Housing Liquidity, Mobility and the Labour Market

Transcription

1 Housng Lqudty, Moblty and the Labour Market Allen Head Huw Lloyd-Ells January 29, 2009 Abstract The relatonshps among geographcal moblty, unemployment and the value of owner-occuped housng are studed n an economy wth heterogeneous locatons and search frctons n the markets for both labour and houses. D erences n labour market condtons between ctes a ect the speed wth whch houses may be sold that s, the lqudty of housng. At the same tme housng market condtons a ect employment decsons and thus the allocaton of labour across ctes. In equlbrum, unemployment rates for home-owners are hgher than for otherwse dentcal renters. Unemployment and home-ownershp rates are, however, negatvely correlated across ctes. In a parameterzed example we nd that, although renters are much more moble than owners, the mpact of home-ownershp on aggregate unemployment s quanttatvely small. Journal of Economc Lterature Class caton: : J61, J64, R23 Keywords: Lqudty, moblty, home-ownershp, unemployment. Queen s Unversty, Department of Economcs, 94 Unversty Avenue, Kngston, Ontaro, Canada K7L 3N6, [email protected]; [email protected]. Ths paper has bene tted from the comments of Morrs Davs, Ahmet Akyol, Rchard Rogerson and semnar partcpants at HEC Lausanne, Rochester, the FRB Chcago Summer Workshop n Money, Payments and Fnance, the Venna Macro Workshop and the Canadan Macroeconomcs Study Group. The Socal Scence and Humantes Councl of Canada provded nancal support for ths research. 1

2 1 Introducton In ths paper, we study the relatonshps among geographcal moblty, unemployment and the value of owner-occuped housng n an envronment characterzed by frctons n the markets for both labour and houses. The prce of a house re ects ts lqudty.e. the speed wth whch t can be transferred to another home owner and ths n turn a ects both moblty and labour market outcomes. Our model s consstent wth recent mcro-evdence on the relatonshp between ownershp and unemployment across ctes and between ndvduals. It s also consstent wth large observed d erences n moblty between renters and owners. Nevertheless, we nd that the mpact of home ownershp on aggregate unemployment s very unlkelyto be economcally sgn cant. In our economy a large number of ex ante dentcal households may choose to lve n ether of two ctes whch d er n regard to the productvty of jobs. Households requre housng and may ether rent n a compettve market or purchase n a market characterzed by a search frcton. All households, whether employed or unemployed, randomly receve o ers of employment n both ther cty of current resdence and the other cty. In order to take a job n the other cty, a household must move and acqure housng there. Mgratng home owners put ther houses up for sale and ntally rent n the other cty whle searchng for a house. In ths envronment, we establsh the exstence and unqueness of a statonary equlbrum characterzed by constant relocaton and housng market actvty. The wllngness of a home owner to accept a job n the other cty depends not only on relatve wages but also on rental rates and the market value of ther current house. Snce the latter depends on how quckly a buyer can be found, the lqudty of housng a ects the dstrbuton of households across ctes and unemployment both at the cty level and n the aggregate. At the same tme, the frequency wth whch households choose to relocate a ects the lqudty of the housng markets. For a range of parameter values, our model can account for some puzzlng and seemngly contradctory recent evdence on the relatonshp between home ownershp and unemployment both across US ctes and at the ndvdual level. 1 In partcular, Coulson and Fsher (2008) nd that across U.S. metropoltan areas home ownershp rates are correlated postvely wth average wages and negatvely wth unemployment. 2 In contrast, usng mcro-data 1 Unlke countres or regons, t seems reasonable to thnk of each metropoltan area as a dstnct labour market. 2 After controllng for other co-varates, Coulson and Fsher (2008, p.26) conclude that, there s a negatve correlaton between unemployment and ownershp and a postve correlaton between wages and ownershp 1

3 and controllng for demographc and locatonal characterstcs, they nd that the margnal mpact of home ownershp on the lkelhood of unemployment for an ndvdual s sgn cant and postve, though modest n sze. At the same tme (and consstent wth the aggregate cross cty results) the lkelhood of unemployment s negatvely correlated wth the rate of home ownershp n an ndvdual s cty of resdence. In our framework, home-ownershp ncreases the lkelhood of unemployment for an ndvdual because, whle separaton rates and o er rates are the same for all, n equlbrum only home owners ever turn down o ers. At the cty level, however, there s a second e ect whch o sets the mpact of home ownershp on unemployment. Because the hgh wage cty has the lower vacancy rate, t has the hgher rate of home-ownershp. It also, however, has the hgher rental rate, makng t unattractve to unemployed renters who may move to the low wage cty even wthout a job o er. In contrast, employed renters n the low wage cty move to the hgh wage (and hgh rent) cty as long as the wage premum s su cent and unemployed home-owners never re-locate to the hgh-wage cty wthout an o er. Dependng on parameters, these factors may combne to generate hgher unemployment n the low-wage cty, where the home ownershp rate s also lower. There s consderable evdence that owners tend to move less than renters, even after controllng for household and locatonal characterstcs (see, for example, Rohe and Stewart, 1996, or Bohem and Taylor, 2002). Recently, a number of commentators have argued that because of ts relatonshp wth moblty, home-ownershp may create frctons n the labour market that lead to ne cent outcomes. 3 Indeed some have gone so far as to conjecture that d erences n home-ownershp rates across countres may be a leadng factor n drvng d erences n unemployment. 4 In general, the avalable evdence based on mcro-data s not partcularly favorable to ths conjecture, ndng at best only a small margnal e ect of ownershp on the lkelhood of unemployment. It s, however, somewhat d cult to nterpret these ndngs. Although uncondtonally the unemployment rate amongst renters s sgn cantly hgher than that amongst owners, ths largely re ects the d erng characterstcs of these households. For example, owners tend to be more educated, older and more lkely to be marred than renters. across US metropoltan areas. 3 See, for example, Blanch ower (2007). 4 Ths argument s typcally based on the observaton of a postve correlaton between unemployment and home ownershp across countres or regons. See Nckell (1998), Oswald (1999), Partrdge and Rckman (1997), Pehkohnen (1999) and Cochrane and Poot (2007). Munch et al. (2006) and Rouwendal and Njkamp (2006) crtcally revew some of ths work. 2

4 These characterstcs also make them less lkely to be unemployed, ndependent of any drect e ect of ownershp on moblty. Obvously, to test the conjecture one must control for all the relevant demographc and locatonal characterstcs some of whch may not be observed. Our framework allows us to solate the e ects of home ownershp per se on both moblty and unemployment. We model all households as ex ante dentcal and so home ownershp a ects moblty, rather than the reverse. Usng a verson of our model calbrated to match aggregate US labour market ows and moblty rates, we nd that the fracton of home owners that turn down hgh wage o ers n the other locaton s substantal. Consequently, n accordance wth the emprcal evdence, the moblty rate for owners s much lower than for renters. Despte these large e ects on moblty, however, we nd that the mpact of ownershp on aggregate unemployment s very small. Moreover, because the mpact of home-ownershp on the lkelhood of unemployment s small compared to the e ect of the rent d erental on the moblty of unemployed renters, unemployment rates and home-ownershp are negatvely correlated across ctes as observed by Coulson and Fsher (2008). These ndngs are robust to several alternatve parameterzatons and generalzatons of our basc model. Others have developed theores of the relatonshp between home ownershp and the labour market. For example, Dohmen (2005) and Munch, Rosholm and Svarer (2006) present models of labour market search n whch home-owners and renters are assumed to behave d erently. Coulson and Fsher (2008) present a theory based on endogenous job creaton that s consstent wth ther evdence on unemployment, but does less well wth regard to wages. In partcular, n ther model home owners receve lower wages than renters as a result of ther mmoblty. All of these theores, however, abstract from both housng choce and transactons n the housng market. Owners are ether smply assumed to be mmoble or to face hgher movng costs than renters. Here, because the prce of housng s endogenously determned, the relatve degree of moblty depends on labour and housng market condtons. Rupert and Wasmer (2008) also develop a theory of the relatonshp between unemployment and housng market frctons. They do not dstngush between ownershp and rentng, and they focus on the trade-o between commutng tme and locatonal decsons wthn a sngle labour market. In contrast, our focus s on the role of housng markets n generatng frctons between labour markets. In ths sense, our paper s complementary to thers. In a generalzaton of our basc model, we consder the role of wthn cty relocaton. Ths extenson s related to Wheaton (1990), who develops a model of housng markets but consders nether lnkages to labour markets nor cross-cty re-locaton. Albrecht, Axelrod, Smth, and 3

5 Vroman (2007) consder a search model n whch the ow values of search to buyers and sellers change over tme. Ther work s related to ours but d erent n that whereas they focus on the relatonshp between prces and tme on the market, we focus manly on moblty and the relatonshp between the labour and housng markets. A substantal lterature also focuses on the relatonshp between the length of resdence spells (whch tend to be hgher for home owners than for renters) and nvestments n socal captal (e.g. see Ross and Weber (1996) and DPasquale and Glaeser (1999)). Emprcally, Coulson, Hwang, and Ima (2002) nd that the fracton of home owners n a neghborhood s assocated wth hgher property values. Whle our model s consstent wth ths observaton (as home ownershp and house prces are both hgher n the hgh-wage cty) as well as the fact that home owners reman n a cty longer than renters, we abstract from nvestment of all knds. Ths remander of the paper s organzed as follows: Secton 2 descrbes the envronment. Secton 3 de nes a symmetrc statonary equlbrum, establshes exstence and unqueness, and characterzes the equlbrum under varous assumptons. Secton 4 consders the mplcatons of the theory for the relatonshp between ownershp, moblty and unemployment at the ndvdual, cty and aggregate level. We also consder a parameterzed verson of the model to assess ts quanttatve mplcatons. In Secton 5 we assess how robust our conclusons are to several generalzatons of the basc model. Secton 6 summarzes and descrbes future work. Proofs, longer dervatons, and the detals of some of the extensons are contaned n appendces. 2 The Envronment Tme s contnuous. The economy s populated by a unt measure of n ntely lved, ex ante dentcal, rsk-neutral households who dscount the future at rate. There are two locatons, called ctes, ndexed by 2 f1; 2g. Households must resde n one and only one cty at any pont n tme. They are free, however, to move between ctes at any tme at no drect cost. Each cty contans two types of resdental dwellngs. At a pont n tme, let R denote the stock of rental housng n Cty and H the stock of owner-occuped housng. Let R denote the ow utlty receved by a household whch lves as a renter n ether cty. Smlarly, H s the ow utlty from lvng n an owned house. We assume that H > R. Frms n Cty produce output y usng labour l accordng to the producton functon y = ( l ) F (1) 4

6 where 2 (0; 1) and represents a cty-spec c productvty parameter. Takng wages as gven, rms may hre as many workers as they lke, provded that they have pad the per perod xed cost F to operate the producton technology. Wthout loss of generalty, we assume that productvty s hgher n Cty 2 than n Cty 1. That s, 2 > 1. In each cty there s labour market whch functons much lke that consdered n Lucas and Prescott (1974). As a consequence of rms demand for labour each cty has a large number of potental employment opportuntes, whch we refer to as jobs. At any pont n tme, each household s ether employed (.e. holdng a job) or unemployed. A household may hold at most one job and that job must be located n ther cty of current resdence. Employed households n Cty receve ow ncome equal to the wage w. Unemployed households receve ow consumpton z. All households, regardless of ther employment status, randomly receve o ers of employment both n ther cty of resdence and n the other cty. These may seen as o ers of admsson to the labour market of a partcular cty, wthn whch wages are determned n a Walrasan fashon. Let denote the Posson rate at whch households receve o ers wthn ther cty of resdence and denote the rate at whch the receve an o er n the other cty. We assume that these rates are symmetrc across ctes and that >. A household (employed or unemployed) whch receves a job o er may ether accept or reject t. Employed households n both ctes lose ther jobs at Posson rate. 5 There are also a large number of rms called real estate managers (REM s) whch are owned by households and perform two functons: They rent out rental housng n cty n a compettve market at rate r, and they ntermedate between buyers and sellers n ctyspec c markets for owner occuped housng. We abstract from costs assocated wth the rental of houses so that r 0. We assume that all households own equal shares n the economy s REM s and receve any pro ts as lump-sum transfer,. Home-owners may sell ther houses at any tme to an REM n a compettve market. Let p denote the prce receved by a household that sells a house to a REM n Cty. REM s receve no servce ow from houses and hold them only for the purposes ether of re-sale or for converson nto rental unts. An REM can convert a formerly owner occuped unt nto a rental unt at a xed per unt cost C R 0. Smlarly, an REM can convert a rental unt to an owner occuped one at cost C H 0. The re-sale market for owner occuped housng s characterzed by a one-sded process 5 It s possble to allow for d erences n o er rates and separaton rates across locatons; we assume symmetry here for smplcty. 5

7 that matches potental buyers wth REM s. Let S denote the stock of houses o ered for sale by REM s n Cty (and therefore vacant). Smlarly, let D denote the stock of potental home buyers n Cty. REM s match wth potental buyers at rate, where 6 = D S = 1; 2: (2) For smplcty, we assume that a REM who matches wth a potental buyer makes a take-t-orleave-t o er, provded that the aggregate match surplus s postve. 7 Let q W and q U denote the prces pad for houses n Cty by employed and unemployed households respectvely. 8 Gven the assumed structure of the markets for owner occuped housng, t takes tme for houses to be transferred from one household to another. Ths frcton results n houses beng llqud, as ther market value depends on the speed wth whch a buyer can be found for a vacant house. Let the value of such a house n Cty be denoted V H. Then V H = E fq W ;q U g maxfq V H ; 0g = 1; 2: (3) In each cty there are four types of households, as each may be ether employed or unemployed and may ether rent or own a house. The measures of households n Cty that are employed-owners, employed-renters, unemployed-owners and unemployed-renters are gven by N W H, N W R, N and N UR each of these states are gven by W H, W R, U H respectvely. The values assocated wth beng n and U R ; respectvely. 6 In our basc model, we assume that unemployed and employed renters nd houses at the same rate. In Secton 6, we show that allowng unemployed renters to match at a lower rate makes lttle d erence. 7 It s straghtforward to generalze the model to allow for a d erent dvson of the surplus. However, t makes no d erence to our man conclusons. Calculatons are avalable from the authors upon request. 8 Snce they earn zero pro ts when purchasng a prevously owned house and make a take-t-or-leave t o er when re-sellng, the role played by REM s n ntermedatng transactons s vrtually equvalent to assumng that mgratng households contnue to own ther vacant house untl they match wth and sell to another household. Assumng that ths functon s performed by REM s greatly smpl es the analyss, however, because t rules out the possblty of a mgratng household returnng to ts prevous locaton and movng back nto ts old house before sellng. Allowng for ths would expand the number of household states and complcate the analyss substantally. Snce n the equlbra we consder these addtonal states would occur wth very low probablty, ths complcaton would add nothng sgn cant to our results. 6

8 3 Statonary, Symmetrc Equlbrum We consder equlbra whch are statonary and symmetrc n that all households of a gven type behave n the same way. In ths case, Cty households value functons satsfy W R = w + R r + + U R W R + max Wj R W R ; 0 + max W H q W W R ; 0 (4) U R = z + R r + + W R U R + max W R j U R ; 0 (5) + max U H q U U R ; 0 W H = w + H + + U H W H + max Wj R + p W H ; 0 (6) U H = z + H + + W H U H + max Wj R + p U H ; 0 (7) where the subscrpt j ndexes the other cty. A statonary symmetrc equlbrum for ths economy s a collecton of ten value functons, eght for the d erent types of households, W R, W H, U R, and U H, for = 1; 2, and one for vacant housng n each cty, V H ; rental prces n each cty, r ; house prces n each cty, q W, q U, and p ; and measures of households n each of the eght states, N W R N, such that:, N UR, N W H, and. Gven wages, rms choose employment levels l to maxmze pro ts. Free entry nto producton mples that pro ts equal zero: ( l ) w l F = 0: (8). Gven prces and the value of houses n each cty, the value functons satsfy (4)-(7).. The rental prces, r 0, clear the markets for rental housng n each cty: N W R + N UR = R = 1; 2: (9) v. House purchase prces n both ctes, p W and p U extract all of households surplus. v. The house sale prce n each cty s equal to the value of a vacant house: =1;2 p = V H = 1; 2: (10) v. The dstrbuton of households over states s consstent wth the populaton: X N W R + N UR + N W H + N = 1: (11) 7

9 v. Rents are dstrbuted equally as dvdends to households: X r R = =1;2 We begn by assumng at least one such equlbrum exsts, and descrbe several characterstcs that t must necessarly have. We then nsh ths secton wth a proposton establshng the exstence and unqueness of the equlbrum, subject to certan restrctons. Pro t maxmzaton by rms n Cty mples that each demands labour l = w : (12) Substtutng (12) nto (8) yelds an expresson for the equlbrum wage n cty : 1 w = F 1 : (13) Thus, n each cty the equlbrum wage s proportonal to local productvty and s una ected by condtons n the housng market. From now on we wll therefore refer to Cty 1 and Cty 2 as the low and hgh wage ctes, respectvely. We restrct attenton to equlbra where employed renters n the low-wage cty (Cty 1) who are o ered a job n the hgh-wage cty choose to relocate, but not vce versa. That s W2 R > W1 R : (14) We also restrct attenton to equlbra n whch all renters, whether employed or not, buy houses when they get the chance. As real estate managers make take-t-or-leave-t o ers to home buyers, ths wll happen as long as the surplus from such a transacton s postve: W H W R = q W > p U H U R = q U > p = 1; 2: (15) Below, we derve the condtons needed for (15) to hold. Fnally, we consder only equlbra that are nteror n the sense that there are unemployed renters n each cty. Because they are moble, n any such equlbrum these households must be nd erent wth regard to ther cty of resdence. That s, U2 R = U1 R = U R : (16) 8

10 Home owners, n contrast, face e ectve movng costs assocated wth the llqudty of housng. Condtons (15) and (14) together mply that employed home owners are also unwllng to move from from the hgh wage cty to the low wage one n equlbrum: W2 H p 2 > W2 R > W1 R : (17) Makng use of the facts that unemployed renters are nd erent between locatons, that real estate managers extract all surplus from households who purchase houses, and that employed renters do not move from the hgh-wage cty to the low-wage one, we may re-wrte the Bellman equatons for renters: W1 R = w 1 + R r (U R W1 R ) + (W2 R W1 R ) (18) U R = z + R r (W1 R U R ) + (W2 R U R ) (19) W2 R = w 2 + R r (U R W2 R ) (20) U R = z + R r (W2 R U R ) + (W1 R U R ): (21) Subtractng (19) from (18) and (21) from (20), and solvng n terms of U R we have expressons for the values of unemployed renters n the two ctes: W1 R = w 1 z +U R : {z } (22) W2 R = 1 w2 z 1 +U R : + + {z } (23) Usng (22) and (23) t s easly shown that 2 Lemma 1. If w 2 > w 1, then 2 > 1 and n a statonary equlbrum, W R 2 > W R 1. Note that equatng (19) and (21), and usng (22) and (23), t s also apparent that r 2 r 1 = (w 2 w 1 ) > 0 (24) + + Thus, the rental rate s hgher n the hgh wage cty by an amount proportonal to the wage d erental that depends on job o er arrval and destructon rates, and the nterest rate. The levels of the rental rates themselves depend on the value of beng an unemployed renter, U R. 9

11 The sale prce of a house n Cty may be shown to satsfy p = (W H + W R ) + (1 )(U H U R ) (25) where N W R =R represents the fracton of renters n Cty that are employed. Snce, n each cty renters consttute the potental buyers n the housng market, we may wrte (2) as: = H R N W H N = 1; 2: (26) Combnng (9), (11) and (26) we may derve a locus of values for 1 and 2 whch are consstent wth equlbrum condtons. (rental market clearng) and v. (aggregaton): R1 + R 2 = R 1 + R 2 + H 1 + H 2 1: (27) 1 2 We depct ths locus (labeled AM) n Fgure 1. 9 As we demonstrate below, one consequence of Lemma 1 s that 2 > 1. Thus, wthout loss of generalty, the equlbra that we study here are all located on the the segment of the AM curve above the 45 o lne. Recall that we consder equlbra n whch employed renters wll move from the low-wage cty to the hgh-wage one f o ered a job, but not vce versa and that n ths case employed home owners wll also not move from the hgh-wage cty to the low-wage one. Assumng that an equlbrum exts, there are two d erent possble cases wth regard to the movement of home owners between ctes: I. Some fracton (possbly all) of unemployed home owners n the low-wage cty move to the hgh-wage cty f o ered a job there, but employed home owners do not move. II. All unemployed home owners and some fracton of the employed home owners (agan, possbly all) n the low-wage cty move f o ered a job n the hgh-wage cty. In Case I, we say that the margnal home owner (.e. a home owner who s nd erent between movng from the low-wage to the hgh-wage cty upon recevng a job o er there) s unemployed. Case II, n contrast, s that n whch the margnal home owner s employed. We consder the two cases separately. We wll see that whch case obtans depends on the magntude of the wage d erental between ctes and s re ected n the relatve lqudty of the housng markets n the two ctes. 9 In the gure, 1 and 2 are asymptotc values below whch the matchng rates n each respectve cty cannot feasbly fall. 10

12 Fgure 1: The AM Curve 3.1 Case I: The margnal home-owner n Cty 1 s unemployed. De ne and W H respectvely as the probabltes wth whch unemployed and employed home owners n Cty move f they receve a job o er n the other cty wthn a unt of tme. Alternatvely, we may thnk of these as the fractons of these households that accept such o ers condtonal on recevng one. Case I equlbra are those n whch 1 2 (0; 1]; W H 1 = 0; and 2 2 (0; 1]: (28) That s, equlbra n whch unemployed home-owners n both ctes accept job o ers requrng relocaton wth some probablty, but employed home owners n the low-wage cty declne to relocate wth probablty one. The steady state ow of households between states n a Case I equlbrum s descrbed 11

13 by (9), (26) and the followng equatons: ( + + )N1 W R = N1 UR + N2 UR N2 (29) N 1 = N W H 1 + N UR 1 (30) N1 W H = N1 W R + N1 (31) ( + ) N2 W R = N2 UR + N1 UR + N1 W R + 1 N1 (32) N 2 = N2 W H + N2 UR (33) N W H 2 = N W R 2 + N 2 : (34) Equaton (29) says that the measure of agents that cease beng employed renters n Cty 1 (by losng ther job, acceptng an o er n cty 2, or buyng a house) equals the measure that become employed renters n that cty (unemployed renters n ether cty who receve o ers n Cty 1 or unemployed home-owners n Cty 2 that receve and accept Cty 1 job o ers). Smlarly, (30) equates the ows nto and out of beng an unemployed home owner n Cty 1, and (31) does the same for employed home owners n that cty. Equatons (32)-(34) represent the analogous condtons for cty Wthn Case I, there are three dstnct possbltes. In what we refer to as the nteror sub-case, a fracton of unemployed homeowners n each cty accept job o ers n the other cty: 1 < 1 and 2 < 1. There are also two cases that we refer to as corners: In Corner X all unemployed homeowners n the low-wage cty accept jobs n the hgh-wage cty: 1 = 1 and 2 < 1. In Corner Y all unemployed home owners n the hgh-wage cty accept jobs n the low-wage cty: 1 < 1 and 2 = 1. Consder Corner X, n whch an unemployed home owner n the low-wage cty accepts a hgh-wage o er wth probablty one. Ths mples that the measure of houses for sale n Cty 1 s at ts hghest wthn Case I (as t could only be hgher f employed home owners began sellng houses n order to relocate.e. n Case II). Thus, the matchng rate for sellers n Cty 1 reaches a mnmum value that we denote X 1. Smlarly, when all unemployed home owners n the hgh-wage cty relocate to the low-wage cty whenever possble (n Corner Y ), the matchng rate n Cty 2 reaches a mnmum value that we denote Y 2. It follows that n the nteror sub-case, the matchng rates for home buyers n each cty must exceed these crtcal levels,.e. 1 > X 1 and 2 > Y 2. Also, n each corner, when the matchng rate for ether cty hts ts crtcal level, the matchng rate n the other cty (whch we denote X 2 Y 1 ) s determned by the AM curve, (27). 10 The asymmetry between (29) and (32) stems from (14). or 12

14 We begn our analyss wth the nteror sub-case. In order an unemployed home owner to leave the low-wage cty for a job n the hgh-wage one wth nteror probablty ( 1 2 (0; 1)), t must be that they are nd erent between the two stuatons. That s W R 2 = U H 1 p 1 : (35) Smlarly, we requre W R 1 = U H 2 p 2 (36) f unemployed home owners n the hgh-wage cty accept jobs n the low-wage cty wth nteror probablty ( 2 2 (0; 1)). Home owners Bellman equatons n ths case are gven by W1 H = w 1 + H + + U1 H W1 H U1 H = z + H + + W1 H U1 H + W2 R + p 1 U1 H W2 H = w 2 + H + + U2 H W1 H U2 H = z + H + + W2 H U2 H + W1 R + p 2 U2 H Combnng these wth (22), (23) and (25) we can derve expressons for the relatonshp between the net surplus from beng an unemployed homeowner n each cty and the value of beng an unemployed renter, U R : (37) (38) (39) (40) U H 1 p 1 = I U1( 1 ; :) + I U1( 1 ; :)U R (41) U H 2 p 2 = I U2( 2 ; :) + I U2( 2 ; :)U R : (42) Here I U1 > 0, I U2 > 0, I U1 2 (0; 1) and I U2 2 (0; 1) are functons of the underlyng parameters of the model (the actual expressons are gven the appendx). The s and s are all decreasng n the values of the applcable, re ectng the fact that n each cty the house sale prce s ncreasng n the matchng rate. Fgure 2 depcts the (lnear) relatonshps represented n (22), (23), (41) and (42). In order for (1) unemployed home-owners to be nd erent wth regard to relocatng and (2) unemployed renters to be nd erent between ctes, 1 and 2 must be such that (22) and (41) ntersect at the same value of U R as do (23) and (42). If, for nstance, 1 were to ncrease, (41) would shft down. Ths would mply that U R 1 < U R 2 nducng unemployed renters to move. Snce nd erence s requred n equlbrum, n ths case 2 would have to ncrease also. 13

15 Fgure 2: Determnaton of demand-sde relatonshp between 1 and 2 The mpled postve relatonshp between 1 and 2 n an nteror Case I equlbrum s n fact lnear (see appendx) and may be wrtten: 2 = I + I 1 (43) where I and I are postve constants. An ncrease n 1 rases house prces n Cty 1, thereby lowerng the cost of relocaton to unemployed home-owners n that cty. To mantan nd erence, (35), ths must be o set by an ncrease n the rental rate n Cty 2. To mantan the nd erence of unemployed renters, (16), ths must n turn be matched by an ncrease n the rental rate n Cty 1. As a result, mgraton from Cty 2 declnes and the consequent declne n houses for sale pushes up 2. Fgure 2 depcts ths relatonshp (labeled VVI) together wth AM and llustrates an nteror Case I statonary equlbrum. We next consder the two corner sub-cases, and these are depcted n Fgure 4. In Corner Y, unemployed home owners n the hgh-wage cty accept jobs (and ntally rent) n the low-wage cty wth probablty one. In ths case, 2 = Y 2 and 1 = Y 1. Dagrammatcally, we can magne movng toward ths case when the w 1 rses toward w 2 (.e. the d erental lessens). In ths case VVI shfts downward and to the rght (see Fgure 4). Intutvely, as the d erence between wages n the two ctes lessens, unemployed home owners n the hgh-wage cty have less ncentve to wat for a hgh-wage job and are thus more lkely to accept an o er of employment n the low-wage cty. Ths ncreases the number of homes for sale n the hgh-wage cty and lowers 2 relatve to 1. The corner occurs at any relatve wage such 14

16 Fgure 3: Case I Interor Equlbrum that VVI les to the rght of Y. Corner X occurs when all unemployed home owners n the low-wage cty accept job o ers n the hgh-wage cty. In ths case 1 = X 1, and 2 = X 2. We approach ths corner as w 1 falls relatve to w 2 (and the wage d erental wdens). In ths case t s more attractve for home owners n the low-wage cty to become renters n the hgh-wage cty, despte not ownng a home and payng hgher rent. Ths results n VVI shftng upward and to the left, ncreasng the number of homes for sale n the low-wage cty and lowerng 1 relatve to 2. The corner occurs when VVI les above and to the rght of X. At ths from ths pont, further ncreases n the wage d erental eventually result n the margnal low-wage cty home owner beng employed.e. n Case II. 3.2 Case II: The margnal home-owner n Cty 1 s employed Case II equlbra are those n whch 1 = 1; W H 1 2 (0; 1]; and 2 2 (0; 1): (44) 15

17 Fgure 4: Case I Corner Equlbra That s, those n whch job o ers n the hgh-wage cty are accepted by home owners n the low-wage cty wth probablty one f they are unemployed and wth strctly postve probablty even f they are employed. We now have two sub-cases: 11 (1) The nteror subcase, n whch employed home owners n the low-wage cty reject hgh-wage job o ers wth postve probablty ( W H 1 2 (0; 1)) and (2) Corner Z n whch they never do. Our analyss of these cases s essentally parallel to that presented n secton 3.1. In Corner Z, the measure of houses for sale n Cty 1 reaches ts absolute maxmum (because all home owners relocate f they get the chance), and ths mples a lower bound on 1 whch we denote Z 1. At the other extreme (for Case II), n Corner X, no employed Cty 1 owners relocate, so that 1 < X 1. Fnally, snce n all Case II equlbra unemployed home owners n Cty 2 reject low-wage o ers wth postve probablty, house sales n Cty 2 are below ther maxmum so that 2 > Y 2 and 1 < Y 1. Thus, n a Case II equlbrum the matchng rate for sellers n the low-wage Cty must satsfy Z 1 < 1 < mn X 1 ; Y 1 11 Note, however, that Corner case X may also be consdered a sub-case of Case II. : (45) 16

18 As before we begn wth the nteror sub-case. As unemployed home owners n the hghwage cty are nd erent wth regard to relocaton f they receve a low-wage o er, (36) contnues to hold as n Case I. Now, however, t s employed home owners, rather than unemployed ones, n Cty 1 who are nd erent wth regard to acceptng a job n Cty 2. Thus, (35) s replaced by W1 H p 1 = W2 R : (46) The Bellman equatons for home owners n Cty 2 and for unemployed owners n Cty 1 reman the same as n Case I. That for employed owners n Cty 1, however, s now gven by W H 1 = w 1 + H + + U H 1 W H 1 + W R 2 + p 1 W H 1 : (47) Followng a smlar procedure as n Case I, one can derve an expresson the net value of beng a employed home owner n Cty 1: W H 1 p 1 = II W 1( 1 ) + II W 1( 1 )U R (48) As before II W 1 > 0 and II W 1 2 (0; 1) (see appendx) are decreasng n 1. To characterze the relatonshp between 1 and 2 n ths equlbrum, we can replace (41) wth (48). As n Case I, we may agan derve a lnear relatonshp between 1 and 2 gven by 2 = II + II 1 (49) where II and II (see appendx) are constants whch depend on labour market condtons. Fgure 5 depcts ths relatonshp (labeled VVII) together wth AM and llustrates Case II statonary equlbra. Note that because II < I and II < I, VVII always les to the rght of VVI. We now turn to Corner Z, n whch home owners n the low-wage cty who receve a job o er n the hgh-wage cty accept t wth probablty one regardless of ther employment status. In ths case 1 = Z 1 and 2 s determned by the AM curve, (27). Intutvely, as w 2 rses relatve to w 1, employed home owners n the low wage cty accept job o ers n the hgh-wage cty wth hgher probablty, ncreasng the number of houses for sale n the low-wage cty and drvng 1 down relatve to 2. Dagrammatcally, the corner occurs when VVII has shfted all the way to the left of AM (See Fgure 5). 17

19 Fgure 5: Case II Equlbra 3.3 Exstence and Unqueness We now establsh the exstence of a unque statonary equlbrum wthn the class that we consder and have descrbed above. As a prelmnary, we rst dentfy a restrcton on parameters su cent to ensure that n any statonary equlbrum the surplus s postve n any match between an REM and a potental home buyer: 12 Proposton 1. If > 1 X 1 then n a statonary equlbrum, both employed and unemployed renters buy houses when they get the chance. That s, (15) holds. We have our man exstence result, whch s proved n the appendx: Proposton 2. Subject to parameter restrctons (see appendx), there exsts a unque statonary equlbrum. Note that statonarty of the equlbrum mples 12 Note that ths restrcton s requred only to ensure that n Corner X unemployed renters wsh to buy houses. In our analyss of calbrated examples below, t s never bndng. 18

20 Corollary 1. In a statonary equlbrum, house sales take place n both ctes. That s, 1 > 0 and 2 > 0: (50) 4 Moblty, Home Ownershp and Unemployment n the Basc Model Implct n our dscusson of the varous equlbrum cases above s the asserton that the rate at whch REM s match wth home buyers s hghest n the hgh wage cty. We now demonstrate ths formally: Proposton 3. If the wage d erental across ctes, w 2 w 1, s su cently hgh, then the matchng rate s hghest n the the hgh-wage cty: 2 > 1. Below, we wll show n examples that the wage d erental may ndeed be very small. In equlbrum, the home ownershp rate n Cty can be expressed as a functon of the matchng rate: h ( ) = : (51) 1 + H R The rate of home ownershp s ncreasng n the rato of the stock of owned to rental housng. Moreover, snce h s ncreasng n t follows that Corollary 2. If the rato of the stocks of owned and rental houses s the same n both ctes, then home ownershp s greatest n the hgh-wage cty: h 2 > h 1. H R Snce home owners and renters receve o ers and are separated from jobs at the same rates, they d er only wth regard to the lkelhood wth whch they accept o ers. As only home owners turn down jobs n equlbrum, the followng result s not surprsng: Proposton 4. The unemployment rate among homeowners exceeds that among renters. Ths result s consstent wth the emprcal ndngs of Coulson and Fsher (2008) who, when controllng for demographc and locatonal d erences between home owners and renters estmate a hgher lkelhood of unemployment for U.S. home owners. Because households n our model are ex ante dentcal, Proposton 4 s not n con ct wth the observaton that n the data unemployment s hgher among all renters than among all home owners. The followng proposton characterzes the tendency of unemployed renters to move to the low-wage cty: 19

21 Proposton 5. There exsts 2 (0; 1) such that f R 1 =R 2 >, then the fracton of renters who are employed s greatest n the hgh wage cty: 2 > 1. In any statonary equlbrum, the majorty of households (all renters and some homeowners) resdent n the low-wage cty 1 that receve hgh-wage job o ers move to accept them. In contrast, only unemployed renters and some fracton unemployed home owners resdent n the hgh-wage cty mgrate to the low-wage cty to accept a job o er. Ths asymmetry tends to drve up the rental rate n the hgh-wage cty relatve to that n the low-wage cty. Ths n turn may nduce unemployed households wth no job o er to reman n or move to the low-wage (and low rent) cty. Consequently, the proporton of renters who are unemployed tends to be hgher n the low-wage cty. Proposton 5 shows that ths s true unless rental housng n the low-wage cty s su cently scarce. At the cty level, then the unemployment rate re ects a trade o between two e ects. The unemployment rate n cty can convenently be expressed as + + = h + 1 (1 h ) + + {z } {z } home ownershp eect rent derental eect where h s gven by (51). The rst term re ects the postve mpact of home-ownershp to the cty s unemployment rate due to the fact that some home owners turn down job o ers rather than relocate. The second term re ects the fact that there s typcally a hgher concentraton of unemployed renters (represented n (52) by ) n the cty wth the lower rental rate. The home ownershp e ect s typcally larger n the hgh-wage cty, as home ownershp s hgher there. In contrast, the rent d erental e ect s typcally hgher n the low-wage cty, as unemployed households to some extent ow there n order to take advantage of relatvely low rent. Overall the relatonshp between unemployment and home ownershp at the cty level depends on whch of these e ects domnate. The aggregate unemployment rate, ; n Case I s gven by = ( + ) ( + + ) ( + ) ( ) + ( + ) + ( + + ) ( + ) ( ) (52) h (53) where h = 1 R 1 R 2 s the aggregate rate of home-ownershp. Thus t may be seen that Proposton 6. Aggregate unemployment s monotoncally ncreasng n the economy wde aggregate home ownershp rate. 20

22 Home ownershp thus contrbutes to aggregate unemployment. Whether or not ths e ect s quanttatvely sgn cant depends on parameters. We take up ths ssue next, usng a parameterzed verson of the basc model. 4.1 A Baselne Example We consder a parameterzed verson of the model n order to llustrate the characterstcs of a typcal equlbrum. As a baselne example, we choose parameters so that n the statonary equlbrum, our economy s consstent wth several observed aspects of the U.S. economy. The parameter values and the relevant targets are gven n Table 1. We base our calbraton on monthly data and, where possble, draw estmates from the lterature whch re ect that frequency. In partcular, target values for the dscount rate, the hrng rate and the separaton rate are taken from Shmer (2005), as are values for the ncome replacement rate and the unemployment rate. Target values for the home ownershp rate and the vacancy rate for owner occuped homes, whch determne the total measure of rental and owner-occuped housng per capta, are taken from the most recent US Census. The d erence between the ow utltes from owned and rented housng s chosen so that the average rent-mortgage d erental n both ctes s negatve as suggested by the estmates of Campbell, Davs, Galln, and Martna (2008). Table 1 Parameter Choces Parameter Value Target Value Source Annual dscount factor Shmer (2005) Monthly separaton rate Shmer (2005) z=w Income replacement rate 0.4 Shmer (2005) w 2 =w Dense/non-dense metro premum 0.1 Glaeser & Mare (2001) R 1 + R Home-ownershp rate 0.68 US Census H 1 + H Homeowner vacancy rate US Census H R :43 0:0145 0: = ; 8 < : Monthly hrng rate Unemployment rate Ann. moblty (countes) 0:44 0:057 0:06 Shmer(2005) US post-war average US Census We assume that the two ctes contan equal housng stocks of each type and that the wage n Cty 2 s 10% hgher than the wage n Cty 1. Our results n ths subsecton are not 21

23 hghly senstve to the partcular values of these parameters. They were chosen, however, based on a useful class caton of US ctes dscussed by Overman and Ioanndes (1999), who appled earler work by Knox (1994). There U.S. ctes are grouped nto four ters: The top ter conssts of 10 nodal centers 13, the second ter conssts of 14 regonal centers 14 and the thrd ter conssts of 19 sub-regonal centers. 15 The remanng 291 ctes are allocated to the fourth ter. For calbraton purposes we take Cty 2 to represent the top 3 ters and Cty 1 to represent the bottom ter. The total populaton of the top three ters s approxmately equal to that of the fourth ter and ths dstrbuton s farly stable over tme. Glaeser and Mare (2001, Table 3) estmate a dense metropoltan wage premum for ctes wth more than 500,000 nhabtants of 0.24 log ponts and a non-dense metropoltan premum of 0.14 log ponts. We use the d erence between these as our estmate of the wage premum between our two ctes. An annual moblty rate (the % of the populaton that change address n a gven year) may be found n the US Census. Although more than 15% of the US populaton change addresses each year, ths ncludes people who move short dstances wthn a county. For our purposes, a more approprate estmate of moblty s that between labour markets. We therefore use as a target the component of the moblty rate assocated wth people who move between countes, whch s roughly 6%. 16 We choose jontly values of ; and to match target values for the monthly hrng rate, the unemployment rate and the cross-county annual moblty rate Labour Market Implcatons For our baselne calbraton the statonary equlbrum s Case I, nteror. Table 2 descrbes the dstrbuton of the total populaton over the eght possble household states. Cty 2 has a larger populaton (slghtly) ncludng more employed renters, employed owners and unemployed owners. Cty 1 has substantally more unemployed renters, re ectng ther ncentve to move to the cty wth the lower rental rate. 13 These are Atlanta, Chcago, Denver, Houston, Los Angeles, New York Cty, Mam, San Francsco, Seattle and Washngton D.C. 14 Baltmore, Boston, Cncnnat, Cleveland, Columbus, Dallas, Indanapols, Kansas Cty MO, Mnneapols, New Orleans, Phladelpha, Phoenx, Portland OR, and St. Lous. 15 Brmngham, Charlotte, Des Mones, Detrot, Hartford, Jackson MS, Lttle Rock, Memphs, Mlwaukee, Moble, Nashvlle, Oklahoma Cty, Omaha, Pttsburgh, Rchmond, Salt Lake Cty, Shreveport, Syracuse and Tampa. 16 Ths s lkely to be an upper bound on moblty. 22

24 Table 2: Allocaton of workers by job and housng status Renter Owner Renter Owner Employed Unemployed The rst column of Table 3 contans statstcs on moblty and unemployment for ths baselne example. Although the parameters have been chosen to match average moblty, the relatve moblty of owners and renters s endogenous. Accordng to the US Census, the uncondtonal moblty rate for renters averages around 10% and for owners t s 2%. Thus, our model overstates somewhat the moblty of renters and understates the moblty of owners. Our economy, however, s populated by ex ante dentcal households, and t s possble that the census gures could understate the d erence between condtonal moblty rates, for a varety of reasons. 17 We have also gnored drect movng costs and these are lkely to a ect moblty. The low-wage cty has a hgher unemployment rate, due entrely to a hgh rate of joblessness among renters. The overall unemployment rate among owners s s only slghtly lower than that for renters, n spte of the fact that unemployed owners turn down opportuntes to relocate at a sgn cant rate (about 30%). Ths s not surprsng as gven observed average moblty, the rate at whch households receve o ers from outsde ther cty of resdence s very low. Thus, although the moblty of renters s much larger than owners, the consequence for ther relatve unemployment rates s very small. 18 Recallng (52), the home ownershp e ect on unemployment s both small and e ectvely equal across ctes. In contrast, the rent d erental e ect s very large (.e. 1 s much larger than 2 ). Overall, ths results n sgn cantly hgher unemployment n the low-wage cty. Thus, as observed by Coulson and Fsher (2008), our model mples a negatve relatonshp between unemployment and homeownershp across ctes (as well as a postve one between wages and homeownershp). 17 For example, t could be that more educated workers are both more lkely to own and to move than less educated ones. 18 Interestngly, however, the mpled ncrease n the lkelhood of unemployed assocated wth home ownershp of the same order of magntude as that estmated by Coulson and Fsher (2008). 23

25 Table 3: Labour Market Statstcs Baselne Rental Asymmetrc Europe Intracty D erental only Ctes Relocaton Matchng Moblty rate of renters of owners Populaton rato Unemployment rate n low-wage cty n hgh-wage cty amongst renters amongst owners Rejecton rate Per capta GDP relatve to baselne Housng Prema Consder a settng n whch there are no frctons n the housng market; that s, n whch households are free to ether rent or own at any tme. In ths case, the followng no arbtrage condton relates the rental rate n cty to the ow cost of ownng a house n that cty: p r = H R + _p (54) We de ne the housng premum n cty, x, as the devaton from (54) due to frctons n the housng markets, measured relatve to the purchase prce of a house: x = r + H R p p : (55) We express housng prema relatve to the prce level, so as to enable comparson to those calculated by Campbell, Davs, Galln, and Martn (2008). These authors estmate qualty adjusted prema for US ctes that vary between 1.84% and 6.45% and average 2.99%. Table 4 presents the values for rents, house prces and housng prema for our baselne example. Whle n our equlbrum housng prema are small relatve to ther estmates, that for Cty 1 s certanly of the rght order of magntude. Overall, our results suggest that a sgn cant fracton of the observed housng prema may be accounted for by the llqudty of housng. 24

26 Table 4: Housng Market Statstcs Baselne Asymmetrc Europe Intracty D erental Ctes Relocaton Matchng Rent (relatve to wage) Prce (rel. to ann. wage) Low Rent mortgage d erental wage Annual Housng Premum 1.6% 4.4% 2.7% 1.4% 2.2% cty Matchng rate Ownershp rate 67% 66.6% 67.8% 67.7% 67.7% Vacancy Rate (quarterly) 3.5% 7.9% 3.2% 3.2% 3.6% Rent (relatve to wage) Prce (rel. to ann. wage) Hgh Rent-mortgage d erental wage Annual Housng premum 0.2% 0.3% 0.7% 0.3% 0.2% cty Matchng Rate Ownershp rate 69% 68.3% 68.3% 68.3% 68.3% Vacancy Rate (quarterly) 0.5% 0.5% 1.0% 0.6% 0.5% The exstence of postve housng premum n our equlbrum s ndcatve of a possble pro t opportunty assocated wth the converson of rental housng to owner-occuped unts. Snce n both ctes the rent-mortgage d erental, r p, s negatve t s not pro table n equlbrum to convert ownable housng to rental, rrespectve of the converson cost. 19 It would, however, be pro table to convert rental unts to ownable housng unless the cost of converson s su cently hgh,.e. C H > p r =. Usng the rent-mortgage d erental n each cty, we can then determne the mnmum converson cost necessary to support the equlbrum of our baselne economy: p r = 0:048 0:047 = 1:02. That s, a (one-tme) converson cost approxmately equal to the average monthly wage s su cent. Another possblty s for REMs to put rented houses on the market for sale, and then convert them to owner-occuped houses (and pay the converson cost) only once they have matched wth a buyer (ths possblty has been excluded up to now) Ths s true whenever H R su cently large. 20 We exclude the possblty of the REM sellng the house mmedately to the current renter. The ow value of 25

27 rentng out a for sale house n Cty s V H (r) = r + (q C H V H (r)) where q s the expected prce at whch the house wll sell once a match s made. It follows that V H (r) = r + (q C H ) + (56) In a statonary equlbrum, the value of an unrented vacant house stats es V H = q + (57) It follows that the REM wll not rent temporarly as long as V H > V H (r) or f C H > r =. In our baselne example, a converson cost approxmately one and a half tmes the the average monthly wage s su cent to prevent REM s from puttng rented houses up for sale. 4.2 Alternatve Parameterzatons Impact of Aggregate Home Ownershp Qualtatvely, as noted above, home ownershp contrbutes to unemployment. Quanttatvely, however, the e ect s typcally tny. For example, n our baselne calbraton a 10 percentage pont ncrement n the aggregate rate of home ownershp results n an ncrease n the unemployment rate of only 0.04 percentage ponts. 21 Ths s the case despte the fact that the model does mply large d erences n moblty between renters and owners. The hgh relatve moblty of renters n our economy s drven by employed renters movng from the low-wage cty to the hgh-wage one. 22 do not move. In the equlbrum we consder, employed owners It s useful to compare the baselne example wth one n whch all housng s rental. That s, suppose that the number of rental unts n each cty equals the total housng stock n the baselne example. In ths case, there s excess supply of rental unts and all the slack n ths ths market arses n the low wage cty. Consequently, the rental rate n cty 1 falls to ts lowest possble value whch, n the absence of mantenance costs, s zero: r 1 = 0. In equlbrum, the unemployed are nd erent between locatons: U R 1 = U R 2 = U R. Consequently, they wll always accept o ers of employment n ether cty. As before, the 21 Ths e ect s thus much smaller than that suggested by some commentators. For example, based on cross country and cross-regonal correlatons, Oswald (1999) suggests that a 10 percentage pont ncrease n home ownershp s assocated wth a 1.3 percentage pont ncrease n unemployment. 22 Whle the fracton of unemployed renters that move (to the low-wage cty from the hgh-wage one) exceeds that of employed renters, there are many more employed renters. 26

28 employed n Cty 1 wll accept o ers from Cty 2 but not vce versa. The ows of workers n the statonary equlbrum therefore satsfy the followng condtons: N W R 1 + N UR 1 = 1 R 2 < R 1 N W R 2 + N UR 2 = R 2 ( + )N1 W R = N1 UR + N2 UR N2 W R = N2 UR + N1 UR + N1 W R Snce r 1 = 0; the ow value of beng an unemployed renter s smply and the rent n cty 2 s gven by r 2 = U R = 2 + R (w 2 w 1 ) : The second column n Table 3 presents labour market statstcs for an example wth rental housng only n whch all other parameters reman at ther baselne values. As can be seen, average moblty ncreases substantally (from 0.06 to 0.16). Unemployment has now shfted consderably to the low wage cty, largely re ectng the ow of unemployed households to the locaton where rents are low. Nevertheless, the consequence for aggregate unemployment s very small, amountng to only a 0.1 percentage pont decrease. Moreover, the consequence of ths change for per capta GDP s also tny Asymmetrc Ctes In our baselne example we assumed that the housng stocks n the two ctes are the same. The thrd column of Table 3 documents the mplcatons of allowng the housng stock n the hgh wage cty to be four tmes the sze of that n the low wage cty, whle keepng the total stock of houses the same. The average moblty rate falls substantally as a result of ths because most of the populaton are employed n the hgh wage cty and are less lkely to leave than those n the low wage cty. Indeed, t s the reducton n the average moblty rate of renters that drves most of ths e ect. Whle asymmetry of ths type reduces the d erence n unemployment rates across ctes, t has no consequence for the d erence between the rates of unemployment for renters and owners. It does, however, have sgn cant consequences for the housng markets (Table 4). Although rents and prces fall n both ctes, they fall by much more n the low wage cty whch also experences a much hgher vacancy rate. 27

29 4.2.3 A Hgh Unemployment / Low-Moblty, European Example Moblty n European countres tends to be much lower than for the US. At the same tme, worker ow rates n European labour markets are markedly d erent from those observed for the US wth lower separaton rates and much lower hrng rates as well. At the same tme, European unemployment rates tend to be hgher than that of the US. To capture these features of a typcal European economy, we adjust the separaton, job o er, and housng market matchng rates (see Table 5) holdng the other parameters at ther values n the baselne example. 23 As n the baselne, the resultng equlbrum s Case I nteror. Labour market statstcs are gven n the fourth column of Table 3. In ths case, 45% of home-owners turn down outsde o ers. As a result, the mpact of home-ownershp on the lkelhood of unemployment s double that of the baselne (an ncrease of 3.1 % rather than 1.4% ). The negatve crosscty relatonshp between unemployment and ownershp s also hgher: Now a 1 percentage pont hgher ownershp rate s assocated wth 2.7 percentage pont drop n unemployment. Fnally, the aggregate e ect of home-ownershp s more than twce as bg as n the baselne example, wth a 10% pont hgher ownershp rate resultng n a 0.1 percentage pont hgher unemployment rate. Ths aggregate e ect s stll, however, quanttatvely small. Table 5: European Parameter Choces Parameter Value Target Value = 0: Monthly separaton rate = = 0:097 = < Monthly hrng rate = 0:10 = 0:005 Unemployment rate = 0:10 ; : = 0:0006 Annual moblty rate = 0:02 5 Generalzatons of the Basc Model In ths secton we consder a number of generalzatons of the basc model so as to demonstrate the robustness of our man results. 23 Ths exercse s meant to be llustratve only, we do not vew t as a calbraton to any partcular economy. 28

30 5.1 Intra cty Relocaton In the basc model, we abstract from housng transactons amongst households who do not mgrate, but reman wthn a cty. Snce all owner-occuped houses wthn a cty are dentcal, there s no reason for a home owner to sell one house n order to move to another. The exstence of ntra-cty relocaton may, however, be mportant for nter-cty mgraton, as t a ects the lqudty of housng. Moreover, most actual relocaton s wthn rather than between ctes (although only a small fracton of ntra-cty moves are job-related). 24 We now consder an extenson of the model along the lnes of Wheaton (1990) whch allows for moves wthn a cty. We show that whle ths results n the housng markets beng more lqud overall, t does not substantally a ect any of our man results. Followng Wheaton (1990), we assume home-owners experence housng taste shocks at rate. On experencng a shock, the servce ow a home owner receves from ther current house falls permanently to H ", whle that potentally avalable to them from other houses remans H. 25 All such msmatched owners mmedately become potental buyers and start searchng for a new house usng the same matchng technology as renters. Once they nd a new house, they mmedately sell ther old house to an REM at the market prce. 26 The REM sells them the new house at a prce whch extracts all of the surplus from the trade: q W H = W H ~W H + p q = U H ~U H + p = 1; 2: (58) where ~ W H and ~ U H denote the values of beng a msmatched owner who s employed and unemployed, respectvely. Let N ~ W H and N ~ denote the stocks of msmatched employed and unemployed owners, respectvely, n cty. Snce the stock of potental buyers now ncludes msmatched owners as well as renters, t follows that the matchng rate n cty s housng market s gven by R + N ~ W H + ~ N ~ = = 1; 2: (59) H N W H N ~N W H The sale prce of a house n Cty now sats es p = ~ ~N (q W R p ) + (1 )(q UR p ) + (1 ) (q W H p ) + q 24 Rupert and Wasmer (2008) document some of these facts. 25 Ths shock could represent a change n tastes or an addtonal chld beng born, etc. 26 As before, ths s equvalent to havng the seller hold the house vacant untl t sells as n Wheaton (1990), but lmts the number of states we must consder. 29 p (60)

31 where = R = R + N ~ W H + N ~ denotes the fracton of buyers that are renters and = N ~ W H = ~N W H + N ~ the fracton of msmatched owners that are employed. We focus on an equlbrum smlar to Case I above. In partcular, parameters are such that the margnal owner n each cty s unemployed and sats ed wth ther current house. 27 To llustrate the mplcatons of ths generalzaton, we agan consder a numercal example. Relatve to the baselne example, we ntroduce two new parameters: " and and retan the baselne values for all others. We set = 0:0015 so that n the statonary equlbrum the fracton of movng owners who reman wthn the same cty (rather than changng ctes) s roughly 60%. Ths corresponds to the fracton of owners who move but reman wthn a county n the US census. It mples that 2% of owners become dssats ed wth ther current house each year. We set " = 0:005, half of the d erence between H and R. 28 As can be seen n Tables 3 and 4 very lttle changes n ths extenson relatve to the baselne economy. In partcular, whle the matchng rates ncrease substantally and house prces n both ctes ncrease re ectng ths, the labour market statstcs are largely unchanged. 5.2 D erental Matchng Rates In the basc model, unemployed renters match wth house-sellers at the same rate,, as employed renters. Snce unemployed owners are nd erent between remanng n one locaton and movng to accept an o er n the other, unemployed renters are always wllng to buy a house f the prce s su cently low. There are, however, a varety of reasons why the unemployed may not be able to buy houses as easly as the employed (e.g. the unwllngness of banks to provde them wth mortgages). Here we generalze our model to capture ths by assumng d erent matchng rates for employed and unemployed renters: w respectvely, where w > u. and u Ths extenson changes the model very lttle and has mnmal e ect on the equlbrum. The probablty that the home buyer n a match s an employed renter s now gven by ^ = w N W R w N W R + u N UR 27 For some parameter con guratons, the margnal home owner could be one dssats ed wth ther current match. Thus, ths extenson of the envronment ntroduce several addtonal equlbrum cases. A full analyss of all the possble cases s omtted for brevty. 28 Our results are largely nsenstve to the exact value of ", provded t s less that H R. (61) 30

32 and the seller s matchng rate s now ^ = w N W R H N W H + u N UR N : (62) The extenson s straghtforward and the qualtatve results from the basc model are essentally una ected. 29 As may be seen n the last columns of Tables 3 and 4, the quanttatve e ects of ths extenson are also small, even f u = 0. There s a small ncrease n the steady state measure of unemployed renters. But because these make up only a small fracton of the populaton, ths has no sgn cant quanttatve e ect. 5.3 Rental Vacances In the basc model, we abstract entrely from frctons n the rental market. As a result, all rental unts are occuped and there are no vacances unless the rental market s slack (and r = 0). In realty, vacancy rates for rental unts are often hgher than for owner-occuped unts, so one may wonder whether ths would a ect the nature of our results. The key ssue, however, s whether vacances n the rental market are assocated wth costs to households of movng and therefore a ect moblty. Vacances n the rental market are more lkely to be symptomatc of the fact that, once a rental unt s vacated, t may not mmedately be avalable to the rental market. For example, mantenance and decoratng may be needed before t s ready to be rented agan. Here we show that t s straghtforward to accommodate rental vacances n the model wthout changng any of our results. We assume that, once t s vacated, a rental unt can only be returned to the market at an exogenous rate. The man consequence of ths frcton s that the e ectve rental stock s less than the actual stock of rental unts, wth the d erence consstng of rental vacances. Spec cally, one can show that n the steady-state and N W R 1 + N UR 1 = R = ^R 1 (63) N2 W R + N2 UR + = R 2 R = ^R 2 : (64) where ^R represents the rental housng that s avalable for rent n Cty. We can therefore smply replace the actual rental stocks, R 1 and R 2 ; wth ^R 1 and ^R 2 respectvely, throughout the analyss. Note nally that f R 1 = R 2, (64) and (63) are symmetrc. 29 Calculatons may be obtaned from the authors upon request. 31

33 6 Concludng Remarks We have developed a two-cty model that allows for nteractons between search frctons n both housng and labour markets. Housng lqudty the tme t takes to sell a house to an approprate buyer determnes the value that the seller can get for the house n the event that he/she wshes to move. Ths determnes the d erent ctes populatons and rates of home-ownershp. These, n turn, determne vacancy rates and, hence, the lqudty of housng n each cty. We show that n equlbrum, homeowners are substantally less moble than renters even though there are no drect barrers or costs to movng. Homeowners turn down job o ers n certan crcumstances, even f they are currently unemployed or are o ered a hgher wage than ther current one, because the prce they can get for ther house s nsu cent to make mgraton worthwhle. In partcular, the lkelhood of unemployment for homeowners exceeds that for otherwse dentcal renters. In contrast, unemployment s negatvely related to ownershp rates across ctes because unemployed renters tend to move dsproportonately to the low rent (low wage) cty, where home ownershp s also lower. A baselne verson of the model, calbrated to match US labour market ows and average moblty, generates relatve moblty rates and unemployment rates for homeowners and renters that accord reasonably well wth the evdence. Moreover, we nd that unemployment s negatvely related to ownershp rates across ctes. Despte large d erences n overall moblty rates between owners and renters, however, we nd that the mpact of ownershp on aggregate unemployment s very small. In a low-moblty calbraton, ntended to capture the lower moblty and hgher unemployment typcal of some European economes, we nd that all of these e ects are magn ed to some extent, but that the relatonshp between home ownershp and aggregate unemployment remans weak. We vew the framework developed here as a useful startng pont to study the nteractons between labour markets, housng markets and the broader economy. It can be bult upon n a number of ways that we plan to consder n future research. These nclude ntroducng varous forms of heterogenety amongst households and allowng for productvty growth, populaton growth and housng constructon. Moreover, extendng the model to ncorporate multple (.e. more than two) ctes would allow for a more exhaustve quanttatve evaluaton. Fnally, snce housng frctons are lkely to play a larger role n transtons than they do n the steadystate, an analyss of the e ects of shocks s lkely to be especally nterestng. 32

34 7 Appendx 7.1 Dervaton of Case I Statonary Equlbrum The soluton to the 10 equaton system descrbed by (9), (26), and (29) (34) can be expressed recursvely as N W R 2 = R 2 + ( + )R N W R (65) 1 = R 1 + ( + )R 2 N2 W R (66) = R N W R (67) H + N W R R (68) H + + N W R R (69) N UR N ( ) = N W H ( ) = The house prces n Cty 1 must satsfy ( ) = R N ( ) : (70) p 1 = 1 1 W1 H p 1 W1 R + (1 1 ) U1 H p 1 U R = 1 1 W1 H p 1 + (1 1 ) U1 H p U R (71) Subtractng (71) from (37) and rearrangng yelds ( ) W H 1 p 1 = w1 + H U R + ( (1 1 ) 1 ) U H 1 p 1 : Smlarly subtractng (71) from (38) and rearrangng yelds ( (1 1 ) 1 ) U H 1 p 1 = (72) 2 + H U R U R + ( 1 1 ) W H 1 p 1 (73) Solvng for W H 1 p 1 and U H 1 p 1 yelds W H 1 p 1 = I W 1 + I W 1U R (74) U H 1 p 1 = I U1 + I U1U R (75) 33

35 where I W 1 = ( ) w 1 + H ( (1 1 ) 1 ) (w 1 2 ( ) + ( + 1 ) ( + + ) 2) (76) I ( + + ) W 1 = 1 + ( ) ( ) + ( + 1 ) ( + + ) I U1 = ( + + ) w 1 + H ( ) (w 1 2 2) ( ) + ( + 1 ) ( + + ) I ( + + ) U1 = 1 + ( ) ( ) + ( + 1 ) ( + + ) Thus, followng the same procedure as for cty 1 we have (77) (78) (79) W H 2 p 2 = W 2 + W 2 U R (80) U H 2 p 2 = U2 + U2 U R (81) where W 2 = ( ) w 2 + H ( (1 2 ) 2 ) (w 2 2 1) ( + 2 ) ( + + ) + ( ) (82) ( + + ) W 2 = 2 + ( ) (83) ( + 2 ) ( + + ) + ( ) U2 = ( 2 2 ) w 2 + H ( ) 2 + H ( + 2 ) ( + + ) + ( ) (84) U2 = ( + + ) 2 + ( ) ( + 2 ) ( + + ) + ( ) (85) Interor Case: In ths nteror soluton < 1; = 1; 2. Usng (68) and (70) ths mples that 1 > X R 1 1 = (86) H 1 (=)N1 W R [( + )=] (= )R 1 and 2 > Y R 2 2 = : (87) H 2 (=)N2 W R [( + )=] (= )R 2 Equatng (75) and (35) yelds the equlbrum value of U R as a functon of Cty 1 s matchng rate U R ( 1 ; I) = w 1 + H ( + 1 ) 2 34 ( ) (w 1 2 ) + + (88)

36 Smlarly equatng (81) and (36) yelds the equlbrum value of U R ( 2 ): U R ( 2 ) = w 2 + H ( + 2 ) 1 ( ) (w 2 2 ) + + (89) Equatng U R ( 1 ; I) = U R ( 2 ; I), yelds the postve, lnear relatonshp between 1 and 2 gven by (43) where I = w 2 w 1 + ( 2 1 ) I = (w 2 2 ) (w 1 2! ) (w 2 2 ) ++ (+)(w 2 2 ) + (+)(w 1 2) (90) (91) Corner Y ( Y = 1): In ths case 1 = Y 1 and 2 = Y 2. Substtutng nto (68) and (69) yelds the equlbrum measures of owners n each state. In ths corner case (35) contnues to hold, but (36) does not. Equatng (75) and (35) yelds U R Y 1 = w1 + H + + Y Y Y 1 (w1 2 ) (92) + + Corner X ( X = 1): In ths case 1 = X 1 and 2 = X 2. Substtutng nto (68) (69) yelds the equlbrum measures of owners n each state. In ths corner case (36) contnues to hold, but (35) does not. Equatng (81) and (36) yelds U R X 2 = w2 + H + + X X X 2 (w2 2 ) (93) Dervaton of Case II Statonary Equlbrum In the case, the steady state ow of workers across states s descrbed by ( + + )N1 W R = N1 UR + 2 N2 + N2 UR (94) ( + ) N1 = N1 W H + N1 UR (95) + W H 1 N W H 1 = N1 W R + N1 (96) ( + )N2 W R = N2 UR + N1 UR + N1 + N1 W R + W H 1 N1 W H (97) N 2 = N2 W H + N2 UR (98) + 2 N W H 2 = N W R 2 + N 2 (99) 35

37 The soluton to these ow equatons can be expressed recursvely as (66), (65), (67) and N1 ( 1 ) = H R 1 N 1 W R R 1 (100) 1 + N1 W H ( 1 ) = H R N W R + 1 R 1 (101) 1 N2 ( 2 ) = H 2 + N 2 W R R 2 (102) 2 N2 W H ( 2 ) = H N 2 W R R 2 (103) 2 W H 1 ( 1 ) = 1 N1 W H R 1 + N 1 W R H 1 + R 1 (104) 1 Followng the same procedure as n Case I, we can derve the followng expresson for the net values of ownershp n cty 1 W H 1 U H 1 p 1 = II W 1 + II W 1U R (105) p 1 = II U1 + II U1U R (106) where W 1 = ( ) w 1 + H ( ( ) ( ) (1 1 ) 1 ) (w 1 2 ) (107) II II W 1 = + 1 (108) II U1 = ( ) w 1 + H ( ) (w 1 2 ) ( ) ( ) (109) II U1 = (110) The Bellman equatons and hence the soluton for Cty 2, (80) and (81), reman the same as n Case I. Interor Soluton: In ths case W H 1 ( 1 ) < 1. Usng (101) and (104), ths mples that 1 > Z 1 = R 1 H 1 (= )R 1 : (111) 36

38 We also requre that W H 1 ( 1 ) > 0; whch mples an upper bound on 1 that s equvalent to X 1. Fnally N 2 Y mples a lower bound on 2 whch s the same as Y 2. Equatng (46) and (105) yelds U R ( 1 ; II) = w 1 + H ( + 1 ) 2 ( (1 1 ) 1 ) (w 1 z) (112) For Cty 2, U R ( 2 ) s the same as n Case I and s gven by (89). Equatng U R ( 1 ; II) = U R ( 2 ), yelds another postve, lnear relatonshp between 1 and 2 that must pertan n ths equlbrum gven by (49) where II = w 2 w 1 + ( 2 1 ) II = (w 2 z) ++! (1 1 )(w 1 z) (w 2 z) ++ It s straghtforward to show that II < I and II < I. (+)(w 2 z) ++ + (w 1 z) +++ (113) : (114) Corner Z ( Z = 1): In ths case 1 = Z 1 and 2 = Z 2. Substtutng nto (100) - (103) yelds the equlbrum measures of owners n each state. In ths corner case (36) contnues to hold, but (35) does not. Equatng (81) and (36) yelds U R Z 2 = w2 + H + + Z Z Z 2 (w2 z) + + (115) 7.3 Proofs of Man Propostons: Proof of Proposton 1: We must show that the followng nequaltes hold n each case: U1 H p 1 > U R (116) W1 H p 1 > W1 R (117) U2 H p 2 > U R (118) W2 H p 2 > W2 R (119) Case I: Inequalty (116) holds n the nteror and corner Y snce U H 1 p 1 = W R 2 > U R. 37

39 In corner X, usng (75), we can express (116) as I U1( X 1 ) + I U1( X 1 )U R ( X 1 ) > U R ( X 1 ): Usng (78) and (79) and re-arrangng we can express ths as w 1 + H + ( + ) 1 Usng (22) and (23) ths can be re-wrtten as w 1 + H + W1 R U R X ( ) + + ( 1 2 ) > U R ( X 1 ) W R 2 W R 1 and so But W1 R = w 1 + R r 1 + W1 R U R + W2 R W1 R r 1 > R H 1 X 1 ( 2 1 ) : + + > W R 1 : Thus, a su cent condton for ths to hold s that > 1 X 1, snce r 1 0 n equlbrum. Inequalty (117) must hold n all sub-cases snce W H 1 p 1 > W R 2 > W R 1. Inequalty (118) must be true n the nteror sub-case and corner X snce W R 1 corner Y, we need that Usng (84) and (85) ths can be expressed as Snce usng (23), ths can be re-wrtten as > U R. In U2 ( Y 2 ) + ( Y 2 )U R > U R (120) w 2 + H + ( + ) 2 > U R w 2 + H + W R 2 U R > W R 2 But W R 2 = w 2 + R r 2 + W R 2 U R and so we requre that r 2 > R H. Snce H R ths must be true snce the rental rate must be postve n equlbrum. Inequalty (119) must be true f (118 ) holds because W H 2 p 2 W R 2 > U H 2 p 2 U R W H 2 U H 2 > W R 2 U R = 2 To see ths note that n the nteror and corner X (U H 2 W H 2 U H 2 = w 2 z + + > 2 38 p 2 = W R 1 ) we have

40 In corner Y (U H 2 p 2 < W R 1 ) we have W H 2 U H 2 = w 2 z + + = = 2 + U H 2 p 2 U R + + W1 R + p 2 U2 H U R + p 2 U2 H + + > 2 Case II: Frst observe that n ths case t s always true that Ths follows from subtractng (38) from (47). W H 1 W R 1 = U H 1 U R (121) In the nteror sub-case, (117) must be true snce W H 1 p 1 = W R 2 > W R 1. From (121) t follows that (116) must also hold n ths case. In corner Z, usng (106), we can express (116) as II U1( Z 1 ) + II U1( Z 1 )U R ( Z 1 ) > U R ( Z 1 ) Usng (109) and (110) ths can be expressed as w 1 + H + ( + ) 1 + ( 2 1 ) > U R Usng (22) and (23) and re-arrangng, ths can be wrtten as w 1 + H + W R 1 U R + W R 2 W R 1 > W R 1 But W1 R = w 1 + R r 1 + W1 R U R + W2 R W1 R and so the condton becomes r 1 > R H, whch must be true n equlbrum. From (121) t follows that (117) must also hold n ths case. Inequalty (118) must be true n all cases snce W1 R reasonng as for Case I. Proof of Proposton 2: > U R, and (119) follows by the same Exstence: We requre that there s su cent rental housng at each locaton to ensure that unemployed renters are the margnal renters overall: R 1 > N W R 1 and R 2 > N W R 2. Usng (65) 39

41 and (66) t s straghtforward to show that a su cent condton for ths s the rato of rental housng n each cty les between two bounds: + + > R 1 R 2 > + ++ : (122) Note that provded that >, the upper bound must exceed 1 and the lower bound must be less than 1. Exstence of case I requres that X 1 < Y 1. That s + R 1 + R 2 + R 2 + N 2 W R + H 1 1 < H 1 N 1 W R + R 1 (123) whch can be re-wrtten as R R 2 + N 1 W R + N 2 W R < 1 (124) That s, the total populaton must be su cently large n comparson wth the stock of rental housng. Exstence of case II requres that Z 1 < mn X 1 ; Y 1. Note from (111) that t must be true that Z 1 < X 1. Hence, (124) s a su cent condton for both cases to exst. If (124 ) does not hold, case II may stll exst f Z 1 < Y 1. That s + R 1 + R 2 + R 2 + N 2 W R + H 1 N < H 1 R 1 whch can be wrtten as 1 + R R 2 + N 2 W R < 1 (125) We also requre that r 1 > 0 and r 2 > 0: Unqueness: Frst note that n all cases, the functon U R (; :) s monotoncally decreasng n. That s: du R ( 1 ; I) d 1 = 1 (w 1 z) + + ( ) < 0 (126) du R ( 1 ; II) d 1 = (1 1)(w 1 z) = 1 2 < 0 (127) du R ( 2 ) d 2 = 2 (w 2 z) = < 0 (128)

42 Now suppose that parameters are such that there exsts an nteror equlbrum (as n Case I), ( 1; 2) such that U R ( 1) = U R ( 2). Observe that snce VVII always les to the rght of VVI, there cannot also exst an nteror equlbrum as n Case II. Now consder corner Y, ( Y 1 ; Y 2 ): Snce U R (; I) s decreasng n Y 1 > 1 ) U R ( Y 1 ; I) < U R ( 1; I) Y 2 < 2 ) U R ( Y 2 ) > U R ( 2) and so U R ( Y 2 ) > U R ( Y 1 ; I). If ths corner case were an equlbrum then W2 R = U1 H p 1 ) U R = U R ( Y 1 ; I) and W R 1 > U H 2 p U R > U2 ( Y 2 ) + ( Y 2 )U R U R > U2( Y 2 ) 1 1 ( Y 2 ) = U R ( Y 2 ) where ( Y 2 ) < 1. Ths mples that U R ( Y 1 ; I) > U R ( Y 2 ): Hence we have a contradcton and ( Y 1 ; Y 2 ) cannot also be an equlbrum. Next consder the corner case ( X 1 ; X 2 ): Note rst that X 1 < 1 ) U R ( X 1 ; I) > U R ( 1; I) X 2 > 2 ) U R ( X 2 ) < U R ( 2) and so U R ( X 2 ) < U R ( X 1 ; I). If ths corner case were an equlbrum then W1 R = U2 H p 2 ) U R = U R ( X 2 ) and W R 2 > U H 1 p U R > I U1( X 1 ) + I U1( X 1 )U R U R > I U1 (X 1 ) 2 1 I U1 (X 1 ) = U R ( X 1 ; I) where I U1 (X 1 ) < 1. Ths mples that U R ( X 1 ; I) < U R ( X 2 ):Hence we have a contradcton and ( X 1 ; X 2 ) cannot also be an equlbrum. Smlar proofs apply to the unqueness of nteror Case II. Proof of Proposton 3: 41

43 (1) Usng (43), snce I > 0, a su cent condton for 2 > 1 s that I > 1. That s Usng (23) to substtute out w (w 1 z) + + > (w 2 z) n the left hand sde yelds the condton that 2 on the rght hand sde and usng (22) to substtute out > ( 2 1 ) + + Snce 2 > 1 ths holds f the wage d erental s su cently large. (2) The home-ownershp rate n cty s h ( ) = N R + N + N W H + N W H = H R + H R R whch s ncreasng n. Snce from (1) 2 > 1 the result follows. Proof of Proposton 4: In aggregate the steady state ows nto and out of the state of beng an unemployed renter must satsfy ( + + ) N1 UR + N2 UR = N W R 1 + N2 W R = R 1 N1 UR + R 2 N2 UR It follows that the unemployment rate amongst renters s gven by R = N 1 UR + N2 UR = R 1 + R : In aggregate the steady-state ows nto and out of beng an unemployed owner must satsfy N1 + N2 + We can wrte ths as 1 N1 + 2 N2 = N W H 1 + N2 W H + N UR = 1 R 1 R 2 N1 N2 ( ) N1 + N2 = (1 R1 R 2 ) + (1 1 )N + N1 UR + N2 UR + N1 + N2 1 + N2 UR N UR 1 + N2 UR N 2 42

44 Dvdng by ( ) (1 R 1 R 2 ) yelds an expresson for the rate of unemployment amongst home-owners H = N 1 + N2 1 R 1 R 2 = (1 1 )N N 2 + N UR 1 + N2 UR + N1 + N2 ( ) (1 R 1 R 2 ) The second term must be postve snce 1, so t follows that H > R. Proof of Proposton 5: From (65) and (66) we can wrte: 2 = N 2 W R R 2 = + ( + )x = N 1 W R = + ( + )=x 2 =x R (129) (130) where x = R 1 =R 2 In order for 2 > 1 we requre that 2 > + ( + )=x 2 =x : Re arrangng and substtutng for 2 usng (129) yelds + ( + )x + + > + ( + )=x (1 + 1=x) If x 1, ths nequalty must hold. It also holds for x < 1 provded x large enough. Dervaton of cty level unemployment rate: The unemployment rate n cty s R N W R + H = N R + N + N UR + N W H = + R + H R R N W R + where the second equalty uses (68) and (69). Dvdng through by R and re arrangng yelds = H R Re-arrangng and usng (51) yelds (52) H R 1 + H R!

45 Proof of Proposton 6: Usng (68) and the aggregate rate of unemployment s gven by = N1 UR + N2 UR + N1 + N2 = R 1 + R 2 N W R 1 N W R 2 + H 1 + H R 1 2 R 2 N 1 W R + N2 W R : Usng (27) we can wrte ths as = R 1 + R 2 N W R 1 N W R R 1 R 2 + N 1 W R + N2 W R Usng (65) and (66) t can be seen that N1 W R + N2 W R = R 1 + ( + )R = (R 1 + R 2 ) : R2 + ( + )R Substtutng and notng that the aggregate home-ownershp rate s h = 1 R 1 R 2 = + 1 h h (131) Thus home-ownershp has two e ects on unemployment. The rst s negatve and comes from the reducton n the number of unemployed renters. The second s postve and comes from the ncrease n the measure of unemployment owners. Re-arrangng (131) yelds (53). 44

46 References [1] Albrect, J., A. Axelrod, E. Smth, and S. Vroman. Opportunstc Matchng n the Housng Market Internatonal Economc Revew 48(2) (May 2007) [2] Battu, H., A. Ma, and Euan Phmnster (2008) Housng Tenure, Job Moblty and Unemployment n the UK Economc Journal, forthcomng [3] Blanch ower, Davd (2007), Trends n European labour markets and preferences over unemployment and n aton," keynote address at the Dresdner Klenwort Semnar: [4] Böhem, R. and M.P. Taylor (2002), Ted down or room to move? Investgatng the relatonshps between housng tenure, employment status and resdental moblty n Brtan," Scottsh Journal of Poltcal Economy, vol. 49 (4), pp [5] Campbell, S. D., M. A. Davs, J. Galln, and R. F. Martna (2008) What Moves Housng Markets: A Varance Decomposton of the Rent-Prce Rato, workng paper. [6] Coulson, Edward N. and Lynn M. Fsher (2002), Tenure choce and labour market outcomes," Housng Studes, vol. 17 (1), pp [7] Coulson, Edward N. and Lynn M. Fsher (2008), Housng Tenure and Labor Market Impacts: The Search Goes On," mmeo, Penn State. [8] Coulson, Edward N., Seok-Joon Hwang, and Susumu Ima (2002) The Value of Owner- Occupaton n Neghborhoods Journal of Housng Research, 13, [9] DPasquale, Dense and Edward Glaeser Incentves and Socal Captal: Are home owners Better Ctzens? Journal of Urban Economcs, 45, (March 1999) [10] Dohmen, Thomas J. (2005), Housng, moblty and unemployment," Regonal Scence and Urban Economcs, vol. 35, pp [11] Flatau, P. Forbes, M., Hendershott, P.H. and G. Wood (2003), Homeownershp and unemployment: the roles of leverage and publc housng, NBER Workng paper # [12] Green, R.K. and P.H. Hendershott (2001), Home-ownershp and unemployment n the US," Urban Studes, vol. 38 (9), pp

47 [13] Harford, Tm (2007), The Renter s Manfesto: Why home ownershp causes unemployment, Slate On-lne, March 17, 2007: yout. [14] Knox, Paul L. (1994) Urbanzaton: An Introducton to Urban Geography, Englewood Cl s, New Jersey: Prentce Hall. [15] Munch, J., M. Rosholm, and M. Svarer (2006) Are home owners Really More Unemployed? Economc Journal, vol. 116, pp [16] Munch, J., M. Rosholm, and M. Svarer (2008) Home Ownershp, Job Duraton, and Wages Journal of Urban Economcs,vol. 63, pp [17] Nckell, S. J. (1998), Unemployment: questons and some answers," Economc Journal, vol. 108 (448), pp [18] Oswald, A.J. (1997) Thoughts on NAIRU (Correspondence) Journal of Economc Perspectves, 11, [19] Overman, H. G. and Y.M. Ioanndes (2001) "Cross-Sectonal Evoluton of the U.S. Cty Sze Dstrbuton," Journal of Urban Economcs, vol. 49(3), pp [20] Partrdge, Mark and Dan Rckman (1997) State Unemployment D erentals:equlbrum Factors vs.d erental Employment Growth, Growth and Change, vol. 28, pp [21] Pehkonen, Jaakko (1999) Unemployment and home ownershp, Appled Economcs Letters, 6, [22] Rouwendal, Jan and Peter Njkamp (2006) Home ownershp and Labor Market Behavor: Interpretng the Evdence workng paper [23] Rupert, Peter and Etenne Wasmer (2008), Housng Markets and Labour Markets: Tme to move and aggregate unemployment, mmeo, UC Santa Barbara. [24] van Leuvenstejn, Mchel and Perre Konng (2000) The E ect of Home-Ownershp on Labour Moblty n the Netherlands Journal of Urban Economcs, vol. 55, pp [25] Wheaton, Wllam. C. (1990), Vacancy, Search and Prces n a Housng Market Match Model, Journal of Poltcal Economy, vol. 98 (6), pp

48 8 Appendx B: Supplemental Calculatons 8.1 Intra-cty Relocaton In each cty there are sx types of households, as each may be ether employed or unemployed, ether rent or own a house and, f they are owners, may ether be matched or msmatched wth ther house. The measures of households n cty that are matched employed-owners, msmatched employed-owners, employed-renters, matched unemployed-owners, msmatched unemployed-owners and unemployed-renters are gven by N W H ; N ~ W H, N W R, N and N UR respectvely. The values assocated wth beng n each of these states are gven by W H, W ~ H, W R, U H ; U ~ H and U R ; respectvely. We let q W R, q UR ; q W H and q denote the prces pad for houses n Cty by employed and unemployed renters and by employed and unemployed owners respectvely. As n the basc model we restrct attenton to equlbra whch are statonary and symmetrc. We also restrct our attenton to the case where the margnal homeowner n both ctes s a matched unemployed owner. Wthn ths case we mpose the followng restrctons and check that they hold n equlbrum: (1) employed renters n the low-wage cty who are o ered a job n the hgh-wage cty choose to relocate, but not vce versa. (2) msmatched owners n both ctes do not become renters ~N ~W H p > W R and ~ U H p > U R = 1; 2: (3) All renters and msmatched owners buy houses when they get the chance. These condtons together mply that employed home owners (matched and msmatched) are also unwllng to move from from the hgh wage cty to the low wage one n equlbrum: W2 H p 2 > W ~ 2 H p 2 > W2 R > W1 R : (132) The steady state ow of workers between states s descrbed by (9), (26) and the followng 47

49 10 equatons: ( + + )N W R 1 = N UR 1 + ( + ) N1 + 1 N1 = N W H ( + ) N W H 1 = N UR 1 + N UR N1 W R + N ~ 1 W H 2 + N ~ N ~ 1 2 N2 (133) (134) + N 1 (135) ( + + ) ~ N 1 = N 1 + ~ N W H 1 (136) ( + ) N ~ 1 W H = N1 W H + N ~ 1 (137) ( + ) N2 W R = N2 UR + N1 UR + N1 W R + N ~ N1 (138) ( + ) N2 + 2 N2 = N W H ( + ) N W H 2 = 2 + N UR N2 W R + N ~ 2 W H 2 + N ~ 2 (139) + N 2 (140) ( + + ) ~ N 2 = N 2 + ~ N W H 2 (141) ( + ) ~ N W H 2 = N W H 2 + ~ N 2 (142) The soluton to ths system can be expressed recursvely as N W R 2 = R 2 + ( + )R N W R N UR N ( ) = (143) 1 = R 1 + ( + )R 2 N2 W R (144) = R N W R = 1; 2 (145) + ( + ) ( + + ~ + ) N W R (146) ~ ( + + ) ( H R ) (147) + ( + ) ( + + ) N ~ W R ( ~ H R ) (148) N W H ( ) = ~N ~N W H ( ) = ~ ( + ) N ( ) = ~ N ( ) + N W H ( ) (149) ( ) + ( + + ) N W H ( ) (150) ( ) = R N ~ ( ) ; (151) N ( ) 48

50 where ~ = ( + ) ( + + ) and = + ( + ) ( + + ~ + ) + + ~ ( + + ) 1 + ~ + ( + ) ~ ( + + ) The ow utltes of owners n ths equlbrum are gven by W H = w + H + + U H W H + ( W ~ H U H = z + H + + W H U H + ( U ~ H U H ) W ~ H = w + H " + + ~U H ~W H U ~ H = z + H " + + ~W H ~U H + U H W H ) ~U H (152) (153) Solvng yelds W H U H ~W H = 1 ^w + H ( ^w ^z) = 1 ^z + H + + ( ^w ^z) + + (154) (155) = W H W " (156) ~U H = U H U " (157) where ^w = w W " and ^z = z U " and W = U = ( ) ( + ) + (158) ( ) ( + ) + (159) The sale prce of housng n cty sats es p = (q W R p ) + (1 )(q UR p ) + (1 ) (q W H = n (W H Solvng for p yelds W R p ) + (1 )(U H U R p ) + (1 ) p ) + (q h (W H p ) ~W H ) + (U H o ~U H ) p = (W H + W R ) + (1 ) U H U R + (1 ) 1 W + (1 1 ) U " (160) 49

51 In cty 1 ths can be expressed as p 1 = (W H 1 1) + (1 1 )U H 1 U R + (1 1 ) 1 W + (1 1 ) U " (161) Snce, n ths equlbrum, matched home-owners n cty 1 are nd erent between stayng or leavng we have that Substtutng for p 1 and solvng for U R we get U H 1 p 1 = 2 + U R U R ( 1 ) = ( ) U H 1 ( ) W H (1 1 ) 1 W + (1 1 ) U " Smlarly, for cty 2 we have U R ( 2 ) = ( ) U H 2 ( ) W H (1 2 ) 2 W + (1 2 ) U " Equatng U R ( 1 ) = U R ( 2 ) yelds a generalzed VVI curve. Usng (59) the generalzed AM curve can be expressed as H 1 +H 2 +R 1 +R 2 1 = R 1 + N ~ 1 W H ( 1 ) + N ~ 1 ( 1 ) + R 2 + N ~ 2 W H ( 2 ) + ~ N2 ( 2 ) : 1 2 The ntersecton of these two curves yelds the equlbrum values of 1 and Rental Vacances In steady state, ows of rental unts out of and nto beng vacant must be equal n each cty. For cty 1 ths mples that R 1 N1 W R N1 UR = ( + ) N1 W R + N1 UR : (162) Re-arrangng yelds (63). For cty 2, employed renters turn down outsde o ers and some unemployed renters move to cty 1 even f they have no job o er. Consequently, the steady state ow condton s gven by R 2 N2 W R N2 UR = N W R 2 + N2 UR + N2 UR + N2 UR (163) where denotes the endogenous rate at whch unemployed renters wth no job o er move from cty 2 to cty 1. In equlbrum N2 UR = N1 W R + N1 UR N2 UR + 1 N1 2 N2 : (164) 50

52 Recall from that N = R ; and so Substtutng nto (163) we get N2 UR + N2 UR = N1 W R + N1 UR + R1 R 2 : (165) R 2 N2 W R N2 UR = N W R 2 + N2 UR + N1 W R + N1 UR + R1 R 2 : (166) Usng (63) and re-arrangng yelds (64). 51