Homeownership and the Scarcity of Rentals

Transcription

1 Homeownershi and the Scarcity of Rentals Jonathan Halket a,b,c,, Matteo Pignatti d,e a University of Essex b Institute For Fiscal Studies c CeMMAP d Confindustria Research Deartment e CeLEG Abstract Using a novel data set of rental listings, we 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. 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 underrovided to long-duration households to discourage short-duration households from searching there, consistent with the data. Ownershi is attractive in art because it cures the rivate information roblem. Keywords: adverse selection, cometitive search, homeownershi JEL Classification: C78, R13, R21, R23. This aer formerly circulated under Housing Tenure Choices With Private Information. We thank Melvyn Coles, Allen Head, Ricardo Lagos, Andreas Muller, Roland Rathelot, Morten Ravn, Thomas Sargent, Randy Wright 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, Society for Economic Dynamics, Vienna Macro Cafe, Paris SAM, HULM-Atlanta and the Nordic Macroeconomics Conferences and to Patrick Killelea, who gathered the Craigslist data. 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. Corresonding author addresses: jonathan@halket.com (Jonathan Halket), matteo.ignatti@nyu.edu (Matteo Pignatti) URL: htt:// (Jonathan Halket), htt://homeages.nyu.edu/~mm272 (Matteo Pignatti) 1

2 1. Introduction Why do some households buy their home while others rent? 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); Fisher and Gervais (2011)), and across locations (Amior and Halket (2014)), among others. 2 In these models, absent the friction, homeownershi rates would be much too low; the data would reject the models. In addition to being emirically necessary for many models, rental scarcity in some segments of the market is also casually intuitive: in arts of the market where homeownershi rates are high, rentals are indeed few in number. This of course is tautological. In this aer we build and examine the evidence for the first theory in which rental 1 e.g. Amior and Halket (2014); Cambell and Cocco (2007); Chambers et al. (2009a,b); Cocco (2005); Diaz and Luengo-Prado (2008); Fisher and Gervais (2011); Gervais (2002); Halket and Vasudev (2014); Iacoviello and Pavan (2013); Kiyotaki et al. (2011); Li and Yao (2007); Rios-Rull and Sanchez-Marcos (2008) 2 Some other models simly assume a warm glow to owning (Iacoviello and Pavan (2013); Kiyotaki et al. (2011)). 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)). 2

3 scarcity is an equilibrium outcome rather than an inut. We use a large, novel data set of for-rent and for-sale listings on 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 also the arts where rentals are relatively chea. This stylized fact, which is consistent with findings from Verbrugge (2008), Verbrugge and Poole (2010) and Bracke (2013) using alternative data sets and markets, is at first counter-intuitive. If the rent-to-rice ratio is exogenous to household demand for homeownershi, then one might exect homeownershi rates to be higher in submarkets where the ratio is higher, not lower. The solution is the correct notion of scarcity. Our data show that vacant rentals in submarkets with low rent-to-rice ratios disaear from the market quickly, the submarkets where homeownershi is high. In other words, households are more likely to search for owneroccuied 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 by the equilibrium suly of rental housing in a submarket. 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. 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, 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. Only when information is rivate does our equilibrium yield that rentals are relatively chea where they are scarce. Moreover this same rivate information friction endogenously rovides that long-duration households search in the scarce markets and also that more of these same households refer to own. 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 ex-ante 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. 3 Homeowners (which may be households or landlords) ost contracts for housing services which secify a (otentially tye-deendent) 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 4 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 long-duration households so as to discourage short- 3 There is a long literature looking at mobility and homeownershi choices. Deng et al. (2003) and Gabriel and Nothaft (2001) find considerable variation across households and Metroolitan Statistical Areas in rental vacancy rates and durations. Boehm et al. (1991), Cameron and Tracy (1997), Haurin and Gill (2002) and Kan (2000) all find relationshis between mobility hazards and homeownershi. 4 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 duration households from searching there. In this sense, rivate information causes housing scarcity in some rental submarkets. 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. When the model is calibrated to match the data, we find that the rivate information roblem leads to a ten ercent reduction in the surlus created by rental housing for most households. 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 heterogenous exogenous cost (a reduced-form way to model the various more wellunderstood 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 5

6 data. 5 To be clear, we intend for this to be the beginning, not the end, of research which uses contracting 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 et al. (2007, 2010); Calin and Leahy (2011); Head et al. (2014); Ngai and Tenreyro (2014); 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. 6 Piskorski and Tchistyi (2011, 2010) examine otimal mortgage design in owner-occuied markets (contracts for loans backed by housing services). Our work looks at contracts to suly housing services when there are search frictions and asymmetric information and thus extends the work of Guerrieri et al. (2010) to include dynamic contracts in a cometitive search equilibrium with adverse selection 7. In our equilibrium, contracts can be dynamic while the markets themselves are in steady-state. 8 5 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. 6 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. 7 Delacroix and Shi (2013) 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. 8 We are therefore abstracting away from the interesting housing dynamics discussed in Head et al. (2014); Ngai and Tenreyro (2014), among others. 6

7 Concurrently and comlementarily, Chang (2011) and Guerrieri and Shimer (2014) examine environments where the markets can change dynamically, however all contracts are one-time exchanges (urchases and sales of assets). The remainder of this aer is as follows: Section 2 examines market tightnesses and rices in rental and owner-occuied markets using data from Craigslist. Section 3 resents the model, first with just renting in ublic and then rivate information and then finally the equilibrium with both owning and renting. Section 4 numerically calibrates the model and section 5 extends the model to fully otimal dynamic contracts. Section 6 concludes. Most roofs are in the Aendix. 2. 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 contain over 2.29 million for-sale ostings and 3.16 million for-rent ostings collected from the Craigslist.org RSS feed of listings for each metroolitan area that Craigslist covered at that time in the USA. The data were collected from ostings starting on January 2, 2010 until June 27, Every ost on the feed that had a listed street address was stored. Craigslist stoed including the street address in the RSS feed in July, In addition to the street address, our data contain an asking rice or rent, an advertised number of bedrooms and the date of the osting as well as 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. 9 Craigslist lists ostings in order of recency which means that searchers on Craigslist see recent ostings first when browsing. Therefore landlords (sellers) have an incentive to reost their advertisement in order to move it to the to of the listings if they are still looking 9 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 raidly removed by Craiglist.org or the osting author. 7

8 for ersective tenants (buyers). We will use the time between first and last ostings for a articular home as our measure of its market tightness: on average, homes for rent or sale in a tight market should have short times between first and last ostings. Time between ostings is clearly an imerfect measure of how long the home is actually vacant and/or vacancy costs to the landlord. We assume in this section that the average vacancy costs er vacant sell in a articular submarket are roortional to the length time between a vacancy s first and last ostings. In order to determine a vacancy s first and last osting, we must first match houses across listings. We use the following algorithm to match houses. 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 re-existing 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). Merlo and Ortalo-Magn (2004) finds that listing rice forecasts well contracted sales rices for the for-sale market. In what follows we use the last osted rent or sale rice as our estimate of 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. T i = lastdate i firstdate i + 3 The three additional days are used so that homes only listed once are still on the market for some time eriod. Our results are robust to changes in this length of time. 10 The average 10 In articular, one might worry that in markets with few Craigslist ostings, landlords would need to reost their advertisement less often and thus that in these markets we should consider them on the market for longer than three days after their last osting. In one robustness exercise, we adjusted the additional days we add so that they vary by market (but not submarket) so that markets with few ostings on Craigslist er 8

9 number of days between ostings for the same home in our rental data is 6 days and 8 days in the for-sale market. In this section, our notion of a submarket is all housing for rent or sale within a zi code with the same number of bedrooms. Piazzesi et al. (2015), using data on for-sale search queries from the San Francisco Bay Area, similarly artition their markets into submarkets based on zi codes and number of bathrooms. 11 US zi codes are five digit ost codes that tend to be cardinally geograhically clustered. For instance, all zi codes with 100 as the first three digits ( three digit zi ) are located in New York City. In this section then a market is a collection of submarkets with the same number of bedrooms and the same first three digits in their zi code. For measures of rent-to-rice ratios, we create rental and sale rice indices for each zi code by number of bedroom cell by taking the mean rental or sale rice for that cell and then taking the ratio of the mean rent to mean rice as the rent-to-rice ratio for that cell. 12 We kee only those cells for which we have at least 10 unique rental and 10 unique for-sale listings in our samle. From the 2010 Census, we have zi code level measures of the number of occuied rentals and owner-occuied homes, the numbers of vacant roerties for-rent and for-sale (though these roerties usually must be vacant for over six months to be recorded as vacant), and the roortion of households with head of house under age 35. The 2000 Census contains information about the rior moving dates of households (conditional on tenure). From the ACS, we have median income in each zi code and homeownershi rates by number of bedrooms for each zi code. Figure 1 shows the densities by cell of various dimensions of day have more additional days; our results did not change. Furthermore, 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. 11 They also artition based on rice to attemt to cature residual quality. 12 In the Aendix, we do a robustness check using rent-to-rice ratios from Zillow.com. 9

10 the rental market. Figure 1: Histograms of time on the market for rentals in Craigslist data and current duration for renters Density Average number of days on the rental market Density Average current duration for renters We are interested in variation within a market at the submarket level. In the model that follows, households choose a submarket from within a market to direct their search to. Our theory connects households exected duration in a home with the submarkets they choose to search and live in and with rental rates and the availability of rental homes. The distribution of exected durations in a market is treated as exogenous, but submarket choice (and therefore the average exected duration in a submarket), rice and availability are all endogenous outcomes. Given that, the regression results resented below should be thought of as evidence of correlations between market outcomes and not anything causal and the standard errors taken with a grain of salt. There are several other caveats worth mentioning. The Craigslist listings are not random selections from their various markets, articularly in the for-sale markets. Furthermore, we cluster the standard errors at the market level and include dummies for at the market and number of bedrooms level but do not otherwise account for satially correlated errors. All of our results are robust to clustering and including dummies at the county or city level instead. Omitted variables are also likely an issue. We use two different roxies for average exected duration: the roortion of households under 35 years old living in the area, and 10

11 the average time since last move for renters. 13 Age is ositively correlated with exected duration (since uncertainty over income and family rosects falls with age, see Halket and Vasudev (2014)), so the roortion under 35 will be negatively correlated with exected duration. Meanwhile (ex-ost) actual duration is almost certainly ositively correlated with a household s (ex-ante) exected duration. We include median income in the zi in our regressions to try and cature some of the differences in unobserved quality across submarkets. Of course, neither age nor erhas realized duration is orthogonal to income. Furthermore exected duration is correlated with tenure decisions due to the high transactions costs of homeownershi. So the homeownershi rate results below should be taken with an additional grain of salt. Finally, our artitions of markets and submarkets are somewhat arbitrary: our theory has nothing to say about whether or why contracts sort along geograhic dimensions nor does it shed any light as to the geograhic size of a submarket. In reality, there may be several heterogenous submarkets within each zi x bedroom cell and some submarkets may otentially overla across cells. Piazzesi et al. (2015) finds multile for-sale submarkets within individual zi codes and that households tyically search in multile zi codes. In this case the elasticities resented in the tables here would then likely understate the true elasticities. Table 1 shows that, in our data, both rental and sale rices are lowest in the submarkets where (resectively) rental and sale time on markets are lowest; a relationshi which will follow naturally from the free entry conditions in our model. For instance, landlords must be comensated with higher rents when renting in submarkets where houses stay vacant for longer. Figure 2 (uer right) shows the same correlation between rent-to-rice ratios and rental vacancy rates using only data on 2 bedrooms houses in King County (Seattle- Tacoma-Bellevue, Washington MSA). 14 Table 1 also shows that younger households with 13 The 2000 Census gives the number of renters that have moved into their current home in the last year, between 1 and 5 years ago, etc... We use the midoint of each cell and take the average. 14 The Seattle MSA has the largest number of rental listings in our data. 11

12 Table 1: Regressions of rices on vacancy durations and durations in home rent rent rent rice rice rice time rent 0.025* (0.010) time sale 0.078** (0.029) ro under 35 yr 0.061** ** (0.023) (0.031) duration in home ** 0.124* (0.014) (0.059) median inc 0.359** 0.364** 0.352** 0.740** 0.652** 0.780** (0.015) (0.015) (0.016) (0.031) (0.032) (0.043) R N All variables are in logs and all regressions include dummies for each market. time rent (time sale) is the T i conditional on being for-rent (for-sale). rent (rice) is the final listed rent (rice) for the roerty. ro under 35 yr is the roortion of households in the zi code with head of house under age 35. SEs, clustered by market, in arentheses. ** < 0.01, * < N is the number of submarkets included in each regression. 12

13 Table 2: Regressions of vacancy duration and homeownershi rates on durations in home time rent time rent time sale time sale homeownershi homeownershi ro under 35 yr 0.048** ** (0.012) (0.011) (0.019) duration in home ** ** (0.013) (0.016) (0.045) median inc * ** 0.379** (0.009) (0.010) (0.013) (0.014) (0.026) (0.021) R N See notes to Table 1 lower exected durations as well as households with shorter actual durations searched in submarkets with higher rental rices but lower sales rices. Rent-to-rice ratios in the submarkets where these households search are therefore higher. Desite the fact that rentals are exensive relative to rices, though, homeownershi rates tend to be lower in these same submarkets (Table 2 and, for Seattle, Figure 2 - uer left)). Verbrugge (2008), Verbrugge and Poole (2010) and Bracke (2013) find similar correlations between rent-to-rice ratios and homeownershi rates, the latter even after controlling for otential unobserved heterogeneity. From Table 2, households with lower exected durations search for rentals in submarkets where time on the market is longer, while time on the for-sale submarket is hardly different. Guasch and Marshall (1985) finds a similar correlation between rental vacancy hazards and rental vacancy durations in a cross-section of Philadelhia rental housing. As we will now show, a negative correlation between exected duration and time on the market is not consistent with a cometitive search equilibrium with ublic information. Instead, the data show that landlords match quickly with long duration households, which will be consistent with the resence of rivate information. 13

14 Figure 2: King County Homeownershi Rate Fitted values Mean time on the market Fitted values Monthly Rent to Price ratio Monthly Rent to Price ratio Mean time on the market Fitted values 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 in King County, Washington (e.g. the mean over 2 bedroom listings in zi code 98001) using the Craigslist data. Homeownershi rates are from the Census data at the same bedroom x zi code level. 3. Model 3.1. 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, synonymously, 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 14

15 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 in tye. 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. 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 The rental market 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-ofstay discount. In section 5 we extend the setting to fully dynamic contracts with risk-averse households and show that our results still follow even with endogenous duration-of-stay discounts. 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 tye-indeendent mechanisms. More recisely, in the rivate information case the equilibrium with contracts is ayoff equivalent to the equilibrium with degenerate mechanisms offering the same rent to each household, which enables us to 15

16 simlify the notation greatly. 15 Because in equilibrium each tye of household directs its search to at most one tye of rental contract (and later when we add owning, at most one tye of owning contract), we will call a submarket all houses with a renting and/or owning contract that attracts a articular tye of household. That is, we will have one submarket for each i. 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. 16 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 is constant: ε(θ) = ε. (θ) dθ 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 17, 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: 15 This esthetic feature is not true in the setting with fully-dynamic contracts in section 5; which is the main reason why we deal with it searately (see our discussion of Lemma 4). 16 Matching is bilateral, thus every household can only aly to one contract, but she can use mixed strategies. 17 These are the values of searching and living in the same market, reeatedly ad infinitum. 16

17 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. 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) ) 1 ψ i (w) 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 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)) ) 1 ψ i (w) α l (θ ρ + γ r(w)) i i I i I with equality if w W r. 17 ψ i (w)r i ρ + γ i ρh

18 (ii) Households otimal search: Then w W I Let Z i r Z i r max w W r with equality if θ r(w) > 0 and ψ i (w) > 0. (iii) market clearing: W r 1 α l (θr(w )) ρ θr(w )(ρ + γ i ) + α l (θr(w )) (h r i) 1 α l (θr(w)) ρ θr(w)(ρ + γ i ) + α l (θr(w)) (h r i) ( 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 W r 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 with ψ i (w B ) > 0 such that ri A = ri B and θ A (w A ) = θ 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. This also ins down the measure of landlords osting the contract w to households of tye i, given by v(w) = 18 r r γ i π i α l (θ r (w))+γ i θ r (w).

19 Characterization A necessary and sufficient condition for a searating cometitive search equilibrium is the following: 18 Proosition 1. For any tye i of households, a osted contract w i r market tightness θ i r and the associated θ r(w i r ) are art of an equilibrium allocation if and only if they solve the following 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 = ε θr i 1 ε + ρ + γ i α l (θr i )(1 ε) The imlicit solution for θ i r is strictly increasing in γ i. (5) Lemma 3. Any cometitive equilibrium with ublic information is a searating cometitive equilibrium. 18 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 Wr 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. 19

20 Proof. Follows immediately from Lemmas 1 and 2 and Proosition 1. The equilibrium values of the flow rent w i r given by: and the household s exected value ρz i r are w i r ρz i r = 1 θ i r r ) = ρ + γ i + α l (θ i ρh α l (θr i ) ε 1 ε ρh 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. 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. Analytically, for the differential-γ case, by differentiating 19 the equilibrium condition (5) 19 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. 20

21 we obtain: dθr dγ = 1 θ 2 r ε θr(ρ + γ) + α l (θr) > 0 dγ = ρh θr(ρ + γ) + α l (θr) > 0 dw r 3.4. Renting with rivate information 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. So we can safely associate any incentive comatible contract w with its associated rent (and thus can assume w [ρh, h]). 20 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 20 Lemma 4 relies on our restriction of the rental contract sace to constant rent rofiles. When we relax this assumtion, as section 5, 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 section 5 s relaxed setting. We also show there that this setting can generate length of stay discounts in rents (e.g. a downward sloing rent rofile with resect to actual duration). 21

22 Proof. Follows from equation 3. The sorting lemma is similar to a standard single-crossing conditions. 21 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] with equality if w W. (ii) households otimal search: Let Z i [ ( 1 + α l (θ(w)) i I max w W ) 1 ] 1 ψ i (w) w ρh ρ + γ i 1 α l (θ(w )) ρ θ(w )(ρ + γ i ) + α l (θ(w )) (h w ) Then w [ρh, h] and γ i Z i with equality if θ (w) > 0 and ψ i (w) > 0. 1 α l (θ(w)) (h w) ρ θ(w)(ρ + γ i ) + α l (θ(w)) (iii) market clearing: W ( ψ i (w) θ(w) + α l(θ(w)) ) dλ(w) = π i γ i i As in the ublic information case, the equilibrium definition imoses conditions on the off-equilibrium beliefs of the landlords. Heuristically, a landlord considering whether to ost 21 It is slightly different. See Guerrieri et al. (2010) for an elaboration. 22

23 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. max Z r (γ i, w, θ) θ R +,w R + α l (θ) ρ + γ i + α l (θ) w ρh and 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: 23

24 (i) For i = 1, w 1 and θ 1 solve R 1 (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, Proof. See Aendix. v i π i u i v i increases; Analytically, for the differential-γ case, then: dθ dγ = 1 ρ + γ dw dγ = (1 ε)ρz α l (θ) [ (1 ε) ρz ] 1 < 0 ρh ε θ [ (1 ε) ρz ρh ε θ dz dγ = θ Z θ(ρ + γ) + α l (θ) < 0 ] 1 > 0 24

25 Contrary to the ublic information case, low-γ tyes search in tighter rental submarkets in equilibrium, and ay lower rents if matched. In this way landlords are able to otimally (with the least cost) searate tyes of households by osting contracts w i first-best otimum w i r tightness θ i. lower than the to those that exect to stay longer, in return for a higher market 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) 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 houses would be owner-occuied. 22 To rovide heterogeneity, we assume that there is an extra friction for owner-occuies which is heterogenous in the oulation. 23 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 22 The i = 1 tye would be indifferent between owning and renting. 23 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 friction is sequential search for owner-occuiers. 25

26 means that, for each tye i, there will be at most one owning contract that in equilibrium attracts searchers, which eases notation. 24 Builders only have to sell a new house. It is imortant to notice that sellers osting an owning contract are not concerned about the buyer s tye, regardless of whether this information is rivate or ublic. 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 an owning contract for sale at rice P with imlied tightness θ 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 will immediately imly that sale rices are negatively correlated with market tightnesses across submarkets. The values of searching as a buyer and living in a submarket with owner-occuied 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). ρz o (γ i, P, θ, χ) = ρz r (γ i, r i, θ) χ 3.6. Equilibrium with both renting and owning In the aendix, we formally define a cometitive equilibrium with rivate information and both renting and owning. The market endogenously segments into submarkets and 24 Fixed tyes and χs are not essential. For instance, we could allow a household s tye i and χ to change whenever it received a searation shock. The economy would be quantitatively exactly the same as long as the distribution over tyes and χ remained constant. 26

27 we can characterize the equilibrium allocation using an equivalent constrained maximization roblem. 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 25 : 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 H α l (θo) i 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 } Z o (γ i, P i, θo, i 0) χ Result 3. The roortion of tye i that are homeowners is increasing in i and thus exected duration. Proof. See Aendix. The equilibrium in the rivate information rental market for the highest-γ tye is the same as in the ublic information case. 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. The otimal market tightness conditional on owning for each tye, θ i o, is identical to the otimal tightness in that tye s rental market with ublic information, θ i r. Homeownershi rates increase in exected duration solely because the rental frictions modeled here are endogenously increasing in exected duration. If we relaxed the assumtion that the distribution of χ was indeendent of i, it would be natural to assume that those with higher exected durations also had lower homeownershi frictions. For instance, an exogenous transactions costs to buying a house would work in such a manner. In such cases, this would reinforce Result We omit the roof, however it is similar to the case with only renting 27