Job Matching and the Theory of TurnoverBoyan Jovanovic The Journal of Political Economy, Vol. 87, No. 5, Part 1. (Oct., 1979), pp. 972-990. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197910%2987%3A5%3C972%3AJMATTO%3E2.0.CO%3B2-Q The Journal of Political Economy is currently published by The University of Chicago Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ucpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact
[email protected]. http://www.jstor.org Wed Jan 30 05:37:30 2008 Job Matching and the Theory of Turnover Boyan Jovanovic Hul( Lnhorctlorzr.~,I nr . , nnd Collrrnh~c~ b ' u i z * ~~. ~i t ) . A long-run equilibriurri theory of turnover is presertted and is shoivrr t o esplaiu thc irnporta~lt regular-ities t1i;tr have l~erll oktser-ved by empirical investigators. r\ \\,or-her's procluctivit) in ;I p;il.ticulnr jot) is 11ot krio\tri ex ante anti Itec-ornes kllo~vn niitre prccisrly its the worker-'s,job tenut-e irlcl-east.9. 'l'ul-no\er is genel-ated I)) the esis- tence oi' ;I noildegerierate distr-ib~ttion of' tlic wor-her's produc-tii~it~ across different jobs. The noxit1ege1lerac.y is caused by the ass~rrrred \a]-i;itiori in the cluality of the svor-ker-e~nplover- tliatch. ?'he objective of this paper is to construct and to interpret a model of pcrn-~anent job separations. A permanent job separation involves a change of employers for the worker. 71- e~~~por ar y separatiolis (con- sisting rnairlly of temporary layoffs) have been the sul-,ject of recent theoretical work by Baily (1974), Azariadis (1975), and Feldstein (1976), anti are not corlsidered here. Recent evidence or] labor turnover falls into two categories: the cr.oss-sectional industry studies (Stoikov and Rarnon 1968: Kur-ton and Parker 1969; Pencavel 1970; Parsons 1972; 'Telser 1972), and the more recent studies using longitudinal data on individuals (Rartel 1975; Rartel and Borjas 1976; E'reernan 1976; Jovanovic and Mincer 1078). The strongest and most c:onsistent finding of all these studies is a negative relationship between quits and layoffs on t he one hand, and job tenure on the other. 'This finding is equally strong for quits as it is for layoffs. Jovanovic and Mincer (1978) find that roughly one- This is a shortened versiorr of n ~ y Ph.1). thesis. I jvo~ild like t o thank R. E, 1-ucas for suggesting this problem ro me ancl foi- his co~~st ari t eiicorir-,igerncr~r and advice thl-oughout the prepararior~ of this work, 1 would also like to thank Gary Beckel- arid Lcster I'elser for their help at various stages of the pr-eparaticin of thc thesis. [[rifrrrini <$Polrlrcai Econo,ni, 19iY. v o l . 87, no. 5, pr I ] 1979 by Thr Uni vei sr~yof Chrrago. 0022-SROX/i9!R751-0004$01..5l JOB MATCHI NG AND TURNOVER 973 half of this negative relationship is explainecl by the negative struc- tural dependence of the separation probability on job ter1ur.e. The rest of the observed clependence is only apparent anct is caused by the fact that within any nonhomogeneous gr-oup a negative col~relation between job tenure anti t he separation probability will exist, simply because those people with a lower propensity to change jobs will tend to have longer job tenure and vice versa. Ot her observed 1.e1ationships are as fbllows: women, young nrorkers, productioii workers. those with less schooling. and those in the private sector tend t o t urn over more, as do those workers not cover-eci by a pension plan atit1 those ~v h ocvork in industries with loxver conce~itratiori ratios ox- with snlaller average firm size. None of these relationships is nearly as strong as that bct~veen job te~iur-e and sep;~t-ation pl-obabilities. Existing ~nodel s of' iurnovel- (that is the "pel-manent separations" component of turnover) all involve irnperkct information. Net\. in- fhrnration arrives either about one's current match or about a possitde alternative match that leads to a job change. In fact, a natural distinc- tion among the rnodels call he made along these lines. In one c;itcgory are models in which turnovet- occurs as a result of tlie arrival of infor-mation about tlie current job match, and the present moctel f,tlls into this category, as do the ~nodel s of Viscusi (1976), M'ilde (I%$), and Jol ~nson (1978).' These are niodels i t 1 which :I job is an "experi- ence good" in the tcl-minology of Nelson (1970); that is. the only \v;iy to determine the quality of' a par~i cul ar rnatcll is to fi)rni the match and "experience it.'' I11 the second category are "pure search-gooct" tnocfels of job change (Kuratani 1973; 1,ucas and PI-escott 1974; Rur- dett 1977; Jovano1-ic 19780; Mortensen 1978: \l.'ilde 1978). In those motlelc, jobs are pure sear-ch goods and matches dissolve because of the an-ivnl of new information about an alternative pl.ospective match. Hil-shleifer (1973) introduces the inol-e appropriate designa- tion "inspection goods." Iti~b~c-tion is ev;llu;ition that car1 take place prior to purctiase, r.~pur.irncuonly after purchase. In this paper, a,job match is treated ;ts a pure experience goort. The paper. makes two separate contributions. First, i t is the only explicitly ec1uilib1-iun~ treatment of tur-nover in its category. An ecjuilibriu~n wage contract is provect to exist itrid a particular wage cotitr;ict is demonstrated to be an equilibriunl one. This particular wage co11- tract has he propert) that at each nioment i t 1 time tlie \ro~.ker- is paitf his ~nargi nal ~ I ' O ~ L I C ~ conditional upon all the :t\.ailal,le inf0rnlation at that time. Second, the c11;tracterization of the inodel's implications I-egarding ' 1 bec.;~niraw;cre of the \\ark o f thew three author..; aftcl- tllc present \v\. c~tk\\.as l ar gcl ~li~lishctl. 974 JOURNAL OF POLITICAL ECONOMY the tenure-turnover relationship arlcl the tenure-wage relationship is more esplicit than that of earlier models, and the predictions are largely consistent I$-it11 the evidence. The rnodel predicts that worker-s -, . rernain on ,jobs in whicl-1 their productivity is reve;~led to be relatively high and that the!. select themselves out of jobs in ivhich their pro- ductivity is I-evealecl to be low. Since wages always eclual expected mai-ginal prodi ~ct s for all \vorker-s, t he nod el generates (011 average) M.age gl-owth as tenure increases. Since .job tenure and lahor market expel-iet~ce are corr-elateel across workers, this also implies wage g r o ~ ~ t h over the life cycle. The model also pi-edicts that each !corker's separation probability is ;I of' his ,job tenure. clec:~-easing f i ~~l ct i on I.oosely speaking, this is 1)ecausc a mismatch betiveen a \t.orker and his employer is likely to he detected early on rattler than late. The learriirlg mechariisrn is such that longel- j ob tellure has a negative st ruct u~al effect on tile \vorker's sepai.ation probability. After cor- recting till- the regression hias that arises bec;ruse of the spurious correlation between job tenure and the separation probability in a hetcrogcneous group of workers, Jovanovic and l l i ncer (1978) find tliat this structui-al tiepe1ldenc.e is very strong. Befor-e (leveloping the model we summarize the major assumptions of the ,jot)-matching approach t o turnover.. E'i1.s~ i t is assumetl that for each worker a nondegerier-ate distribution of productivities exists across different ,jobs. The same is true Lhr. thc employer-~vorkers di f kr in their productivities in a given task that the ernployel- needs to have performed. The problenl is one of optirnallp assigning ~vorkers to ,jobs. The second assumption is that employers c;tn contract \vith workers on an iildiviclual basis. The employel- is then able to re\v;irti a ivorker with \ vho~n he matches well by paying the \corker r.elatively more. 111dividtt:tI contracting creates a structure of re~vai-ds th;it PI-ovictes pr ope~. signals fbr the attainment of optimal rn:itches. !In extreme exa~riple of individual coritr;~cting is a piece-rate ivage scale. A less extreme and a widely prevalent esanlple is ;i s)stenl of pr-ornotion or dela)-eci pay increases based on the quality of the woi-ker's perfi)l-- Itlance on the ,jol, ovel- a past period of time of sorrle given length. These are examples \vtlere workers' pay is c or ~t i nge ~~t on their- per- forni;~nce. 'I'he third major assumption of the ,job-matching approach is that imperfect information exists on both sides of' the rnarket about the exact location ~f one's optirnal iissignn~erlt. Follo~vi~rg an initial as- si g~~rri ent , inforrn~ition lteco~nes availal,le, and I-eassignnient be- r m v comes optirnal in certain cases. The job-matching model ger-rer.ates turnover. as the phenomenou of optiinal reassigrt~nent cztused 11)- the accumulation of ttette~. iiiforn~ation with the passage of time. JOB MATCHI NG AND TUKNOVER 975 The Model Assume that firms' production functions exhibit constarit returns to scale and that labor is the only factor of prod~~ct i ori . Cnder conipeti- tive conditiorls, the size of firm is tllen intieterminate. Each worker's output is assumecl to be obser\,ed instantaneously by the worker and by the ernployet- so that infbrmational asymmetries do not arise. Let S( t ) be the contribution by a worker to the total output of' the firni over- a period of length t , and let X( t \ = pt + m( t ) (for each t > 0) ( 1 ) where p and cr al e constants and a > 0, and where z(t) is a standal-d normal \.rtriable with meall 7ero and variance t (a stanclartl Wiener process with indepentient iricrernerits so that cov lz(t), z(t')] = nrin [t, If]). Then X(t) is nor-n~ally distrib~ttecl with mean yt arid ~ . i t h variance CT' ) ~. Assun~e that (T is the same l or each firm-worker rnatch while in general p cliffel-s across matches. The interpretation of p is not one of I\-1-ker- ability but a nieasure of the cjuality of the match. When the ~rratclr is forrned, p is unknown. As the nratch continues, further intormation (in the f i ~r ni of'output as given by eq. [ I ] ) is generated. A "gootl match" is one possessing a large p. Let p he nornlally distrib- uted XI-ctss matches, with niean m ant1 with variance J , and assurne that job dianging involves drawing a new value of p from this dis- tl-ihution and the successive drawings are independent. ?'he latter assu~nption guarantees that the worker's prior history is of' no rele- vance i r i assessing his p on a newly formed match. The only way to learn about p is to observe the worker on the,jol> for a period of time. I' his inclependeuce assunlption also means that the i~iforrnatiorlal capital thus generated is con~pletely nratch specific and is analogous to the conrept of firm-specific human capital.' For a worker with ,job tenure t and cumulative output X(t) = x the above assumptions irnply that the available information on p on his current job can be characterized by a posterior distribution that is norriial (see Chel-noff 1968, p. 266) with posterior mean -E,.,,(p) = (wzs-' + xa-')(s-' + t a- ~~) - ' (2) posterior v;~riarice = Si t ) = (. ?-I + tcr-')-'. .l'lie pair- [X(t), t ] is t hu-efi ~re a sufficient statistic for the information co~ttainect in the entire posterior distribution. ('Ihis property is essen- tially due to the independent increments property of the Wiener ''1.0 elaborate: Cl'hcn tlealing wit11 rarrtion~ variables the corlcept of inforrrration spccificit) is associatet! with the conccpt of i~ltlependcnce while perfect informational generalits is associatecl wit11 perfect correlation. 976 JOURNAL 01: POLITICAL. ECONOhlY process.) Furthermore, Il:.Y(ttt ( p) is 110r1na11y distributed with mean m :irid variance .\ - S( t ) ((:her-noff 1968). Firms are assumed to be risk neutral and to nlaxirnize the rnatlie- matical expectation of revenues discounted by the rate of interest. r. The): cornpete t br workers by offering wage contracts. In a long-~-un equilibrium the payments practices of each firm ~ ~ ~ o u l d be well under- stood and would not need to be explicitly written. An implicit contract equilibrium is studied here. The present model al~stracts entir-el) from the cctnsideration of' shocks stemming f'rom the product market. A11 firms face the same product price. uorrnalized at unity, so that a mairitaineti h!-pothesis of' the model is that demand conctitiorls are stationary. Assume that the firm's wage policy can be characterized by a wage function ul [X( t ) ,t ] . -Phis is the wage paid to the I\-orker with tenure t if' his cumulative output contrihution is ecjual to X(t ), I f the firm tvishes to fire a certain worker, rather than doing so directly the firm is assumed t o l o~ver his wage by an arnount sufficient to itrctuce him tct quit. ,411 the job separations are therefore at t he .rvor-ker's initiatke, but since sorne of the separations are disguised layoffs their empir-ical counterpart is really total separations (quits pl ~i s layoffs). ?Yorkers are assumed to live for-ever," ancl this assumption justifies the exrlusio~i of age as an explicit. argument from the wage function. As long as he remains with the firm, the ~vorker receives payment according to tlie wage functiorl w( . ) . He has the option of' quitting at any time. L.et Q be the present value of' quitting a job and then pursuing the best a1ternatix.e. The infinite hori mn, constant discourit rate, and the independence of' the successive drawings o f p imply that Q is a constant.' 1,et a(Q.[ic!]) be the present value to the worker of ohtainilig a job with a finn which offers ) ;IS its \\.age contr;ict and when the value of quitting is Q. Then ifc represents tIre direct ancl the foregone earnings costs of job changing, The constant c is assunled to be parametrically given for each \vorker, although i t may vary ;icross workers. Let T he the quitting tirne and let H(x,t I [w], Q) = prob ( X[ t ]s x and 7' > f given [a!] and Q) and F(t / [ u ~ ] , (2) = prob (7' G t given [as]and Q). I' hen F is t he probability that the 15,orker quits befi>re tenure t , while N is the probability that he does not quit before tenure I , and that by that time his cu~nulative output :' klorc gcrierall), otie c.oulil ;rsstlrnc. thaf \\.orlers' lifetinres arc csponc~nti,+ll\ tlistrib- utrrl implvitry: the ;tt)srtice of aging+)nc. \\trultl not r~litkerl diff' cr~tlt pledic-tion ,ihouc the Irt~gth of tlw t.rtn;iining litc of a \\or-ket- tvho has ;iIrc,~dv liveti d l or ~giirr~cth;~n for a wor ker who tias onI\ li\ztl ;t shor-I tirne. 4 ' 1 he c-oristarlc-\ o f Q over t i t ~l r tnr:;r~-rs that (lie war-ker never returns to a jot) from ~vhich he once separarect. 111 other- \vo~.tls. if i t esistetl, the optiori of I-cc;~ll W O L I I ~ rrever. be exrrcisrd b\ ttle wet-kcr . .JOB MATCHI NG AND TURNOVER 977 doe4 not exceecl x. I'hen define the appropnate ctens~ties h (r,t 1 [ ul ] ,Q) and f ( t 1 [zo], Q) b) h = dNI dv a n d j = dFl dt Both f and h are chosen b~ the ~voi kei In respon5e to a wage function X I ( . ) and ,I p~esent value of quitting Q. -1 hen Equ;ition (4) holds at the optimall?. chosen f~inctions h and f . Since f integrates to a number not exceeding unity, aa/ aQ = S;, pi r' f i 1t < 1 . Then it is easi1)- seen that for given functions h , f', and zu, equations (3) and (4) possess exactly one solution for the pair of' scalars (a,(2). All new workers look alike to the firm, and each ~vorker is offered the same wage cor~t ract . ~ In differential form, equation (1) reads dX i t ) = pdt + cdz(t). Letting E,,, be the mathematical expectation operation conditional on X(t) = x at t , the disco~~nteci revenue from the output of a single worker is ~f : p- " d~( t ) = = Ef Te-vfl;v, , , fd~(t) EfreF'" E,,,,,(p)dl + Ef Tu -' . ' ~E~, , , , di(t). 'l'he stochastic integrals are It6 inte- grals (see fiushner [1971] for their definition) anct the last integral is therefore zero, b y the indepenclent increments property of the Wiener process, so that Ef:e-'$ix(t) = ~ f ~ ~ ' - ~ ~ ~ ~ ( ~ ~ ~ ( ~ ) d t f;crt = f z_E, . , (p)hix,t / [XI], Q)dxrl t = P(Y, [il.~]). Firnis are aware of the work- er's optiinal quitting response to the wage contract {zu) , and this is re- flected in the above equation. Now let n(Q, fur)) be the discounted expected net revenue from the employment of a given worker who is offerect the contract {ui ) and who has a present value of quitting equal to Q. Then where y = ~ f T a - ~ y 1 [ul],Q) dt . ( t In maximizing n(Q, [ X I ] ) over functions [ w] , the film treats Q as gihen, since Q is determined by the wage policies of other firms. Let 13 be the set of competitive equilibrium wage contracts, and for an, 7 ~ ( . ) let Q([ro]) denote the unique solution for Q from equation (3). ''hen, if us(.) E B, (E 1) each worker fc>llo.clr., his optimal quitting 'Sirnilarl), all f i r - l r i \ look alike to the worket- ex ante. Straightf'ortvarii estensions of the nod el t o the case where there art. observable differences in characteristics anlong workers are outlined at the enti of the paper. Salop (1973) takes up the search problem when the fcorher is able to distil~guish among firnrs ex ante arld has partial inful-niatioti riot only about the wage offerrd by the firm hut also about the likelihootl that he will receive an emplovmeiit offrl- Yrotn the firrn in the event that he saniples it. 1x1 Salop's analssis the most attractive opportunities are saniplect first, arid the job seehet- lowers Iris acceptance wage with his iiuration of unernplo~rnent as his ~retnair~ing opportunities \\~OrSCrl. Y?a JOURNAL OF POLITICAL ECONOMY poll~v in lesponw to zu(.) anti to Q([w]); (E2) T{Q([zL~I), [zL*]) 3 n-{Q([IP]),[GI)for all in(.) 1 711 (. ), so that ZLI(. ) maximizes expected profits; (E3) n-{Q([w]),[ul])= 0 (zero expected profit ~ortstraint). Let 1 , ) = , t o r 1 ( t ) contlact stntes tl-l,it the ~ o r k e l I hi\ \+,\.,ige \\111 be paid hls eupecter-i (1n,11 glrlnl) p~octrlc t .it each rnomerlt 111 ttrne. 1x1Qv =Q([itx4]). ? hr o? ~~n I -70X E R Ploof -E:S I \ tiearl\ sat15fled b> 711" 1o pl ol e E,1 anti E2. \ilppo\e t x ( ont r a d~t t l o~~ that F2 1s ttot \,ltlshed b\ wv 40 th,it t h e ~ c e\ilst\ sorne 711 E H such that a de~l antf t r m offers ~t while the .ilo~ kel must be tfolng at least a\ tvcll '1s uncle1 711": (The value of'quittirlg the deviant firm is unchanged at Q*.) From (5), ?'hen equations (6) ;illri (7) imply that the left-hand side of (8) is strictly positive. But the right-hand side of' (8)is equal to JTe rlJ",w* (x,t){h(~,t I [itl],Q*) - h ( ~ , t I [w*], Q*))d~dt + Q*J;p-"~f(t / [ i ~] , Q*) - /'(t 11711], p ) >c l t , and this expression cannot be positive since the quit- ting policb implied by { h ( . ~ . t / [ ai * ] , (I*), j ( t / Q*)) is optimal fhr [ z r l * ] , the workers when facet1 with t ) anti the the wage contract ~L!*(Y, present value of' quitting Q*. Q.E.D. Since workers and firrrls are risk neutral, ul*(x, f ) is rlot a unique equilibrium contract. any random variable [ possessing the property I?,,([) = u!*(x,t) would also qualify. A pure piece-rate wage involving a payment ofX(i + At) -X(t) over the interval ( t ,t + At ) theretbl-c also qualifies as eqttilihriurri since E,.,ldX(t) = / ) dt + crE,,clz(t) ilr*(x, ~L,"(.Y, = t ) t i t . Ally such contract leads to idetitical turnover behavior as under ZL~*(X, t ) . Ever1 within the class of functiorls of s and t alone, u,*(x. t) may not be unique. ?'he following theorern guarantees, however, that tul-nover behaviol- is unique. ~ ' ~ P O T P W ~ 2.-If 71, E H the11h{x, t 1 [n*], Q([i/l])) = h{~, t 1 [ill*], Q([uI*])}, and j {t / Q ( [ ~ L ' ] ) ) = 1 [711], Q([af*])). [~LI], J'{f Proof'.-See Jovanovic 19780. The proof' is lengthy and rlot par- ticularl), instructive. Theoren, 2 states that the separation policy of the worker is unique even though the wage contract leading to it is not. This turnover behavior is identical with that which results in a situa- tion i r l which each firm offers a wage corttract zom(x, t ) = E,,(p). Purc.to optirnnlity rft~irri'ozlrr.-Sint:e all the agents are risk tleutl-al, the 979 JOB MAI'CHING AND TURNOVER correct optirnality cr-iter-ion is the maximization of the discounted expectation of' aggregate output. Theorem 2 inlplies that whatever the prevailing equilibrium wage contract, the worker behaves so as to maximize his own expected discounted output. He collects all of the rent associateti with the match, arld the decision about \\-tlether or- not to terminate the match rests with hirn (although the tirni is equally involveti in the sepal-ation decision since i t lowers the worker's wage to the point where i t knows the worker will quit). Therefore, a separa- tion occur-s if and only if' the rent associated with the match falls to rero. A central planner could improve on this situation onlv if he krielv zcjhich workers and it~hirhfirms would make good matches. Assume that the worker is faced with the wage contract zir*(x, t ) = E,,(p) and a present value of quitting Q. The sufficient statistics (state variables) areX(t) and t . It is more convenient to use instead w ( t )and t as the t wo state variables, where ul(t) = EX,,,,(p). Since ~ ( t ) is normally tlistributed with mean 7n i-tnd variance .r - S ( t ) for all t , it satisfies the stochastic differential equation so that the worker's wage folIows a driftless random process with ever-decreasing incremental variance that tends to zero as tenure tends to infinity. Let V ( w ,t) be the ("current") value of the game to the worker rvho has tenure t and wage ~ ~ ( t j = w. Then letting E,rt denote the nlathernatical expectation operator conditioned upon zc and t , 6 ~ ' ( Z U ,t ) = i ~l At+ P- ' ~ ' E, ~ Y + At], t ) + o(At). (~~t [t (10) Subtracting \ ' (XI, t ) from both sides, dividing through by At , taking the limit as At tends to 7er0, and applying ItO's Lemma (5ee Kushner 197i ) j ieIdc As with most optimal stopping problems involving Mai-kov processes, the space of points ( w, t ) can be divided into a continuation region and a stopping region (see Shiryaev 1973). The continuation regiori con- sists of those wage-tenure cornbinations at which it is optimal for the worker to remain with tllc firm. E;quatioris (10) and (1 1) hold for all " ( ) ( St )represertts ttarlns rentling to zero f'aster than At does. Note that the optiort of stopping or1 ( t . t + At ) (in wtlich casc a rcwartl Q is rollectecl) is exercised wirh a prob;ibilitr. that hehaves esser~tiall\ as does I ( At ) ' " ;1 1 - <- ;I '\ 1 - q v 5 z CXP 1- *(At), 2 1 = ( ) ( At ) (At)' - (see Feller I!)6t5. p 171!, xvhrr-e thc. inec1ualitv follows by a ~vell-knowt~ on the result hlill's I-atio atitl whel-r s is eclnal to 11' - @( I ) . 980 JOURNAL OF POLITICAL ECONOMY wage-tenure combinations that belong to the continuation region. Let [O(t), t] be the boundary of the continuation region so that along the boundary V[O(t), t] =Q, and O(t) may be thought of as the reservation wage at which the worker quits the firm. Evaluating equation ( 1 1 ) at = O(t), O( t ) = rQ - [ s(1)~! 2~r~] l ' , , , . [ e(t ), - V,[e(t), I]. A welI-kno.ilm t ] "smooth-fit" condition of optimal stopping (see Shiryaev 1973) states that along the boundary: V,[O(t), t ] = aQ!at = 0, implying that $ ( t ) = rQ - -V,,., 0)' [ $i t ) , t ] . 2aZ In the interior of the continuation region V(u8, t ) > Q. Since at the reservation wage V[O(t), t] = Q: and since V,,.[O(t), t] = 0, this implies that V,,,,.[O(t), t] 3 0. Note that S ( t ) declines monotonically to zero which suggests that H(t) should be rnonotonically increasing up to rQ. It is possible to prove [see the Appendix) that H( t ) <rQ for all t , that (IOldt 3 0, and that !im O(t) = rQ so that the reservation wage increases up to its limit froni below. The reason fhr the increase in the reserva- tion wage is the decrease of the incremental variance of the wage process as tenure increases. A large incremental variance implies a large dispersion in possible future wages. If wages t urn out to be very high the worker does not quit. If they become very low, the worker partially avoids this adverse outcorne by quitting attd collecting Q. In the absence of the opportunity to quit, the risk-neutral t\.orker's welfare would be unaffected by changes in the incremental variance. The limit of the reservation wage is rQ. This is because the wage tends to a constant as t tends to infinity. Ther e is nothing further- to be learned, and at the point of indifference between staying and quitting the capitalized value of this constant .tr,age must be equal to the present value of quitting, Q. To obtain an approximation to the probability of job separation by tenure, set H(t) = rQ for all t . Then for this approxinlation to the reservation wage? ' An infbrnral proclf is as fbllo~vs: V( W,1 ) = Q + ji',,V,.fv. /)rlv is rnaxinrircd with respect to @(/) (the reservatiotl wage at t ) . Therefore dtfferentiating both sicies with I-espect to @(/),setting the result equal to zero, anti taking thc limit as uptends to @(0, one obtains that V',.[@(t), t ] = 0, which in turn implies V, [@( t ) , = 0 since V[ %( t ) , = (2 = / I t i constant. In the Appendix it is shown that B( t ) <rQ for all t , implying that V,,.. > 0 along the boundar-y. where it is also true that E ' , . = 0. So, if it was true that the continuation region was boundeci from above, this would imply that V <Q for some point in the interior of the continuation region suf'ficiently close to the boundary, which cannot be true. Therefore, H(t) is single valued and it bounds the continuation region from below so that the optimal policy does hal e the reservation wage property. This is not surprising since it is known (Rothchild 1974, p. 709) that optimal search rules from normal distributions with unknown mearls and known variance have the reservation price property when the prior distribution is also normal. ' The wage is a standard Wiener process in the s - S ( t ) scale (see the discussion JOB MATCHING AND TURNOVER where iY(s) = ( ~T) - "?S&"~dz iwhere p(t) = s - S ( t ) is the precision. 'l'he unique mode of this distribution is (171 -7.0,)" After the mode, the prohal,ility of turnovei- ciecliries rapidly to zero. Sonle ivorkers never change jobs, since lim F(t / . ) < 1. r+= 1.0detel-mine thi- p! eciicted behavior of' the separation probability by tenure: consider the hazard rate, 4(t)-f!(l - F).Then +( t ) is the density of separation conditional upon an attained level of tenure, t . The rnodel predicts ;I nonrnonotollic relationship: first [4' (t)] > 0 and then 4' (t) < 0 as t gets relatively large. That $ ( t ) must eventually decline fi>llows since limf(t) = 0, while 1 - F(t) is bounded away from I--, zero. The precise inarheliiatical expression hl-the tenure level t X at which 4'(t) changes sign and finally becomes negative cannot be obtained in closed form, but sincef' > 0 implies +' > 0 clearly t* 2 m - rQ = the model off. If the mode off is close to zero, 4' ( t )is likely to become riegative early on, as appears to he the case empirically (see Jovanovic and Mincer 1978). The tenure-wage profile (defined as the conditional expectation of the wage given that the worker has attained tenure t ) may also he calculatedl"and is equal to 6 ( t )= (?n + (nr - ~ - Q) %~ ' ( - N[ s- S(t)]-""11 - 212'{-n[s -S ( t ) ] ' ! ' ) ) , Note that GI([) increases nionotonically from ~ I I when tenure is zero up to [ m + ( m - rQ)'LS(-crs-'I" )!l - 2:Y(-(r,\C's2)] when tenure tends to i~lfiriity. Therefore, as low-wage workers quit arid high-wage workers stay, the model iniplies that the average wage of a coho^-t of workers increases with tenure, eventually at a decreas- ing rate. In the limit, as tenure becomes indefinitely large, the average wage of those members of the cohort who have not quit approaches a constant as the wage of each worker becomes constant and equal to his true productivity. .I'his then is an alternatike explanation for -ivage gr t ~ot hon the ,joi,. preceriirig eq. 191). Therefore the fhrmula represents the first passage probability for a Wiener pt - t ~es r through a linear I,ottndar) (Cox and Miller- 1965, p. 221). '" 'l'he prolmhility that a Wiener process will rlot c-ross a linear hotindary by a partic-uiar time and that it will etrd up at a particular value at ttiitt time is also aiailable in closeti form (see Cox and Lfiller 1965, p. 221, eq. 71). .4f'ter appt-opriate adjustment t he conditional density of M-ages ( by tertur-e le\.el) is obtained. atid ri.(l) is the rr~athemarical expectation of this distt-ibution, 982 JOLTRNAL OF POLITICAI. ECONOMY A mismatch leads to a lobver \rage and an early separation. 7' hus, holding constant market experience, average past earnings are likely to be l o~ver t or a worker ivho has experienced many job separations." 'This prediction appeii1-s to be consistent ~vi t h evidencc from t he National l.ongitudinal Stud); (NI-S) mature men's sarnple (see Bal-tel arid Borjas 1976). Job durations over t he life cycle itre identically and independerltly distributed ~. andoni vitriahles. 'The turnover generated 1)); t he model therefore fi)~-ms a pure renebval process (see Feller- 1966, chap. 11). Let y denote t he ivorker's labor market experience and 11(3') + o ( AJ) denote the pl-oljability that the worker experiences a job separation on the market experience interval ( y , . .u + Ay) . -1'hen R( J )is the renewal (lensit!, whictt satisfies t he equation Jovanovic and Sfincer (1978) prove that a monotorrically declining $ ( t ) irnplies a rrlonotonicall): cleclining K (J). I n ot her isorcis, a mono- tonically declining separatior~ pr-obattility hy t enure isI)j itvclf'sufficicnt to cause turnover t o tieclinc monotonic;~lly ove1. the life cycle.I2 Last, t he model ge~ier;~lizes stl-aightfol-avaiicilyt o incorpor;tte pel.- miinent cliffel-ences in rvor-kcrs' cfiiiractel~istics such as l ewl of school- itlg, ability, race. sex, ant1 so on. l ' he pal-alnetel-s of'the rnotiel (S, 771, a" J . ) can then be 1.egal-der1 as fi~nctions of these ~ari ahl es, with each distinct group of \vat-kers treated as though they belonged to a distinct lnarkct of' \vorke~-s of that type. The entire ;tiialysis I-emains valiti co long as infi)rmational synirnetr\- l,et~veen wol.ker.s and ernplo?ers is rnaintainetl, so that issues of sigr~aling artrl self-selection are side- stepped. '1-he riatul-e of' t he assurrietl f'unctional deperitlencc bet ~vecn w, t - , r n, and , $ on t he one hanti, anti the \vot.kers' persorl;tl character-is- tics or 1 t hc ot her, will dr:terminc t he preclicted relationships het\vcer~ turnover and these personal charactel-istics. Thi s is not pursuer1 here, hut is a11 ir~ter.esiing probleln fi)r future research. " Holding evcr)thing else c.onst,lrtt. This statentrtlt shoulti ilot I,e inter-pretcti ;is s;r\ing t1i:lt \vithir~;I group ol'obsc~.\,ition:iIl). t-qui\;tletit prople t l ~ o e that have changed jot~s often in tlic p;i.;t hat e had lv\\.cr. a\rt.age past r, i rt ~i ng\ th;in those rhai h,i\t. not changed ,jobs often. I n other wortis, the rnotiel does [tot inrpl! rllat "tnovers" st~ould do \$or-se than "\t;t\rrs" c3veit though ernpit-ic.,11ly thi\ ;ippe;tt-s t o be trut.. " A sirnil;rt- rrlaiionship holtfc fhr wage: 1.t.tI.(.;) he t hr rrtarhcr~t:~tical c.rpec-tation of' the wagc ;II a giben level of lahoi- ni;ir-krt expc.r.irtice J . I'hert I.(\.) s;iti\fies the equ;ition L(Y\.)= i >( j ) i l - I . ' ( , Y) ~+ [ ~ : / ( O L ( ~ - tjdt. I':ci. (13) & kr~o\rtt ;is the retiew;il ecluation which, fbr- all! giveti continuocis tlcr~sit! I ( / ) , possessea a utticlue solurion K ( J ) (Feller 1966) sucft that K ( 0 ) = f (0) ailti lim Ii(\. ) =[I;//(0d!ir1. 0- ' i JOB hl ATCHING AND TLTRNOVEK gH3 Appendix L V t x now pro\-<, the asser.tiol~s niadr in the text following equation ( 12)ahout @( I ) , the bourlci;~~-) of the optinlal contirtuatiori region. We pr-ove that H ( t ) <1-Q tbl. all I , that @ ( I ) is r~ontlecre;~sirig. and that i t approaches t-Qas t tends to irifinity. Sorric transfol.rrl;itior~s of the original problcrri \\ere rlccessary before these ;~ssc.r.tions co~tl(l Ilc proved, anti since tllese tr;insforrnatic,rls move orle a\t.;t\ fr.orn rhct ecor~omics of the prot,lerri, it seenied prefer.al)le to incliitle these proofs ill t hc ;\pper~diu. Suppose {hat a probahilit\. space (0,F'. P) is giverr, with w heirig the elenier~taryeverits (wE0;t ).For any real-valued F-n~e;isurable function f ( w) , the rri;ttherrlatical expc.ct;itiori operator E is tlefir~eti as E V ( w) ]= Jj ( w) dP. Let S ( t ) E K1be a hIarkov process cletincd on the above space. A particular sample path of' the process is written as [ X( t , o) ] : Tf . Let E,, he the expectation oper,trcx cortdrt~on,rl upon Y( 1) = \ ( on5ltler the follot~ rng pr ohlenr of optir~i,lll\ stopplng t( t ) Lcr '1 utrlrt\ fi~rlctror~ I>e g i ~ c r ~ , to the t i ( \ ) \the11 u ( \ ) deliotes the Irr\rnnt<lneous p,i\oft 111.1\c.r at trnw 1 ~f the p~irrrc~ 1s st111 111 progless ~ 1 t t a~rctr f \ ( 1 ) = 1.Let Cr (1) be the rc.1 rr1111,rl p, ~\ off furrct~oli cler~oting the utrllt\ to the plarer if the g~lrne13 stopped c\,lctl\ at I ,rritl S ( I ) = \ I he player's otqectl\e 1s to rnn\lrrirle h ~ s e\pectt,cl tlrscourrtcd i ~ t ~ l r t \ frorir pla\ ~ n q ) 11it. t l ~s co~l ~r t rlre garrre ( ~ t ~ t h = late) over- F-rrleasural~lr stopping tirrle functions 7' ( w) . X flirther restl-icrior~ or1 T ( w) is that i t must not anticipate the future. A rigorous discussion of this ~.c.quir-enrent ;ippears in S1tirv;iev ( 1973) . For rnost stoppirrg problems, itrtd cert;linl\. fill. the problems discussed hclo\v. this reqitireirlent niearis that thc solutiorl to the optinral stoppiirg problerrl car1 bc charac terizeci by a co~itiiiu;i- tiori regiorl for thc pr-ocr:>s X(1) so th;it the first exit tinre fronr the I-egioli is the oy)tirir;il stoppirig tirnc for X( r ) . L.et C( n, I ) he the value of t h ~ gaiitt. to the pIa\c.r. ;lt I , corrtlitio~r;il uporr X( t ) = s. 't'herr where ?' *(a) is the optinial stopping policy and C( x. 1 ) is the current value furlction. L.ct 1xt q(\, t ) = (,(L) - L7(\, t ) , for '111 ( 1 , / ). ancl let ,/,(\. / ) = ~ ~ ~ ~ - r r 7 * ( ~ ) - t l w ~ , T * ( ~ ) ) , g i ~ i ~ * ( w ) , ('14) ;~rtd co~~sitler- the problem of' irtaxirnizing F 6, "'"' g ( Y [ I ( w) , 01, 7 ( w) }= I:' g * { X[ T (a),w] , 7 ( w) ) i A.5) over s~oppirig-tirrle t'urictions 7. (w). I.er f ( w ) be the optirnal policv for rhis pr~ohlerr~. 'llierr the follo~virig tt~eorern t ~ol ds. 7. h~orrm3.-If EJ: I , - ' . ~] ul . Y(t , w) , / I ~{ I I < r,then f ( w ) = 7' *(w). a t d g84 JOURNAL OF POLITICAL ECONOMY Proof.-Shiryaev 1973, p. 101. Theorern 3 asserts that st oppi ng problenis such as ( A l )which itivo1ve an instantaneous utility obtainable ~r hi l e t he gitrnc is plavetl cart be transfot.med into problenis such as (A.5) ~\.hich involve ol -rl a trrrninal pa\of'f function g(x, /). Note that C' (r, t ) is t he current value of t he policy "never stop t he garne no mattel- wh;~t h;ippe~-rs to X(t ). " Let X( t ) satisfv t he stochastic Ith equation (.A 7 ) (or (1X(t)= ( t [ X( i ) , t l dt + b[X( t ) , t ] dv( t ) in differential form). Her e r?(t)is t he st ai ~t l ard \Vierter procrss a r ~ d . X( / ) is ;I l.lat.kov pl-occss j\.ith instantaneous nrean n ( . ) and i nst ant aneor~s variarice [ b ( .) I 2 . -1 he following theoretn cotrtains t he basic resulls associ:rted \?it11 t hc proh- Icni of optirnall! stopping . Y(i ) when X( / ) is tiefined hv ecjuatiorl ( A7) . 7'Jzcot.cnr ?.-Let X( t ) he itefi~ietl b\ ecjuation ( A' i ) , a nd let t he st oppi ng 1"-oblem be given 1)y ei1uatio1-r(A5).Let 7, ,< -c be given, at ~t l in atf(litior-1 to t he ot her requirements on T( w) ,let T( wj E 10, TI,] for i l l wEIZ. 1,et.J = {( t , x) : t E[O. 7' , , l . sER1),arid let V( x, t )= supE,.&{X[T(w),wj, -/ ' (w)), where t he s up is tal\eri over t he atlrnissahle filnctio~is T ( 0 ) .Assume that t he firtictiorts c r ( . ) , h ( . ) anti ((.) ar e :dl t~vice contitluouslv dif'ferential>le in x ant1 once it1 I ,:tnti t1i;tt for all 1 ' - ( x . t ) ~ J , i!t!l+ it,rl+ ~t.,.,~is k t l + 1 . ~ ; ) ' . / ~ . , , , ~ l + 111,r,, 1 + I ( ( , ! + ~ I I , ~ ! &( I + 1 .Y / )" and that / a, 1 + h, / k where ancl k ar e positive (.onstants. 1,etD = [(t , x):V> 51 and A + (z(.)tx(.) > 0) . = { ( t , x ) : t t ( . ) + (112)[8(.)j2[,,(.) The n t he follo.rvirtg pi-opositions holtl: ( 1 ) V 3 (onJ . (2) I f V is diff'eretltiahle, t hen Z7,(.) + n( . ) C' , ( ' ) + ( 1 / 2 ) [ b ( ~) ~V, , ( . ) = 0 for ( t , s ) E] . ( 3) Th e first exit time of t he process Lt , X( t ) ]from D is a n optimal st oppi ng time. Therefin-e L) is t he region of t he continueti observations. and. al ong its bourirlary. I,,' = 5.(4) .4 C D. (5) If ,4 is connected, so is I). Proc?f:-Miroshriichenko 1975, p. 387. C;onsider riow t he worker's problem. Let i i ~*[X(i ). t ] = Ex,,,, ( p) = IV*(t) be t he basic l l arkov process defined on (a, F. P) . 'The worker rllaxirnizes discou~itect expected earriings. His instal-i- tatleous utility is It ' *(/ ),while t he tcrrninal p a ~ o f f ftinctiorl is a c.olrst;int, (2. ' I' heref(,re t he counterpart of ccluatiori 011) is 7' heprocess I.V*(/)has ~ e r o drift. ' I' heretorc t he courlterpart of' I . ' ( x, t ) is E, *, J,P-''s-"w*(.s, w)dc = r-lLl.r*. 7' herefitre. g(x. t ) = Q - r-'lV*. Since E J ~ P - "I IV*(t, w) 1 tit < x, t . heo~ern3 ma) he applieti to t he pro1,lenr to cortclutle tltat t he solution t o t he worl er. ' ~ prohlenr o f r r ~axi n~i ~i r r g t he espr-cs- sion in ( A8 ) is itientic;~l ~r i t h t he solution to t he problern of' rnasirrri7ing If T*( w) is t he opt i ~nal solution, t hen equation (A6) \.ielcls wht.rc (:(I\'*, t ) is t he worhcr' s currellt value function. No w let Cl'(r) he the stant1;ircl \Viener- process, with I,l.' (O) = 1 1 1 ; 12'":t) is ;I stanti;irtl \\'icner pt.octlss in t h e \ - S ( t )scale (Chernoff 1968, p. 22ti). Lc r t i n g ~- a - S(/)+I = rr21(\ - V ) -~ ~ ~ - '1, art0 k'*(w) -- \ - S[' / ' "( w) ]. JOB MATCHING AND TURNOVER 985 ~vhere 7'*(w)E[O,m) -+ Y*(w)E[O, . r). ?'he prohlern has therefore been trans- fortned illto orie of' stopping a stal~tlard \.t:ierler process, M'(J), on the illterval LO, s), \kith only a terminal payoff' function 1 heoren] 4 m,iI no\\ be appllecl to tht\ problem ~\ r t h ( I ( . ) = 0, b ( . ) = 1 Lct V(iZ. J ) bc the ptesent \ , ~l ~i e funttton for thts prohlenl ticfinetl b~ 1'1o~x)~tt10t14 of the theorern nssel ts that , I C I ) hrre I1 15 the contitluatlon regtot] for the proce\s L1Z' (I), 1J I.ct [$(I), I ] he [he t)ouritlai\ of the corrttnu,%- tlon rcglori. I hen [B(?), 3) Ff A + 0 ) I for t!lO, 0 (.I15) -1 he 8( >)i i ~t r eresct\,itiotl \\,lge I ? the (It' , \ ) \p'i<c Let 6it) be the ~cs er \ ~at i ot ~ \\age in the (bt' . t ) sp.ice 1 hen B(t ) = 01, - \([)I. r hr o~pm5 4 0 ) < rQ f o ~)!LO, 5 ) Proof -Along the boundar) , l ' l @O1, % I = ). I I . (:II ti) In view of (.415), it is sufficietlt to prove that B(J) # rQ fitr ariyjE[O, s). By contradictiol~, suppose that fitr rotnejO!LO,s). f)(jO) rQ. Equation (A16) then = implies chat V[f)(y0), y o i = [().(i, so) = 0. C:onsitier ria\\. thc value of the fol l o~-i ng polic) ;it ( ~( 1, yo): For some 8 such that y o + 6 <\ , continue thc game ~t r i ~i l y o + 8. ?'hen i f' 71(yo + 8) <42, stop the galilc at yo + 6, anit collect 5[z1(11" + &) , yo + 61 >0. If7?(j"+ 8) >rQ, cot~till~ie = 5 , ancl collect a the ganir urttilr pi~koff ccliial to zero. But putb I v( y0+ 6) < JQgiver] that 71(y0) = r Q] = 112, allti so there is a positive cxpectcti pavof't'und(;.r this policv. Since this policy is feasible. C'[0(j0), yo] rllust also be positive. This colnpletes the proof of the theorern. Let F(y) be the probability that the \vorkrr-'s optinlal policv will lead hirn tit quit beforej. 17henF (yo)= prob {inf [Ctr(j) - 0 ( ~ ) ] 0). I,etf(y) he the density. Then O = s , ' . y " (A 17) Let cu ancl ,R be two par;tmerers. Assun~e rllat the evolution of X(t) is not af'f'ected t q (1 anti @. Let u(.v, t , a) be the instantar~eous utility firnctiori in present valrrc terms, and let G(x, I , p) be the tel-rninal payoff' function also in present value terms. Let I f ) ( / ;a, p),I ] be the optimally cletermitled bourtciary of the contintration region for tile process IX(t), 11. Th e function O( t , a, P) is assumed to he sir~gle valuect. Let h(s, 1 , a, p) be the probability (density) that the game will not have been stopped befi~re t , and that X(t) = r, and letf(/, a, 986 JOURNAL OF PO1.ITICAL ECONOhiY p) 1)rthe piob,rb~lrt~ (clen,it\) t h, ~t the ganie it111 be stopped el,ic t l \ nt t It rs t leal th,~tH( nit11 olle ), ht . ) , and f ( ) ,lie In one-to-orlc cor ~e$po~l t l e~l t e , ~r l ot he~ ,inti ilroultl be thought of d5 decrsrori $,$I ~~t bi cs Let I be the hort~oi?. 0 <: I -:r I ct I ( a,p) be the \<iluc of the g~trr~t. . ~ t tlllte rel o I her1 1l i r o t ~ n ~ 1l ~ ( o r ~ t n ) a ,lnd /3 do riot ' i f f t ~t the e~olutron ot Y(1). 6 ( l .117'(~/0po -11 .ind r f I ( ( . ) . (,j ), h ( ), f (.I aild V(.) ~ I C \ \ ~ t l rrespcct to a and 8, cl~fft~ierlt~able tiler, clV;ria = iJV/i)a = f , ,1-1u, , ( \ , 1 , cr)h (\ , 1. (I P)d\cil 'lnd ((1 ,rip = OVia,8 = f,',(..~e(c,( 2. p). t . PI/ ((,CY. p)dt Proof-Cnless s t , ~t t ~l ), (,(.), A t . ) . f ( ), ,~nci 0( ) all other~tr\c.,z c ( e\aiu<~ted ,it ( \ , 1 , a, p) Ful the? rilo~e. sllrce the proof f o ~ a I \ ,111trost ~dent ~cal \\lth the proof f o ~p, oul\ thc l'ltter is p e n the theol.e111 will have Iteen pi.oveti. Since tht: worket-'s policy i t r r.csporlse to a ar ~d /3 is optiliial, fol a1101p * 0 Subtldc tlng \'(a. /3) from borh s~cics of ( Ayl ) , tli\lil~rlg thlough In (la, auri taLlrig the llrnlt JS d p -0 , the ierult i\ A change i r ~ /3 i l r~pl i e~, a charrge in the optirrial stopping polic!. in genei.;~l, But the polic\ which lvas optinral prior t o the shift i l l ,8i.ernains a feasible polic! . 'l'hcreforc Kcluations ('-123),tntl ( AI Y) ~rnply that and (121)and ( Al t f ) 1111pl\ that ( A20) holds -+ tlV/cl@ = i3Vla@ .ilrd t he theor em rs p i o ~ e d I he results of I heoiern f i al e nou used to obt'iin ciualrrati\e 11rfo1 nr'itloii about the t l e~~~, i t l ve\ ( . ( kt \lnce of the uorker's cuixcnt \ al ~i e f u l ~t l o r ~ ' , 1 ) IV' *(0) = rn, JOB MATCHING AND TURNOVER BL the envelopr theorem. slntr- [[Ot s) ,\ 1 = 0. Since/ ( v ) is a (iensit?, i t is rionneg;itivc., \vhilr theor-em 5 implies that &H( Y) , J/ > O foryElO,s). Therefore (ac/a. s)(ni,0) > 0. But the state ( m, 0 ) isarbitrarj-.If the state is ( I t - , r ) , wher-eY = \ - S(/),the ar~aloguc. of the right-hand side of' ec~u~t i on replac cd b\ S ( 0 . I he onlp \$.a\. in which the (.42.i) n.ou1t-i holtl. r%.ith,\ r\rol.Ler's \\clf;u-e is af'fecteci b\, the nler-e passage ot time is thl-ough the drcr-case in S( t ) . Since a( : ( i V4: . I)ia. S(i) > 0, I' he envelope tlieor-em cannot be clirectly applied it1 ( X 2 5 ) to calculate il(;li3rti bt>c;iuseni is the s t ar t i ~~g point of the stanctarci \tiietler process f l r ( y ) , and if' i t is changed it changes the probabilities of' reaching a given bourltlar-v H ( v ) . However, / ( s o)is the derivative of F( y 0)which i r ~turn is tlehried h!. -1his means that if l o(!),f( I ) ] was a feasible policl pair- prior to the ch;+rigth in )a, then the new feasible policy pair is [ H ( T ) + dtn,f ( , ~ ) ] . In other words, after the change in m, the boundary [ @( j )+ rlm] induces the same first-pilssag-e (tensity/ ( Y) as did the bi)unda~-y @(?)prior to the change in r n , and this holds for all boul~dal-ies Nv ) . I herefore. the c h a n ~ e ft-on] rn to rtl + ilrri car1 be \ 1 7 considereti as having no effect on the feasibilit) of reaching a boundar), but sir~lpl\as changing the form of the pa).of'ffunction frorn 5(11', Y ) to [(It' + iim, y ) . Application of ~heot-cnl (5 then viclcls and sit~cef(v) is ;I density. a(:latri > 0. iI#ain, the ?rate (171,o)is arbitrar-y, ancl a siniilar I.CSLIII tloltis f i ~r (irCiilll')(l.t', >). 1.ettirigf ' ( 1 ) -/ ( r ) ( d ~ Mt ) he the first- passage pr.obabilitv in the ol.iginaI rirne scale, 7' h~ot - ~t n is norlctecreasing in t. 7.-$(t) Prorj.-By contraciiction, suppose that at t*. H(t) is decreasing. Then there exists ari E > 0 sufficier~tly small such that the points [H(l*),t* + T] for T E [ O , !1 all lie in the continuation region. Therefore, since C > Q in the continuation region, ~ [ $ ( t * ) , 1" + !1 > c [ @( l *) , / *I = Q. (A31) 988 JOURNAL OF POLITICAL ECONOMY But In vie\\. of (A27). S~nce (A32) is a contradiction to (A31), the theorem ici proved. 7 I ~ ~ o 7 ~ m 6( t ) = r (2 8 -11m 1-z Proof.--Since 6( t ) = B[s - S(t)] = %(y), it is sufficierlt to prove tllal lim O(y) = rQ. (A33) V -s Bp contradiction, suppose that lim #(y) = q and that q <rQ. Now choose6 > 0 u-L such that q + 8 < rQ. By theorem 7, H(y) is nondecreasing iny. Tllerefore the point (q + 6, s - E ) 111ust lie in the continuation region for all)- E > 0. In terms of the present value function V(\,tr,J ) and the present value of the payoff function <( W,J ) , this 111eans that where f'fq + 8, .( - ~ , y ) is the probability (density) that the game will end aty E [r - E , J ) given that M'fs - E ) = r) + 6. Since { is decreasing in Mr and decreasing in j.arid since H(y) is nondecreasing, 4[8 (s - E ) , s - !1 > <[H(J),y] for y E (.r - E , 5 ) . *I'herefi~re V(q + 8. s - t ) < <[H(s - !1, s - !1 S" f + 6, s - E J (A351 S-f Furthermore, f ( q + 6, s - t , y) is the first-passage density of the standard Wiener process (originating at q + 6 at s - t ) through the boundary #(y) on the interval [A - t , s). Then the integral on the right-hand side of (A35) is smaller than the probattilit)- that the same standard Wiener process will cross the threshold sup B(y) = 7). From Feller (1966, p. 171) this latter probability is \C=,,--e ,\ equal to 2[1 - T(t -' '8)] where.V(.u) = I:m(2.rr)-''2exp[- 112u2] ~l z c z .Therefore Equat~ons (,.236), (A34), and (A12) then imply that [Q - r - ' H ( c - ~) ] 2[ 1- A'(E I %)] > [Q - r ' (q + 6)] > 0. (A37) But since 60) 1s nondecreasing, and since by assumption linl O(y) = q < rQ, u-8 (Q - rq)2[1 - A'(t-"L6)] > [Q - r-I(q + 6)]. f.438) The right-hand side of (A38) is positive and does not depend on E. Therefore t may be chosen sufficiently snlall such that the inequality in (A38) does not hold. ?'he theorem is proved. References Az;rriadis, Costar. .'Implicit (:ontracts and Underemployment Equilibria." I.P.E. 83, no. 6 (December 1975): 1183-1202. JOB 3fATCHING AND TURNOVER g89 Baily, Martin N. "14'ages and Employriierlt untier Uncertain Dctn;ind." Rrcb. Ecorr. Stzcdirc 4 1 , no. 1 (Januar) 1974): 37-50. Aartel. A. P. ‘Job L'fobility ant1 Earnings (;t-o\~.th." \\'orking Papel., Nat. Bur. Econ. Res.. 197.5. Barrel, A. P., and Bor:i:is, G. , I . "Zliddie-Age Job Xiobilit! ." \\'orking Paper, Sat . Bur. Econ. Res., 1976. Bnrdett, Kerlncth. "'Thcor> of Ernplo!.ee Search: Quit Katcs." z*l.b:.Ii. 68 (blar-ch 1978): 2 12-20. Rurtori, J ohn F.. and ParLer-. soh11 E. "Inter-i~rdustr? I'ariations in \'oluntary Lahor I\Iot)ility." Ir~dus. clnd 1.rrhot. Rr l <~t i o ~l (j;~rlu;ir? 1960): Nri l . 22, rio. I 199-2 16. ~: t i er ~i of l , H. "Optirnal Stochastic Control." Snrrlrlrjrr, Ser. .A, 43, no. 2 (June 1968): 1 11-42. (;ox, David K.. and hfiller-, H. I). 7 % ~ 7%t~or!of Stocitcrstic I'rocrc.r~s. Sew York: i\'iley, 196.5. Feldstein, hiartin S. ".I'enlporar> l.a!ofZ's in the 'Theory of' I~rlcriiploymelit." J.P.E. 84, no. 5 (,October 1976): 937-55. Feller, Williarn. '4tr I~rtrodzrctio7~to Probcrhilitj Throry iitrcl Its A/~~>lzc.(ttiort.\. 1'01.2. 2d ed. New York: It'iiey, 1966. Freeman, K. B. "Exit Voice Tradeof'f'in the I.ahol- I\l;~rket: I:nionisln, Quits, anti J o b Te n ~~r e . " Unpnhlished paper-. Harvard Univ., 19715. Hirslrleif'er, Jack. "iCller-e Are \Ve in the Theory of' Info~-~i i ; ~t i ol i ?' .4.E.li. 87 (XIay 1973): 31-39. Johrison, W. .'.A L\I'heor> of' J o b Shopping.'' Ql1.E. 92 (bra! 1978): 26 1-77. Jo\.anovic.. Bo)ari. "Joh Xfatching and the Thcor? of Turnover." Ph.D. dis- ser.tation, I,'riiv. Chicago. J u n e 1978. ( ( 1 ) . "1,abor I'urnovet. Whel-r J o t ~ s Are Pure Se ~~r c h (;oods." I:np~tblisheci paper, Colurribia Liniv., Fet~ruary 1978. ( h ) .Jovanovic, Ro)an. and Minter, J;lcoh. "1,;tl)or Mobility and \.17;~ges." I:npub- listied paper. <;olumbia Utliv., Julie 1978. Kuratani. 31. "Theory of' l' raining, E:;n.riings, arid En~plo!rnent: .An r2pplic;1- tion to Japan." Ph.11. disser-tation, (;olumbia Uni \ . , 1973. Kushner, Harold. introductinn to . S t o ~h ( t . ~t ~~ Co~ltr-01.KC\\. York: 1folt, Kincti;t~.t 8r \Viriston, 1971. l.ucas, Robert E. , JI-., arid Prescott. Edward C. "Equiliht-iurn Search and Unemployment." J . Erotr. TI z ~wy 7, no. 2 (Fet)rua~.y 1974): 188-209. Zliroshnicheriko, .T. P. “Optimal Stopping of' an Integral of' a M'ienel. Pro- cess." Thro? c i j Probnhiii/y unci It.\ Appl. 9, no. 4 (July 197.5): 35.5-62. klortensen, 1);tle '1'. "Specific I-l~trnan Capital Bargztining anti Laltor TLII-11- over." Disc us s i o~~ Paper, Sor.tti\cestern mi\ .. .\la~.ch 1978. Selson, Phillip. "Infor~nariorl and Consurnel- Behavior." ,].P.E. 78. no. 2 (XI~lat-chiAp~.il 1970): 31 1-29. p. ,tlsons, Donald. 0."Specific Hurnan Capital: An .Applicatiori to Quit Rates .. rid Layoft' Kates.",].P.E. 80, no. 6 (Novernbcr-/l)cce~~~l>er 1972): 11 20-43. Penc;~vcl, J ohn H. i l n Ancilj.\i\ i f tlrr Qtrit Kcltr in .4tnrriccrr~ .Lf~ln~rfnclurilzg I ~~( i i ~. {t ry. PI-inceton, N.J.: Pr i nc~t on Cnik. Press. 1970. Kothschild, h'lichael. "Searching for the Lowest Price When the I>istribution of Prices Is Ilnknown." J. P. K. 82, 110. 4 (Jul>!hugust 1974): 689-71 1. Salop, Steven. "S)stcnratic J o b Search arid Cne~nplo!-~nerit." Rp-0. Eron. Studir>t 40 (,.April 1973): 191-202. Shiryaev, AI'Bert N. Stcrti,\ticnl S ~ q ~ ~ r ~ l t i n I Opt i n1~~1 Provi- d 4 ~ ~ ( ~ l j ~ i ( : S~o~~pi r r g~Kz i l r , ~. dence, K. I . : Anlerican I\lathern;~tical Society, 1973. 9g0 JOURNAL OF PO1, ITICAL ECONOMY Stoikov. Tlaclin~ir, and Ramon, R. I,. "Deterrninn~~ts of' the 1)iffel-ences in t he Quit Rate anlorig Incit~stries."A.t.:.R. 58, no. 5 (Decert~ber 1968): 1280-98. '1-else!., 1.este1. G. C) ~l p~t i r i ( ) , t , (;cinrr Aldine (;ollu.tiorr m 1 2 d Ti l ror).. (:hirago: A121erto11,1972. Viscusi, k.':Jot) Ha/artls ;tnti tVorker Quit Katcs: ,411 Analysis of' Adal)ti\e Workel- ISchavior.." Unpublished paper, North\vester-11 Uriix ., 19'76. M'ilde, 1.. " \ n 11lfi)rniatiori-theol-etic- Approach to Jot) Quits." Social Science Lt'or-king P:tpc~- no. 150, (:alifo~-nia Illst. Tec-linol., 1977. You have printed the following article: Job Matching and the Theory of Turnover Boyan Jovanovic The Journal of Political Economy, Vol. 87, No. 5, Part 1. (Oct., 1979), pp. 972-990. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197910%2987%3A5%3C972%3AJMATTO%3E2.0.CO%3B2-Q This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. [Footnotes] 5 Systematic Job Search and Unemployment S. C. Salop The Review of Economic Studies, Vol. 40, No. 2. (Apr., 1973), pp. 191-201. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28197304%2940%3A2%3C191%3ASJSAU%3E2.0.CO%3B2-N References Wages and Employment under Uncertain Demand Martin Neil Baily The Review of Economic Studies, Vol. 41, No. 1. (Jan., 1974), pp. 37-50. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28197401%2941%3A1%3C37%3AWAEUUD%3E2.0.CO%3B2-4 Systematic Job Search and Unemployment S. C. Salop The Review of Economic Studies, Vol. 40, No. 2. (Apr., 1973), pp. 191-201. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28197304%2940%3A2%3C191%3ASJSAU%3E2.0.CO%3B2-N http://www.jstor.org LINKED CITATIONS - Page 1 of 1 - NOTE: The reference numbering from the original has been maintained in this citation list. 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