2

3 long-duration households should be aealing tenants and yet the data indicates that rental housing for them is harder to find. We build a cometitive search economy with adverse selection and choice of contracting to exlain why rent-to-rice ratios are low but rentals are scarce and thus homeownershi rates are high for long-duration households. Figure 1 illustrates the basic stylized correlations between rent-to-rice ratios, homeownershi rates and rental vacancy rates using zi codes from the Seattle area. In the figures, each dot is a submarket, which we define as all housing in our data set with a articular number of bedrooms in a articular zi code. Rent-to-rice ratios are the ratio of the mean rent to mean rice in a submarket (e.g. the mean over 2 bedroom listings in zi code 49820) using the Craigslist data (see Section 5 for a comlete descrition). Homeownershi rates are from the Census at the same bedroom x zi code level. Time on the market is the average number of days that a rental roerty is osted for rent in our Craigslist data. Figure 1: Greater Seattle Area Homeownershi Rate Fitted values Mean time on the market Fitted values Mean time on the market Fitted values Monthly Rent to Price ratio Monthly Rent to Price ratio Homeownershi Rate Rent-to-rice ratios and time on the rental market are the ratio of the mean rent to mean rice and mean number of days that a roerty is osted for rent for two bedroom houses in a zi code (e.g. the mean over 2 bedroom listings in zi code 49820) using the Craigslist data. Homeownershi rates are from the Census data at the same bedroom x zi code level. The Greater Seattle Area refers to zi codes starting with 980. Rent-to-rice ratios are negatively correlated with homeownershi rates. Verbrugge (2008), Verbrugge and Poole (2010) and Bracke (2013) find similar correlations, the latter even after controlling for otential unobserved heterogeneity. Figure 1 also shows a new finding: that rental roerties stay on the market longer in high rent-to-rice ratio submarkets and in submarkets where homeownershi is low. To our knowledge, this is the first data on rental vacancies where a within-market analysis of the variation in submarket rental vacancy duration is ossible. In section 5, we show that these same submarkets - the ones where rentals stay vacant longer er sell - have renters who move out sooner on average. The remainder of the aer concerns writing down a theory of equilibrium heterogeneous rental scarcity and homeownershi rates. Since owning and renting are just labels for different (erhas many different) contracts to 3

4 rovide housing services, we model the homeownershi decision and the availability of rental housing as outcomes of a contracting roblem and a search roblem. In the baseline model, houses are exante identical and households differ only according to their cost of owning and their exected duration of stay in a house, which may be rivate information 4. Homeowners (which may be households or landlords) ost contracts for housing services which secify a (otentially tyedeendent) rice for housing services as well as whether, after eventual searation, the current owner or the eventual occuant is resonsible for finding the next tenant (a rental or owning contract, resectively). Within the housing market in this economy, households can direct their search to a secific tye of contract (so that each tye of contract is its own submarket) and are bilaterally matched to houses within that submarket subject to the frictions from cometitive search theory (Moen (1997) and Shimer (1996)). In equilibrium, vacancies in a articular submarket adjust so that the exected return to adding a new house in any submarket is the same. Our main results are twofold. First, when households exected durations in a house are unobservable, an incentive roblem in rental markets distorts market tightnesses 5 comared to the ublic information benchmark. In the economy where households exected durations are ublic information, households with low vacancy hazard rates (long-duration households) ay lower rental rates and search in less tight submarkets than households with high hazard rates. However, when exected durations are rivate information, long-duration households search in tighter submarkets than short-duration households, and thus send more time on average searching for a house (er searation sell), but ay even lower rental rates once matched. (The unique equilibrium is searating.) The intuition for the result is that in equilibrium housing is under-rovided to longduration households so as to discourage short-duration households from searching there. In this sense, rivate information causes housing scarcity in some rental markets. The data are consistent with the resence of an information roblem. Under ublic information, submarkets with higher surlus matches (due to the high exected duration of the matches) should have less tight markets for rental housing and lower rents. Instead the data show that while rents are lower, tightnesses are higher in submarkets where high surlus households live. In our economy, owning a house solves the rivate information roblem as households internalize their searation hazards in their otimal search roblems. However, owning comes at some 4 There is a long literature looking at mobility and homeownershi choices. Deng, Gabriel and Nothaft (2003) and Gabriel and Nothaft (2001) find considerable variation across households and Metroolitan Statistical Areas in rental vacancy rates and durations. Boehm, Herzog Jr. and Schlottmann (1991), Cameron and Tracy (1997), Haurin and Gill (2002) and Kan (2000) all find relationshis between mobility hazards and homeownershi. 5 Markets are less tight if households on average take less time to find a house, or equivalently if landlords take longer on average to fill a vacancy. 4

5 heterogenous exogenous cost (a reduced-form way to model the various more well-understood frictions in the owner-occuied market). Our second result is that households that exect to stay in their house long enough are more likely to choose to own rather than rent. The distortions from the incentive roblem in the rental market ile-u: the deviations from the ublic information benchmark due to rivate information are larger in markets where the long-duration households search. Meanwhile the owning contract is always incentive comatible. If a household has a high enough exected duration, the distortions in the rental market due to the information roblem are more likely to dominate her idiosyncratic cost of owning so that she refers to own. This too is consistent with the correlations we find in the data: homeownershi rates are high in submarkets where rentals are scarce. In equilibrium, free entry imlies that submarkets with scarce rentals must have low rents. So rent-to-rice ratios are low where homeownershi rates are high, as in the data. 6 A olicy of rent control redictably leads to a lower suly of rental housing and tighter markets in the regulated market in both ublic and rivate information cases. With rivate information however, the effects on the regulated market sill into the unregulated market, leading to lower suly and tighter markets there as well. This haens even though there is no excess demand in any market (as in e.g. Fallis and Smith (1984)); all markets are in equilibrium. Instead, by worsening the allocation for low-duration households, rent control exacerbates the information roblem, making it more costly for higher-duration households to screen the low-duration households. In the final art of our aer, we give the economy access to a technology which ermits the building of non-conforming, customized houses; which we model as giving a higher utility flow at some cost to the matching robability. We show that customization aeals most to longduration households. So, unlike rent control, the customization technology offers an additional way to relax the incentive comatibility constraints in the rental market; thus there may be overcustomization in the rental market relative to the ublic information benchmark. And yet, since the aeals of owning and customization are each increasing in exected duration, more owners than renters may customize. If customization is observable to an econometrician using hedonics, than owner-occuiers will aear to live in houses with more amenities, otherwise they will aear to get a warm glow from owning (that is, they would aear to get a higher utility flow from the same observable set of house attributes). To be clear, we intend for this to be the beginning, not the end, of research which uses contract- 6 Variations in households marginal rates of substitution across submarkets could also otentially exlain the correlation between exected duration and ownershi rates (as in Sinai and Souleles (2005)) but only if the marginal rates of transformation between rental and owner-occuied housing varied similarly across submarkets. In our aer, rent-to-rice ratios vary even though the marginal rates of transformation are constant. 5

6 ing roblems in rental markets to construct equilibria consistent with the data on homeownershi rates and rices in various markets. There are many et theories on why so many eole choose to own; each may lay some role in artly exlaining why ownershi occurs. We are following a growing literature by looking at housing in a search or matching framework (e.g. Albrecht, Anderson, Smith and Vroman (2007); Albrecht, Gautier and Vroman (2010); Calin and Leahy (2008); Ngai and Tenreyro (2009); Piazzesi and Schneider (2009); Wheaton (1990)). To our knowledge, we are the first to look at both renting and owning in such a framework and the first to look jointly at renting and owning with adverse selection 7. Our work looks at contracts to suly housing services 8 when there are search frictions and asymmetric information and thus extends the work of Guerrieri, Shimer and Wright (2010) to include dynamic contracts in a cometitive search equilibrium with adverse selection 9. In our equilibrium, contracts can be dynamic while the markets themselves are in steady-state. Concurrently and comlementarily, Chang (2011) and Guerrieri and Shimer (2012) examine environments where the markets can change dynamically, however all contracts are one-time exchanges (urchases and sales of assets). Our work on customization is a sort of comanion to House and Ozdenoren (2008). In their model of durable goods, goods that are more durable are endogenously more homogeneous due to resale concerns. They cite McMansions (which are redominately owner-occuied) as an examle of a generic durable good. In our model, durable goods are endogenously heterogeneous or homogeneous based on the exected duration of the match (rather than the duration of the good). The tyical owner-occuied house is actually relatively varied comared to rental housing in our economy since, endogenously, owner-occuiers exect to be matched longer with their house. The remainder of this aer is as follows: Section 2 resents economies of renting with ublic and then rivate information; Section 3 resents the owning technology and the equilibrium with owning and renting; Section 4 resents a numerical examle and the effects of rent control; Section 5 examines market tightnesses and rices in rental and owner-occuied markets using data from Craigslist while Section 6 resents the customization technology. Section 7 concludes. Most roofs are in the Aendix. 7 Hubert (1995); Miceli and Sirmans (1999) have models with renters and adverse selection in which long-term tenants have declining rent schedules while Barker (2003) shows that if households have inelastic demand for housing, those that exect to stay longer do not usually get discounts on their rent. Brueckner (1994) resents a model with adverse selection and evidence that banks use menus of mortgage oints and interest rates to obtain information on a household s exected mobility. 8 and in this sense comliments the work on otimal mortgage design in owner-occuied markets (contracts for loans backed by housing services) by Piskorski and Tchistyi (2011, 2010) 9 Delacroix and Shi (2007) and Albrecht et al. (2010) have adverse selection roblems where the side osting the rice has full information. Here, as in Guerrieri et al. (2010), the side directing its search has the suerior information. 6

7 2 The Rental market For simlicity, we will first model an economy with only renting. The qualitative differences between the cometitive equilibrium with ublic versus rivate information will remain unchanged when later we add in owner-occuation. 2.1 Preferences and technology Time is continuous and the horizon is infinite. There is a measure one of households indexed by their tye i I = {1, 2,.., I} and a large set of landlords or builders. Let π i be the fraction of households of tye i in the oulation, for all i. If a landlord decides to articiate in the market, she ays a cost H in units of utility to build a house but then houses are costless to maintain; if she doesn t articiate, she gets a ayoff equal to 0. Households receive a flow utility of h when they occuy a house and 0 when they do not. Households and landlords each discount at the same rate ρ < h/h. Households that are currently occuying a house searate with it at a hazard rate γ : I Γ R +, at which oint a searated household no longer receives any utility from living in that articular house. Without loss of generality, we assume that γ is strictly decreasing. We will often refer to a household of tye i as having a hazard γ i = γ(i). We denote γ γ 1 and γ lim i I γ i so that Γ = [γ, γ]. We will also derive some analytical and comutational results for the secial case where {γ i } i=1 is dense in Γ. We refer to this secial case as the differential-γ case. A rental contract w W secifies a flow rent, ossibly contingent on tye, aid by the household to the landlord if matched. The contract ends in the case of searation. For simlicity, we will restrict our attention to rental contracts with a fixed flow rent. Barker (2003) and Guasch and Marshall (1987) find that most rental contracts do not have a duration-of-stay discount. In the Aendix we extend the setting to fully dynamic contracts with risk-averse households and show that our results still follow. We consider two cases. In the first, a household s tye is ublicly observable and so contracts are also free to have tye-secific rents. However, we will show that in equilibrium, only one tye is lured by each contract. In the second case, a household s tye is rivate information. In this case, by the revelation rincile, we assume that landlords ost a contract which contains direct revelation mechanisms for each tye, without loss of generality. Following Guerrieri et al. (2010), we will show that we can assume without loss of generality that landlords ost contracts with tyeindeendent mechanisms. More recisely, in the rivate information case the equilibrium with contracts is ayoff equivalent to the equilibrium with degenerate mechanisms offering the same 7

8 rent to each household, which enables us to simlify the notation greatly. 10 The matching rocess between households and landlords is frictional. At any given time landlords ost a single contract at zero cost and households direct their search to the most attractive contracts. 11 Associated with any contract w, let u be the measure of households directing their search to w and v be the measure of landlords osting w. Define θ = u/v as the market tightness associated with contract w, θ : W R +. Households find a house at rate α h (θ) where α h : R + R + and α h is decreasing in θ. Landlords fill a vacancy at rate α l (θ), where α l : R + R + is increasing in θ. We assume that α l (θ) = θα h (θ), that is equivalent to constant returns to scale in matching, and α h (0) = α l ( ) = and α h ( ) = α l (0) = 0. We assume that the elasticity of α l (θ), ε(θ) θ dα l (θ) α l (θ) dθ is constant: ε(θ) = ε. Let ψ i (w) be the share of households of tye i alying to any given contract w, so that ψ(w) = {ψ 1 (w), ψ 2 (w),..., ψ I (w)} I, where I is the I-dimensional unit simlex, ψ : W I. The market tightness θ(w) and the share of households alying to w, ψ(w) are determined in equilibrium. Let V r (γ i, r, θ) and Z r (γ i, r, θ) be the exected values of living in a house and searching for a house 12, resectively, to the households of tye i alying to any given contract, w with rental ayment for that tye of r. θ = θ(w) is the market tightness associated with the contract w. Then: ρv r (γ i, r, θ) = h r + γ i (Z r (γ i, r, θ) V r (γ i, r, θ)) (1) ρz r (γ i, r, θ) = α l(θ) (V r (γ i, r, θ) Z r (γ i, r, θ)) (2) θ Let Y r (γ i, r, θ) and X r (w, θ) be the exected values of an occuied house when matched with a tye i and a vacant house, resectively, to the landlord: ρy r (γ i, r, θ) = r + γ i (X r (w, θ) Y r (γ i, r, θ)) ρx r (w, θ) = α l (θ) i I ψ i (w)(y r (γ i, r i, θ) X r (w, θ)) where ψ i (w) is the share of households of tye i alying to the contract w, secifying rent r i for that tye, and θ is the market tightness associated with that contract. 10 This esthetic feature is not true in the setting with fully-dynamic contracts in the Aendix; which is the main reason why we relegate that art to the Aendix (see our discussion of Lemma 4). 11 Matching is bilateral, thus every household can only aly to one contract, but she can use mixed strategies. 12 These are the values of searching and living in the same market, reeatedly ad infinitum. 8

9 Solving for the flow value of searching ρz r (γ i, r, θ) and osting ρx r (w) gives: α l (θ) ρz r (γ i, r, θ) = (h r) (3) θ(ρ + γ i ) + α l (θ) ( ρx r (w, θ) = 1 + α l (θ) ) ψ i (w) 1 α l (θ) ψ i (w)r i (4) ρ + γ i ρ + γ i i I i I Notice that ρz r (γ i, r, θ) < 0 if r > h, i and θ > 0, thus no household would aly to a contract that imoses a flow rent r higher than the flow utility from housing h. Similarly, X r (w, θ) < H if r i < ρh for all i for which ψ i (w)θ > Equilibrium with ublic information A cometitive search equilibrium satisfies the following conditions in every submarket: (i) landlords maximize exected rofits; (ii) free entry (new entrants earn zero rofits in exectation); (iii) households direct their search to the most convenient osted vacancy; (iv) θ = θ(w) is consistent with rational exectations in equilibrium but also for any ossible deviation w. More recisely, a landlord offering w w exects that households aly until the market tightness θ imlies an exected value for the household equal to the outside otion Z r, that is taken as given by the (atomistic) firm. Formally: Definition 1. A steady-state cometitive search equilibrium with renting and ublic information is a vector {Z i r } i I, a set of contracts W r W I each of which secifies a rent r i for each i I, a function θ r : W I R +, a measure λ on W I with suort W r, and a function ψ : W I I satisfying, for each i I: (i) Landlords rofit maximization and free entry: ( 1 + α l (θr(w)) ) ψ i (w) 1 α l (θ ρ + γ r(w)) i i I i I ψ i (w)r i ρ + γ i ρh with equality if w W r. (ii) Households otimal search: Let Z i r max w W r 1 α l (θr(w )) ρ θr(w )(ρ + γ i ) + α l (θr(w )) (h r i) Then w W I Z i r with equality if θ r(w) > 0 and ψ i (w) > 0. 1 α l (θr(w)) ρ θr(w)(ρ + γ i ) + α l (θr(w)) (h r i) 9

10 (iii) market clearing: W r ( ψ i (w) θr(w) + α l(θr(w)) ) dλ(w) = π i γ i i I The equilibrium definition imoses restrictions on the off-equilibrium beliefs of the landlords. The otimal search value of any tye-i household is defined over the set of contracts osted in equilibrium W r only, but under any deviating contract w / W r, landlords exect market tightness θ r(w ) to adjust to make all tyes of households weakly worse off. We can distinguish cometitive equilibria according to whether there are contracts which attract more than one tye in equilibrium. Definition 2. A searating cometitive equilibrium is any cometitive equilibrium where for all w Wr and for all i, ψ i (w) > 0 imlies ψ i (w) = 1. A ooling equilibrium is any cometitive equilibrium that is not searating. Two cometitive equilibria (indexed by A and B) are allocatively equivalent if for all i I and w A Wr A, ψ i (w A ) > 0 imlies there exists a w B Wr B ψ i (w B ) > 0 such that ri A = ri B and θr A (w A ) = θr B (w B ) and vice versa. Lemma 1. If there exists a ooling cometitive equilibrium with ublic information, then there exists an allocatively equivalent searating cometitive equilibrium. Proof. See Aendix. In searating cometitive equilibria, the market endogenously segments into submarkets, one for any different tye i of households. Thus without loss of generality we can assume that a contract w in a searating cometitive equilibrium contains a menu of rents where there is only one rent r i < h and thereafter label w = r i. contract w to households of tye i, given by v(w) = Characterization with This also ins down the measure of landlords osting the γ i π i α l (θ r (w))+γ i θ r (w). A necessary and sufficient condition for a searating cometitive search equilibrium is the following: 13 Proosition 1. For any tye i of households, a osted contract w i r tightness θ i r and the associated market θ r(w i r ) are art of an equilibrium allocation if and only if they solve the following 13 See e.g. Acemoglu and Shimer (1999) for a roof, with one caveat to the roof of sufficiency: in our setting, even if mechanisms in W r are searating, other mechanisms in W I can be ooling. It is straightforward to use the argument in the roof of Lemma 1 to show that if the sufficiency conditions are met for a searating cometitive search equilibrium with searating-only mechanisms then they will be met here too. 10

11 constrained maximization roblem, R i : α l (θ i ) max w i,θ i θ i (ρ + γ i ) + α l (θ i ) (h w i) s.t. α l (θ i ) ρ + γ i + α l (θ i ) w i ρh The equilibrium allocation maximizes the exected value of search of any tye-i household conditional on the firms making non-negative rofits. Proosition 2. A solution to R i exists for each i. The solution is unique. Proof. See Aendix. Lemma 2. In the solution to R = {R i } i I, for all i, j I with i j, θ i r θ j r Proof. Using the constraint with equality to substitute for w i r, the first order condition imlies the following equilibrium condition for the market tightness: h ρh = θr i ε 1 ε + ρ + γ i α l (θr i )(1 ε) (5) The imlicit solution for θ i r is strictly increasing in γ i. Lemma 3. Any cometitive equilibrium with ublic information is a searating cometitive equilibrium. Proof. Follows immediately from Lemmas 1 and 2 and Proosition 1. by: The equilibrium values of the flow rent w i r and the household s exected value ρz i r are given w i r = ρ + γ i + α l (θr i ) α l (θr i ρh ) ε 1 ε ρh ρz i r = 1 θ i r We have the following comarative static results as γ i varies: Result 1. In equilibrium, as the searation hazard γ i increases: (i) the market tightness θ i r increases; (ii) the flow rent w i r increases; (iii) the exected value to households Z i r decreases. 11

12 Proof. See Aendix. Thus, households with lower exected durations have lower surluses from matching with a house and thus face tighter markets and higher rents once matched and as a consequence have lower search values. Searching for a new tenant is costly, so more surlus is created when a landlord is matched with a high exected duration tenant. In equilibrium, absent other frictions, this extra surlus should be allocated towards some combination of lower rents and more houses, decreasing both the cost of housing and search costs for households. That is, absent other frictions, long duration households should find housing faster and landlords should take longer to match with long duration households. obtain: Analytically, for the differential-γ case, by differentiating 14 the equilibrium condition (5) we 2.3 Renting with rivate information dθr dγ = 1 θr 2 ε θr(ρ + γ) + α l (θr) > 0 dwr dγ = ρh θr(ρ + γ) + α l (θr) > 0 dzr dγ = θ Z r r ε(θr(ρ + γ) + α l (θr)) < 0 The equilibrium allocation in the ublic information case imlies that every tye j < I strictly refers to search in a higher (i > j) tye s market if she was offered the higher tye s contracted rent. In this subsection, we assume that the tye of the household, i, is known only by the household. So, the ublic information allocation will not be incentive comatible under rivate information. A mechanism in this setting would be a set of rents {r} i I. However, from the household s value of being matched (1), it is clear that the only mechanism comatible with truth telling offers the same rent to any reorted tye. Lemma 4. A contract is incentive comatible if and only if it offers the same rent to any reorted tye. Proof. Follows from the household s value of being matched to a contract. 14 We need to exlicitly define our notion of differentiation. Let f : N R and g : range(f) G R. Define g f g(q ) g(q) q = lim q q q q where q, q range(f). The total derivative is defined analogously. 12

13 So we can safely associate any incentive comatible contract w with its associated rent (and thus can assume w [ρh, h]). 15 Lemma 5. Sorting: i, w [ρh, h], θ 0, and ϵ > 0, there exists a coule (w, θ ) B ϵ (w, θ(w)), with w < w and θ > θ, such that Z r (γ j, w, θ ) > Z r (γ j, w, θ), γ j γ i and Z r (γ j, w, θ ) < Z r (γ j, w, θ), γ j > γ i Proof. Follows from equation 3. The sorting lemma is sufficient to have a searating equilibrium and differs from the (assumed) condition in Guerrieri et al. (2010) in that it involves local erturbations in both the contract w and the market tightness θ. We can now define the equilibrium, following and extending the definition in Guerrieri et al. (2010) to a dynamic setting. Definition 3. A steady-state cometitive search equilibrium with renting and rivate information is a vector {Z i } i I, a set of rents (i.e. incentive comatible contracts) W [ρh, h] I, a measure λ on [ρh, h] with suort W, a function θ : [ρh, h] R + and a function ψ : [ρh, h] I satisfying: (i) landlords rofit maximization and free entry: for any w [ρh, h] [ ( 1 + α l (θ(w)) ) ψ i (w) 1 ] 1 w ρh ρ + γ i i I with equality if w W. (ii) households otimal search: Let Z i Then w [ρh, h] and γ i max w W Z i with equality if θ (w) > 0 and ψ i (w) > 0. 1 α l (θ(w )) ρ θ(w )(ρ + γ i ) + α l (θ(w )) (h w ) 1 α l (θ(w)) ρ θ(w)(ρ + γ i ) + α l (θ(w)) (h w) (iii) market clearing: W ( ψ i (w) θ(w) + α l(θ(w)) ) dλ(w) = π i γ i i 15 Lemma 4 relies on our restriction of the rental contract sace to constant rent rofiles. When we relax this assumtion, as in the Aendix, the lemma no longer holds. However lemma 4 is not necessary for a searating equilibrium and it is straightforward to see that such an equilibrium exists in the Aendixes relaxed setting. 13

14 As in the ublic information case, the equilibrium definition imoses conditions on the offequilibrium beliefs of the landlords. Heuristically, a landlord considering whether to ost a deviating contract w imagines an initial market tightness θ = 0. If no household is willing to aly, then θ = 0 and the deviation is not rofitable. Otherwise, some households aly, increasing market tightness θ, until only one tye of household i is indifferent about the deviating w and all others j (weakly) refer their equilibrium contracts. This in turn ins down the share ψ i of households alying to that contract Equilibrium and Characterization The characterization of the equilibrium with rivate information is equivalent to the ublic information equilibrium with an extra incentive comatibility constraint that imoses that no other tyes of households j are attracted to the contract w i. In the next roosition, we show that at the otimum, for all i > 1, only the marginal incentive comatibility constraints IC(i 1, i) bind: every tye (i 1) is indifferent between his own contract and the contract offered to the tye i with marginally higher exected duration. Proosition 3. Let the roblem (P R) be defined by the following constrained maximization roblem (P R i ), for any i I: s.t. and max Z r (γ i, w, θ) θ R +,w R + α l (θ) ρ + γ i + α l (θ) w ρh Z r (γ j, w, θ) Z r (γ j, w j, θ j ) for all j i [IC(j, i)] where w i, θ i is an otimal solution for i. The solution of (P R) exists and is unique. Moreover, only the marginal incentive comatibility constraints IC(i 1, i) bind, for all i > 1: Proof. See Aendix. Z r (γ i 1, w i, θ i ) = Z r (γ i 1, w,i 1, θ,i 1 ) and Z r (γ j, w i, θ i ) < Z r (γ j, w j, θ j ) j i, i 1 Thus, for the tye with the highest searation hazard, γ 1 = γ, the equilibrium allocation is the same as the one with ublic information. Then, the roblem is solved iteratively for all other tyes: (i) For i = 1, w 1 and θ 1 solve R 1 14

15 (ii) For each i > 1, w i and θ i are the solutions to s.t. and max Z r (γ i, w, θ) θ R +,w R + α l (θ) ρ + γ i + α l (θ) w ρh Z r (γ i 1, w i, θ i ) Z r (γ i 1, w,i 1, θ,i 1 ) We are now ready to rove the existence and uniqueness of the equilibrium and characterize the equilibrium allocation: Proosition 4. There exists a unique searating equilibrium. A set of contracts {w i } I, [ρh, h] and market tightnesses {θ i } I, θ i w i θ (w i ) θ i associated with their resective tyes γ i are art of the equilibrium allocation if and only if they solve the roblem P R. Proof. See Aendix. We have the following comarative static results as γ i varies. Result 2. In equilibrium, as the searation hazard γ i increases: (i) the market tightness θ i decreases; (ii) the flow rent w i increases; (iii) the exected value to households Z i decreases; (iv) the vacancy rate, v i π i u i v i increases; Proof. See Aendix. Analytically, for the differential-γ case, then: dθ dγ = 1 [ (1 ε) ρz ρ + γ dw dγ = (1 ε)ρz α l (θ) dz ] 1 < 0 ρh ε θ [ (1 ε) ρz ρh ε θ θ dγ = Z θ(ρ + γ) + α l (θ) < 0 ] 1 > 0 Contrary to the ublic information case, low-γ tyes search in tighter markets in equilibrium, and ay lower rents if matched. In this way landlords are able to otimally (with the least cost) searate 15

16 tyes of households by osting contracts w i lower than the first-best otimum wr i to those that exect to stay longer, in return for a higher market tightness θ i. Households that exect to stay longer are less affected by a higher market tightness (and thus longer exected search times) because they exect to searate from the house and ay the search cost less frequently. On the other hand, those that exect to stay longer are more affected by a lower rent w because they exect to be matched a higher fraction of time for any given market tightness θ. The combination of these two factors imlies that the second best allocation dictates tighter markets for those that exect to stay longer, contrary to the first best allocation. These tighter markets imly a lower vacancy rate (defined as vacancies er unit of housing). 3 Owning market An owning contract simly secifies an u-front ayment P aid by the household to the seller, which may vary across submarkets. Households derive the same flow utility h if they own or rent the house, and landlords (i.e. builders) ay the same building cost H. As will become clear below, absent some further friction, owning would efficiently solve the rivate information roblem and all markets would be owner-occuied markets 16. To rovide heterogeneity, we assume that there is an extra friction in the owning market which is heterogenous in the oulation 17. We assume that each household draws a friction χ [χ, χ] R + from a robability distribution F, which is a fixed effect for the household. For simlicity, we assume that χ is indeendent of tye i and that the friction additively taxes the value of searching and living in a house. Indeendence could be easily relaxed while additivity means that there will be a single owning submarket for each tye i, which eases notation. Builders only have to sell a new house. It is imortant to notice that the owning submarket is not affected by the rivate information friction, because a household that buys the house, an owner, fully internalizes the exected search cost eventually aid in the case of searation, contrary to a renter. The builder s exected value of osting in an owning submarket with tightness θ a contract for sale at rice P is simly given by: X o (P, θ) = α l(θ) ρ + α l (θ) P (6) Notice that (6) is indeendent of γ i. In equilibrium X o (P, θ) = H for any owning submarket with ositive ownershi rates. This immediately imlies that sale rices are negatively correlated with 16 The i = 1 tye would be indifferent between owning and renting. 17 There are lenty of otential candidate frictions. For instance, a (ossibly heterogeneous) financing cost or additional transactions costs. See Halket and Pignatti (2012) for an examle where the extra cost in the owning submarket is sequential search. 16

17 market tightnesses across the for-sale submarkets. The values of searching as a buyer and living in a submarket with market tightness θ and rice P for a household of tye i and cost χ, resectively, are given by: ρz o (γ i, P, θ, χ) = α h (θ)(v o (γ i, P, θ, χ) Z o (γ i, P, θ, χ) P ) χ ρv o (γ i, P, θ, χ) = h + γ i (X(P, θ) + Z o (γ i, P, θ, χ) V o (γ i, P, θ, χ)) χ Solving for the flow value of searching as a buyer gives: where r i = P ρ+γ i+α l (θ) ρ+α l (θ). 3.1 Equilibrium with only owning ρz o (γ i, P, θ, χ) = ρz r (γ i, r i, θ) χ Conditional on market tightnesses, neither builders nor owners when selling care about the tyes of the buyers in the submarket in which they have osted. So the equilibrium does not deend on whether households tyes are ublic or rivate information. The equilibrium definition of an economy with only owning markets is similar to the equilibrium in economy with only rental markets with rivate information in that contracts (rices) are not tye-secific: each submarket offers just one contract rice (see the Aendix for a definition). The market endogenously segments into submarkets and we can characterize the equilibrium allocation using an equivalent constrained maximization roblem. As in the economies with renting, the equilibrium in the owning economy can be found by solving a constrained otimization roblem iteratively by tye. 18 The otimal market tightness conditional on owning for each tye, θ i o, is the same as the otimal tightness for the ublic renting market θ i r. 3.2 Equilibrium with both renting and owning We are now ready to study the equilibrium roblem in the housing market with rivate information. Landlords/builders are free to enter in both the rental and the owning market. If they enter, they ay a building cost H and ost a contract in one submarket. Households have rivate information over their mobility hazard rate γ and direct their search to their referred ostings. In the aendix, we formally define a cometitive equilibrium with rivate information and both renting and owning. The equilibrium with both renting and owning can be characterized by the iterative solutions to a roblem analogous to those with only owning or renting 19 : 18 That is, a similar version of either Proosition 1 or 3 holds. Furthermore, it is also easy to show that a similar tye of incentive comatibility constraint as the one in Proosition 3 never binds. 19 We omit the roof, however it is similar to the case with only renting 17

18 Z i o(χ) { max Z i {rent,own} max θ i R +,w i [ρh,h] s.t. Z r (γ i, w i, θ i ), α l ( θ i ) ρ + γ i + α l ( θ i ) w i ρh P i = ρ + α l(θo) i α l (θo) i H max θ i o R +,P i [H,h/ρ] Z i 1 o (χ) Z r (γ i 1, w i, θ i ) for all i > 1 Result 3. The roortion of tye i that are homeowners is increasing in i Proof. See Aendix. } Z o (γ i, P i, θo, i 0) χ The equilibrium in the rivate information rental market for the highest-γ tye is the same as in the ublic information case while the maximands to the owning art of the characterization are identical to the solutions for the owning-only economy. For households with lower γ s, the equilibrium in the (rivate information) rental market is increasingly distorted with resect to the first best (ublic information) equilibrium. This immediately imlies that the tye-secific cutoff χ such that Z i = Z i o (χ ) is increasing in i. 4 Examle and alication to rent control As a arametrization, we set ρ =.05, h =.1, H = 1, α l = θ ε with ε =.5. We allow for γ [.2,.7] so that exected durations are between between 1.4 and 5 years, aroximately. The disutility to owning, χ, only affects homeownershi rates and has no effect on vacancy rates or rices. For simlicity then we assume it is distributed uniformly on [0,.9Z 1 r ]. The left anel of figure 2 lots the market tightness, or queue length, and the homeownershi rate and the center anel lots the flow rent (or housing rice) as a function of γ in the three economies: renting with ublic information, renting with rivate information and owning. The queue length increases as γ increases in the case of renting with ublic information and in the owning economy (and it is shorter in the latter case), while it decreases as γ increases in the renting economy with rivate information. In both renting economies, the flow rent increases with γ: it increases faster in the rivate information case to offset the ositive effect of the longer queue length faced by low γ-tyes on landlords rofits. The housing rice in the owning economy markets, exressed in flow terms (ρp ), decreases slightly as γ increases; rices are lower in markets where houses sell quickly, as follows from free entry condition for the owning market. 18

19 Figure 2: Market tightness, rices and values Market tightnesses 1 PUBLIC INFO & OWNING PRIVATE INFO OWNERSHIP RATE Prices PUBLIC INFO PRIVATE INFO OWNING PRICE Values PUBLIC INFO PRIVATE INFO Queue Length Ownershi Rate Rent Exected Value Searation Rate Searation Rate Searation Rate Finally, the right anel of figure 2 shows the exected value of searching for a house as a function of γ when renting with ublic information and renting with rivate information. The value of renting with ublic information is always higher than the other cases (and it coincides with the rivate information renting for the highest value of γ). The exected value increases as γ decreases in both markets but it increases less in the rivate information renting market. As we know from the theoretical results above, for any arametrization, the model with rivate information gives the following qualitative atterns: Scarce rentals and low time on the market in submarkets with lower rents. Households with high exected durations search in rental markets with scarce housing and lower rents if they search for rental housing. Households with high exected durations search in for-sale markets with higher rices if they search for owner-occuied housing. Areas with high ownershi rates also have low rent-to-rice ratios. As we show in Section 5, the data generate these same atterns. Quantitatively, if we equate exected duration in the model with the average observed duration in the data, the model could generate a similar elasticity of time on the rental market with resect to exected duration by setting ε.1. However as we later note, our estimates in Section 5 may underestimate the true correlations due to not observing the true submarket delineations. In which case, the imlied model ε would be larger. 4.1 Rent control We continue the examle by analyzing the same economy but with the addition of a very stylized rent control olicy. Here rent control is just a simle rent ceiling (which we set to 90 ercent of the 19

20 highest rent in the uncontrolled economy). Figure 3 shows the queue length, rents and exected value of searching in the controlled economy. Figure 3: Market tightness, rices and values with and without rent control 40 Market tightnesses 1 PUBLIC INFO & OWNING PUBLIC INFO RC PRIVATE INFO PRIVATE INFO RC OWNERSHIP RATE OWNERSHIP RATE RC Prices PUBLIC INFO PUBLIC INFO RC PRIVATE INFO PRIVATE INFO RC OWNING PRICE Values PUBLIC INFO PUBLIC INFO RC PRIVATE INFO PRIVATE INFO RC Queue Length Ownershi Rate Rent Exected Value Searation Rate Searation Rate Searation Rate In the case of ublic information, the rent control olicy distorts the markets for the shortestduration households the most; their queues lengthen considerably and the suly of regulated rental housing falls, increasing the ownershi rate only for those tyes whose rental submarkets have rents at the ceiling rate. All rental markets that had rents above the ceiling in the uncontrolled economy now have rents at the ceiling rate. However, because markets segment erfectly with ublic information, rent control does not affect the uncontrolled rental markets that already had low rents. With rivate information, all rental markets are affected by the ceiling even though only the low-duration households have rents at the ceiling rate. That the controlled market affects the uncontrolled markets is not due to some households leaving the controlled market for an uncontrolled one, as in e.g. Fallis and Smith (1984) (where there is excess demand in the controlled market) and Weibull (1983) (where there is no excess demand) nor to misallocation of high-quality housing (as hinted at in Glaeser and Luttmer (2003) and examined more broadly for the case of China by Wang (2011)). Here, there is no excess demand; the controlled market is in equilibrium, albeit an inefficient one 20. Instead, rent control exacerbates the rivate information roblem by making the low-duration households worse-off in their own market, tightening the incentive comatibility constraint. Queues in all rental markets are higher and the suly of rental housing is everywhere lower. The lower exected value of searching in the rental market also leads to more ownershi, which, unlike in the case of ublic information, occurs with any binding rent ceiling. Obviously rent control is not a welfare-imroving olicy in our economy. In fact, the Pareto oti- 20 There is no excess demand or suly at the controlled rent, and in that sense the controlled market is in equilibrium (as in Weibull (1983)). However landlords would enter into the market offering a higher rent and lower imlied market tightness, if they could. Therefore neither the ublic nor rivate information controlled market allocations are cometitive equilibriums as defined above. Rather they are cometitive equilibriums to economies with the added restriction that w [ρh, w] I, where w is the rent ceiling. 20

21 mal olicy would be a system of market deendent lum-sum taxes and transfers to households that effectively shares the surlus that the longer-duration households have over the shorter-duration ones in the ublic information economy 21. Rather than focusing on these olicies though, we will examine in Section 6 an economy with a customization technology which in equilibrium hels screen low-duration tyes. 5 Rental Markets in the data We merge data from the 2000 and 2010 U.S. Census and 2011 American Community Survey (ACS) with a novel data set constructed from rental and for-sale advertisements osted on Craistlist.org. The Craigslist data san and contain over 2.29 million for-sale ostings and 3.16 million for-rent ostings with listed addresses on Craigslist websites for metroolitan areas within the U.S.A. In addition to the street address, the data contain an asking rice or rent, an advertised number of bedrooms and the date of the osting 22. We match houses across listings according to the following algorithm. We searately sort both the rental and for-sale data by date. We assume that a listing is a new listing and not a continuation of a re-existing listing if there is not an existing listing for an earlier date at the same address osted within 31 days of this listing and with an asking rent (rice) within 15 ercent of the reexisting listing. Matching ostings using the date and rice of the osting is necessary because unfortunately not every listing contains an aartment number even when the home is clearly an aartment and there are several instances where multile aartments in the same building seem to be concurrently for rent (or sale). 23 In what follows we use the last osted rent or sale rice as the contracted rice. Using our matched data, we construct a measure of how long each home is on the market, T i, by using the time san in days between its first and last ostings lus three days. 21 This otimum can otentially relicate the first best queues and rents if the masses of long-duration households are large enough. 22 The data also contain the URL of the osting and sometimes contain additional information such as the number of bathrooms, whether the residence is a single-family home, and some contact information for the listing agent. To the best of our knowledge, our data set contains all ostings on Craigslist with a listed address during the time eriod excet for listings that were removed by Craiglist.org or the osting author. 23 Some rental listings may not be genuine vacancies but rather may just be a stock ad used by a real estate agency fishing for customers to call them. Others may be a stock ad used by a management comany for a building with multile vacancies in its building. To account for these ossibilities as best as ossible, we censor our data by droing any listing which we record as T i > 75. The results resented below are robust to alternative censoring thresholds as well as alternative windows for matching listings. Using the 75 day threshold, 3 ercent of our samle is droed. 21