Joint-Search Theory: New Opportunities and New Frictions

Transcription

1 Joint-Seach Theoy: New Oppotunities and New Fictions Bulent Gule Fatih Guvenen Giovanni L. Violante Febuay 17, 2010 Abstact Seach theoy outinely assumes that decisions about the acceptance/ejection of job offes (and, hence, about labo maket flows between jobs o acoss employment states) ae made by individuals acting in isolation. In eality, the vast majoity of wokes ae somewhat tied to thei patnes in couples o families and decisions ae made jointly. This pape studies, fom a theoetical viewpoint, the joint job-seach and location poblem of a household fomed by a couple (e.g., husband and wife) who pefectly pools income. The objective, in the spiit of standad seach theoy, is to chaacteize the esevation wage behavio of the couple and compae it to the single-agent seach model in ode to undestand the amifications of patneships fo individual labo maket outcomes and wage dynamics. We focus on two main cases. Fist, when couples ae isk avese and pool income, joint seach yields new oppotunities simila to on-the-job seach elative to the single-agent seach. Second, when the two spouses in a couple face job offes fom multiple locations and a cost of living apat, joint seach featues new fictions and can lead to significantly wose outcomes than single-agent seach. Indiana Univesity; bgule@indiana.edu Univesity of Minnesota, Fedeal Reseve Bank of Minneapolis, and NBER; guvenen@umn.edu New Yok Univesity, CEPR and NBER; gianluca.violante@nyu.edu 1 Electonic copy available at:

2 1 Intoduction In the yea 2000, the labo foce paticipation ate of maied women stood at 61%, and in one-thid of maied couples wives povided moe than 40% of household income (US Census, 2000; Raley, Mattingly, and Bianchi, 2006). Fo these households which make up a substantial faction of the population economic decisions ae suely taken jointly by the two spouses. Among such decisions, job seach, boadly defined, is aguably one of the most cucial to the economic well-being of a household. Macoeconomics is apidly shifting away fom the stylized bachelo model of the household to models that explicitly ecognize the elevance of within-household decisions fo aggegate economic outcomes. 1 Supisingly, instead, since its inception in the ealy 1970s, seach theoy has almost entiely focused on the single-agent seach poblem. The ecent suvey by Rogeson, Shime, and Wight (2005), fo example, does not contain any discussion on optimal job seach stategies of twopeson households acting as the decision units. This state of affais is athe supising given that Budett and Motensen (1977), in thei seminal piece entitled Labo Supply Unde Uncetainty, lay out a two-peson seach model and sketch a chaacteization of its solution, explicitly encouaging futhe wok on the topic. Thei pioneeing effot, which emained vitually unfollowed, epesents the stating point of ou theoetical analysis. In this pape, we study the job seach poblem of a couple who faces exactly the same economic envionment as in the standad single-agent seach poblem of McCall (1970) and Motensen (1970) without on-the-job seach, and Budett (1978) with on-the-job seach. A couple is an economic unit composed of two identical individuals linked to each othe by the assumption of pefect income pooling. The simple unitay model of a household adopted hee is a convenient and logical stating point. It helps us to examine moe tanspaently the ole of the labo maket fictions and insuance oppotunities intoduced by joint-seach, and it makes the compaison with the canonical singleagent seach model especially stak. Fom a theoetical pespective, couples would make a joint decision leading to choices diffeent fom those of a single agent fo seveal easons. We stat fom the two most natual and elevant ones. Fist, the couple has concave pefeences ove pooled income. Second, the couple can eceive job offes fom multiple locations, but faces a utility cost of living apat. In this latte case, deviations fom the single-agent seach poblem occu even fo linea pefeences. As summaized by the title of ou pape, in the fist envionment joint seach intoduces new oppotunities, wheeas in the second it intoduces new fictions elative to single-agent seach. One appealing featue of ou theoetical analysis is that it leads to two-dimensional diagams in the space of the two spouses wages (w 1, w 2 ), whee the esevation wage policies can be easily analyzed and intepeted. In the fist envionment we study, couples have isk-avese pefeences and have access to a 1 Fo example, see Aiyagai et al. (2000) on integeneational mobility and investment in childen, Cubeddu and Rios-Rull (2003) on pecautionay saving, Blundell et al. (2007) on labo supply, Heathcote et al. (2008) and Lise and Seitz (2008) on economic inequality, and Gune et al. (2009) on taxation. 2 Electonic copy available at:

3 isk-fee asset fo saving but ae not allowed to boow. A dual-seache couple (both membes unemployed) will quickly accept a job offe in fact, moe easily than a single unemployed agent. The dual-seache couple can use income pooling and joint seach to its advantage: it initially accepts a lowe wage offe (to smooth consumption acoss states) while, at the same time, not giving up completely the seach option (to incease lifetime income), which emains available to the othe spouse. Once a woke-seache couple (one spouse unemployed, the othe employed), the pai will be moe choosy in accepting the subsequent job offes. We fomally show that the gap between the esevation wage of the woke-seache couple (a function of the employed spouse s wage) and that of the dual-seache couple (a constant) depends on the degee of absolute isk avesion in pefeences and on how absolute isk avesion changes with the level of consumption. Futhemoe, if the seaching spouse eceives and accepts a job offe, this may tigge a quit by the employed spouse to seach fo a bette job, esulting in a switch between the beadwinne and the seache within the household. As is well known, this endogenous quit behavio neve happens in the coesponding single-agent vesion of the seach model. We call this pocess of quit-seach-wok that allows a couple to climb the wage ladde even in absence of on-the-job seach the beadwinne cycle. Theefoe, one can view joint seach as a costly vesion of on-the-job seach, even in the fomal absence of it. The cost comes fom the fact that in ode to keep the seach option active, the pai must emain a woke-seache couple, and must not enjoy the full wage eanings of a dualwoke couple as it would be capable of doing in the pesence of on-the-job seach. Oveall, elative to singles, couples spend moe time seaching fo bette jobs, which esults in longe unemployment duations, but it eventually leads to highe lifetime wages and welfae. We uncove two equivalence esults between single-agent seach and joint-seach outcomes. The fist envionment equies the pesence of on-the-job seach with equal seach effectiveness on and off the job. The second equies exponential (i.e., constant absolute isk avesion) pefeences and loose boowing limits. In both cases, a isk-avese couple acts like two sepaate single agents. These equivalence esults follow diectly fom the value added of joint seach in tems of climbing the wage ladde and of smoothing consumption, as discussed above. Finally, we also show an intuitive and useful esult: the joint-seach model is exactly isomophic to a model whee a single agent seaches fo jobs and she has the possibility of holding multiple jobs. Ou second envionment featues multiple locations and a flow cost of living apat fo each of the spouses in the couple. The couple has to choose esevation functions with espect to inside offes (jobs in the cuent location) and outside offes (jobs in othe locations). Even with isk-neutal pefeences, the seach behavio of couples diffes fom that of single agents in impotant ways. Fist, the dual-seache couple is less choosy than the individual agent because it is effectively facing a wose job offe distibution, since some wage offe configuations ae attainable only in diffeent locations hence, by paying the cost of living apat. Second, thee is a egion in which the beadwinne cycle is optimal fo the couple. Fo example, a couple who gets a vey geneous job offe fom the outside location could be bette off if the cuently employed spouse quits and follows the spouse with the job offe to the new location. It should be noted that we also obtain 3

4 these two esults couples being less picky than singles and the beadwinne cycle in ou pevious envionment, but fo completely diffeent easons. The model allows us to fomalize what Mince (1978) called tied-stayes i.e., wokes who tun down a job offe in a diffeent location that they would accept as single and tied-moves i.e., wokes who accept a job offe in the location of the patne that they would tun down as single. Oveall, the disutility of living sepaately effectively naows down the job offes that ae viable fo couples, who end up choosing among a moe limited set of job options. The elevance of a multiple-location joint-seach model of the labo maket is suppoted, fo example, by Costa and Kahn (2000) who document that highly educated dual-caee couples have inceasingly elocated to lage metopolitan aeas in the United States since the 1960s (moe so than compaable singles): cities offe a geate and moe divese set of job oppotunities, theeby mitigating the fictions associated with joint seach. Finally, we also show that this multiple-location model, togethe with the assumption that women have highe job quit ates than men, can explain why men s unemployment duation is falling in the wage of thei spouses wheeas the opposite is tue fo women a supising finding that Lentz and Tanaes (2005) labelled the gende asymmety puzzle. The set of popositions poved in the pape fomalizes the new oppotunities and the new fictions in tems of compaison between the esevation wage functions of the couple and the esevation wage of the single agent. We also povide some illustative simulations to show that the deviations of joint-seach behavio fom its single-agent countepat can be quantitatively substantial. Fo example, in the one location model with CRRA utility and a coefficient of elative isk avesion equal to fou, joint-seach geneates a job quit ate of 2% pe month a clea sign of the beadwinne cycle in action and each spouse in a couple eans a lifetime income that is 3% highe than a compaable single agent. In the multiple-location model with isk neutality, when the disutility cost (of living sepaately) is equal to 15% of a dual-woke couples aveage eanings, moe than 50% of all households moving acoss locations involve a patne who is a tied-move, and the lifetime income of each spouse in a couple is 6.5% lowe than compaable singles. Only vey ecently, a handful of papes have stated to follow the lead offeed by Budett and Motensen (1977) into the investigation of household inteactions in fictional labo maket models. Gacia-Peez and Rendon (2004) numeically simulate a model of family-based job-seach decisions to tease out the impotance of the added woke effect fo consumption smoothing. Dey and Flinn (2008) study quantitatively the effects of health insuance coveage on employment dynamics in a seach model whee the economic unit is the household. Gemici (2008) estimates a ich stuctual model of migation and labo maket decisions of couples to assess the implications of joint location constaints on labo outcomes and the maital stability of couples. Relative to these contibutions, ou pape is less ambitious in its quantitative analysis, but it povides a moe focused and systematic study of joint-seach theoy. Fom a theoetical pespective, ou analysis of the one-location model has useful points of contact with existing esults in seach theoy applied to at least thee sepaate contexts. Fist, stating fom 4

5 the static analysis of Danfoth (1979), a numbe of papes have studied the ole of isk-fee wealth in shaping dynamic job-seach decisions (e.g., Andolfatto and Gomme, 1996; Geenwood, Gomes, and Rebelo, 2001; Pissaides, 2004; Lentz and Tanaes, 2005; Bowning et al., 2007). The income of the spouse diffes cucially fom isk-fee wealth because it is isky (in the pesence of exogenous sepaations) and because it can be optimally contolled by the job-seach decision itself. Second, Albecht, Andeson, and Voman (2009) study a diffeent type of joint-seach decision, that of a committee that votes on an option which gives some value to each membe. The authos ae inteested in dawing a compaison between single-agent seach and committee seach, in the same spiit as ou execise. 2 Thid, as we explain in the main text, thee is an analogy with some seach models of maiage fomation whee the flow value of the maiage is a concave function of the sum of the spouses endowments (e.g., Vissches, 2006). The est of the pape is oganized as follows. Section 2 descibes the single-agent poblem which povides the benchmak of compaison thoughout the pape. Section 3 develops and fully chaacteizes the baseline joint-seach poblem. Section 4 extends this baseline model in a numbe of diections: nonpaticipation, on-the-job seach, exogenous sepaations, and access to boowing. Section 5 studies an economy with multiple locations and a cost of living apat fo the couple. Section 6 povides some illustative simulations on both models. Section 7 concludes the pape and discusses possible extensions and applications. The Appendix contains the poofs of all ou popositions. 2 The Single-Agent Seach Poblem We begin by fist pesenting the sequential job-seach poblem of a single agent the well-known McCall-Motensen model (McCall, 1970; Motensen, 1970). This model povides a useful benchmak against which we compae the joint-seach model that we intoduce in the next section. Fo claity of exposition, we begin with a vey stylized vesion of this optimal stopping poblem and then conside seveal extensions in Section 4. Economic envionment. Conside an economy populated by single individuals who all paticipate in the labo foce: agents ae eithe employed o unemployed. Time is continuous and thee is no aggegate uncetainty. Wokes maximize the expected lifetime utility fom consumption, E 0 e t u (c (t)) dt, 0 whee is the subjective ate of time pefeence, c (t) is the instantaneous consumption flow at time t, and u ( ) is the instantaneous utility function, stictly inceasing, stictly concave, and smooth. 2 The similaities, though, stop hee moe o less. Fo example, Albecht, Andeson, and Voman (2009) also find that committees ae less picky than single agents. In ou one-location model, this esult is due to a consumptionsmoothing agument. In thei envionment, it is due to the negative extenality that committee membes impose on each othe (e.g., voting against when dawing a paticulaly low value). 5

6 An unemployed woke is entitled to an instantaneous benefit, b, and eceives wage offes, w, at ate α fom an exogenous wage offe distibution, F (w) with suppot [0, ). Thee is no ecall of past wage offes. The woke obseves the wage offe, w, and decides whethe to accept o eject it. If she ejects the offe, she continues to be unemployed and to eceive job offes. If she accepts the offe, she becomes employed at wage w foeve, i.e., thee ae no exogenous sepaations and no new offes on the job. All individuals ae identical in tems of thei labo maket pospects, i.e., they face the same wage offe distibution and the same aival ate of offes, α. Finally, we assume that individuals have access to isk-fee saving but ae not allowed to boow. As will become clea below, in the pesent famewok individuals face a wage eanings pofile that is nondeceasing ove the life cycle (without exogenous sepaation isk), and, theefoe, consumption smoothing only equies the ability to boow but does not benefit fom the ability to save. As a esult, individuals will optimally set consumption equal to thei wage eanings evey peiod even though they ae allowed to save. 3 Value functions. Denote by V and W the value functions of an unemployed and employed agent, espectively. Then, using the continuous time Bellman equations, the poblem of a single woke can be witten in the following flow value epesentation: 4 V = u (b) + α max {W (w) V, 0} df (w), (1) W (w) = u (w). (2) This well-known poblem yields a unique esevation wage, w, fo the unemployed such that fo any wage offe above w, she accepts the offe and below w, she ejects the offe. Futhemoe, this esevation wage can be obtained as the solution to the following equation u (w ) = u (b) + α [u (w) u (w )] df (w) (3) w = u (b) + α u (w) [1 F (w)] dw, w whee the second equality comes fom integation by pats. This equation shows that the instantaneous utility of accepting a job offe paying the esevation wage (left-hand side, LHS) is equated to the option flow value of continuing to seach in the hope of obtaining a bette offe in the futue (ight-hand side, RHS). Since the LHS is inceasing in w wheeas the RHS is a deceasing function of w, and they ae both continuous, equation (3) uniquely detemines the esevation wage, w. 3 The Joint-Seach Poblem We now study the seach poblem of a couple facing the same economic envionment descibed above. Fo the puposes of this pape, a couple is defined as an economic unit composed of two 3 Nonpaticipation, on-the-job seach, exogenous job sepaation, and boowing ae intoduced in Section 4. 4 Below, when the limits of integation ae not explicitly specified, they ae undestood to be those of the suppot of F (w). 6

7 individuals who ae ex ante identical in thei pefeences and in the labo maket paametes they face. The two individuals pefectly pool income to puchase a maket good which is jointly consumed by the couple. As in the single-seach case, households simply consume thei (total) income in each peiod. Couples make thei job seach decisions in ode to maximize thei common welfae. A couple can be in one of thee labo maket states. Fist, if both spouses ae unemployed and seaching, they ae efeed to as a dual-seache couple. Second, if both spouses ae employed (an absobing state), we efe to them as a dual-woke couple. Finally, if one spouse is employed and the othe is unemployed, we efe to them as a woke-seache couple. As can pehaps be anticipated, the most inteesting state is the last one. Value functions. Let U denote the value function of a dual-seache couple, Ω (w 1 ) the value function of a woke-seache couple when the woke s wage is w 1, and T (w 1, w 2 ) the value function of a dual-woke couple eaning wages w 1 and w 2. The flow value in the thee states becomes T (w 1, w 2 ) = u (w 1 + w 2 ), (4) U = u (2b) + 2α max {Ω (w) U, 0} df (w), (5) Ω (w 1 ) = u (w 1 + b) + α max {T (w 1, w 2 ) Ω (w 1 ), Ω (w 2 ) Ω (w 1 ), 0} df (w 2 ). (6) The equations detemining the fist two value functions (4) and (5) ae staightfowad analogs of thei countepats in the single-agent seach poblem. When both spouses ae employed, thei flow value is simply equal to the total instantaneous wage eanings of the household. When they ae both unemployed, thei flow value is equal to the instantaneous utility of consumption (which equals the total unemployment benefit) plus the expected gain in case a wage offe is eceived. Because both agents sample new offes at ate α, the total offe aival ate of a dual-seache couple is 2α. 5 The value function of a woke-seache couple is somewhat moe involved. As can be seen in equation (6), upon eceiving a wage offe (which now aives at ate α, since only one spouse is unemployed) the couple faces thee choices. Fist, the unemployed spouse can eject the offe, in which case thee is no change in the value. Second, the unemployed spouse can accept the job offe and both spouses become employed, which inceases the value by T (w 1, w 2 ) Ω (w 1 ). Thid, the unemployed spouse can accept the wage offe w2 and the employed spouse simultaneously quits his job and stats seaching fo a bette one. In this case, the gain to the couple is Ω (w 2 ) Ω (w 1 ). This latte case is the fist impotant diffeence between the joint-seach poblem and the singleagent seach poblem. In the single-agent seach poblem, in a stationay envionment, once a job offe is accepted, the woke will neve choose to quit. In contast, in the joint-seach poblem, the esevation wage of each spouse depends on the income of the patne. When this income gows fo example, because of a tansition fom unemployment to employment the esevation wage of 5 Because time is continuous, the pobability of both spouses eceiving offes simultaneously is negligible and is hence ignoed. 7

8 the peviously employed spouse may also incease, which could lead to execising the quit option. Below, we etun to this endogenous nonstationaity implicit in the joint-seach poblem. 3.1 Chaacteizing the Couple s Decisions Befoe we begin chaacteizing the solution to the poblem, we state the following useful lemma. We efe to Appendix A fo all the poofs and deivations. Lemma 1 Ω (w) is a stictly inceasing function of w. We ae now eady to chaacteize the couple s seach behavio. Fist, fo a dual-seache couple, the esevation wage which is the same fo both spouses by symmety is denoted by w and is detemined by the equation Ω (w ) = U. (7) Because U is a constant and Ω is a stictly inceasing function (Lemma 1), w is a singleton. A woke-seache couple has two decisions to make. The fist decision is whethe to accept the job offe to the unemployed spouse (say, spouse 2) o not. The second decision, conditional on accepting, is whethe the employed spouse (spouse 1) should quit his job o not. Let the cuent wage of the employed spouse be w 1 and denote the wage offe to the unemployed spouse by w 2. 6 Accept/eject decision. Let us begin by supposing that it is not optimal to execise the quit option upon acceptance, Ω (w 2 ) < T (w 1, w 2 ). In this case, a job offe with wage w 2 will be accepted when T (w 1, w 2 ) Ω (w 1 ). Fomally, the associated esevation wage function φ (w 1 ) solves T (w 1, φ (w 1 )) = Ω (w 1 ). (8) Now, in contast, suppose that it is optimal to execise the quit option upon acceptance, Ω (w 2 ) T (w 1, w 2 ). Then, the job offe will be accepted when Ω (w 2 ) Ω (w 1 ). Since Ω is invetible, the associated esevation wage function solves φ (w 1 ) = w 1. (9) In this case, φ (w 1 ) coincides with the 45 0 line and the optimal ule is vey simple: accept the new offe w 2 (and the othe spouse will simultaneously quit) wheneve it exceeds the cuent wage w 1. In sum, the woke-seache esevation wage function φ ( ) is piecewise, being detemined by (8) and (9) in diffeent anges of the domain fo w 1. The kink of this piecewise function, which always lies on the 45 0 line in the (w 1, w 2 ) space, plays a special ole in chaacteizing the behavio of the couple. We denote this point by (ŵ, ŵ), and it satisfies: T (ŵ, φ (ŵ)) = Ω (ŵ) = Ω (φ (ŵ)). Since T (ŵ, ŵ) = u (2ŵ), ŵ solves u (2ŵ) = Ω (ŵ). (10) 6 To bette undestand the optimal choices of the couple, it is instuctive to teat the accept/eject decision of the unemployed spouse and the stay/quit decision of the employed spouse as two sepaate choices (albeit made simultaneously by the couple). 8

9 Stay/quit decision. A quit will neve take place if the wage offe w 2 is ejected, as the pai would be wose off. Theefoe, the stay/quit decision is nontivial only when w 2 is accepted. The simplest way to undestand this decision is to envision it as the accept/eject decision of the cuent wage w 1, conditional on etaining the job offe w 2. Fomally, the associated indiffeence condition is T (φ (w 2 ), w 2 ) = Ω (w 2 ). The use of the same φ function is not accidental: a compaison with equation (8) and the symmety of T imply that the stay/quit decision is chaacteized by the same function φ defined by (8). In othe wods, in the (w 1, w 2 ) space, this decision ule is the mio image of φ with espect to the 45-degee line. In light of this, ŵ is the highest wage level at which the unemployed spouse of a woke-seache couple is indiffeent between accepting and ejecting an offe and, at the same time, he spouse is indiffeent between keeping and quitting his job. To emphasize this featue, we efe to ŵ as the double indiffeence point. 7 Taking stock. In light of what we established above, a dual seache couple accepts any wage offe above w ; a woke-seache couple accepts any wage offe w 2 above φ (w 1 ) and chooses to quit wheneve the cuent wage w 1 is below φ (w 2 ). Since a low wage w 1 on the cuent job makes quitting moe attactive, it is immediate that the piece of the φ function coesponding to the 45 0 line is elevant in the ange w w 1 < ŵ, wheeas the piece of the φ function defined by the indiffeence condition (8) will be elevant only in the ange w 1 ŵ. Oveall, these diffeent esevation ules divide the (w 1, w 2 ) space into fou egions: one in which both spouses wok, one whee both spouses seach, and the emaining two egions in which spouse 1 (2) seaches and spouse 2 (1) woks. Chaacteizing the optimal stategy of the couple means the following: (i) studying the conditions unde which w < ŵ, a necessay inequality to activate the esevation ule φ (w 1 ) = w 1 ; (ii) analyzing the shape of the function φ beyond ŵ; and (iii) anking ŵ elative to the single-agent esevation wage w, which is useful fo compaing single-agent seach to joint-seach stategies. Poposition 2 tackles (i). Poposition 3 tackles (ii) and (iii). With this chaacteization in hand, we offe an intuitive gaphic epesentation of optimal joint-seach in the (w 1, w 2 ) space. 3.2 Risk Neutality To povide a benchmak, we begin by pesenting the isk-neutal case, then tun to the esults with isk-avese agents. Poposition 1 [Risk Neutality] With isk-neutal pefeences, i.e., u = 0, the joint-seach poblem of the couple educes to two independent single-agent seach poblems fo the two spouses, 7 Fo any wage above ŵ, eithe the employed spouse stictly pefes etaining his job o, if he quits, his unemployed spouse is always stictly bette off accepting the job offe than emaining unemployed. So, thee can be no longe indiffeence in both choices. 9

10 Figue 1: Resevation Wage Functions with Risk Neutality with value functions T (w 1, w 2 ) = W (w 1 ) + W (w 2 ), U = 2V, Ω (w 1 ) = V + W (w 1 ). The esevation wage function φ ( ) of the woke-seache couple is constant and is equal to the esevation wage value of a dual-seache couple (egadless of the wage of the employed spouse), which, in tun, equals the esevation value in the single-agent seach poblem, i.e., φ (w 1 ) = w = ŵ = w. Figue 1 shows the elevant esevation wage functions in the (w 1, w 2 ) space. Thoughout the pape, when we discuss woke-seache couples, we will think of spouse 1 as the employed spouse and display his wage w 1 on the hoizontal axis, and think of spouse 2 as the unemployed spouse and display he offe, w 2, on the vetical axis. As stated in the poposition, the esevation wage function of a woke-seache couple, φ (w 1 ) is simply the hoizontal line at w. Similaly, the esevation wage fo the quit decision is the mio image of φ (w 1 ) and is shown by the vetical line at w 1 = w. The intesection of these two lines geneates fou egions, and the couple displays distinct behavios in each. No wage below w is eve accepted by a dual-seache couple in this model. Theefoe, a woke-seache couple will neve be obseved with a wage below w. If the unemployed spouse eceives a wage offe w 2 < w, she ejects the offe and continues to seach. If she eceives an offe highe than w, she accepts the offe. At this point the employed patne etains his job, and the couple becomes a dual-woke couple. Fo things to get inteesting in this one-location model, isk avesion must be bought to the foe. In Section 5, we will also see that when the job-seach pocess takes place in multiple locations 10

11 and thee is a cost of living sepaately fo the couple, then even in the isk-neutal case thee ae impotant deviations fom the single-agent seach poblem. 3.3 Risk Avesion We stat with a key implication of isk avesion summaized by the following poposition. Poposition 2 [Beadwinne Cycle] If u is concave, the esevation wage value of a dualseache couple is stictly smalle than the double-indiffeence point: w < ŵ. The fact that the esevation wage of a dual-seache couple is stictly smalle than the doubleindiffeence point activates a egion whee φ (w 1 ) = w 1 which, in tun, gives ise to endogenous quits and to dynamics which we label the beadwinne cycle. To undestand how this happens, suppose that w 1 (w, ŵ) and the unemployed spouse eceives a wage offe w 2 (w 1, ŵ). Because w 2 > w 1 = φ (w 1 ), the unemployed spouse accepts the offe and becomes employed. Howeve, accepting this wage offe also implies w 1 < φ (w 2 ) = w 2. Because the theshold fo the fist spouse to keep his job now exceeds his cuent wage, he will quit. As a esult, spouses simultaneously switch oles and tansit fom a woke-seache couple into anothe woke-seache couple with a highe wage level. This pocess epeats itself ove and ove again and the identity of the employed spouse (i.e., the beadwinne) altenates until the employed spouse stictly pefes etaining his job and the pai becomes a dual-woke couple. We etun to the beadwinne cycle below HARA utility To obtain shape pedictions on the esevation wage function φ (w 1 ) beyond ŵ, we impose futhe stuctue on pefeences. Specifically, we now estict attention to concave pefeences in the HARA (hypebolic absolute isk avesion) class. This class encompasses seveal well-known utility functions as special cases. Fomally, HARA pefeences ae defined as the family of utility functions that have linea isk toleance: u (c) /u (c) = ρ + τc, whee ρ and τ ae paametes. 8 This class can be futhe divided into thee subclasses depending on the sign of τ. Fist, when τ 0, then isk toleance (and hence absolute isk avesion) is independent of consumption level. This case coesponds to constant absolute isk avesion (CARA) pefeences, also known as exponential utility u (c) = ρe c/ρ. Second, if τ > 0 then absolute isk toleance is inceasing and theefoe isk avesion is deceasing with consumption, which is the deceasing absolute isk avesion (DARA) case. A well-known special case of this class is the constant elative isk avesion (CRRA) utility: u (c) = c 1 σ / (1 σ), which obtains when ρ = 0 and τ = 1/σ > 0. Finally, if τ < 0 isk avesion inceases with consumption, and this class is efeed to as inceasing absolute isk avesion (IARA). A special case of this class is quadatic utility: u (c) = (ρ c) 2, which obtains when τ = 1. 8 Risk toleance is defined as the ecipocal of Patt s measue of absolute isk avesion. Thus, if isk toleance is linea, isk avesion is hypebolic. 11

12 Poposition 3 [HARA Utility] With HARA pefeences, fo w 1 ŵ: (i) The esevation wage function satisfies (0, 1) if u is DARA φ (w 1 ) : = 0 if u is CARA < 0 if u is IARA. (ii) The double indiffeence point satisfies > w if u is DARA ŵ : = w if u is CARA < w if u is IARA. Befoe discussing the implications of the poposition, it is inteesting to ask why it is the absolute isk avesion that detemines the popeties of joint-seach behavio, as opposed to, fo example, elative isk avesion. The eason is that individuals ae dawing wage offes fom the same pobability distibution egadless of the cuent wage eanings of the couple. As a esult, the uncetainty they face detemined by the dispesion in the wage offe distibution is fixed, making the attitudes of a couple towad a fixed amount of isk and theefoe, absolute isk avesion the elevant measue. 9 While Appendix A contains a fomal poof of this poposition, it is instuctive to sketch the agument behind the poof hee. To this end, begin by conjectuing that above a cetain wage theshold it is neve optimal to execise the quit option. 10 In this wage ange, equation (6) simplifies to Ω (w 1 ) = u (w 1 + b) + α [T (w 1, w 2 ) Ω (w 1 )] df (w 2 ). φ(w 1 ) Substituting out Ω and T, using equations (4) and (8), shows that the esevation wage function fo the unemployed spouse must satisfy u (w 1 + φ (w 1 )) u (w 1 + b) = α Divide both sides by the left-hand side: 1 = α φ(w 1 ) φ(w 1 ) [u (w 1 + w 2 ) u (w 1 + φ (w 1 ))] df (w 2 ). (11) [ u (w1 + w 2 ) u (w 1 + φ (w 1 )) u (w 1 + φ (w 1 )) u (w 1 + b) ] df (w 2 ). (12) Next, applying a well-known esult on HARA pefeences established by Patt (1964, Theoem 1), it can easily be shown that the ight-hand side of equation (12) is stictly inceasing (deceasing) in w 1 unde the DARA (IARA) specification, and independent of w 1 unde the CARA specification. Also, note that the ight-hand side of that equation is stictly deceasing in φ (w 1 ). Theefoe, fo the equality to hold in equation (12), φ (w 1 ) should be stictly inceasing (deceasing) with DARA (IARA) pefeences, and constant with CARA pefeences. 9 If, fo example, individuals wee to daw wage offes fom a distibution that depended on the cuent wage of a couple, this would make elative isk avesion elevant as well. 10 In the Appendix, we fomally pove that this theshold is ŵ fo the CARA and IARA cases. Fo the DARA case, even though the esevation wage function φ tuns out to be stictly inceasing, a finite suppot fo the wage offe distibution and the fact that φ < 1 will ensue that such a theshold always exists. 12

13 Figue 2: Resevation Wage Functions with CARA Pefeences CARA case. Figue 2 povides a visual summay of the contents of this poposition fo the CARA case in the wage space. The eason why the φ function is constant and equal to the esevation wage value of the single agent is that, with CARA utility, attitude towad isk does not depend on the consumption (and hence wage) level. As the wage of the employed spouse inceases, the couple s absolute isk avesion emains unaffected, implying a constant esevation wage fo the unemployed patne. Combining the esults of Popositions 2 and 3, we conclude that the dual seache couple is less choosy than the single agent (w < w ). With isk avesion, the optimal seach stategy involves a tade-off between lifetime income maximization and the desie fo consumption smoothing. The fome foce pushes up the esevation wage, the second pulls it down as isk-avese agents paticulaly dislike the low income state (unemployment). The dual-seache couple can use income pooling to its advantage: it initially accepts a lowe wage offe (to smooth consumption acoss states) while, at the same time, not giving up completely the seach option (to incease lifetime income) which emains available to the othe spouse. In contast, when the single agent accepts his job he gives up the seach option fo good, which induces him to be moe picky at the stat. Notice that joint seach plays a ole simila to on-the-job seach in the absence of it, pecisely though the beadwinne cycle. DARA and IARA cases. Figue 3 illustates gaphically the esevation wage policies in these two cases. Unde DARA (IARA) pefeences, the esevation function of the woke-seache couple is inceasing (deceasing) with the wage of the employed spouse fo wage levels highe than ŵ. This is because with deceasing (inceasing) absolute isk avesion, a couple becomes less (moe) concened about smoothing consumption as household esouces incease and, consequently, becomes moe (less) picky in its job seach. 13

14 Figue 3: Resevation Wage Functions fo DARA (left) and IARA (ight) Pefeences An impotant featue of DARA pefeences one that complicates the poof of Poposition 3 is that the beadwinne cycle is obseved ove a wide ange of wage values of the employed spouse compaed to the CARA and IARA cases. As can be seen in Figue 3 (left panel), φ is stictly inceasing in w 1. As a esult, even when w 1 > ŵ, a sufficiently high wage offe not only will be accepted by the unemployed spouse but will also tigge the employed spouse to quit. At fist blush, it may seem supising that in the DARA case one cannot ank w and w by combining Popositions 2 and 3. Afte all, the logic used to explain why w < w in the CARA case is based on the elative stength of consumption smoothing and income maximization motives. But the agument is moe subtle. To see why, conside the one-peiod gain when deciding whethe to accept o eject an offe w. The couple compaes u (b + w) to u (2b), wheeas the single agent compaes u (w) to u (b). The couple makes this compaison at a highe level of consumption and, because of DARA, the couple is less isk avese. This foce tends to push w above w and does not allow a geneal anking. 11 Howeve, when b is vey small, w < w even in the DARA case, which allows us to state the following esult. Lemma 2 Suppose u is DARA and u (0) >. Then, if b = 0, w < w. To povide a bette sense of how the beadwinne cycle woks, Figue 4 plots the simulated wage paths of a couple when spouses behave optimally unde joint seach (lines with makes) and fo the same individuals when they act as two unelated singles (dashed lines). 12 To make the compaison meaningful, the paths ae geneated using the same simulated sequence of job offes fo each individual when he/she is single and when they act as a couple. Fist, the beadwinne cycle is seen clealy hee as spouses altenate between who woks and who seaches depending on the offes 11 We veified, by simulation, that with CRRA utility thee ae paamete configuations whee w > w. 12 Pefeences ae assumed to be of the CRRA (hence DARA) class. 14

15 Figue 4: Simulated Wage Paths fo the Same Individuals as Couple and Singles Wage Single 1 Spouse Time (weeks) Wage Single 2 Spouse Time (weeks) eceived by each spouse. Instead, when faced with the same job offe sequence, the same individuals simply accept a job (agent 1 in peiod 33 and agent 2 in peiod 60) and then neve quit. Second, in peiod 29, agent 2 accepts a wage offe of 1.02 when she is pat of a couple but ejects the same offe when acting as single, eflecting the fact that dual-seache couples have a lowe esevation wage than single agents. The opposite happens in peiod 60 when agent 2 accepts a job offe of 1.08 as single but tuns it down when maied, eflecting the fact that woke-seache couples ae moe picky in accepting job offes than single agents. It is also easy to see that in the long un, the wages of both agents ae highe unde joint seach thanks to the beadwinne cycle, even though it may equie a longe seach pocess. Below we povide some illustative simulations to show that, on aveage, joint seach always yields a highe lifetime discounted value of income. 3.4 Consumption as a pivate good within the couple Some goods consumed by the household have featues of public goods (e.g., house), othes of pivate goods (e.g., food). In the baseline model we have chosen the fome specification. Suppose we take the exteme view that consumption, instead, is a fully pivate good within the couple, i.e. pe-capita inta-peiod household utility is u ( y 1 +y 2 ) 2. One can easily adapt all the poofs and show that all the esults stated so fa ae obust to this extension, the only exception being that, in the CARA 15

16 case, ŵ < w. 13 Hence, ou chaacteization of optimal joint-seach behavio emains fundamentally valid, independently of the degee to which consumption is pivate within the household An Isomophism: Seach with Multiple Job Holdings The joint-seach famewok analyzed so fa is isomophic to a seach model with a single agent who can hold multiple jobs at the same time. To see this, suppose that the time endowment of a woke can be divided into two subpeiods (e.g., day shift and night shift). The single agent can be (i) unemployed and seaching fo his fist job while enjoying 2b units of home poduction, (ii) woking one job at wage w 1 while seaching fo a second one, o (iii) holding two jobs with wages w 1 and w 2. It is easy to see that the poblem faced by this individual is exactly given by the equations (4), (5), and (6) and theefoe it has the same solution as the joint-seach poblem. 15 Consequently, fo example, when the agent woks in one job and gets a second job offe with a sufficiently high wage offe, he will accept the offe and simultaneously quit the fist job to seach fo a bette one. Hee, it is not the beadwinnes who altenate, but the jobs that the individual woks at. 4 Extensions The baseline joint-seach model analyzed so fa was intended to povide the simplest deviation fom the well-known single-seach poblem. Despite being highly stylized, this model illustated some new and potentially impotant mechanisms that ae not opeational in the single-agent seach poblem. In this section, we enich this basic model in fou empiically elevant diections. Fist, we allow fo nonpaticipation. Second, we add on-the-job seach. Thid, we allow fo exogenous job sepaations. Fouth, we allow households to boow in financial makets. 4.1 Nonpaticipation We now extend the two-state model of the labo maket we adopted so fa to a thee-state model whee eithe spouse can choose nonpaticipation. Nonpaticipation means that the individual does not seach fo a job oppotunity. Consistent with the est of the pape, whee we intepet b as income, we model the benefit associated to nonpaticipation z (with z > b) in consumption units (e.g., though home poduction). We now edefine some of the value functions. Fist, conside the two configuations whee (i) neithe spouse paticipates in the labo foce, and (ii) one spouse 13 When consumption is a pivate good within the couple, equation (11) becomes ««w1 + φ (w 1) w1 + b u = u + α» w1 + w «2 w1 + φ (w 1) u u df (w 2) 2 2 φ(w 1 ) 2 2 and, by exploiting the popeties of CARA utility, the esult follows immediately. 14 Incidentally, it is also easy to see that even if consumption is a pivate good, in the DARA case it is not geneally tue that w < w as long as b > Thee is a futhe implicit assumption hee: the aival ate of job offes is popotional to the nonwoking time of the agent (that is, 2α when unemployed and α when woking one job). 16

17 does not paticipate and the othe is employed at wage w. Because of the absence of andomness, both of these states ae absobing, as is the dual-woke state. Theefoe, in the fist case we have T (z, z) = u (2z), and in the second case we have T (z, w 2 ) = u (z + w 2 ). This fomulation shows that nonpaticipation is equivalent to a job oppotunity which pays z (and entails foegoing seach) and is always available to the woke. The value of a dual-seache couple becomes U = u (2b) + 2α max {T (z, w) U, Ω (w) U, 0} df (w), (13) which shows that upon eithe spouse finding an acceptable job, the othe one has the choice of eithe continuing to seach o dopping out of the labo foce. The value when one spouse does not paticipate and the othe is unemployed is Ω (z) = u (z + b) + α max {T (z, w 2 ) Ω (z), Ω (w 2 ) Ω (z), 0} df (w 2 ), (14) which makes clea that when spouse 2 accepts a job offe, spouse 1 can eithe emain out of the labo foce, o stat seaching. Similaly, the value of a woke-seache couple whee spouse 1 is employed is Ω (w 1 ) = u (w 1 + b) + α max {T (w 1, w 2 ) Ω (w 1 ), Ω (w 2 ) Ω (w 1 ), 0} df (w 2 ). (15) The choices available to the couple when spouse 2 finds an acceptable job offe ae eithe spouse 1 emains employed at w 1 o spouse 1 quits into unemployment. This state will aise only fo w 1 > z, since z is always available. 16 As clea fom this equation, once the couple eaches this state, nonpaticipation will neve occu theeafte. This obsevation is impotant, since it means that ou definitions of w, ŵ, and φ (w) emain unchanged and these functions ae independent of z. Poposition 4 [Joint Seach with Nonpaticipation] With eithe CARA o DARA pefeences, the seach behavio of a couple can be chaacteized as follows: (i) If z w, the seach stategy of the couple is unaffected by nonpaticipation, since the latte option is neve optimal. (ii) If w < z < ŵ, dual seach is neve optimal, and wheneve a spouse is unemployed, the othe is eithe employed o a nonpaticipant. The esevation wage of a nonpaticipant-seache couple is z, and the esevation function of a woke-seache couple is the same function φ (w) as in the absence of nonpaticipation. (iii) If z ŵ, nonpaticipation is an absobing state fo both spouses, and seach is neve optimal. 16 Moe pecisely, thee is a thid option in the max opeato which is, theoetically, available to spouse 1: quitting into nonpaticipation and accepting z foeve with a gain T (z, w 2) Ω (w 1) fo the couple. Howeve, the wage gain associated with spouse 1 keeping his/he cuent job, T (w 1, w 2) Ω (w 1), must be lage, since peviously spouse 1 has accepted w 1 when z was available. 17

18 Since nonpaticipation is like a job offe at wage z that is always available, if z < w such offe is neve accepted by a dual-seache couple, and nonpaticipation is neve optimal. When w < z < ŵ, then consumption-smoothing motives induce the jobless couple to move one of its membes into nonpaticipation say, spouse 1 while spouse 2 is seaching with esevation wage φ (z) = z. As soon as a wage offe w 2 lage than z aives, the unemployed spouse accepts the job and spouse 1 switches into unemployment, since seach is equivalent to being employed at φ (w 2 ) ŵ > z. The fist inequality follows fom the CARA o DARA assumption unde which φ is a nondeceasing function. If z ŵ, instead, then both spouses exit the labo foce ight away and no seach occus. As soon as one chooses not to seach, the othe spouse esevation wage becomes φ (z), which is always smalle than z in this egion. As a esult, nonpaticipation is attactive fo the othe spouse as well. 17 The joint-seach poblem is, once again, diffeent fom the single-agent seach poblem. Fo example, in the CARA case whee ŵ = w, we can establish that unde configuation (ii), a single agent would be always seaching and nonemployment would neve aise, wheeas a jobless couple would choose to move one spouse out of the labo foce fo consumption-smoothing puposes. Finally, it is easy to show that the couple will neve be in a state whee one spouse woks and the othe is a nonpaticipant: this state can only occu in the pesence of wealth effects on labo supply, uled out by ou pefeences, o in the pesence of asymmeties between spouses. Howeve, with IARA pefeences, the woke-nonpaticipant configuation may be optimal fo the couple. Intuitively, since φ (w) < 0 (ecall Figue 3), a wage offe w could aive say, to a dual seache couple that is, high enough to induce the couple to accept the offe and set the new esevation wage fo the unemployed membe to φ ( w) < z. Thus, the unemployed membe immediately exits the labo foce. 4.2 On-the-Job Seach Suppose that agents can seach both off and on the job. Duing unemployment, they daw a new wage fom F (w) at ate α u, wheeas duing employment they sample new job offes fom the same distibution F at ate α e. What we develop below is, essentially, a vesion of the Budett (1978) wage ladde model with couples. The flow value functions in this case ae U = u (2b) + 2α u max {Ω (w) U, 0} df (w), (16) Ω (w 1 ) = u (w 1 + b) + α u max {T (w 1, w 2 ) Ω (w 1 ), Ω (w 2 ) Ω (w 1 ), 0} df (w 2 ) (17) + α e max { Ω ( w 1) Ω (w1 ), 0 } df ( w 1), 17 In ode to save space, we do not epesent gaphically this vesion of the model. Fo the CARA case, fo example, it is immediate to see that one can geneate the gaph with nonpaticipation coesponding to configuation (ii) by ovelapping a squaed aea with coodinates (x, y) = (z, z) to Figue 2. This aea would substitute the dual-seache couple with the nonpaticipant-seache couple. 18

19 T (w 1, w 2 ) = u (w 1 + w 2 ) + α e + α e max { T ( w 1, ) w 2 T (w1, w 2 ), 0 } df ( w 1 ) max { T ( w 1, w 2) T (w1, w 2 ), 0 } df ( w 2). (18) We continue to denote the esevation wage of the dual-seache couple as w and the esevation wage of the unemployed spouse in the woke-seache couple as φ (w 1 ). We now have a new esevation function, that of the employed spouse (in the dual-woke couple and in the wokeseache couple) which we denote by η (w i ). It is intuitive (and can be poved easily) that unde isk neutality the joint-seach poblem coincides with the poblem of the single agent egadless of offe aival ates. Below, we pove anothe equivalence esult that holds fo any isk-avese utility function but fo the special case of symmetic offe aival ates α u = α e, i.e., when seach is equally effective on and off the job. Poposition 5 [On-the-job Seach with Symmetic Aival Rates] If α u = α e, the jointseach poblem yields the same solution as the single-agent seach poblem, even with concave pefeences. Specifically, w = w = b, φ (w 1 ) = w and η (w i ) = w i fo i = 1, 2. To undestand this equivalence esult, notice that one way to think about joint seach is that it povides a way to climb the wage ladde fo the couple even without on-the-job seach: when a dual-seache couple accepts the fist job offe, it continues to eceive offes, albeit at a educed aival ate. Theefoe, one can view joint seach as a costly vesion of on-the-job seach. The cost comes fom the fact that, absent on the job seach, in ode to keep the seach option active, the pai must emain a woke-seache couple and cannot enjoy the full wage eanings of a dual-woke couple as it would be capable of doing with on-the-job seach. As a esult, when on-the-job seach is explicitly intoduced and the offe aival ate is equal acoss employment states, it completely neutalizes the benefits of joint seach and makes the poblem equivalent to that of a single agent. The solution is then simply that the unemployed patne should accept any offe above b and the spouse employed at w 1 any wage above its cuent one. The peceding poposition povides an altenative benchmak to the baseline model, which had α u > α e 0. The empiically elevant case is pobably in between these two benchmaks, in which case joint-seach behavio continues to be qualitatively diffeent fom single seach (fo example, the beadwinne cycle will be active). We povide some simulations in Section 6.1 below to illustate these intemediate cases. 4.3 Exogenous Sepaations As discussed above, in the absence of exogenous sepaations, agents optimally choose not to accumulate assets, so a simple no-boowing constaint ensues that agents live as hand-to-mouth consumes. This is no longe tue when exogenous sepaation isk is intoduced, because in this 19

20 case accumulated assets can be used to smooth consumption when agents lose thei jobs. This saving motive, howeve, significantly complicates the analysis. Thus, to establish some geneal theoetical esults, we ule out savings in this section. Suppose that jobs ae exogenously lost at ate δ and that upon a job loss, wokes ente unemployment. Once again, unde isk neutality it is easy to establish that the joint-seach poblem collapses to the single-agent poblem. With isk avesion, howeve, this is not the case anymoe. The modifications to the value functions ae immediate, so we omit them. The following poposition states ou main esult fo this famewok. Poposition 6 [CARA/DARA Utility with Exogenous Sepaations] With CARA o DARA pefeences, no access to financial makets, and exogenous job sepaation, the seach behavio of a couple can be chaacteized as follows: (i) The esevation wage value of a dual-seache couple satisfies: w < ŵ (with w < ŵ), which implies that the beadwinne cycle exists. (ii) The esevation wage function of a woke-seache couple has the following popeties: fo w 1 < ŵ, φ (w 1 ) = w 1, and fo w 1 ŵ, φ (w 1 ) is stictly inceasing with φ < 1. Two emaks ae in ode. Fist, fo DARA pefeences, the existence of exogenous sepaations has qualitatively no effect on joint-seach behavio, as can be seen by compaing Popositions 3 and Second, and pehaps moe inteestingly, fo CARA pefeences φ (w 1 ) is no longe independent of the employed spouse s wage but is now inceasing with it. 19 In the context of joint-seach, the sepaation isk has two sepaate meanings. Conside the poblem of the woke-seache couple with cuent wage w 1 contemplating a new job offe with wage w 2. Fist, thee is the isk associated with the duation of the new job offeed to the seaching spouse. Second, thee is the isk of job loss fo the cuently employed spouse. 20 The fist effect of exogenous sepaations is also pesent in the single-agent seach model: if the expected duation of a job is lowe (high δ), the unemployed agent educes he esevation wage fo all values of w 1. Howeve, the lage the wage w 1 of the employed spouse, the smalle this effect, since the utility value fo the household of the additional wage deceases in w 1. Since φ (w 1 ) is weakly inceasing in the case δ = 0, with δ > 0 we obtain φ (w 1 ) > The only diffeence is that hee we explicitly ule out saving, wheeas pevious popositions did not equie this assumption, as explained befoe. Apat fom this stonge assumption, the seach behavio with DARA utility is the same with and without sepaations. 19 Based on the poof of Poposition 6, it is possible to show that unde CARA pefeences if income duing unemployment fo each spouse is fully popotional to the wage eaned in the last employment spell (an appoximation to a UI system with eplacement ate used, fo example, by Postel-Vinay and Robin, 2002), then the esevation function φ is still constant, even with exogenous sepaations. 20 In a model with spouse asymmeties in sepaation ates, this would be even moe clea, since we would have a pai (δ 1, δ 2) in the value functions as opposed to just δ. 20

21 The second effect is elated to the event that the cuently employed spouse might lose his job. If the couple tuns down the offe at hand and the job loss indeed occus, its eanings will fall fom w 1 + b to 2b fo a net change of b w 1 < 0. Clealy, this income loss (and, theefoe, the fall in consumption) is lage, the highe is the cuent wage of the employed spouse. If instead the couple accepts the job offe and spouse 1 loses his job, eanings will change fom w 1 + b to b + w 2, fo a net change of w 2 w 1. On the one hand, setting the esevation wage to φ (w 1 ) = w 1 would completely insue the downside isk of spouse 1 losing his job (because then w 2 w 1 0). At the same time, letting the esevation wage ise this quickly with w 1 educes the pobability of an acceptable offe and inceases the pobability that the seache will still be unemployed when spouse 1 loses his job. As a esult, the optimal policy balances these two consideations to povide the best self-insuance to the couple and, consequently, have φ (w 1 ) ise with w 1, but less than one fo one: φ < Boowing in Financial Makets With few exceptions, seach models with isk-avese agents and a boowing-saving decision do not allow analytical solutions. 22 have access to a isk-fee asset. One such exception is when pefeences display CARA and agents This envionment has been ecently used in pevious wok to obtain analytical esults in the context of the single-agent seach poblem (e.g., Acemoglu and Shime (1999), and Shime and Wening (2008)). Following this tadition, we stat fom the CARA famewok studied in Section 3.3.1, extended to allow fo boowing. Befoe analyzing the jointseach poblem, it is useful to ecall hee the solution to the single-agent poblem. Single-agent seach poblem. Let a denote the asset position of the individual. Assets evolve accoding to the law of motion, da = a + y c, (19) dt whee is the isk-fee inteest ate, y is income (equal to w duing employment and b duing unemployment), and c is consumption. single agent ae, espectively: W (w, a) = max c The value functions fo the employed and unemployed {u (c) + W a (w, a) (a + w c)}, (20) max {W (w, a) V (a), 0} df (w), (21) V (a) = max {u (c) + V a (a) (a + b c)} + α c whee the subscipt denotes the patial deivative. These equations eflect the non-stationaity due to the change in assets ove time. Fo example, the second tem in the RHS of (20) is (dw/dt) = (dw/da) (da/dt). And similaly fo the second tem in the RHS of (21). 21 This mechanism is closely elated to Lise (2007), in which individuals climb the wage ladde but fall to the same unemployment benefit level upon layoff. As a esult, in his model, the savings ate inceases with the cuent wage level, wheeas this inceased pecautionay savings demand manifests itself as delayed offe acceptance in ou model. 22 Some examples in which the decision make is an individual ae Costain (1999), Lentz and Tanaes (2005), Rendon (2006), Bowning, Cossley, and Smith (2007), Lise (2007), Rudanko (2008), Kusell et al. (2009), and Lentz (2009). 21

22 We begin by conjectuing that W (w, a) = u (a + w). If this is the case, then the fist-ode condition (FOC) detemining optimal consumption fo the agent gives u (c) = u (a + w), which confims the conjectue and establishes that the employed individual consumes his cuent wage plus the inteest income on the isk-fee asset. Let us now guess that V (a) = u (a + w ). Once again, it is easy to veify this guess though the FOC of the unemployed agent. Substituting this solution back into equation (21) and using the CARA assumption yields w = b + α [u (w w ) + ρ] df (w), (22) w which shows that w is the esevation wage and is independent of wealth. Theefoe, the unemployed woke consumes the esevation wage plus the inteest income on his wealth. This esult highlights an impotant point: the asset position of an unemployed woke deteioates and, in pesence of a debt constaint, she may hit it. As in the est of the papes cited above which use this setup, we abstact fom this possibility. The implicit assumption is that boowing constaints ae loose, and by this we mean they do not bind along the solution fo the unemployed agent. Joint-seach poblem. When the couple seaches jointly fo jobs, the asset position of the couple still evolves based on (19), but now y = 2b fo the dual-seache couple, b+w 1 fo the woke-seache couple, and w 1 + w 2 fo the employed couple. The value functions become T (w 1, w 2, a) = max c {u (c) + T a (w 1, w 2, a) (a + w 1 + w 2 c)}, (23) max {Ω (w, a) U (a), 0} df (w), (24) U (a) = max {u (c) + U a (a) (a + 2b c)} + α c Ω (w 1, a) = max {u (c) + Ω a (w 1, a) (a + w 1 + b c)} (25) c + α max {T (w 1, w 2, a) Ω (w 1, a), Ω (w 2, a) Ω (w 1, a), 0} df (w 2 ). Solving this poblem equies chaacteizing the optimal consumption policy fo the dual-seache couple c u (a), fo the woke-seache couple c eu (w 1, a), and fo the dual-woke couple c e (w 1, w 2, a), as well as the esevation wage functions, now potentially a function of wealth too, which must satisfy, as usual: Ω (w (a), a) = U (a), T (w 1, φ (w 1, a), a) = Ω (w 1, a), and Ω (φ (w 1 ), a) = Ω (w 1, a). The following poposition chaacteizes the solution to this poblem. Poposition 7 [CARA Utility with Boowing-Saving] With CARA pefeences, access to isk-fee boowing and lending, and loose debt constaints, the seach behavio of a couple can be chaacteized as follows: (i) The optimal consumption policies ae: c u (a) = a + 2w, c eu (w 1, a) = a + w + w 1, and c e (w 1, w 2, a) = a + w 1 + w 2. (ii) The esevation function φ of the woke-seache couple is independent of (w 1, a) and equals w, so thee is no beadwinne cycle. 22

23 (iii) The esevation wage w of the dual-seache couple equals w, the esevation wage of the single agent. The main message of this poposition could pehaps be anticipated by the fact that boowing effectively substitutes fo the consumption smoothing povided within the household, making the latte edundant. Each spouse can implement seach stategies that ae independent fom the othe spouse s actions and, as a esult, each acts as in the single-agent model. Of couse, to the extent that boowing constaints bind o pefeences deviate fom CARA, the equivalence esult no longe applies. 5 Joint Seach with Multiple Locations The impotance of the geogaphical dimension of job seach is undeniable. Fo the single-agent seach poblem, accepting a job in a diffeent maket could equie a elocation cost high enough to induce the agent to tun down the offe. In the joint-seach poblem, this spatial dimension intoduces an additional and inteesting seach fiction with impotant amifications as we show in this section. Basically, a couple is likely to suffe fom the disutility of living apat if spouses wok in diffeent locations. This cost can easily ival o exceed the physical cost of elocation, since it is a flow cost as opposed to the latte, which is aguably bette thought of as a one-time cost. To analyze the joint-seach poblem with multiple locations, we extend the famewok poposed in Section 2 by intoducing a fixed flow cost of living sepaately fo a couple. The intoduction of location choice leads to impotant changes in the seach behavio of couples compaed to a single agent, even with isk neutality. To make this compaison shape, we focus pecisely on the iskneutal case. Futhemoe, many of these changes ae not favoable to couples, which seves to show that joint seach can itself ceate new fictions as opposed to the new oppotunities analyzed in the fist pat of the pape. 23 To keep the analysis tactable, we fist conside agents that seach fo jobs in two symmetic locations and povide a theoetical chaacteization of the solution. In the next subsection, we examine the moe geneal case with L(> 2) locations that is moe suitable fo a meaningful calibation, and povide some esults based on numeical simulations. 5.1 Two Locations Envionment. A couple is an economic unit composed of a pai of isk-neutal spouses (1, 2). The economy has two locations. Couples incu a flow esouce cost, denoted by κ, if the two spouses live apat. Denote by i the inside location, i.e., the location whee the couple esides, and by o the 23 This fiction aises the issue of whethe the couple should split. While the inteaction between labo maket fictions and changes in maital status is a fascinating question, it is beyond the scope of this pape. Hee we assume that the couple has committed to stay togethe o, equivalently, that thee is enough idiosyncatic non-monetay value in the maiage to justify continuing the elationship. 23

24 outside location. Unemployed individuals eceive job offes at ate α i fom the cuent location and at ate α o fom the outside location, e.g., job seach in the inside location is moe effective with α i > α o. The two locations have the same wage offe distibution F. We assume away moving costs: the aim of the analysis is the compaison with the single-agent poblem, and such costs would also be bone by the single agent. A couple can be in one of fou labo maket states. Fist, if both spouses ae unemployed and seaching, they ae efeed to as a dual-seache couple. Second, if both spouses ae employed in the same location (in which case they will stay in thei jobs foeve) we efe to them as a dual-woke couple, but if they ae employed in diffeent locations we efe to them as a sepaate dual-woke couple (anothe absobing state). Finally, if one spouse is employed and the othe one is unemployed, we efe to them as a woke-seache couple. Because of symmety in locations, couples with seaches have no advantage fom living sepaately, so they will choose to live in the same location. Let U, T (w 1, w 2 ), S (w 1, w 2 ), and Ω (w 1 ) be the value of these fou states, espectively. Then, we have T (w 1, w 2 ) = w 1 + w 2 (26) S (w 1, w 2 ) = w 1 + w 2 κ (27) U = 2b + 2 (α i + α o ) max {Ω (w) U, 0} df (w) (28) Ω (w 1 ) = w 1 + b + α i max {T (w 1, w 2 ) Ω (w 1 ), Ω (w 2 ) Ω (w 1 ), 0} df (w 2 ) (29) + α o max {S (w 1, w 2 ) Ω (w 1 ), Ω (w 2 ) Ω (w 1 ), 0} df (w 2 ). The fist thee value functions ae easily undestood and do not equie explanation. The value function fo a woke-seache couple now has to account sepaately fo inside and outside offes. If an inside offe aives, the choice is the same as in the one-location case, since no cost of living sepaately is incued. If, howeve, an outside offe is eceived, the unemployed spouse may accept the job, in which case the couple has two options: eithe it chooses to live sepaately incuing cost κ, o the employed spouse quits and follows the newly employed spouse to the new location to avoid the cost. The decision of the dual-seache couple is entiely chaacteized by the esevation wage w. Fo the woke-seache couple, let φ i (w 1 ) and φ o (w 1 ) be the esevation functions coesponding to inside and outside offes. Once again, these functions ae piecewise with one piece coesponding to the 45-degee line. By inspecting equation (29), it is immediate that, as in the one-location case, the same functions φ i (w 2 ) and φ o (w 2 ) chaacteize the quitting decision. Single-agent seach. Befoe poceeding futhe, it is staightfowad to see that the single-seach poblem with two locations is the same as the one-location case, with the appopiate modification to the esevation wage to account fo sepaate aival ates fom two locations. In the isk-neutal 24

25 Figue 5: Resevation Wage Functions fo Outside (Left) and Inside (Right) Offes case, we have w = b + α i + α o [1 F (w)] dw. (30) w Recall that in the one-location case, isk neutality esulted in an equivalence between the singleseach and joint-seach poblems. As the next poposition shows, this esult no longe holds in the two-location case, wheneve thee is a positive cost κ of living apat. Poposition 8 [Two-Location with Risk Neutality] With isk neutality, two locations, and κ > 0, the seach behavio of a couple can be chaacteized as follows. Thee is a wage value ŵ S = b + κ + α i [1 F (w)] dw + α o [1 F (w)] dw ŵ S κ ŵ S and a coesponding value ŵ T = ŵ S κ such that: (i) w (ŵ T, ŵ), wheeas w (ŵ, ŵ S ). Theefoe, w < w, which implies that the beadwinne cycle exists. (ii) Fo outside offes, the esevation wage function of a woke-seache couple has the following popeties: fo w 1 < ŵ S, φ o (w 1 ) = w 1, and fo w 1 ŵ S, φ o (w 1 ) = ŵ S. (iii) Fo inside offes, the esevation wage function of a woke-seache couple has the following popeties: fo w 1 < ŵ, φ i (w 1 ) = w 1, fo w 1 (ŵ, ŵ S ), φ i (w 1 ) is stictly deceasing, and fo w 1 ŵ S, φ i (w 1 ) = ŵ T. The fist useful esult is that the dual-seache couple is less choosy than the individual agent because it is effectively facing a wose job offe distibution: some wage offe configuations ae attainable only in diffeent locations, hence by paying the cost of living apat. Figue 5 gaphically 25

26 show the esevation wage functions fo outside offes and inside offes, espectively. As seen in these figues, the esevation wage functions fo both inside and outside offes ae quite diffeent fom the coesponding ones of the model with one location (Figue 1). In paticula, the esevation wage functions fo both inside and outside offes now depend on the wage of the employed spouse at least when w 1 (w, ŵ S ). This has seveal implications. Conside fist outside offes fo a woke-seache couple whee one spouse is employed at w 1 < ŵ S (left panel). The couple will eject wage offes below w 1, but when faced with a wage offe above w 1, the employed woke will quit his job and follow his spouse to the outside location. The new wage offe is too high to be foegone, but the cost κ is too lage to justify living apat while being employed at such wages. In this egion, the beadwinne cycle is active acoss locations. In contast, when w 1 > ŵ S if the couple eceives a wage offe w 2 > ŵ S, it will bea the cost of living sepaately in ode to maintain both high wages. Compaing the ight panel fo inside offes to the left panel (outside offes), it is immediate that the ange of wages fo which inside offes ae accepted by a woke-seache couple is lage, since no cost κ has to be paid. Inteestingly, the esevation function φ i (w 1 ) now has thee distinct pieces. Fo w 1 lage enough, it is constant, as in the single-agent case. In the intemediate ange (ŵ, ŵ S ) the function is deceasing. This phenomenon is linked to the esevation function fo outside offes φ o, which is inceasing in this ange: as the wage w 1 fom employment in the inside location ises, the expected gains fom seach accuing though outside offes ae lowe (it takes a highe outside wage offe w 2 to induce the employed spouse to quit) and the esevation wage fo inside offes falls. Fo w 1 small enough, the esevation function φ i (w 1 ) is inceasing and equal to the wage of the employed spouse. In this egion, the beadwinne cycle is again active. Howeve, if the wage offe is high enough, the couple accepts it and etains its cuent wage becoming a dual-woke couple. In this multiple location model, we obtained two esults that wee also pesent in ou pevious envionment with one location and isk-avesion: (i) the couple being less picky than the individual, and (ii) the beadwinne cycle. As explained, the analogy stops hee, since the economic intuition is completely diffeent in the two models. Tied-moves and tied-stayes. In a seminal pape, Mince (1978) has studied empiically the job-elated migation decisions of couples in the United States (duing the 1960s and 1970s). Following the teminology intoduced by Mince, we efe to a spouse who ejects an outside offe that she would accept when single as a tied-staye. Similaly, we efe to a spouse who follows he spouse to the new destination even though he individual calculus dictates othewise as a tied-move. Using data fom the 1962 Bueau of Labo Statistics (BLS) suvey of unemployed pesons, Mince estimated that 22 pecent o two-thids of the wives of moving families would be tied-moves, while 23 pecent out of 70 pecent of wives in families of stayes declaed themselves to be tied-stayes (page 758) Moe pecisely, Mince (1978) defines an individual to be a tied-staye (a tied-move) if the individual cites his/he spouse s job as the main eason fo tuning down (accepting) a job fom a diffeent location: Mince wote (page 758): 26

27 Figue 6: Tied-Stayes and Tied-Moves in the Joint-Seach Model Figue 6 e-daws the esevation wage functions fo outside offes and indicates the egions that give ise to tied-stayes and tied-moves in ou model. Fist, if the wage of the employed spouse, w 1, is highe than w, then the unemployed spouse ejects outside offes and stays in the cuent location fo all wage offes less than φ i (w 1 ). In contast, a single agent would accept all offes w 2 above w, which is less than φ i (w 1 ) by Poposition 8. Theefoe, an unemployed spouse who ejects an outside wage offe w 2 (w, φ i (w 1 )) is fomally a tied-staye (as shown in Figue 6). Thee is a egion in which the employed spouse is a tied-move. Suppose the wage of the employed spouse, w 1, is between w and ŵ S, and the unemployed spouse eceives an outside wage offe highe than w 1, then the unemployed spouse accepts the offe, the employed spouse quits the job, and both move to the othe location. The employed spouse would not move to the othe location if she wee single, since she would not be seaching any longe, so the employed spouse is a tied-move (see Figue 6). Both sets of choices involve potentially lage concessions by each spouse compaed to the situation whee he/she wee single, but they ae optimal fom a joint decision pespective. This featue opens the possibility of welfae costs of being in a couple vesus being single with espect to job seach, an aspect of the model which we analyze quantitatively, though simulation, in the next section. Finally, we note that the isomophism to the single-agent seach model with multiple job holding extends to this set up as well. It is enough to think of κ as a commuting cost the agent would incu when holding two jobs in diffeent locations. The unemployed wee asked whethe they would accept a job in anothe aea compaable with the one they lost. A positive answe was given by 30 pecent of the maied men, 21 pecent of the single women, and only 8 pecent of the maied women. Most people who said no cited family, home, and elatives as easons fo the eluctance to move. Howeve, one quate of the women singled out thei husbands job in the pesent aea as the majo deteent facto. 27

28 6 Some Illustative Simulations In this section, ou goal is to gain some sense about the quantitative diffeences in labo maket outcomes between the single-agent seach and the joint-seach economy. We begin with the onelocation model and then we tun to the multiple location model. 6.1 Single Location Model Ou benchmak is the case of CRRA utility (moe common than CARA o IARA in macoeconomics) and exogenous sepaations. Late we add on-the-job seach. The economy is chaacteized by the following set of paametes: {b,, ρ, δ, F, α u, α e }. When on-the-job seach is not allowed, we simply set α e = 0 and α α u. We fist simulate labo maket histoies fo a lage numbe of individuals acting as singles, then compute thei optimal choices and some key statistics: esevation wage w, mean wage, unemployment ate, and unemployment duation. Second, we pai individuals togethe and teat them as couples solving the joint-seach poblem in exactly the same economy (i.e., same set of paametes). We use the same sequence of wage offes and sepaation shocks fo each agent in both economies. The inteest of the execise lies in compaing the key labo maket statistics acoss economies. Fo example, it is not obvious whethe the joint-seach model would have a highe o lowe unemployment ate: fo the dual-seache couples, w < w, but fo the woke-seache couple φ (w) is above w at least fo lage enough wages of the employed spouse. Calibation. We calibate the model (with singles) to eplicate some salient featues of the US economy. The time peiod in the model is set to one week of calenda time. The coefficient of elative isk avesion ρ will vay fom zeo (isk neutality) up to eight in simulations. The weekly net inteest ate,, is set equal to 0.001, coesponding to an annual inteest ate of 5.3%. Wage offes ae dawn fom a log-nomal distibution with standad deviation σ = 0.1 and mean µ = σ 2 /2 so that the aveage wage is always nomalized to one. We set δ = , which coesponds to a monthly employment-unemployment (exogenous) sepaation ate of Fo each isk avesion value, the offe aival ate, α u, is ecalibated to geneate an unemployment ate of oughly Fo the model with on-the-job seach, we set the offe aival ate on the job, α e, to match a monthly employment-employment tansition ate of Finally, the value of leisue b is set to 0.40, i.e., 40% of the mean of the wage offe distibution. Table 1 epots the esults of ou simulation. The fist two columns confim the statement in Poposition 1 that unde isk neutality the joint-seach poblem educes to the single-seach poblem. Let us now conside the case with ρ = 2. The esevation wage of the dual-seache couple is almost 25% lowe than in the single-seach economy. And this is eflected in the much 25 As isk avesion goes up, w falls and unemployment duation deceases. So, to continue matching an unemployment ate of 5.5%, we need to decease the value of α u. Fo example, fo ρ = 0, α u = 0.4 and fo ρ = 8, α u =

29 Table 1: Single vesus Joint Seach: CRRA Pefeences ρ = 0 ρ = 2 ρ = 4 ρ = 8 Single Joint Single Joint Single Joint Single Joint Res. wage (w o w ) Res. wage (φ (1)) n/a Double ind. (ŵ) Mean wage Mm atio Unemp. ate 5.5% 5.5% 5.4% 7.6% 5.4% 7.7% 5.3% 5.6% Unemp. duation Dual-seache Woke-seache Quits/Sepaations 0% 11.1% 5.5% 0.7% EQVAR- cons. 0% 4.5% 14% 26% EQVAR- income 0% 1.1% 2.8% 0.7% shote unemployment duations of dual-seache couples. At the same time, though, the esevation wage of woke-seache couples is always highe than w. In the second ow of the table, we epot the esevation wage of the woke-seache couple at the mean wage offe. Indeed, fo these couples, unemployment duation is highe. Oveall, this second effect dominates and the jointseach economy displays a longe aveage unemployment duation 12.6 weeks instead of 9.7 and a consideably highe unemployment ate, 7.6% instead of 5.4%. Compaing the mean wage tells a simila stoy. The job-seach choosiness of woke-seache couples dominates the insuance motive of dual-seache couples, and the aveage wage is highe in the joint-seach model. The ability of the couple to climb highe up the wage ladde is eflected in the endogenous quit ate (leading to the beadwinne cycle), which is sizeable, 11.1% of all sepaations ae quits. Indeed, the egion in which the beadwinne cycle is active is athe big, as measued by the gap between w and ŵ, which is equal to 2.7 times the standad deviation of the wage offe. The next fou columns in Table 1 display how these statistics change as we incease the coefficient of elative isk avesion. As is clea fom the fist ow, in the case when ρ = 0 the diffeence between w and w is zeo. As ρ goes up, both esevation wages fall. Clealy, highe isk avesion implies a stonge demand fo consumption smoothing, which makes agents accept job offes moe quickly. Howeve, the gap between w and w fist gows but then shinks. Indeed, as ρ, it must be tue that w = w = b, so the two economies convege again. As fo φ (1), it falls as isk avesion inceases, which means that fo highe values of ρ, the woke-seache couple accepts job offes moe quickly, thus educing unemployment. Indeed, at ρ = 8 the unemployment ate and the mean wage ae almost the same in the two economies. We also epot a measue of fictional wage dispesion, the mean-min atio (Mm), defined as the 29

30 Table 2: Single vesus Joint Seach: CRRA Pefeences and On-the-Job Seach ρ = 0 ρ = 2 ρ = 2 ρ = 4 α u = 0.2 α u = 0.1 α u = 0.11 α u = 0.11 α e = 0.03 α e = 0.1 α e = 0.02 α e = 0.02 Single Joint Single Joint Single Joint Single Joint Res. wage (w o w ) Res. wage (φ (1)) Double ind. (ŵ) Mean wage Mm atio Unemp. ate 5.4% 5.4% 5.4% 5.4% 5.3% 5.8% 5.3% 5.4% Unemp. duation Dual-seache Woke-seache EU Quits/Sepaations 0% 0% 0.9% 0.2% EQVAR-cons. 0% 4.6% 4.1% 15% EQVAR-income 0% 0% 0.2% 0.1% atio between the mean wage and the lowest wage, i.e., the esevation wage. Honstein, Kusell, and Violante (2009) demonstate that the sequential seach model with homogeneous wokes, when plausibly calibated, geneates vey little fictional wage dispesion. The fifth ow of Table 1 confims this esult. It also confims the finding in Honstein et al. that the Mm atio inceases with isk avesion. What is novel hee is that the joint-seach model geneates moe fictional dispesion: the esevation wage fo the dual-seache couple is lowe, but the couple can climb the wage distibution faste which tanslates into a highe aveage wage. Next, we discuss two sepaate measues of the welfae effects of joint seach in the simulated economy. Recall that the jointly seaching couple has two advantages: fist, it can smooth consumption bette, and second, it can get highe eanings. The fist measue of welfae gain is the standad consumption-equivalent vaiation and embeds both advantages. The second is the change in lifetime income fom being maied, which isolates the second aspect the novel one. 26 The consumption-based measue of welfae gain is vey lage, not supisingly. What is emakable is that also the gains in tems of lifetime income can be vey lage fo example, aound 2.8% fo the case ρ = 4. As isk avesion goes up, the welfae gains fom family insuance keep inceasing, but as explained above, the ones stemming fom bette seach oppotunities fade away. Table 2 pesents the esults when on-the-job seach is intoduced into this envionment. The fist fou columns simply confim the theoetical esults established in pevious sections. Fo example, 26 To make the welfae compaison between singles and couples meaningful, we assume that each spouse consumes half of the household s income (as opposed to all income assumed in the theoetical analysis). Recall that, with DARA pefeences, this altenative assumption does not affect any of ou theoetical esults. 30

31 when agents ae isk neutal, on-the-job seach has no additional effect, and both the single-agent and joint seach poblems yield the same solution egadless of paamete values. Similaly, as shown in Poposition 5 when on-the-job seach is as effective as seach duing unemployment (α e = α u ), then, again, single-agent and joint seach coincide. Oveall, compaing these esults to those in Table 1 shows that the effects of joint seach on labo maket outcomes ae qualitatively the same as befoe, but they become much smalle quantitatively. This is pehaps not supising in light of the discussion in Section 4, whee we agued that joint seach is a patial substitute fo on-the-job seach (o a costly vesion of it). Theefoe, once on-thejob seach is available, having a seach patne is not so useful any longe to obtain highe eanings. But it obviously emains an effective means to smooth consumption, as evident fom the last two lines of the table. 6.2 Multiple Location Model The two-location case seves as a convenient benchmak to illustate all the key mechanisms. Fo the simulation execise, we extend the famewok descibed above to multiple locations and allow exogenous sepaations. Specifically, conside an economy with L geogaphically sepaate symmetic labo makets. Fims in each location geneate offes at flow ate ψ. A faction θ of total offes ae distibuted equally to the L 1 outside locations and the emaining (1 θ) is made to the local maket. 27 The value functions coesponding to this economy ae povided in Appendix B and ae staightfowad extensions of the value functions in (26) (29). The numbe of locations, L, is set to nine epesenting the numbe of US census divisions and θ is set to 1 1/L, implying that fims make offes to all locations with equal pobability. The emaining paamete values ae exactly the same as in the one-location model with isk neutality (see Section 6.1). Table 3 pesents the simulation esults. A compaison of the fist two columns confims that the single-agent and joint seach poblems ae equivalent when thee is no disutility fom living apat (κ = 0). The thid and fouth columns show the esults when κ = 0.1 and κ = 0.3, espectively epesenting a flow cost equal to 10% and 30% of the mean offeed wage. Fist, the esevation wages ae in line with ou theoetical esults in Poposition 8: ŵ T < w < w < ŵ S. Second, the pesence of the cost κ makes outside offes less appealing, inducing the couple to eject some offes that a single peson would accept. As a esult, the unemployment ate is highe in the joint-seach economy. Fo example, when κ = 0.3 the unemployment ate is 13.7% compaed to 5.5% in the single-agent model. Howeve, the aveage duation of unemployment is not necessaily longe unde joint seach: when κ = 0.1 the aveage duation falls to 9.8 weeks fom 9.9 weeks in the single agent case, but ises to 13 weeks when κ is futhe aised to 0.3. The next two ows decompose 27 The assumption that thee is a vey lage numbe of individuals in each location, combined with the fact that the envionment is stationay (i.e., no location-specific shocks), implies that we can take the numbe of wokes in each location as constant, despite the fact that wokes ae fee to move acoss locations and acoss employment states depending on the offes they eceive. 31

32 Table 3: Single vesus Joint Seach: Nine Locations and Risk-Neutal Pefeences κ = 0 κ = 0.1 κ = 0.3 Single Joint Joint Joint Res. wage (w o w ) ŵ T Double indiff. point (ŵ) ŵ S Res. wage (φ i (1)) n/a Mean wage Mm atio Unemployment ate 5.5% 5.5% 6.9% 13.7% Unemployment duation Dual-seache Woke-seache Moves (% of population) 0.5% 0.5% 0.7% 1.3% Stayes (% of population) 1.1% 1.1% 1.5% 3.4% Tied-moves/Moves 0% 29% 56% Tied-stayes/Stayes 0% 11% 23% Quits/Sepaations 0% 23% 50% EQVAR-income 0% 0.8% 6.5% aveage unemployment duation into the component expeienced by dual-seache couples and by woke-seache couples. The duation of the fome goup is shote than that of single agents (since w < w ) and gets even shote as κ inceases (falls fom 6.5 weeks to 3 weeks in column 4). Howeve, because woke-seache couples face a smalle numbe of feasible job offes fom outside locations, they have much longe unemployment spells: 12.9 weeks when κ = 0.1 and 28 weeks when κ = 0.3, compaed to 9.3 weeks when κ = 0. Oveall, thee ae moe people who ae unemployed at any point in time, and some of these unemployed wokes those in woke-seache families stay unemployed fo much longe than they would have had they been single, while tying to esolve thei joint-location poblem. We next tun to the impact of joint seach on the migation decision of couples. In ou context, we define to be moves dual-seache and woke-seache couples who move to anothe location because one of the spouses accepts an outside job offe. 28 Similaly, we define a couple to be a staye if eithe membe of the couple tuns down an outside job offe. Using this definition, the faction of moves in the population is 0.52% pe week when κ = 0; it ises to 0.74% when κ = 0.1 and to 1.26% when κ = 0.3. Pat of the ise in the moving ate is mechanically elated to the ise in the unemployment ate with κ: because thee is no on-the-job 28 Howeve, conside a dual-woke couple in which spouses live in sepaate locations. If one of the spouses eceives a sepaation shock and becomes unemployed, she will move to he spouse s location. In this case, the household is not consideed to be a move, since the move did not occu in ode to accept a job. 32

33 Figue 7: Resevation Wage Functions fo Outside (Left) and Inside (Right) Offes When a Wife Has a Highe Sepaation Rate than He Husband seach, individuals only get job offes when they ae unemployed, which in tun inceases the numbe of individuals who accept offes and move. Notice also that while the faction of moves appeas high in all thee cases, this is not supising given that we ae completely abstacting fom the physical costs of moving. Pehaps moe stiking is the fact that almost 56% of all moves ae tied-moves when κ = 0.3, using the definition in Mince (1978) descibed above. The faction of tied-stayes is also sizeable: 21% in the high-fiction case. The faction of employment-to-unemployment tansitions that ae due to voluntay quits is as high as 50% when κ = 0.3. Finally, a compaison of lifetime wage incomes shows that the fiction intoduced by the spatial dimension in joint seach can be substantial: it educes the lifetime income of a couple by about 0.8% (pe peson) compaed to a single agent when κ = 0.1 and by 6.5% when κ = 0.3. Oveall, these esults show that with multiple locations, joint-seach behavio can deviate substantially fom the standad single-agent seach A Solution to the Lentz-Tanaes Gende Asymmety Puzzle? Lentz and Tanaes (2005) and Lentz (2009) have estimated empiically, fom Danish data, how the unemployment duation of spouses in maied couples depends on the eaned income of thei patnes. One would expect a positive elationship. The data, instead, eveal a supising gende asymmety : while the unemployment duation of the wife (and theefoe, the couple s esevation wage) is inceasing in the husband s wage, the unemployment duation of the husband is deceasing in the wife s wage. In this section, we show that the joint-seach famewok with multiple locations is able to qualitatively eplicate this patten of the data to the extent that maied women have a highe exogenous job sepaation ate than maied men. The multiple location model has the potential of 33