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 15, 2008 Abstact Seach theoy outinely assumes that decisions about acceptance/ejection of job offes (hence, about labo maket movements 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 into couples and families, and decisions ae joint. 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 pool income. The objective of the execise, vey much 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 deive the implications 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 couple faces offes fom multiple locations and a cost of living apat, joint-seach featues new fictions and can lead to wose outcomes than single-agent seach. Guvenen thanks the NSF fo financial suppot unde gant SES Univesity of Texas at Austin; bgule@mail.utexas.edu. Univesity of Minnesota and NBER; guvenen@umn.edu New Yok Univesity, CEPR and NBER; gianluca.violante@nyu.edu. 1

2 1 Intoduction In 2000, ove 60% of the US population was maied, the labo foce paticipation of maied women stood at 61%, andin35% of couples, wives eaned at least as much as thei husbands. Put it diffeently, thee is a lage shae of individuals whose labo maket choices ae intimately tied to those of anothe individual. Fo these couples, jobseachisveymuchajointdecision-making pocess. Supisingly, 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 two-peson households acting as a single decision units. This state of affais is athe supising given that Budett and Motensen (1978), in thei seminal piece on Labo Supply Unde Uncetainty, lay out a twopeson seach model and sketch a chaacteization of its solution, explicitly encouaging futhe wok on the topic. This pioneeing effot, which emained vitually unfollowed, epesents the stating point of ou theoetical analysis. Only vey ecently, a enewed inteest seems to have aisen in the study of household inteactions in the context of fictional labo maket models (see, e.g., Dey and Flinn, 2007). Ou theoetical analysis focuses on the 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 by two individuals linkedtoeachothebytheassumptionofpefect income pooling. Thee is an active and gowing liteatue that attempts to undestand the household decision making pocess, and emphasizes deviations fom the unitay model we adopt hee, e.g. Chiappoi (1992). While we agee with the impotance of many of those featues, incopoating them into the pesent famewok will make it hade to compae the outcomes of single-seach and joint-seach poblems. The simple unitay model of a household adopted hee is a convenient stating point, which helps to examine moe tanspaently theoleofthelabomaketfictions and insuance oppotunities intoduced by joint-seach. Fom a theoetical pespective, thee ae numeous easons why couples would make a joint decision leading to choices diffeent fom those of a single agent. We have stated fom the most obvious and natual 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 (hee, deviation fom the single-agent seach poblem occus even fo linea pefeences). Ou theoetical analysis has the appeal of leading to two-dimensional gaphs in the space of the two spouses wages (w 1,w 2 ) whee the esevation wage policies can be easily analyzed and intepeted. As summaized by the title of ou pape, elative to single-agent seach, joint seach induces new oppotunities and new fictions. Fist, the model with concave utility shows that joint-seach woks 2

3 similalytoon-the-jobseachbyoffeing additional job oppotunities to the couple to climb the wage ladde: the esevation wage of the dual-seache couple is lowe than that of the single agent, but (with CARA o DARA utility) the esevation function of the woke-seache couple is always highe. As a esult, the unemployment duations ae longe in geneal, but the mean wage is also highe in the joint-seach economy. A symbol of this popety of joint-seach is what we labelled the beadwinne cycle : spouses may altenate in quitting/seaching and woking accoding to whom eceives the best job offe. In this economy, couple ae always bette off than singles not justbecauseofmoeeffective consumption smoothing. Second, the model with multiple locations and a cost of living apat shows that joint-seach induces new fictions: the model can geneate what Mince called tied-stayes (wokes who tun down a job offe in a diffeent location that would accept as single) and tied-moves (wokes who accept a job offe in the location of the patne who would tun down as single). In this economy, with isk neutality, couple ae always wose off than singles. The set of Popositions poved in the pape fomalizes the new oppotunities and the new fictions in tems of compaison between esevation wage functions of the couple and esevation wage of the single agent. 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: on-the-job seach, exogenous sepaations, access to boowing and saving, and symmeties in labo maket chaacteistics between husband and wives. Section 5 studies an economy with multiple locations, and a cost of living apat fo the cople. Section 6 concludes the pape. All the poofs of the lemmas and popositions in the pape ae in the Appendix. 2 The single-agent seach poblem To wam up, we fist pesent the sequential job seach poblem of a single agent the well-known McCall-Motensen (McCall, 1970; Motensen, 1970) model. This model povides a useful benchmak against which we compae the joint-seach model, which we intoduce in the next section. Fo claity of exposition, we begin with a vey stylized vesion of the seach poblem, and then conside seveal extensions in Section 4. Economic Envionment. Conside an economy populated with 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 3

4 whee isthesubjectiveateoftimepefeence,c (t) is the instantaneous consumption flow at time t, andu ( ) is the instantaneous utility function. 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, ). Thewokeobseves the wage offe, w, and decides whethe to accept o eject it. If he accepts the offe, he becomes employed at wage w foeve. If he ejects the offe, he continues to be unemployed and to eceive job offes. 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. Thee is no access to financial makets, no stoage, so consumption will be equal to the wage eanings. thee ae no exogenous sepaations, and no on the job seach. 1 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: 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. 2 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 0 (w)(1 F (w)) dw, w which equates the instantaneous utility of accepting a job offe paying the esevation wage (left hand side, LHS) 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, the above equation 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. A couple is an economic unit composed by two individuals who ae linked to each othe by 1 Access to financialmakets,onthejobseachandandexogenous job sepaation ae intoduced in Section 4. 2 In the equations above, and in what follows, when we abstain fom specifying the uppe o/and lowe limits of the integal, it is implicit that they should be the uppe o/and lowe bound of the suppot of w. 4

5 the assumption that they pefectly pool income. Given the absence of stoage, households simply consume thei total income in each peiod which is the sum of the wage o benefit income of each spouse. Couples make thei job acceptance/ejection/quit decisions jointly, because each spouse s seach behavio affects the couple s joint 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 one 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,andT (w 1,w 2 ) the value function of a dual-woke couple eaning wages w 1 and w 2.Theflowvalue 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-seach poblem. In the fist case, both spouses stay employed foeve, and the flow value is simply equal to the total instantaneous wage eanings of the household. In the second case, the 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 eceive wage offes at ate α, thetotaloffe aival ate of a dual-seache couple is 2α. Once a wage offe is eceived by eithe spouse, it will be accepted if it esults in a gain in lifetime utility (i.e., Ω (w) U>0), othewise it will be ejected. The value function of a woke-seache couple issomewhatmoeinvolved. Ascanbeseenin equation (6), if a couple eceives a wage offe (which now aives at ate α since only one spouse is unemployed) thee ae thee choices facing the couple. 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 job offe and the employed spouse simultaneously quits his job and stats seaching fo a bette one. As we shall see below, this thid case is the fist impotant diffeence between the joint-seach poblem and the single-agent seach poblem. In the single-seach poblem, once an agent accepts ajoboffe, she will neve choose to quit he job. This is because an agent stictly pefes being employed to seaching at any wage offe highe than the esevation wage. Because the envionment 5

6 is stationay, the agent will face the same wage offe distibution upon quitting and will have the same esevation wage. As a esult, a single employed agent will neve quit, even if he is given the oppotunity. 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, also the esevation wage of the peviously employed spouse may incease, which could lead to execising the quit option. We etun to this point below and discuss it in moe detail. 3.1 Chaacteizing the couple s decisions 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 the couple makes them simultaneously). 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 is a stictly inceasing function, i.e., Ω 0 (w) > 0 fo all w [0, ). 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,andis 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. 3 A woke-seache couple has two decisions to make. The fist decision is whethe accepting 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. Let s begin by supposing that it is not optimal to execise the quit option. The couple must decide whethe to accept o eject the job offe w 2. The offe 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) 3 Note that no wage below w will eve be accepted by the couple, and theefoe, obseved in this model, which means that we can focus attention on the behavio of value functions and esevation functions fo wages above w. Theefoe, the statements we make below about the popeties of cetain function should be intepeted to apply to those functions only fo w>w, and may o may not apply below that level. 6

7 Suppose now instead that it is optimal to execise the quit option. Then, the job offe will be accepted when Ω (w 2 ) Ω (w 1 ) which implies the esevation ule Ω (φ (w 1 )) = Ω (w 1 ). (9) Given the stict monotonicity of Ω, the esevation wage ule is vey simple: accept the new offe and quit the old job wheneve w 2 w 1. The woke-seache esevation wage function φ ( ) is theefoe piecewise, being composed of (8) and (9) in diffeent anges of the domain fo w 1. The kink of this piecewise function, which always lies on the 45 degee line of the (w 1,w 2 ) space, plays a special ole in chaacteizing the behavio of the couple. We denote this point by (ŵ, ŵ). 4 Since T (ŵ, ŵ) =u (2ŵ), ŵ solves u (2ŵ) =Ω (ŵ). (10) It emains to chaacteize the quitting decision. If T (w 1,w 2 ) Ω (w 2 ) it is optimal fo the employed spouse to quit his job while the unemployed spouse accepts he job offe (that is, this choice yields highe utility than him staying at his job and the couple becoming a dual-woke couple). This inequality implies the indiffeence condition: T (w 1,ϕ(w 1 )) = Ω (ϕ (w 1 )). (11) Two impotant popeties of ϕ should be noted. Fist, ϕ is not necessaily a function, it may be a coespondence. Second, ϕ is the invese of that piece of the φ function defined by (8). Thisiseasilyseen. BysymmetyofT,fom(8) we have that T (φ (w 1 ),w 1 )=Ω(w 1 ),o T w 2,φ 1 (w 2 ) = Ω φ 1 (w 2 ) which compaed to (11) yields the desied esult. Since ϕ = φ 1 then ϕ will also coss the function φ onthe45degeelineatthepointŵ. Theefoe, ŵ is the wage whee the unemployed spouse is indiffeent between accepting and ejecting he offe and the employed patne is indiffeent between keeping and quitting his job. To emphasize this featue, we efe to ŵ as the double indiffeence point. In what follows, we chaacteize the optimal stategy of the couple in the (w 1,w 2 ) space, the wage space. This means establishing the anking between w and ŵ, especially in elation to the single-agent esevation wage w, and studying the function φ. Once we have chaacteized the shape of φ, thatofφ 1 follows immediately. Oveall, these diffeent esevation ules will divide the (w 1,w 2 ) into fou egions. One whee both spouses wok, one whee both spouses seach and the emaining two egions whee spouse one (two) seaches and spouse two (one) woks. 4 Atthisstagewehavenotpovedthatŵ is unique, but it will tun out that it is. 7

8 3.2 Risk-neutality As it will become clea, isk avesion is cental to ou analysis. 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 00 = 0, the joint-seach poblem educes to two independent single-seach poblems. Specifically, the value functions ae: 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-seach poblem, i.e., φ (w 1 )=w = w. Figue 1 shows the elevant esevation wage functions in the (w 1,w 2 ) space. Without loss of geneality we denote the cuent wage of the employed spouse (in a woke-seach couple) by w 1 and display it on the hoizontal axis, and denote the wage offe eceived by the unemployed spouse by w 2 and display it 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 invese (mio image with espect to the 45 degee line) of φ (w 1 ) and is shown by the vetical line at w 1 = w. The intesection of these two lines gives ise to fou egions, in which the couple display distinct behavios. No wage below w is eve accepted in this model. Theefoe, a woke-seache couple will neve be obseved with a wage below w. As a esult, the only wage values elevant fo the employed spouse ae above the φ (w 1 ) function. 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, 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 and thee is a cost of living sepaately fo the couple, then even in the isk neutal case thee is an impotant deviation fom the single-agent seach poblem. 3.3 Risk-avesion To intoduce isk avesion into the pesent famewok we employ pefeences in the HARA (Hypebolic Absolute Risk Avesion) class. This class encompasses seveal well-known utility functions as 8

9 Figue 1: Resevation Wage Functions of a Risk-Neutal Couple. Seach behavio is identical to the single-seach economy. special cases. Fomally, HARA pefeence ae defined as the family of utility functions that have linea isk toleance: u 0 (c) /u 00 (c) =a + τc, whee a and τ ae paametes. 5 This class can be futhe divided into thee sub-classes 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 exponential utility u (c) = e ac /a andisalsoefeedtoas constant absolute isk avesion (CARA). Second, if τ>0then 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) case: u (c) =c 1 ρ / (1 ρ), which obtains when a 0 and τ =1/ρ > 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) = (a c) 2,whichobtains when τ = 1. The esults deived in this section ae elated to Danfoth (1979) who shows that, in the pesence of saving and no exogenous job sepaation, depending on the degee of absolute isk avesion of the utility function, the esevation wage is eithe inceasing o deceasing in wealth. 5 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. 9

10 3.3.1 CARA utility We fist chaacteize the seach behavio of a couple unde CARA pefeences and show that it seves as the wateshed fo the desciption of seach behavio unde HARA pefeences. The following poposition summaizes the optimal seach stategy of the couple. Poposition 2 (CARA utility) With CARA pefeences, the seach behavio of a couple can be completely chaacteized as follows: (i) The esevation wage value of a dual-seache couple is stictly smalle than the esevation wage of single agent: w <w =ŵ. (ii) The esevation wage function of a woke-seache couple is piecewise linea in the employed spouse s wage ( w 1 if w 1 [w,w ) φ (w 1 )= w if w 1 w. Figue 2 povides a visual summay of the contents of this poposition in the wage space. Thee impotant emaks ae in ode. Fist, 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 lowe wage offes (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 he to be moe picky at the stat. Notice that joint-seach playsaolesimilatoon-the-jobseachinthe absence of it. We etun to this point late below. Second, fo a woke-seache couple eaning a wage geate than w,theesevationwage function is constant and equal to w, the esevation wage value of the single unemployed agent. This is because with CARA utility agents attitude towads isk does not change with 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. While the appendix contains a fomal poof of this esult, it is instuctive to sketch the agument behind the poof. To this end, fist suppose that the employed spouse neve quits when his wage w 1 exceeds w. In this case, the esevation wage function fo the unemployed spouse would have to satisfy: u (w 1 + φ (w 1 )) = u (w 1 + b)+ α [u (w 1 + w 2 ) u (w 1 + φ (w 1 ))] df (w 2 ). φ(w 1 ) 10

11 Figue 2: Resevation Wage Functions with CARA Pefeences. With exponential utility we have: u (w 1 + w 2 )= u(w 1 ) u (w 2 ),whichsimplifies the pevious condition by eliminating the dependence on w 1 : u (φ (w 1 )) = u (b)+ α (u (w 2 ) u (φ (w 1 ))) df (w 2 ). φ(w 1 ) Notice that, since the dependence on the employed patne wage w 1 ceases, this condition becomes exactly the same as the one in the single-seach poblem (equation 3) and is thus satisfied by the constant esevation function: φ (w 1 )=w. Moeove, when φ is a constant function, its invese φ 1 (w 1 )=, and thus thee is no wage offe w 2 that can exceed φ 1 (w 1 ) and justify quitting, which in tun justifies ou conjectue that the employed spouse does not quit in the wage ange w 1 >w. Beadwinne cycle. A thid emak, and a key implication of the poposition, is that the esevation wage value of a dual-seache couple w being stictly smalle than w activates the egion whee φ (w 1 ) is stictly inceasing, and in tun gives ise to the beadwinne cycle. Suppose that w 1 (w,w ) and the unemployed spouse eceives a wage offe w 2 >w 1 = φ (w 1 ),whee the equality only holds in the specified egion (w,w ). Because the offe is highe than the woke-seache couple s esevation wage, the unemployed spouse accepts the offe and becomes employed. Howeve, accepting this wage offe also implies w 2 >φ 1 (w 1 )=w 1 which, in tun, implies w 1 <φ(w 2 ). This means that the theshold fo the fist spouse to keep his job now exceeds 11

12 his cuent wage, and he will quit. As a esult, the spouses simultaneously switch oles and tansition fom a woke-seache couple into anothe woke-seache couple with highe wage level. This pocess epeats itself ove and ove again as long as the employed spouse s wage stays in the ange (w,w ), although of couse the identity of the employed spouse (i.e., the beadwinne) altenates. Once both spouses have in hands job offes beyond w, the beadwinne cycle stops and so does the seach pocess DARA utility As noted ealie, DARA utility is of special inteest, since it encompasses the well-known and commonly used CRRA utility specification u (c) =c 1 ρ / (1 ρ). Moe geneally, the coefficient of absolute isk avesion with DARA pefeences is u 00 (c) /u 0 (c) =ρ/(c + ρa), which deceases with the consumption (and hence the wage) level. The following poposition chaacteizes the optimal seach stategy fo couples with DARA pefeences. Poposition 3 (DARA utility) With DARA pefeences, the seach behavio of a couple can be completely 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 0 < 1. Figue 3 povides a gaphical epesentation of the esevation wage functions associated to the DARA case. Unlike the CARA case, the esevation function of the woke-seache couple does not featue a constant piece. It depends on the wage of the employed spouse at all wage levels. This is because with DARA utility, absolute isk avesion falls with household esouces. Theefoe, as the wage of the employed spouse inceases, the couple becomes less concened about smoothing consumption and becomes moe picky in its job seach. The poposition also shows that the beadwinne cycle continues to exist. In contast to the CARA case, now the beadwinne cycle is obseved ove a wide ange of wage values of the employed spouse. This is because, as can be seen in Figue 3, φ is stictly inceasing in w 1, so its invese is not a vetical line anymoe but is itself an inceasing function. As a esult, even when w 1 > ŵ, asufficiently high wage offe one that exceeds φ 1 (w 1 ) will not only be accepted by the unemployed spouse but it will also tigge the employed spouse to quit. One way to undestand this esult is by noting that the employed spouse will quit if his esevation wage upon quitting is highe than his cuent wage. If w 2 >φ 1 (w 1 ), this implies that upon quitting the job, the 12

13 Figue 3: Resevation Wage Functions with DARA Pefeences (CRRA is a Special Case). esevation wage fo the cuently employed spouse becomes φ (w 2 ) >φ φ 1 (w 1 ) = w 1. Since this esevation wage is highe than his cuent wage, it is optimal fo the employed spouse to quit the job. Finally, note that only if the wage offe is w 2 φ (w 1 ),φ 1 (w 1 ), the job offe is accepted without tiggeing a quit IARA utility We now tun to IARA pefeences, which display inceasing absolute isk avesion as consumption inceases. One well-known example fo IARA utility is quadatic utility: (a c) 2 whee c a. Poposition 4 (IARA utility) With IARA pefeences, the seach behavio of a couple can be completely chaacteized as follows: (i) The esevation wage value of a dual-seache couple satisfies: 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 deceasing. 13

14 Figue 4: Resevation Wage Functions with IARA Pefeences (Quadatic Utility is a Special Case). The poof of the poposition is vey simila to the DARA case, and is theefoe omitted fo bevity. 6 Figue 4 gaphically shows the IARA case. The esevation wage function φ of a woke-seache couple deviates fom the CARA benchmak in the opposite diection of the DARA case. In paticula, beyond wage level ŵ, the esevation function φ (w 1 ) is deceasing in w 1, wheeas it was inceasing in the DARA case. As a esult, if the unemployed spouse eceives a wage offe highe than φ 1 (w 1 ), she accepts the offe, the employed stays in the job and both stay employed foeve. If the wage offe instead is between φ (w 1 ) and φ 1 (w 1 ), then the job offe is accepted followed by a quit by the employed spouse. This behavio is the opposite of the DARA case whee high wage offes esulted in quit and intemediate wages did not. Moeove, now the beadwinne cycle neve happens at wage levels w 1 > ŵ. This is a diect consequence of inceasing absolute isk avesion which induces a couple to become less choosy when seaching as its wage level ises. 6 The logic of the poof is as follows. Guess that at some wage w 1 the employed woke neve quits, and veify the guess by using the popety of IARA equivalent to (30), but with the inequality evesed. The est of the poof is exactly as fo the DARA case. 14

15 4 Extensions The basic famewok in the pevious section was intended to povide the simplest possible deviation fom the well known single-seach poblem in the diection of intoducing couples jointly seaching fo jobs. Despite being highly stylized, this simple famewok illustated some new and potentially impotant mechanisms diving a couple s seach behavio that ae not opeational in the singleagent seach poblem. In this section, we enich this basic model in thee empiically elevant diections. Fist, we add on-the-job seach. Second, we allow fo exogenous job sepaations. Thid, we allow households to access financial makets. We ae able to establish analytical esults in some special cases. We also simulate a calibated vesion of ou model to analyze the diffeences between a single-agent seach economy and the joint-seach economy in moe geneal cases. 4.1 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 while 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 become: U = u (2b)+2α u max {Ω (w) U, 0} df (w) (12) Ω (w 1 ) = u (w 1 + b)+α u +α e max {T (w 1,w 2 ) Ω (w 1 ), Ω (w 2 ) Ω (w 1 ), 0} df (w 2 ) (13) max Ω w 0 1 Ω (w1 ), 0 ª df w 0 1, T (w 1,w 2 ) = u (w 1 + w 2 )+α e max T w1,w 0 2 T (w1,w 2 ), 0 ª df w1 0 +α e max T w 1,w2 0 T (w1,w 2 ), 0 ª df w2 0 (14) We keep denoting 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 woke-seache 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. Below, we pove anothe equivalence esult fo 15

16 the special case of symmetic offe aival ates α u = α e, i.e. when seach is equally effective on and off the job. Poposition5(On-the-jobseachwithsymmeticaivalates)If α u = α e, the joint-seach poblem yields the same solution as the single-agent seach poblem, egadless of pefeences. Specifically, w = w = b, φ (w 1 )=w and η (w i )=w i fo i =1, 2. To undestand this equivalence esult, note 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 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 thus it not enjoy the full wage eanings of a dualwoke 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 fully neutalizes the benefits of joint-seach and makes the poblem equivalent to that of a single-agent. The solution is simply that the unemployed patne should accept any offe above b and the spouse employe at w 1 any wage above its cuent one. 4.2 Exogenous sepaations Once again, unde isk neutality it is easy to establish that the joint-seach poblem collapses to that of the single agent. Howeve, when isk avesion is intoduced new economic foces stat playing a ole. Without exogenous sepaation, the futue wage eanings of the employed spouse ae simply a deteministic income steam constant at least as long as the seaching patne emains unemployed. When a woke-seache couple employed at w 1 sets its esevation wage φ (w 1 ),this wage steam acts as a isk-fee asset in the household s potfolio leading to a wealth effect whose stength depends on the degee of absolute isk avesion. With exogenous sepaations, instead, the employment status becomes stochastic, which makes the futue wage steam of the employed patne effectively a isky asset in the household s potfolio. Fo CARA and DARA pefeences, we can pove the following esult. Poposition 6 (CARA utility with exogenous sepaations) With CARA o DARA pefeences and exogenous job sepaation, the seach behavio of a couple can be completely 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. 16

17 (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 0 < 1. Qualitatively, this case is simila to the DARA case without exogenous sepaations. When pefeences ae CARA, what is that twists the slope of the φ function fom zeo to positive? With exogenous sepaations, the job becomes a isky asset. As w 1 inceases, the gap w 1 b ises and the sepaation isk bone by the couple goes up accodingly. Because of CARA utility, the degee of isk avesion emains constant, but since isk goes up the couple wants to optimally ebalance its potfolio towads the safe asset, which is unemployment. This choice calls fo a ise in the esevation wage φ (w 1 ) DARA utility and the gende asymmety puzzle Lentz and Tanaes (2005) document empiically, fom Danish data, that while the unemployment duation of the wife (hence thei esevation wage) is inceasing in the husband s wage, the unemployment duation of the husband is deceasing in the wife s wage, a fact that they tem the gende asymmety puzzle. By simulation, we can show that the extent that maied women have a highe exogenous sepaation ate than maied man, the joint-seach famewok is able to geneate this phenomenon. Gende-specific diffeences in sepaation ates could aise due to unexpected shocks to household s home poduction needs (such as childeaing, etc.) that may equie the wife to quit he job (moe so than the husband), o to women being oveepesented in moe volatile occupations o sectos. Figue 5 plots the esevation wage functions fo a couple unde this assumption Access to boowing and saving With few exceptions, seach models with isk-avese agents and a saving decision ae typically not amenable to theoetical analysis. 8 One such exception is when pefeences ae of CARA type and agents have access to a isk-fee asset, an envionment that has been used in some pevious wok to obtain analytical esults (Danfoth (1979), Acemoglu and Shime (1999), Shime and Wening (2006)). Following this tadition, we conside the CARA famewok studied in Section extended to boowing and saving. Befoe analyzing the joint seach poblem, it is useful to ecall hee the solution to the single-agent poblem. 7 Weekly male sepaation ate is assumed to be 0 while female sepaation ate is It is theefoe not supising that most studies of seach models with isk-avesion and savings estict attention to quantitative analyses. Fo examples whee the decision make is a household, see Costain (1999), Bowning, Cossley and Smith (2003), Lentz (2005), Lentz and Tanaes (2005), Rendon (2006) and Lise (2006). 17

18 Figue 5: Gende asymmety puzzle. Single-agent seach poblem. Let a denote the asset position of the individual. Assets evolve accoding to the law of motion da dt = a + y c, (15) whee is the isk-fee inteest ate, y is income (equal to w duing employment and b duing unemployment), and c is consumption. The value functions fo the employed and unemployed single agent ae, espectively: W (w, a) = max{u (c)+w a (w, a)(a + w c)}, (16) c V (a) = max{u (c)+v a (a)(a + b c)} + α max {W (w, a) V (a), 0} df (w), (17) c whee the subscipt a denotes the patial deivative with espect to wealth. These equations eflect the non-stationaity due to the change in assets ove time. Fo example, the second tem in (16) is (dw/dt) =(dw/da) (da/dt). And similaly fo the second tem in (17). We begin by conjectuing that W (w, a) =u (a + w). If this is the case, then the FOC detemining optimal consumption fo the agent gives u 0 (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 s now guess that V (a) =u (a + w ). Once gain, it is easy to veify 18

19 this guess though the FOC of the unemployed agent. Substituting this solution back into equation (17) and using the CARA assumption yields w = b + α ρ w [u (w w ) 1] df (w) (18) which shows that w is the esevation wage, which 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 the est of the papes cited above which use this set up, 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 seach jointly fo jobs, the asset position of the couple still evolves based on (15), but now y =2bfo 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: {u (c)+t a (w 1,w 2,a)(a + w 1 + w 2 c)}, (19) T (w 1,w 2,a) = max c U (a) = max{u (c)+u a (a)(a +2b c)} + α c Ω (w 1,a) = max c +α max {Ω (w, a) U (a), 0} df (w)(20), {u (c)+ω a (w 1,a)(a + w 1 + b c)} (21) 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 dualseache couple c u (a), fo the woke-seache couple c Ω (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 and access to financial makets) 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 Ω (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. 19

20 (iii) The esevation wage w of the dual-seache couple equals w, the esevation wage of the single-agent poblem. The main message of this poposition could pehaps be anticipated by the fact that boowing and saving effectively substitutes fo the consumption smoothing povided within the household, making the latte edundant. Consequently, each spouse in the couple can implement labo maket seach stategies that ae independent fom the othe spouse actions: each spouse acts as in the single-agent model. 4.4 Some illustative simulations In this section we wish to gain some sense about the quantitative diffeences in labo maket outcomes between the single-seach and the joint-seach economy. We stat fom the case of CRRA utility and exogenous sepaations. Late we add on-the-job seach. Thus the economy is chaacteized by the following set of paametes: b,, ρ, δ, F, α u and α e when we have on the job seach. We fist simulate labo maket histoies fo a lage numbe of individuals acting as singles, compute thei optimal choices and some key statistics: the esevation wage w, the mean wage, unemployment ate and unemployment duation. Second, we pai individuals togethe and we teat them as couples solving the joint-seach poblem in exactly the same economy (i.e., same set of paametes {b,, ρ, δ, F, α u,α e }). 9 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 lage o smalle unemployment ates: 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 to eplicate salient featues of the US economy. The time peiod in the model is set to one week of calenda time. The shot duation of each peiod is meant to appoximate the continuous time stuctue in the theoetical models (which, among othe things, implies that the pobability of both spouses eceiving simultaneous offes is negligible). 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 offesaedawnfomalognomaldistibutionwithmeanμ and standad deviation σ =0.1 and mean μ = σ 2 /2 so that the aveage wage is nomalized to one. We set δ =0.0054, which coesponds to a monthly employment-unemployment (exogenous) sepaation ate of Theoffe aival ate α u, is set to diffeent values depending on the isk avesion to match unemployment 9 To educe the simulation vaiance, we use the same sequence of sepaation shocks and wage offes in the two economies. 20

21 Table 1: A Compaison of Single- vesus Joint-Seach with CRRA Pefeences ρ =0 ρ =2 ρ =4 ρ =8 Single Joint Single Joint Single Joint Single Joint es. wage w /w es. wage φ (1) n/a double ind. ŵ mean wage Mm atio uate 5.5% 5.5% 5.4% 7.6% 5.4% 7.7% 5.3% 5.6% u duation dual-seache woke-seache job quit ate 0% 11.1% 5.55% 0.74% EQVAR- cons. 0% 4.5% 14% 26% EQVAR- income 0% 1.1% 2.8% 0.7% duation and 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 s now conside the case 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 shote unemployment duations fo the dual-seache couples. At the same time, though, the esevation wage of the 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 joint-seach economy displays longe aveage unemployment duation, 12.6 weeks instead of 9.7, and consideably highe unemployment ate, 7.6% instead of 5.4%. Compaing the mean wage tells a simila stoy. The job-seach choosiness of the woke-seache couples dominates the insuance motive of the 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%. Indeed, the egion whee the beadwinne cycle is active is athe big, as documented by the gap between w and ŵ which is equivalent to 2.7 of the standad deviation of the wage offe. 10 As isk avesion goes up w falls, and unemployment duation deceases so to keep matching an unemployment ate of 5.5% we decease the value of α u. Fo example, fo ρ =0, α u =0.4 and fo ρ =8, α u =

22 The next six columns display how these statistics change as we incease the coefficient of elative isk avesion. As clea by looking at the fist ow, in the case ρ =0the 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 the agent accept a job offe moe quickly. Howeve, the gap between w and w fist gows but then it shinks. Indeed, as ρ,itmustbe 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 couples ae less demanding which educes unemployment. Indeed, at ρ =8the 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 atio between the mean wage and the lowest wage, i.e. the esevation wage. Honstein, Kusell and Violante (2006) demonstate that a lage class of seach models, in paticula those without on the job seach, when plausibly calibated geneate vey little wage dispesion. The fifth ow of Table 1confims 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 inequality: 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. This esult is consistent with the finding in Honstein et al. that single-agent seach models with on the job seach fae bette in tems of esidual wage dispesion. We also epot two measues of welfae gains fom being in a couple vesus single in ou economy. Recall that the jointly seaching couple has two advantage: fist, it can smooth consumption bette, second it can get highe eanings. The fist measue of welfae gain is the standad consumptionequivalent vaiation and embeds both advantages. The second is the change in lifetime income fom being maied which isolates the second aspect the novel one. 11 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 we intoduce on-the-job seach into this envionment. The fist fou columns suppot ou theoetical esults. Regadless of paamete values, if agents ae isk-neutal, then on-the-job seach has no additional effect and both the single-seach poblem and joint-seach poblem yield the same solution. We also poved that if the offe aival ates ae equal duing employment and unemployment, then again, both economies will have the same solution. All the qualitative diffeence between single-seach and joint-seach models that we have high- 11 Fo the sake of compaison, when we compae the consumption o the income of a maied individual to he single countepat, we take household consumption o income and divide it by two. 22

23 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 e =0.03 α u =0.1,α e =0.1 α u =0.11,α e =0.02 α u =0.11,α e =0.02 Single Joint Single Joint Single Joint Single Joint es. wage w /w es. wage φ (1) double ind. ŵ mean wage Mm atio uate 5.4% 5.4% 5.4% 5.4% 5.3% 5.8% 5.3% 5.4% u duation dual-seache woke-seache EU quit ate 0% 0% 0.93% 0.19% EE tansition 0.45% 0.45% 1.03% 1.03% 0.49% 0.47% 0.51% 0.49% EQVAR-cons. 0% 4.6% 4.1% 15% EQVAR-income 0% 0% 0.2% 0.05% lighted though the analysis of Table 1 emain tue hee. Howeve, the diffeences between economies ae much smalle quantitatively. As we agued in Section 4, joint-seach and seach on the job shae many similaities so they ae somewhat substitute: once on the job seach is available, having a seach patne is not so useful any longe to obtain highe eanings albeit it emains obviously a geat way to smooth consumption, as evident fom the last two lines of the table. 5 Joint-seachwithmultiplelocations The impotance of the geogaphical dimension in job seach is undeniable. Fo the single-agent seach poblem, accepting a job in a diffeent maket could equie a elocation cost that may be high enough to induce the agent to tun down the offe. In the joint-seach poblem, the spatial dimension intoduces a new inteesting seach fiction. In addition to migation costs that also apply to a single agent, a couple is likely to suffe fom the disutility of living apat if spouses accept jobs in diffeent locations. To analyze the joint-seach poblem with multiple locations, we modify the famewok intoduced in Section 2 by intoducing a fixed flow cost of living sepaately fo a couple. As we shall see below, the intoduction of multiple locations leads to seveal impotant changes in the seach behavio of couples compaed to a single agent, even in the isk-neutal case. Futhemoe, many of these changes ae not favoable to couples, which seves to show that joint-seach can itself ceate new fictions. This is in contast to the analysis pefomed so fa which only showed new 23

24 oppotunities of joint seach. 12 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. As befoe, we define a couple as an economic unit composed by two individuals (1, 2) who ae linked to each othe by the assumption that income is pooled and household consumption is a public good. No stoage is pemitted. Thee ae no exogenous sepaations. The economy has two locations. Couples incu a flow esouce cost, denoted by κ, if they live apat. Denote by i the inside location and by o the outside location. Offes aive at ate α i fom the cuent location and at ate α o fom the outside location. The two locations have the same wage offe distibution F. 13 We assume away moving costs: the point of the analysis is the compaison with the single-agent poblem and such costs would be equally 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 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. As explained, individuals in a dual seache couple have no advantage fom living sepaately, so they will be 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, 12 This bings to the table the issue of whethe two individuals foming a couple would be bette off as single and should theefoe 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 match to justify continuing the elationship. 13 Moe specifically, one should think of the two locations as offeing the same job oppotunities. Howeve, being in one paticula location allows to get moe contacts. This set up implies that once a spouse in a dual-seache couple finds a job in the outside location, the othe spouse will follow as thee is no paticula advantage fo job-seach in emaining in the old location while the couple must pay the cost κ of living apat. 24

25 we have T (w 1,w 2 ) = u (w 1 + w 2 ) (22) S (w 1,w 2 ) = u (w 1 + w 2 κ) (23) U = u (2b)+2(α i + α o ) max {Ω (w) U, 0} df (w) (24) Ω (w 1 ) = u (w 1 + b)+α i max {T (w 1,w 2 ) Ω (w 1 ), Ω (w 2 ) Ω (w 1 ), 0} df (w 2 ) (25) +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 tun down the offe o 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 45 degee line. By inspecting equation (25) it is immediate that, also as in the one location case, the coespondences φ 1 i (w 1 ) and φ 1 o (w 1 ) chaacteize the quitting decision. Single-agent seach. Befoe poceeding futhe, it is staightfowad to see that the singleseach poblem with two locations is the same as the one-location case, except fo the slight adjustment to the esevation wage to account fo sepaate aival ates fom two locations. In the isk neutal case, we have: w = b + α i + α o w [1 F (w)] dw. (26) Recall that in the one-location case, isk neutality esulted in an equivalence between the single-seach and joint-seach poblems. As the next poposition shows, this esult does not hold in the two-location case anymoe, as long as thee is a positive cost κ of living apat: Poposition 8 (Two locations) With isk neutality, two locations and κ>0, the seach behavio of a couple can be completely chaacteized as follows. Thee is a wage value ŵ S = b + κ + α i ŵ S κ [1 F (w)] dw + α o 25 ŵ S [1 F (w)] dw

26 Figue 6: Resevation Wage Functions fo Outside Offes with Risk-Neutal Pefeences and Two Locations and a coesponding value ŵ T =ŵ S κ such that: (i) The esevation wage of a dual seache couple 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,andfow 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,fow 1 [ŵ, ŵ S ), φ i (w 1 ) is stictly deceasing and fo w 1 ŵ S, φ i (w 1 )=ŵ T. Figues 6 and 7 gaphically 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 offes 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 (Figue 6). 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 the othe spouse to the outside location. The 26

27 Figue 7: Resevation Wage Functions fo Inside Offes with Risk-Neutal Pefeences and Two Locations cost κ is too lage to justify living apat while being employed at such wages. 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 eceive such high wages. Compaing Figue 7 fo inside offes to Figue 6, 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 w 1 ises the gains fom seach coming fom outside offesaelowe(ittakesavey high outside wage offe w 2 to induce the employed spouse to quit), hence 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, so wheneve the wage offe is highe than the employed spouse s wage but smalle than ϕ i (w 1 ), the couple goes though the beadwinne cycle. Howeve, if the wage offe is high enough, the potential negative impact of the outside wage offes induces the couple to become a dual-woke couple. Using the same easoning we applied to the ange (ŵ, ŵ S ), the esevation wage fo being a dual-woke couple deceases as w 1 inceases. Tiedmovesandtiedstayes. In a seminal pape, Mince (1978) has studied empiically 27