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Chapter 39

N E W D E V E L O P M E N T S I N M O D E L S O F S E A R C H I N T H E L A B O RM A R K E TDALE T. MORTENSENNorthwestern Universi tyCHRISTOPHER A. PISSARIDESLondon School c f Economics, Centre fo r E conomic Performan ce and C entre Jbr E conomic Pol icy ResearchC o n t e n t sA b s t r a c tJ E L c o d e s1 I n t r o d u c t i o n2 M o d e l i n g m a r k e t s w i t h f r i c t io n

2.1 The stopping problem2.2 Two-sided search and wage determination2.3 Matching technology2.4 Search equilibrium

3 E q u i l i b ri u m u n e m p l o y m e n t3.1 Exogenou s jo b destruction3.2 Job and worker flows3.3 Social efficiency

4 A l t e r n a t i v e m o d e l s o f w a g e d e t e r m i n a t i o n4.1 Competitive search equilibrium4.2 Monop oly union4.3 Strategic bilateral bargaining4.4 Rent sharing with turnover costs4.5 Insider wage4.6 Efficiency wage

5 L a b o r m a r k e t p o l i c y a n al y s is5.1 Modeling labor market policy5.2 The qualitative effects of polic y5.3 The quantitative effects of policy5.4 A call for research

6 W a g e p o s ti n g g a m e s6.1 The Diamond paradox6.2 Wa ge dispersion: differential costs of search6.3 Wag e dispersion: more than one offer

2 5 6 82 5 6 82 5 6 82 5 7 12 5 7 12 5 7 42 5 7 52 5 7 72 5 7 82 5 7 82 5 8 32 5 8 72 5 8 92 5 8 92 5 9 12 5 9 22 5 9 42 5 9 72 5 9 82 6 0 12 6 0 22 6 0 32 6 0 42 6 0 72 6 0 72 6 0 82 6 1 02 6 1 2

Hand book o f Lab or Economics, Volume 3, Edi ted by O. Ashenfel ter and D. Card 1 999 Elsevier Science B.V. Al l rights reserved.

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25686.4 Sea rch on the .job6 .5. W orke r and emp loye r he te rogene i ty6.6. Struc tura l est im ation7 . W a g e p o s t i n g i n a m a t c h i n g m o d e l7 .1. Sea rch and ma tch ing7 .2 . W age pos t ing7 .3. Endog enous p roduc t ive he te rogene i ty

8 . S u m m a r yR e f e r e n c e sA b s t r a c t

D. T. Mortensen and C. A. Pissarides

2 6 1 32 6 1 52 6 1 72 6 1 92 6 2 02 6 2 02 6 2 22 6 2 32 6 2 4

Eqnil ibf ium mo dels o f labor markets character ized by search and recru i ting f r ic t ion and by the needto rea l locate w orkers f rom t ime to t ime across a l ternat ive produ ct ive ac t iv i ties rep resent the segm entof the research front ier explored in th is chapter . In th is l i tera ture, unem ploym ent spell and job spel ldurat ions as wel l as w age offers are t rea ted as endogen ous ou tcom es of forward lookin g job crea t ionand job des t ruc t ion dec i sions made b y the worke rs and em ploye rs who popu la te the mode ls . Thesolutions studied are dynamic stochastic equilibria in the sense that time and uncertainty are expli-c i t ly modeled , expecta t ions are ra tional , p r iva te ga ins f rom trade are explo ited, and the ac t ions takenby a l l agents are mu tual ly consis ten t. W e argue tha t the f ram ew ork provides a usefu l set ting in wh ichto study the effects of alternative w age setting institutions and differen t labor mark et polic y regimes. 1999 Elsevier Science B .V. All r ights reserved .JEL codes : D58 ; E24 ; J31 ; J41 ; J64

1 . I n t r o d u c t i o n

E q u i l i b r i u m m o d e l s o f l a b o r m a r k e t s c h a r a c t e r i z e d b y s e a r c h a n d r e c r u it i n g f r i ct i o n a n d b yt h e n e e d t o r e a l l o c a t e w o r k e r s f r o m t i m e t o t im e a c r o s s a l t e r n a t i v e p r o d u c t i v e a c t i v i t ie sr e p r e s e n t t h e s e g m e n t o f t h e r e s e a r c h f i ' o n t ie r e x p l o r e d i n t h is c h a p t e r . I n t h i s l i te r a tu r e ,u n e m p l o y m e n t s p e l l a n d j o b s p e l l d u r a t i o n s a s w e l l a s w a g e o f f e r s a re tr e a t e d a s e n d o -g e n o u s o u t c o m e s o f f o r w a r d l o o k i n g jo b c r e a t io n a n d j o b d e s t r u c t io n d e c i si o n s m a d e b yt h e w o r k e r s a n d e m p l o y e r s w h o p o p u l a t e t h e m o d e l s . T h e s o l u t i o n s s t u d i e d a r e d y n a m i cs t o c h a s t i c e q u i l i b r i a i n t h e s e n s e t h a t t i m e a n d u n c e r t a i n t y a r e e x p l i c it l y m o d e l e d , e x p e c -t a t i o n s a r e r a t i o n a l , p r i v a t e g a i n s f r o m t r a d e a r e e x p l o i t e d , a n d t h e a c t i o n s t a k e n b y a l la g e n t s a r e m u t u a l l y c o n s i s t e n t . I n c o n t r a s t t o th e e a r l i er l i t e ra t u r e o n i n d i v i d u a l w o r k e r j o bs e a r ch d e c i s io n s , f o r e x a m p l e , m u c h o f th a t r e v i e w e d b y M o r t e n s e n ( 1 9 86 ) , t h e e q ui li -b r i u m s e a r c h a p p r o a c h e x p l i c i tl y a c c o u n t s f o r a n d in d e e d e m p h a s i z e s t h e r o le o f e m p l o y -e r s o n t h e d e m a n d s id e o f t h e l ab o r m a r k e t . A s a c o n s e q u e n c e , w e a r g u e , th e f r a m e w o r kp r o v i d e s a r i c h a n d u s e f u l s e t t in g i n w h i c h t o s t u d y t h e e f f e c t s o f a l t e r n a t iv e w a g e s e t t in gi n s t it u t i o n s a n d d i f f e r e n t l a b o r m a r k e t p o l i c y r e g i m e s .

T h e n e e d f o r a r i c h e r e q u i l ib r i u m f r a m e w o r k f o r l a b o r m a r k e t a n a l y s i s t h e n t ha tp r o v i d e d b y t h e f r i c t i o n l e s s c o m p e t i t i v e m o d e l i s b o t h e m p i r i c a l a n d c o n c e p t u a l . L a r g e

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Ch 39: New Developments in Models of Search in the Labor Market 2569numbers of workers and jobs flow between inactivity and market production at the aggre-gate level. At the level of individual workers and employers, worker flows between labormarket states and job creation and job destruction flows are reflected in activity spellsfound in panel data whose durations reflect the time spent searching for work, filling avacancy, and working in a particular job. These movements are concealed in existingmodels of employment that focus on stocks. The emphasis on mobility makes the types ofmodels reviewed here part of the so-called flows approach (see Blanchard and Diamond,1992). to labor market analysis. 1

Still another empirical reason for interest in the framework is wage dispersion acrossobservably identical workers. These differentials have led many observers to questionand some to reject perfectly competitive wage theory. Search and matching frictionsinevitably generate match specific rents that the wage must divide between worker andemployer. Because the precise way in which these rents might be shared is indetermi-nate, the framework requires some alternative to the marginal productivity theory ofwages, at least in its simplest form. Although the natural and usual specification is expost bargaining in the models reviewed, alternatives such as a monopoly union speci-fication, an insider-outsider story, and efficiency wage theories can all be accommo-dated and studied within the framework. The principal alternative to the wage as abargaining outcome, that studied most extensively in the literature, is the assumptionthat employers post wage offers.

What new lessons can be learned about the effects of policy using the new approach?Because unemployment has an economic role in the flows framework, welfare state-ments about the effects of policy on unemployment and on the cost of unemploymentexperienced by those who bear it are possible. Also, the total effect of a policy can bedecomposed into effects on unemployment duration and on unemployment incidence.As a consequence of this fact and the two sided nature of the models, multiple channelsof influence arise. For example, unemployment benefits influence both worker incen-tives to accept employment and the wage. Because the wage affects employer incen-tives to create vacancies and recruit workers, the total impact on unemploymentduration is a consequence of decisions made on both sides of the labor market. Asthe wage impacts job destruction as well, there are a least three different channelsthrough which unemployment insurance benefits might be expected to affect unemploy-ment.

Consider the effects of different forms of employment subsidies as another example offruitful policy application. In conventional static models, a subsidy is treated as a reduc-tion in labor costs which increases the demand for labor. Whether the subsidy is paid toemployers on a per employee hired basis or is proportional to the employment stock isimmaterial. Because these two alternative subsidy forms have different effects on the job

We have also written a companionpaper, Mortensen and Pissarides (1998), on the macroeconomic mplica-tions of the flows approachfor the forthcomingHandbook of Macroeconomics. For that reason, the focus of thischapter is more microeconomic.

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2570 D. Z Mortensen and C. A. Pissaridesc r e a t io n a n d j o b d e s t r u c ti o n d e c i s i o n s t h a t d e t e r m i n e w o r k e r a n d e m p l o y e r fl o w s , q u e s -t i o n s t h e f o r m t h a t s u b s i d i e s s h o u l d t a k e c a n b e a n a l y z e d w i t h i n t h e e q u i l i b r i u m f l o w sf r a m e w o r k .T h e c o m m e n t s a b o v e s u g g e s t t h a t t h e e x i s ti n g l it e r a tu r e o n e q u i l i b r i u m s e a r c h f o r m s aun i f i ed who le . A l thoug h r e l a t ed , t he re a r e tw o qu i t e d i f f e r en t b r an ches o f t he s ea rchequ i l i b r ium l i t e ra tu re , each w i th i ts own p r im ary con ce rns . T he goa l o f t he f i rs t i s t oe x p l a i n w o r k e r a n d j o b f l o w s a n d l e v e l s o f u n e m p l o y m e n t w i t h i n t h e r a t io n a l f o r w a r dl o o k i n g a g e n t p a r a d i g m . F u n d a m e n t a l i s t h e i d e a t h a t t w o - s i d e d f r i c t i o n s e x i s t i n t h ep r o c e s s o f m a t c h i n g t r a d i n g p a r t n e rs a n d t h a t a g e n t s o n b o t h s i d e s o f a m a r k e t m a k ei n v e s t m e n t s i n o v e r c o m i n g t h e m . A s a re s u lt , t h e j o b c r e a t io n f l o w d e p e n d s o n t h en u m b e r s o f u n e m p l o y e d w o r k e r s a n d v a c a n t j o b s a v a i l a b l e a n d o n th e i n te n s i ti e s w i t hw o r k e r s s e a r c h a n d e m p l o y e r s r ec r u it , a r e l a t io n s h i p w h i c h h a s b e c o m e k n o w n a s th em a t c h i n g f u n c t i o n . T h e e f f e c ts o f m a r k e t f r i ct io n o n t h e i n c e n t i v e s t o i n v e s t i n s e a r c h ,r e c r u it i n g , t r a i n i n g a n d o t h e r f o r m s o f m a t c h s p e c if i c c a p i t a l w h i c h i n t u r n d e t e r m i n e t h ee q u i l i b ri u m l e v e l o f e m p l o y m e n t a r e t h e p r i m a r y c o n c e r n s i n t h e l i te r a t u re b a s e d o n t h e" m a t c h i n g a p p r o a c h " t o la b o r m a r k e t a n a ly s is .

C o n t r i b u t o r s t o t h e s e c o n d l it e r at u r e s h o w t h a t w a g e d i s p e r s io n c a n b e a n e q u i l i b r iu mo u t c o m e i n m a r k e t s w i t h f i ' i c t i o n . B y a s s u m p t i o n , w a g e o f f e r s a r e s e t b y e m p l o y e r s i n an o n - c o o p e r a t i v e s e t ti n g w h i l e w o r k e r s s e a r c h f o r t h e b e s t a m o n g t h e m . H e r e , s e a r c hf r i c t i o n i s r e g a r d e d a s s i m p l y t h e t i m e r e q u i r e d f o r w o r k e r s t o g a t h e r i n f o r m a t i o n a b o u tw a g e o f f er s . T h e o u t c o m e s o f t h e s e s tr a te g i c " w a g e p o s t i n g " g a m e s a r e s t u d ie d a s e x p l a -n a t i o n s o f w a g e d i f fe r e n ti a ls t h a t a r e n o t a s s o c i a t e d w i t h o b s e r v e d w o r k e r s k il l.

A l t h o u g h a r e v i e w o f r e c e n t d e v e l o p m e n t s i n s ea r c h e q u i l i b r i u m i s a p r i n c i p a l p u r p o s eo f t h e c h a p t e r , i t i s n o t t h e e x c l u s i v e o n e . A n o t h e r g o a l i s t o s h o w h o w t h e g e n e r a la p p r o a c h h a s a n d c a n b e u s e d t o s t u d y t h e e m p l o y m e n t e f f e c ts o f d i f fe r e n t w a g e d e t e r m i -n a t i o n m e c h a n i s m s a n d c a n b e a p p l i e d to l a b o r m a r k e t p o l i c y a n a l y si s . W e a l s o s h o w h o wt h e t w o b r a n c h e s o f th e s e a r c h e q u i l i b r iu m l it e r at u r e c a n b e r e i n t e g r a t e d a n d s u g g e s t s o m eof t he r ew ards t ha t such a syn thes i s o f f e r s .

T h i s c h a p t e r i s c o m p o s e d o f s e v e n s ec t io n s . T h e t o o l s u s e d a n d th e c o n c e p t s a p p l i e d b yc o n t r i b u t o r s t o th e l i t e ra t u r e o n s e a r c h e q u i l i b r iu m m a r k e t m o d e l s a r e b r i e f ly in t r o d u c e d i nS e c t i o n 2 . F o r m a l m o d e l s o f j o b - w o r k e r m a t c h i n g , l a b o r m a r k e t fl o w s , a n d e q u i l i b r iu mu n e m p l o y m e n t a r e t h e t o p i c s o f S e c t i o n 3 . I n t h e s e m o d e l s , w a g e s a r e d e t e r m i n e d b y as p e c i f i c r e n t s h a r i n g r u l e t h a t c a n v i e w e d a s t h e o u t c o m e o f a N a s h b a r g a i n b e t w e e nw o r k e r a n d e m p l o y e r e n g a g e d i n w h e n t h e y m e e t . S e c t io n 4 r e v i e w s v a r i a ti o n s o f th em a t c h i n g m o d e l c h a r a c t e r i z e d b y d i ff e r e n t w a g e d e t e r m i n a t i o n m e c h a n i s m s . I n S e c t i o n 5,a p p l i c a t i o n s o f th e m a t c h i n g a p p r o a c h t o t h e a n a l y s i s o f la b o r m a r k e t p o l i c y a r e re v i e w e da n d i l lu s t ra t e d . F o r m s o f w a g e d i s p e r si o n t h a t a r is e s a s e q u i l ib r i u m o u t c o m e s o f w a g ep o s t i n g g a m e s a r e th e p r i n c i p a l t o p i c s o f S e c t i o n 6 . T h e i m p l i c a t i o n s o f a s y n t h e s i s o f th em a t c h i n g a n d w a g e p o s t i n g a p p r o a c h e s t o m o d e l i n g l a b o r m a r k e t e q u i li b r iu m a r e s k e tc h e di n S e c t i o n 7 . F i n a ll y , a v e r y b r i e f s u m m a r y c o n c l u d e s t h e c h a p t e r.

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Ch 39: N ew D evelopments in M odels of Search in tile Labor Market2 . M o d e l i n g m a r k e t s w i t h fr i c t io n

2571

M a r k e t f r i c t io n , the costly delay in the process of finding trading partners and determiningthe terms of trade, is ignored in the standard theory of perfectly competitive markets.Friction is explicitly modeled in the work reviewed in this chapter. The central problem ofthe theory of markets with friction is to find a useful way to make the behavior ofindividual agents both individually rational and mutually consistent. In this section, wesurvey the concepts introduced in the recent literature on the problem. 22 .1 . T h e s t o p p i n g p r o b l e mThe tools of dynamic optimization applied in the equilibrium search literature are intro-duced first. We do so in the process of reviewing the sequential job search model, theworkhorse of the literature, which is based on the decision theoretic optimal stoppingproblem.

A distribution of payoffs character ized by a c.d.f. F(W) is postulated which is known tothe searcher. A sequential sample of realizations can be drawn with replacement at aconstant per observation cost denoted by a. Only one of the realizations can be acceptedand acceptance is a sequential decision without recall. A search strategy determines whento accept, i.e., it is a stopping rule. An o p t i m a l s t o p p i n g s t r a t e g y maximizes the expectedpresent value of the realization accepted net o f the accumulated costs of search.

Application of the model to the job search problem in which workers are not fullyinformed about the terms of available employment offers is simply a matter of interpreta-tion of this structure. Think of the sampling process as that of sequentially applying forjobs selected at random and let each realization of W represent the value of an offeredemployment contract, either the wage or more generally the present value of a worker'sfuture utility stream conditional on accepting the offer. In discrete time, the stoppingdecision is easily formulated as a dynamic programming problem. If a single sample istaken in every period until the process stops and past realizations cannot be recalled, thenthe value of searching in each period, Ut, is generated by the Bellman equation

b - a + 1 ~ m a x { W , U ~ + l } d F ( W ) ,U t - 1 + ~ 1 + ~ t = 1,2, .... (1 )where r is the discount or risk free interest rate, b is income flow received contingent onunemployment, and a represents the cost of search per period. Namely, the optimalstrategy involves comparing the observed current realization of the sampling process W,with the value of continued search Ut+ 1 in the next period. I f the former exceeds the latter,then the search process stops, i.e., the optimal strategy satisfies a r e s e r v a t i o n p r o p e r t y . Inthe infinite horizon case, the value o f continued search is the stationary solution to Eq. (1),

2Some of the original papers that raised the issues discussed in this section include Diamond (1981, 1982a,b),Mortensen (1982a,b), and Pissarides (1984a,b).

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2 5 7 2 D . T . M o r t e n s e n a n d C . A . P i s s a r i d e si.e., Ut = U for all t where

b- a + 1 [max {W,v - 1 + ~ 1 + ~ U } d F ( W ) , (2 )As the r ight side is a contraction map for all 0 < r < 1, 3 call it T(U), a unique finitesolution U = T ( U ) exists provided that the c.d.f. F has a first moment.

Virtually all the literature on equilibrium search is cast in continuous rather thandiscrete time. Although this fact Js partially a historical accident, continuous time tech-niques can often reduce the apparent complexity of sequential search and recruitingproblems. For example, allowing for a stochastic time interval between offer arrivals isone realistic extension easier to formalize in continuous time. Because arrival dates areseparated in continuous time, decisions are revised only after arrivals. Hence, the analysisreduces to a dynamic programming formulation in which the time intervals betweendecision dates is a strictly positive random variable with a know duration distribution.

Characterize the distribution of random waiting time between offer arrivals by itsgenerally duration dependent hazard function A(t), i.e., the probability that an offer willnot arrive before T, the associated survivor function of the waiting time distribution, isexp[- fS A(t)dt]. Taking account of the waiting duration, the Bellman equation for theextended model becomesU ( t ) = E r { ( b - a ) exp(-rs)ds + e xp [- r( r - 0] max{W, U ( T ) } d F ( W ) }t

= ( (b - a ) e x p ( - r s ) d s + ex p[- r(T - t)]tx fm a x { W ,U ( r ) } d F ( W ) ) a ( r ) e x p [ - [ r a ( t ) d t ] d r , (3 )J 0 1where U ( t ) is the value of search at time t and T > t is the future random date at which thefirst offer arrives. Given an exponential waiting time distribution, a constant hazardA(0 --= A, the value of search is stationary and solves

b - aU = (1 - exp (-rT )) + ex p( -r T max{W, U } d F ( W Aexp(-AT)dT0 /"_ b - a + _ _ l f m a x { W , U } d F ( W ) , (4 )r + h r + h J

where A is the Poisson offer arrival rate. Note that Eq. (4) is a simple generalization of Eq.(2) that accounts for the duration of the waiting period, 1/h in the exponential case.

The full power of the continuous time formulation is suggested by the followingequivalent "asset pricing" representation of Eq. (4):

3 I t s a t i s fi e s B l a c k w e l l s u f f i c ie n t c o n d i t i o n s ( s e e L u c a s a n d S t o k e y , 1 9 8 9 , p . 5 4 ).

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Ch 39: New D evelopments in Mo dels of Search in the Labor Market 2573r U : b - a + A f [ m a x { W , U } - U ] d F ( w ) . (5 )U r e p r e s e n t s t h e " a s s e t " o r " o p t i o n " v a l u e o f s e a rc h a c t i v it y . G i v e n t h i s in t e r p r e ta t i o n ,E q . ( 5 ) s i m p l y p r i c e s t h e o p t i o n b y r e q u i r i n g t h a t t h e o p p o r t u n i t y c o s t o f h o l d i n g i t, t h e l e f th a n d s i d e, is e q u a l t o t h e c u r r e n t i n c o m e f l o w , b - a , p l u s t h e e x p e c t e d c a p i t a l g a i n f lo w ,t h e p r o d u c t o f t h e a r r i v a l f r e q u e n c y h a n d t h e e x p e c t e d c a p i t a l g a i n g i v e n a n o f f e r ar r iv a l.

I n th e g e n e r a l c a s e, o n e c a n s h o w t h a t U ( t ) m u s t b e a s o l u t i o n t o t h e g e n e r a l i z e d a s s e tp r i c i n g e q u a t i o n , t h e d i f f e r e n t i a l e q u a t i o n

f d U ( t )r U ( t ) = b ( t ) - - a ( t ) + h ( t ) [ m a x { W , U ( t) } - U ( t ) ] d F ( W ) + d - - ~ ' ( 6 )

w h e r e t h e d u r a t i o n d e r i v a t i v e d U / d t i s th e p u r e r a t e o f c a p i ta l g a i n o r l o s s a t t r ib u t a b l e t ow a i t i n g a n o t h e r i n s t a n t f o r a n o f f e r a r r iv a l . T h i s e q u a t i o n c a n b e o b t a i n e d d i r e c t l y b yd i f f e r e n t ia t i n g b o t h s i d e s o f E q . ( 3 ) w i t h r e s p e c t t o t. U s i n g t h e f a c t t h a td U ( t ) l i m { U ( t + d t ) - U ( t ) }

d t d t~0 d t 'o n e c a n a l s o w r i t e

1 [ (b( t) - a ( t )) d t + h ( t ) d t f m a x { W , U } d F ( W ) + [1 - h ( t ) d t ] U ( t + d t ) ]U ( t ) - - l + r d~a s a n a p p r o x i m a t i o n t o E q . ( 6 ) f o r a ll s u f fi c i e n tl y s m a l l v a l u e s o f t h e p e r i o d l e n g t h d t > 0 .O b v i o u s l y , t h i s re l a t i o n s h i p h a s a n a t u r a l i n t e r p r e ta t i o n a s a B e l l m a n e q u a t i o n i n a d i s c r e t et i m e f o r m u l a t i o n o f t h e p r o b l e m w h e r e h ( t )d t i s th e p r o b a b i l i t y o f a n o f f e r a r r iv a l d u r i n gt h e p e r i o d ( t, t + d t ) a n d r d t i s th e d i s c o u n t r a t e f o r th e s p e c i f i e d p e r i o d l e n g t h d t . O fc o u r s e , t i m e p a t h s f o r t h e v a l u e o f a n o p t i m a l s e a r c h s t r a t e g y m u s t a l s o s a t i s f y th e t r a n s-v e r s a l i t y c o n d i t i o n l i m t ~ o o U ( t ) e x p ( - r t ) = 0 . T h e g e n e r a l f a c t t h a t t h e o p t i o n v a l u e o fs e a r c h s o l v e s a g e n e r a l a s s e t p r i c i n g e q u a t i o n a n d t r a n s v e r s a l i t y c o n d i t i o n o f t h is f o r mp r o v i d e s a v e r y q u i c k a n d p o w e r f u l c h a r a c t e ri z a t io n o f o p t i m a l i t y c o n d i t i o n s i n e q u il i-b r i u m s e a r ch m o d e l s i n c o n t i n u o u s t im e . 4

I n t e r e s t a m o n g e m p i r i c a l l a b o r e c o n o m i s t s i n s e a r c h t h e o r y w a s g e n e r a t e d i n i t i a l l y b yt h e fa c t th a t it a d d r e s se d o b s e r v a t i o n s o n u n e m p l o y m e n t s p e ll d u r a t i o n l e n g t h s a n d s u b s e -q u e n t a c c e p t e d w a g e d i s t r ib u t i o n i n p a n e l d a t a. A s t h e u n e m p l o y m e n t s p e ll h a z a r d ish ( t ) [ 1 - F ( U ( t ) ) ] a n d t h e c o n d i t i o n a l a c c e p t a b l e w a g e d i s t r ib u t i o n c o n d i t i o n a l o n d u r a t i o nis F ( W ) / [ 1 - F ( U ( t ) )] , t h e m o d e l c a n b e f o r m a l l y a p p l i e d t o in t e r p re t a v a i la b l e d a t a a n d t og e n e r a t e t e s t a b l e e m p i r i c a l h y p o t h e s e s . T h e r e i s n o w a s u b s ta n t i a l l i te r a t u r e t h a t d o e s j u s t

4 The value of continued search is not stationary if (a) the horizon is finite, (b) payo ff realizations revealsinformation abou t its distribution to the searcher, or (c) the environm ent is non -stationary. Alth oug h all of thesecases ha ve bee n studied in the partial equilibrium literature, (c) is the principal source o f non-stationarityconsidered in the equilibrium literature.

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Ch 39: New Developments in Models of Search in the Labor Market 2 5 7 5i s a m a j o r i s s u e i n t h e c o n te x t o f s e a rc h e q u i l ib r i u m m o d e l l i n g . U n l i k e c o m p e t i t i v e t h e o r yw i t h o u t f r i c ti o n , a n e x is t in g m a t c h w i ll a l w a y s c o m m a n d q u a s i -r e n t s e x p o s t b e c a u s e i t isc o s t l y i n t i m e a n d r e s o u r c e s f o r e i t h e r p a r t y i n t h e p a i r t o s e e k t h e n e x t b e s t a l t e r n a t i v e .G i v e n t h e e x i s t e n c e o f t h e s e q u a s i - r e n ts , t h e " m a r k e t w a g e " i s n o t u n i q u e i n th i s e n v i r o n -m e n t . A n y d i v i s i o n t h a t s a t is f ie s i n d i v i d u a l r a t i o n a l i t y i s a f o r m a l p o s s i b i l i t y . H o w e v e r , t h em o s t c o m m o n s p e c i f i c a t i o n f o u n d i n t h e l it e r a tu r e i s t h e a s s u m p t i o n t h a t r e n ts a r e d i v id e dw i t h t h e w o r k e r ' s s h a r e / 3 r e g a r d e d a s a f r e e p a r a m e t e r . P o s s i b l e j u s t i f i c a ti o n s a s w e l l a sa l te r n a ti v e s a s s u m p t i o n s a b o u t w a g e d e t e r m i n a t i o n a r e c o n s i d e r e d s u b s e q u e n t ly .2 . 3 . M a t c h i n g t e c h n o l o g yA m a t c h i n g t e c h n o l o g y , l i k e a p r o d u c t i o n t e c h n o l o g y , is a d e s c r i p t i o n o f th e r e l a ti o nb e t w e e n i n p u t s, s e a r c h a n d r e c r u i t i n g a c t i v i t y , a n d t h e o u t p u t o f t h e m a t c h i n g p r o c e s s ,t h e f lo w r at e a t w h i c h u n e m p l o y e d w o r k e r a n d v a c a n t j o b s f o r m n e w j o b - w o r k e r m a t c h e s .B e c a u s e a n e m p l o y e r j o i n s a w o r k e r w h e n a w o r k e r j o i n s a n e m p l o y e r to f o r m a m a t c h , a n" a d d i n g u p " c o n d i t i o n h o l d s t h a t n e e d s to b e m a d e e x p l ic i t. T h e f l o w r a te a t w h i c hu n e m p l o y e d w o r k e r s m e e t v a c a n c i e s is id e n t i c a ll y e q u a l t o th e f lo w r a te a t w h i c h e m p l o y -e r s w i t h v a c a n t j o b s m e e t a p p l i ca n t s. F o r m a l l y , t h e a s s u m p t i o n t h a t e a c h s e a r c h i n g w o r k e rm e e t s p r o s p e c t i v e e m p l o y e r s a t f r e q u e n c y h i m p l i es t h a t t he e x p e c t e d a g g r e g a t e r a te a tw h i c h u n e m p l o y e d w o r k e r s m e e t v a c a n t j o b s i s e q u a l t o h u w h e r e u d e n o t e s th e m e a s u r eo f u n e m p l o y e d w o r k e r s . S i m i l a r l y , t h e a s s u m p t i o n t h a t e a c h v a c a n c y i s v i s it e d b y w o r k e r sa t f r e q u e n c y r / i m p l i e s t h a t t h e a g g r e g a t e ra t e a t w h i c h v a c a n c i e s m e e t a p p l i c a n t s i s rj vw h e r e v r e p r e s e n t t h e m e a s u r e o f v a c a n c i e s . T h e s e t w o f l o w s a r e i d e n t i c a l l y e q u a l .O b v i o u s l y , s i n c e t h e v a c a n c y a n d u n e m p l o y m e n t p a i r ( v , u ) c a n b e a n y t h i n g , t h e i d e n t i t yh u =-- r / v r e q u i r e s t h a t t h e a rr i v a l f r e q u e n c i e s a r e f u n c t i o n s o f th e m e a s u r e s o f p a r ti c i p a -t i o n , u a n d v .

T h e g e n e r a l s o l u t i o n t o t h i s p r o b l e m f o u n d i n t h e l i t e r a t u r e i s t o i n v o k e a m a t c h i n g. funct ion , d e n o t e d a s m ( v , u ) , w h i c h c h a r a c t e r i z e s t h e a g g r e g a t e m e e t i n g r a t e . T h e nAu ~ m (v , u) =-- ~ v ( 1 0 )i m p l i e s t h a t t h e t w o m e e t i n g r a t e f u n c t i o n s , A = m( v , u ) / u a n d ~ = m( v , u ) / v , r e p r e s e n t t h ea v e r a g e r a t e s a t w h i c h u n e m p l o y e d w o r k e r s a n d v a c a n c i e s m e e t p o t e n t i a l p a r t n e r s . T h em a t c h i n g f u n c t i o n s u m m a r i z e s a ll t he d e ta i ls o f t h e m e e t i n g p r o c e s s i s a m a n n e r a n a l o g o u st o t h e w a y a n a g g r e g a t e p r o d u c t i o n f u n c t i o n s u m m a r i z e s a p r o d u c t i o n p r o c e s s . N a m e l y , i ti s t h e " o u t p u t " o f th e m e e t i n g p r o c e s s e x p r e s s e d a s a f u n c t i o n o f i ts i n p u t s , a s r e f l e c t e d int h e m e a s u r e s o f a g e n t s o f e a c h t y p e p a r t i c i p a t i n g i n t h e p ro c e s s . 7

A t t h e m i c r o l e v e l , d i f f e r e n t m a t c h i n g f u n c t i o n s c a n b e d e r i v e d f r o m s p e c i f i c s p e c i f i c a -t i o n s o f t h e m e e t i n g p r o c e s s . F o r e x a m p l e , i f e a c h a g e n t o n o n e s i d e o f th e m a r k e t h a s a llt h e t e l e p h o n e n u m b e r s o f u n m a t c h e d a g e n t s o n t he o t h e r si d e an d e a c h m a k e s c o n t a c t b yp h o n i n g a n u m b e r c h o s e n a t r a n d o m f r o m t i m e t o ti m e , t h e n o n e c a n s h o w t h a t th e m e e t i n g

Although the matching function was no t stated and estimated explicitly until the early 1980s, it features inearlier search models, mo st notably Phelps (1968) and Bow den (1980).

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2576 D. 72 Mortensen and C. A. Pissaridesfunc t ion i n con t in uous t ime t akes t he l i nea r fo rm m = f u + gv whe re f r ep re sen t s t hec a l l i n g f r e q u e n c y o f t h e ty p i c a l u n e m p l o y e d w o r k e r a n d g i s t h e a v e r a g e f r e q u e n c yw i t h w h i c h e m p l o y e r s w i t h v a c a n c i e s m a k e c a l l s . F o r m a l l y , f u c a l l s a r e m a d e p e r p e r i o do f l e n g th d t b y t h e s e a r c h i n g w o r k e r s. A s t h e e x p e c t e d n u m b e r o f c a ll s m a d e b y w o r k e r sp e r e m p l o y e r w i t h a v a c a n c y p e r p e r i o d o f l e n g t h d t i s f ud t / v a n d t h e a c t u a l n u m b e r o f c a l lsr e c e i v e d b y a n y o n e e m p l o y e r i s a P o i s s o n r a n d o m v a r i a b l e , t he p r o b a b i l i t y th a t a p a r t i -c u l a r e m p l o y e r i s n o t c a l l e d d u r i n g t h e i n t e r v a l i s e x p ( - ( f u / v ) d t ) . H e n c e , t h e n u m b e r o fe m p l o y e r s w h o r e c e i v e o n e o r m o r e c a l l s is v(1 - exp(-OCu/v)dt ) . A n a l o g o u s l y , t h en u m b e r o f w o r k e r s w h o r e c e i v e a t l e a s t o n e c a l l f r o m s o m e e m p l o y e r w i t h a v a c a n c y i su (1 - e x p ( - ( g v / u ) d t ) . T h e a g g r e g a t e c o n t a c t r a t e p e r u n it p e r i o d i s t h e s u m o f t h e s e tw on u m b e r s d i v i d e d b y t h e p e r i o d l e n g t h d t . T a k i n g t h e l i m i t a s t he l a t t e r t e n d s t o z e ro , o n eob ta in sl im ~ v[1 - e x p ( - ( f u l v ) d t ) ] + u[1 - e x p ( - ( g v l u ) d t ) ]d~O ( dt

= l i m { f u e x p ( - q~t/v)dt) + g v e x p ( - (gv/u)dt ) } = fu + gvdt~0prov ided tha t ( u , v ) > 0 .

H o w e v e r , i f t h e t e l e p h o n e b o o k i n c l u d e s a l l a g e n t s o n t h e o t h e r s i d e o f th e m a r k e t ,m a t c h e d a n d u n m a t c h e d , t h e n t he a g g r e g a t e r at e a t w h i c h u n m a t c h e d a g e n ts o f t h e t w otypes meet i s m - - - - uv/k + gvu/ l , w h e r e I a n d k r e p r e s e n t t h e t ot a l n u m b e r o f w o r k e r a n dj o b s , s i n c e v /k i s th e p r o b a b i l i t y th a t a r a n d o m l y s e l e c t e d e m p l o y e r w i l l h a v e a v a c a n c ya n d u/1 i s t h e p r o b a b i l i t y t h a t a r a n d o m l y s e l e c t e d w o r k e r w i l l b e u n e m p l o y e d . A l t h o u g h i nb o t h c a s e s t h e a g g r e g a t e m a t c h i n g r a t e is i n c r e a s i n g ( a n d c o n t i n u o u s ) i n i t s a r g u m e n t s, ac o n d i t i o n a n y r e a s o n a b l e m e e t i n g p r o c e s s s h o u l d h a v e , t h e m a t c h i n g f u n c t i o n i s h o m o -genous o f deg ree one i n t he f i r s t c a se bu t exh ib i t s i nc reas ing r e tu rns i n t he s econd in t hes e n s e t h at d o u b l i n g t h e n u m b e r s o f p a r t i c i p a t e s o f t h e t w o t y p e s q u a d r u p l e s t h e m e e t i n gra te . 8

T h e m o s t c o m m o n s p e c i f ic a t i o n i n t he a p p l i e d l i t e r a t u r e i s t h e l o g l i n e a r o r C o b b -D o u g l a s m a t c h i n g f u n c t i o n w i t h c o n s t a n t r e t u r n s , i . e . ,re(v, u) ~ mo vl-c~u~,wi th 0 < a < 1 . A l th ough the na tu re o f the spec i f i c ma tc h ing p roce s s t ha t migh tg e n e r a t e s u c h a f u n c ti o n i s n o t k n o w n , P i s s a r i d e s ( 1 9 96 ) a n d B l a n c h a r d a n d D i a m o n d(1989) p r ov i de em pi r i ca l j u s t i f i ca t i on fo r t h i s f o rm. Indee d , t he i r r e su lt s sugges t t ha t ane l a s t i c it y p a r a m e t e r a i s i n th e n e i g h b o r h o o d o f o n e - h a l f is c o n s is t w i th a g g r e g a t edata . 9

8 D iamond and Maskin (1979) call the first the "linear" case and the second the "quadratic" case.9 However, more recent research based on micro panel data suggest important elements of increasing returns toscale. For example, see Warren (1996), Co les and Smith (1994), and Munich et al. (1997).

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Ch 39: New Developments in Models of Search in the Labor Market 25772 . 4 . S e a r c h e q u i l i b r i u m

A s p a r t i c i p a t i o n i s v o l u n t a r y , a f u ll d e f i n it i o n o f s e a r c h e q u i l i b r i u m m u s t s p e c i f y t h em e a s u r e s o f s e a r c h p a r ti c i p a n t s , u a n d v , a s w e l l a s t h e p a r t i c i p a t i o n v a l u e s f o r w o r k e ra n d e m p l o y e r , U a n d V . T h e f o r m o f t h e m a t c h i n g f u n c t i o n a n d t h e s p ec i fi c m o d e l o fu n e m p l o y m e n t a n d v a c a n c y d e t e r m i n a t i o n h a v e i m p o r t a n t c o n s e q u e n c e s f o r t h e e x i s te n c ea n d u n i q u e n e s s o f se a r c h e q u i l i b ri u m b e c a u s e t h e y c h a r a c te r i z e t h e w a y s i n w h i c h t h ed e c i s i o n s t o p a r t i c i p a t e s t r a t e g i c a l l y i n te r a c t. F o r e x a m p l e , s u p p o s e t h a t s e a r c h p a r t ic i p a -t i o n is " e s s e n t i a l " i n t h e m e e t i n g p r o c e s s i n t h e u s u a l s e n s e o f p r o d u c t i o n t h e o r y , i .e .,r e ( O , u ) = r e ( v , 0 ) = 0 . N o p a r t i c i p a t i o n b y a l l, (v , u ) = ( 0 , 0 ) , i s a lw a y s a n e q u i l i b r i u m int h e n o n - c o o p e r a t i v e g a m e th e o r e t i c s e n s e i n t h is c a s e . F o r m a l l y , v = 0 ~ A = m ( 0 , u ) / u =0 ~ U = ( b - c ) / r ~ b / r f o r a l l u > 0 b y E q . ( 4 ) a n d u = 0 ~ ~ = m ( v , O ) / v = 0 ~ V =- c / r 0 b y E q . ( 7 ) . A s a w o r k e r ' s v a l u e o f n o t p a r t i c i p a t i n g i s b / r a n d a ne m p l o y e r ' s v a l u e o f n o t p a r t ic i p a t in g i s 0, n o o n e o n e i t h e r s id e o f t h e m a r k e t h a s a ni n c e n t i v e t o p a r t i c i p a t e i f n o o n e p a r t i c i p a t e s o n t h e o t h e r s i d e o f th e m a r k e t .

W i t h t h e m a t c h i n g t e c h n o l o g y d e t e r m i n i n g t h e a rr i v a l r a te s , E q s . ( 2 ), ( 7 ), ( 9 ), a n d ( 1 0 )i m p l y t h a t th e v a l u e o f u n e m p l o y e d s e a r ch U , a n d j o b v a c a n c y V s ol v e th e s y s t e m o fe q u a t i o n s

b - a + [ m ( v , u ) l u ] ~ f m a x { X , U + V } d F ( X ) ( 1 1)r + { r e ( v , u ) / u ]U ~

a n dV = - c + [ m ( v , u ) / v ] (1 - f i ) f m a x { X , U + V } d F ( X ) ( 1 2 )r + [ m ( v , u ) / v ]B e c a u s e t h e m a t c h i n g f u n c t i o n , m ( v , u ) , i s i n c r e a s i n g i n b o t h a r g u m e n t s , t h e u n e m p l o y e dw o r k e r s a n d v a c a n t j o b s a r e c o m p l e m e n t s i n t h e s e n s e t h a t a n i n c r e a s e i n t h e m e a s u r e o fe i t h e r in c r e a s e s t h e v a l u e o f p a r t i c i p a t i o n f o r a g e n t s o n t h e o t h e r s i d e o f th e m a r k e t . O nt h e s a m e s i d e o f t h e m a r k e t t h e r e i s a c o n g e s t i o n e f f e c t , i .e ., g r e a t e r n u m b e r o f e it h e rt y p e r e d u c e t h e v a l u e o f t h e i r o w n p a r t ic i p a t i o n . T h i s , h o w e v e r , r e q u i r e s t h a t t h e m e e t -i n g r a t e m ( v , u ) / u f o r u n e m p l o y e d w o r k e r s a n d m ( v , u ) / v , d e c r e a s e s w i t h t h a t t y p e ' sp a r t ic i p a t io n m e a s u r e ; a p r o p e r t y p o s s e s s e d b y b o t h t h e " l i n e a r " a n d th e " q u a d r a t i c "f o r m s o f t h e fu n c t i o n d e r i v e d i n t h e p r e v i o u s s e c t io n a n d b y a l l c o n s t a n t r e tu r n s m a t c h -i n g f u n c t i o n s .

T h e r e a r e s e v e r a l a lt e r n at i v e a p p r o a c h e s t o m o d e l l i n g t h e s p e c if i cs o f u n e m p l o y m e n ta n d v a c a n c y d e t e rm i n a t i o n . T h e s i m p l e st s u p p o s e s a n u n l i m i t e d s u p p l y o f p a r ti c ip a n t s o fb o t h t y p e s . I n t h i s c a se , w o r k e r a n d j o b s e n t e r u n t il t h e n u m b e r s a r e s u c h t h a t e a c h isi n d i f f e r e n t b e t w e e n p a r t i c i p a t i n g a n d n o t p a r t i c i p a t in g . B e c a u s e w o r k e r s e a r n b p e r p e r i o dw h e n n o t p a r t i c i p a t in g a n d t h e r e a r e n o p u r e p r o f it o p p o r t u n i t i e s f o r e m p l o y e r s , a s e a r c he q u i l ib r i u m u n e m p l o y m e n t - v a c a n c y p a i r ( u , v) i s a n y s o l u t i o n t o U = b / r a n d V = 0 . A s ~c o n s e q u e n c e o f t h e p r o p e r t i e s o f E q s . ( 1 1 ) a n d ( 1 2 ) , i f t h e r e a r e c o n s t a n t r e t u r n s i n t h em a t c h i n g t e c h n o l o g y , o n l y a n o n - p a r ti c i p a t i o n e q u i l i b r iu m o r a c o n t i n u u m o f eq u i li b ri a

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2578 D. T. Mortensen and C. A. Pissarides(u, v) = (0, 0) exists since both m(u,v)/u and m(u,v)/v are function of the ratio u/v. Indeed, asolution pair (u*,v*)> 0 exists which is also "stable" in the sense that the value ofparticipation diminish with the participation measures if and only if the matching technol-ogy exhibits decreasing returns to scale in some range. 10 Finally, if the matching functionexhibits increasing returns for small values of u and v as well, multiple equilibria arepossible.

These results follow from other strong specification assumptions. The value of a match,X, is independent of the number of matches is one. The supply of participants is unboundedfor any positive value of participation is another. Partly motivated by these observation butalso by the empirical fact that the aggregate labor supply does not change very much overtime for endogenous reasons, much of the labor market literature has adopted the assump-tion that the total labor supply is constant. Given the fixed size, a natural normalization isunity, so u becomes the unemployment rate and v the vacancy rate. The assumption ofunlimited entry of vacancies has, however, been retained, so the equilibrium level ofvacancies v solves the no profit condition V = 0. A steady condition determines unem-ployment u. Once again, in most of the existing analyses X is independent o f employment,i.e., there are no diminishing returns to scale in production. Constant returns to scale in thematching technology, in the sense that m(v,u) is homogenous of degree one, is alsotypically assumed.

A third approach to the determination of v and u is that adopted by Diamond (1982b)and Blanchard and Diamond (1989). They fix both the labor force size and the totalnumber of jobs. The equilibrium pair in this case is found from two steady state conditions,one for unemployment and one for vacancies.

3. Equilibrium unemp loymentIn this section we review the theory of equilibrium unemployment for labor marketscharacterized by frictions developed in Pissarides (1990). The flow of newly createdjobs is the outcome of a matching process in which both workers and employers partici-pate. Wages are determined by a generalized Nash bargain after worker and employermeet. The labor force size is constant but firms create job vacancies until any incrementalprofit is exhausted.3.1. Exogenous job destructionConsider first a simple environment characterized by identical workers and employerswith no uncertainty about match product, a matching function that determines the flow ofnew jobs created, and a fixed job destruction rate (see Pissarides, 1990, for a detailedanalysis of this case). Jobs are created at the rate m(v,u), where, as before, v is the vacancy

10 We leave a proof of this assertion as an excercise for the reader.

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Ch 39: New Developments" in Models o f Search in the Labor Market 2579ra te and u is the unemployment ra te . The matching funct ion is increas ing, concave, andhomogen ous o f degree 1. All jobs have the same produc t iv i ty deno ted as p . Employ er andworker negot ia te a wage when they meet and subsequent ly produce unt i l an idiosyncrat icshock arr ives that des troys the job match. At separat ion, the f i rm leaves the m arket and thework er joins u nem plo ym ent to look for another job. Th e a lTival ra te of the idiosyncrat icshock is a cons tant 6 .

The evo lu t ion o f unem ploy men t is g iven by= 6(1 - u ) m ( v , u ) . (13)

Und er the assum ption that the matching tec hnolo gy exhibi ts cons tant re turns, th is equationhas a unique s table s teady s ta te solut ion for every vaca ncy ra te v , i.e. ,

6u - - 6 + m(v /u , 1) 6 + A( 0)' (14)where the vacancy to unemployment ra t io 0 = v /u s ignals market t i g h t n e s s and A(0) =m ( v / u , 1) represents the unemployment spel l hazard.

Wh en Eq . (14) i s d rawn in vac ancy-u nemp loym ent space it genera tes a B e v e r i d g ec u r v e , a negat ive re la t ion between vacancies and unemployment . I t i s convex to theorigin by the propert ies of the ma tching funct ion. One can a lso express the re la t ion-ship in terms borro wed f rom the empir ica l l i terature on job creat ion and job des truc-t ion . Div ide bo th job c rea t ion and job sepa ra t ion f lows by emp loym ent 1 - u togenerate the j o b c r e a t io n r a te m ( v , u ) / ( l - u ) and the j o b d e s t r u c t i o n r a te 6 . Eq.(14) gives the unemployment ra te that equates the endogenous job creat ion ra tewith the cons tant job des truct ion ra te .

Equi l ib r ium unem ploym ent depends on the pa ramete rs o f the mode l th rough the depen-dence o f endog enous m arke t t ightness on them. M arket t ightness , in turn, uniquely deter-mines the durat ion o f unemployme nt , u / m ( v , u ) . Equil ibr ium market t ightness is obta inedf rom prof i t maximiza t ion g iven the wage ba rga in . The f i rm main ta ins a job vacancy byincurr ing a f low cos t of recrui t ing a worke r c . Appl icat ions arr ive a t the ra te m ( v , u ) / v whichwe denote by

re(O, 1) a(0)n(o ) - - (15)0 0Since A(0) is increa sing and con cav e, 3"/(0) < 0 with elasticity 1 0A/(0)/A(0) betw eenzero and one.

Wh en any unemployed w orker and employe r wi th a vacancy mee t , wage ba rga in ingtakes place . The outco me is a wa ge w that divides the quas i-rents associa ted with a matchbetween worker and employer . Given an arbi t rary wage w, the associa ted value of a f i l ledjob to the employer , J , solves the asset pr ic ing equat ionrd---- p - w - 6J , (16)w he re p represents the output o f the match. As the value is los t in the event o f des truction,

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2580 D.T. Mortensen and C. A. PissaridesEq. (16) is equivalent to the continuous time Bellman equation

1J - - I + r d t ( ( p - w ) d t + (1 - 8 d t ) J ) ,where 8 d t represents the probability of job destruction during any sufficiently small timeinterval of length dt. Analogously, the value of the job to the worker satisfiesr W = w - 8 ( W - U ) . (17)The difference in the workers case is that job destruction generates unemployed searchwhich has value U.11

We seek the generalized Nash bargaining wage outcome, that which solves

w = a r g m a x ( W - U ) ~ ( J - V ) l - P = a r g m a x -rTt_ ~ ] \ r 2 ~ 8

where V equals the employers value of holding the job vacant and U is the value ofcontinued search, i.e., the values of the agents outside options. The maximization impliesthat the worker's share of match surplus is the constant/3, formallyw - g =/3(W + J - U - V). (19)Substitution from Eqs. (16) and (17) into (19) gives the implied wage equationw = r U + [ 3 [p - r U - ( r + 8 ) V ] . (20)The wage outcome compensates the worker for the loss of his return to unemployment r U ,and pays in addition a fraction/3 of the net flow return of the match, which is the matchproduct p, net of the worker 's and firm's reservation incomes, r ( U + V ) , and the lost valueof the job site in the event of destruction, 8 V .

The value of unemployment U satisfiesr U = b + A(0)(W - U). (21)where here b represents the value of leisure or home production foregone when employed,less any cost of search denoted as a above but suppressed here. Similarly, the value of anew vacancy, V, solvesr V = - c + r l ( O ) ( J - V ) , (22)where here c represents the flow cost of recruiting a worker expressed as a fraction of theworker's productivity once employed. Profit maximization and free entry require that allrents from new vacancy creation are exhausted, i.e., the j o b c r e a t i o n c o n d i t i o n

11 Alternatively, if 8 were interpreted as the worker "death" rate, the employer would have a vacant job in theevent of separation which has value V. The simplifying assumption here is that all separations are the consequenceof job destruction rather than worker quitting behavior. One can easily generalized the value equations to accountfor both reasons for separation.

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Ch 39." New Developments in Models of Search in the Labor Market 2581V = 0 (23)holds .A s teady s ta te search equilibrium f o r th i s s i m p l e e c o n o m y is v e c t o r (u,w,O,V,U) tha tsa t is f ies Eqs . (14) , (20) , (23) , (22) , and (21) . Subs t i tu t ion f rom E qs . (16) and (23) in to (22)y i e l ds t he a l t e rna t i ve fo rm fo r t he j ob c r ea t i on cond i t i on

c p-w- - ( 2 4 )~7(0) r + 6 '

N o t e t h a t E q . ( 2 4 ) is a d y n a m i c d e m a n d f o r l a b o r co n d i ti o n . T h e e x p e c t e d d u r a t io n o f th evaca ncy i s 1 /~7(0 ), so w i t h a f l ow r ec ru i t i ng cos t c , t he exp ec t ed h i r i ng cos t f o r t h i s j ob i s c~~7(0) . Thus , t he cond i t i on r equ i r e s t ha t t he expec t ed h i r i ng cos t equa l t he p r e sen td i s c o u n t e d v a l u e o f th e d i f f er e n c e b e t w e e n t h e f u t u re f lo w s o f m a r g i n a l p r o d u c t a n dw a g e p a y m e n t s w h e r e t h e d i s c o u n t r a te i s t h e s u m o f t h e i n te r e s t a n d j o b d e s t ru c t io nra tes . Because Eqs . (22) , (21) , (19) and (23) imply

(1 - f l ) ( r U - b )r V = - c + = 0 ,~0subs t i tu t ion in to Eq. (20) for rU g i ves t he equ i l ib r i um wage equationw = (1 - fl)b + ~(p + cO). (25)T h e w a g e i n c re a s e s w i t h t h e u n e m p l o y m e n t b e n e fi t, j o b p r o d u c t i v it y , a n d m a r k e t t ig h t -

N Wage Cu rvet

Fig. 1. Equilibriummarket tightness and wage rate.

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2582 D . T . M o r t e n s e n a n d C . A , P i s s a r i d e sn e s s . F i n a l l y , t h e e q u i l i b r i u m i s f u l l y d e s c r i b e d b y t h e w a g e a n d m a r k e t t i g h t n e s s p a i r( w , O ) t ha t s a t i s f i e s Eqs . (24 ) and (25 ) .

T h e t w o e q u i l i b r i u m c o n d i t io n s h a v e u s e f u l d e s c r i p t i v e p r o p e r t ie s s h o w n i n F ig . 1 . T h e yi n t e r s e c t o n l y o n c e , h e n c e e q u i l i b r i u m i s u n i q u e . A n i n c r e a s e i n t h e w o r k e r ' s u n e m p l o y -m e n t i n c o m e b s h i f ts t h e w a g e c u r v e u p a n d s o i n c re a s e s w a g e s a n d r e d u c e s m ~ k e tt ig h t n e s s. A n i n c r e a s e i n t h e w o r k e r ' s s h a r e p a r a m e t e r / 3 h a s s i m i l a r e ff e c ts . I n c o n t ra s t ,a n i n c r e a s e i n m a t c h p r o d u c t p s h if ts th e jo b c r e a t i o n c o n d i t i o n u p , i n c r e a s i n g b o t h w a g e sa n d m a r k e t t i g h tn e s s . M o r e t u r b u l e n c e i n t h e l a b o r m a r k e t i n t h e s e n s e o f a n i n c r e a s e in t h ea r r iv a l r a t e o f n e g a t i v e r e a l l o c a t io n s h o c k s , 3 , s h i f ts t h e j o b c r e a t i o n c o n d i t i o n d o w nr e d u c i n g b o t h w a g e s a n d t i g h t n e s s .

G i v e n t h e s o l u t io n f o r ti g h t n e ss o b t a i n e d f r o m F i g . 1 , w e c a n n o w d r a w t h e B e v e r i d g ed i a g r a m t o d e r i v e e q u i l ib r i u m u n e m p l o y m e n t a n d v a c a n c i e s a s il l u st r a te d i n F ig . 2 . W i t hf i x e d p r o d u c t i v i t y , t h e s o l u ti o n f o r 0 i s in d e p e n d e n t o f u n e m p l o y m e n t , s o e q u i l i b r i u m 0 i nt h e B e v e r i d g e d i a g r a m i s s h o w n a s a s tr a i g h t l in e t h r o u g h t h e o r i g in . C a l l i t t h e j o b c r e a t i o nc o n d i ti o n . E q u i l i b r i u m u n e m p l o y m e n t i s a t t h e p o i n t w h e r e t h e j o b c r e a ti o n c o n d i t i o ni n t e rs e c t s t h e B e v e r i d g e c u r v e ,

F o l l o w i n g o n f r o m o u r p r e v i o u s a n a l y s i s , a n i n c r e a s e i n e i t h e r t h e w o r k e r ' s s h a r ep a r a m e t e r o r u n e m p l o y m e n t i n c o m e r o t a te s t h e j o b c r e a t i o n li n e d o w n i n F i g. 2 a n d s oi n c r e a s e s e q u i l i b r iu m u n e m p l o y m e n t . A h i g h e r j o b p r o d u c t i v i t y ro t a t e s it u p a n d r e d u c e su n e m p l o y m e n t . H i g h e r a r r i v a l r a t e o f i d i o s y n c r a ti c s h o c k s s h i ft s t h e B e v e r i d g e c u r v e o u ta n d r o t a te s t h e j o b c r e a t io n l in e d o w n , s o i t i n c r e a s e s u n e m p l o y m e n t b u t h a s a m b i g u o u se f f e c ts o n v a c a n c i e s .

riot2

j0 u

Fig. 2. Equilibrium vacancies and unemployment.

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Ch 39: New Developments in Models' of Search in the LaborMarket 2583 The contrasting results that we obtained for different productivity levels and different

arrival rates of idiosyncratic shocks form the basis of much of the discussion aboutwhether changes in unemployment (and hence the business cycle) are driven by aggregateshocks or reallocation shocks. Papers that have explored this contrast include Jackman etal. (1989) who find that the rise in UK unemployment in the 1970s and to some extent inthe 1980s was not due to aggregate shocks and Abraham and Katz (1986) and Blanchardand Diamond (1989), who attributed cyclical shocks in the United States largely to aggre-gate shocks, changes in p.

Andolfatto (1996) and Merz (1996) adopt the search equilibrium approach as a char-acterization of the labor market in a dynamic stochastic general equilibrium macroeco-nomic model of capital accumulation. In these models, a household sector decides how toallocate current production between consumption and saving as well as how to divide atime endowment between work, search and leisure. See Merz (1997) for a review of thecontributions of these hybrid models found in the "real business cycle" literature. Theseapplications include studies of the tradeoff between the insurance benefits provided andthe allocation distortions induced by unemployment insurance systems by Andolfatto(1996), Costain (1995), and Valdivia (1995).3.2. Job an d worker flow sEmpirical evidence shows that the job destruction rate, 6 in the notation of the precedingsection, is not constant, especially over business cycle frequencies (see Davis et al., 1996).In this section we generalize the model to variable job destruction flow and derive theequilibrium conditions (see Mortensen and Pissarides, t994).

The model builds on that of the preceding section by allowing future job productivity totake more than two values. We write job productivity aspx where x E [0, 1] represents therelative value of a job's specific service or product. Suppose that the product or serviceprovided by the match is a irreversible decision made at the time the job is created. Anidiosyncratic shock to the productivity of a match, a new value of its relative value x,arrives at a finite constant rate 6 and is distributed according to the c.d.f. F(.), assumedindependent of previous realizations. Thus, the idiosyncratic shock process to the value ofworker product has persistence but no memory. Profit maximization at the time the job iscreated requires that product or service be of highest relative value, x = 1, given thatfuture values are determined by the first order Markov process assumed above. The modelof the previous section can now be re-interpreted as one where the support of the distribu-tion of a future idiosyncratic product ivity shock is the unit interval.The value of a filled jobwith idiosyncratic productivity x is denoted J(x) and satisfies the functional equationrJ(x) = px w(x ) + ~[ f max (J(~), O)dF(Yc) - J(x) ], (26)where the expression on the right accounts for the fact that the employer will end the matchif its future expected present value falls below zero. Job creation takes place as before, so

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Ch 39: New Developments in Models of Search in the Labor Market 2585equa t ions , ( 27 ) and (29 ) . Equ i l i b r ium i s aga in un ique , a p rope r ty t ha t we i l l u s t r a t e wi th ad i a g r a m a f t e r d e r i v a t i o n s o f m o r e e x p l i c i t e x p r e s s io n s f o r t h e jo b c r e a t i o n a n d j o b d e s t r u c-t i on cond i t i ons .T h e j o b v a l u e e q u a t i o n , E q . ( 2 6 ) , c a n b e r e w r i tt e n a s( r + 6 ) J ( x ) = (1 - f i ) ( p x - r U ) + J ( 2 ) d F ( 2 ) (35)ew h e n t h e w a g e c o n t r a c t s at is f ie s (3 1 ) a n d t h e r e s e r v a t i o n p r o d u c t s o l v e s ( 32 ) . A s f ( x ) =( 1 - f i ) p / ( r + 3 ) b y i m p l i c a t i o n a n d J ( R ) = 0 , i t f o l l ow s tha tJ ( x ) = (1 - / 3 ) ~ p , f o r a l l x . ( 36 )He nce , fo r t he case o f x = R , Eqs . (32 ) , (35 ) and (36 ) im p ly tha t t he r e se rv a t ion p rod uc ts o l v e s

R + ( x - R ) d F ( x ) p = r U . (37)r + 6 RT h e r e s e r v a t i o n p r o d u c t i v i t y , p R , f a l ls sho r t o f t he wo rk e r ' s r e tu rn t o s ea rch , t he t e rm ont h e r ig h t s i d e, b y t h e o p t i o n v a l u e o f c o n t i n u in g t h e m a t c h , t h e s e c o n d t e r m o n t h e l e f t si de .T h e o p t i o n v a l u e i s p o s i t i v e b e c a u s e t h e r e l a t iv e v a l u e o f a n e x i s ti n g j o b m a y i n c r e a s e i nthe fu tu re .

Eqs . (23 ) , ( 27 ) , (30 ) and (36 ) imp ly tha t t he j ob c r ea t ion co nd i t i on can be wr i t t en a s[ "~- Rc = ( 1 - f i ) ~ ( O ) ~ ) p . (38)T h e r e l a t i o n b e t w e e n t h e t w o u n k n o w n s i m p l i e d b y t h e j o b c r e a t i o n c o n d i t io n i s n e g a t i v eb e c a u s e t h e v a c a n c y h a z a r d , ~ 7( 0), d e c r e a s e w i t h t ig h t n e s s a n d b e c a u s e a t a h ig h e r r e s e r -v a t i o n p r o d u c t i v i t y t h e e x p e c t e d l i f e o f a n e w j o b i s s h o rt e r, a n d s o t h e e x p e c t e d p r o f itf r o m a n e w j o b i s l o w e r . F e w e r j o b v a c a n c i e s a r e cr e a t e d , r e d u c i n g m a r k e t ti g h tn e s s , a s R ,i n c r e a s e s g i v e n t h e fr e e e n t ry c o n d i t io n . T h e n e g a t i v e r e l a t io n b e t w e e n R a n d 0 im p l i e d b yEq . (38 ) i s i nd i ca t ed in F ig . 3 by the j ob c r ea t ion cu rve , J C in the sequel . F inal ly , the f lowr e t u rn t o u n e m p l o y m e n t s e a r c h i s a n in c r e a s i n g l i n e a r f u n c t io n o f m a r k e t t ig h t n e s s

/3c0r U = b + - - (39)1-/3by v i r t ue o f Eqs . ( 29 ) and (30 ) . Th ese t h r ee equ a t ions , ( 37 ) , ( 38 ), and (39 ) , cha rac t e r i zes e a r c h e q u i l i b r iu m i n t h e m o d e l o f j o b c r e a t i o n a n d j o b d e s t r u c ti o n f l o w s .

F o r g i v e n 0 , E q s . ( 3 7 ) a n d ( 3 9 ) i m p l y t h a t t h e r e s e r v a t i o n p r o d u c t i v i t y f a ll s w i t h 6 a n dr i s es w i th r , b e c a u s e o f th e e f f e c t s th a t e a c h h a s o n t h e o p t i o n v a l u e o f c o n t i n u i n g t h e j o b -w o r k e r m a t c h . I t r is e s w i th u n e m p l o y m e n t i n c o m e b a n d f a l ls w i t h m a t c h p r o d u c t pb e c a u s e o f t h e i r e f f e c t s o n t h e f o r g o n e r e l a t i v e i n c o m e w h e n t h e w o r k e r i s u n e m p l o y e d .F i n a l ly , t h e r e s e r v a t i o n p r o d u c t i v i t y i n c r e a s e s w i t h m a r k e t t i g h t n e ss b e c a u s e t h e e x p e c t e d

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R

. Jo b D e s l ruc t ion

D. T . Mor tense n and (2. A . P i s sar id es586

Fig. 3. Equi l ib rium market t ightness and reserv at ion product ivi ty.

re tu rns to unem ployed sea rch inc reases wi th 0 . The dependence o f the re se rva t ion p roduc-t ivi ty on marke t t ightness is i l lust ra ted in F ig. 3 by the job des truct ion curv e referred to asJD in the sequel. The eq uilibriu m solution for the pai r (R, 0), l ies at the unique intersectionof the two curves JC and JD drawn in Fig. 3.

As an i l lus tra t ion of the new resul ts obta ined w ith endoge nous job des truct ion f low,cons ide r the impl ica t ions o f two pa ramet r ic sh i f t s , h igher unemployment income b andgreater marke t " turbulence ' 6 . Because a higher b represents an increase in the opportuni tycos t o f employmen t , the JD curve shif ts up in F ig. 3. Equi l ibr ium R r ises and equi l ibr ium 0fal ls in response . Thu s , the job des truct ion ra te, wh ich is equal to 8F(R), unambiguous lyrises . At a give n unem plo ym ent ra te , job creat ion fa l ls beca use the fa l l in 0 imp lies a fa l l invacancies . Bec ause the job creat ion ra te has to increase to match the higher job des truct ionrate , the s teady s ta te emplo ym ent fa lls and the une mp loy me nt ra te r ises . We also see thisin the Beveridg e dia gram in F ig. 2 , where the fa l l in 0 rota tes the job creat ion l ine downand the r ise in R shif ts the Beveridge curve out , increas ing s teady s ta te unemployment .F inal ly , wages m ay r ise or fa l l in equi l ibr ium because , on the one hand, w ages r ise with bbu t , on the o ther , they fa ll wi th 0 . In sum, h igher un emp loym ent income impl ies more jobdes truct ion and less job creat ion.

Consider now the implicat ion o f fas ter arrival of idiosyncrat ic prod uct ivi ty shocks , anincrease in 6 . In response , the opt ion value of cont inuing a match increases so that thereservat ion produc t ivi ty must fa l l g iven 0 from Eq. (37), i .e. , the JD curve in Fig. 3 shiftsdown. Because higher 6 reduces the expected l i fe of a job and so leads to less vacancy

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Ch 39: New Developments in Models of Search in the Labor Market 2587creation from Eq. (38), the J C curve shifts to the left. The equilibrium reservation produc-tivity R falls but the diagram does not give a clear answer about the change in 0. Howeverby completely differentiating the equilibrium conditions, one can show that 0 also falls.The implications of an increase in 6 for unemployment and for the job destruction andjob creation rates are not clear-cut. The job destruction rate, 6F(R), may rise or fallbecause the direct effect from an increase in 6 can in principle be offset by the fall inthe equilibrium value of R. Although the fall in 0 initially implies less job creation, theresulting change is unemployment is also ambiguous. In the Beveridge diagram, the fall in0 rotates the job creation line down, but the ambiguity about the change in job destructionintroduces an ambiguity about the direction of the shift in the Beveridge curve.

Mortensen and Pissarides (1994), Mortensen (1994b) and Cole and Rogerson (1996) allfind that an extension of the model that regards p as a stochastic process characterizing anaggregate shock are consistent with the time series characteristics o f job creation and jobdestruction series reported by Davis et al. (1996).3.3. Social efficiencyThere are two offsetting external effects of the decision to participate in any labor marketwith search friction. Because the expected time required to fill a vacancy, v/m(v, u), isincreasing in v, the marginal vacancy has a congestion effect on other competing vacan-cies. However, the marginal vacancy also decreases the time a worker can expect to spendsearching for a job, u/m(v, u). For a class of related models, Hosios (1990) establishes thatthese two externalities just offset one another in the sense that search equilibrium issocially efficient if and only if the matching function is homogeneous of degree oneand the worker's share of surplus /3 is equal to the elasticity of the matching functionwith respect to unemployment. This same condition is both necessary and sufficient in thecase of the Mortensen and Pissarides (1994) model as well as its extension by Merz (1995).

Hosios' efficiency condition characterize the solntion to a utilitarian social planner'sproblem. Namely, in the case of a linearly homogenous matching function

um ,(v , u) 0A/(0)13 -- 1 (40)m(v, u) A(0)is necessary and sufficient for a search equilibrium to maximize the expected present valueof future aggregate income where A(0) = m(O, 1) = re(v, u)/u = 0~/(0) is the unemploy-ment hazard rate. We provide a derivation of this result below.

Recall that new matches produce at rate p, subsequently productivity shocks arrive atrate 6, the new value is p x with x distributed F(x), and only jobs that have realization x >-R continue. Hence, gross market output rate evolves according toy ' = A ( O )d tu p + ~dt(1 - u)p f'R xdF (x) + [1 - 6dt]y, (41)where y~ = y ( t + dt) is the market output rate at the end of the time interval [t, t + dt) and

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2588 D.T. M o r t e n s e n a n d C . A . P i s s a r i d e s

y = y ( t ) is the rate at the beginning of the interval. As before, the path of unemploymentsolvesu ~ = 8 d t F ( R ) ( 1 - u ) + [1 - A ( O ) d t ] u , (42)where u ~ = u ( t + dt) and u = u ( t ) . The aggregate net income flow during the interval is( y + b u - c v ) d t after account is taken of unemployment income and recruiting costs.

The utilitarian social planner's problem is to choose a future time path for the decisionvariables, the reservation product and tightness pair ( R , O ) , that maximizes the expectedpresent value of future aggregate income, defined as market output net of recruiting costsplus unemployment income. The value function for the problem solves the Bellmanequation

1 ) }L ( y , u ) = ma x' (- - [(y + b u - c O u ) d t + L ( y ' , d ) ] . (43)r,o ( \ l + r d tAs the right side is a contraction map for all r d t > 0, a unique solution exists for the valuefunction L ( y , u ) . Because the right sides of Eqs. (41) and (42) are linear in y and currentincome is also linear in both given the decision variables, the contraction maps the set oflinear functions, a compact metric space, to itself. Hence, the solution is necessarily linearby the contraction map theorem.

The first order necessary conditions required of an optimal choice of ( R , O ) a r e[L, - L y p R ] S d t F ' ( R ) ( 1 - u ) = O ,{ ( L I P - L.)A/(0) - c ] u d t = O ,where L y and L . a r e the constant partial derivatives of the value function. Note that thesecond order conditions are satisfied at any solution if m ( v , u ) is concave in v and homo-genous of degree one since then A"(0) < 0. By the envelope theorem, the partial deriva-tives of the value function must satisfy( ' )y = ~ [ d t +Ly( 1 - d t S ) ] ,

f'. = [ b d t + L.(1 - 6 d t F ( R ) - A(0)dt) + L y ( A ( O ) - 8 x d F ( x ) ) p d t ] .eBecause the first equation in each group together imply L y = 1/ (r + 8) andL . = p R L y = p R / ( r + 8 ) , appropriate substitution into the remaining two equations yieldsthe necessary and sufficient conditions characterizing the stationary optimal decision pair:c = A~(0)p(1 - R) (44)r + 6and

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2590 D , T . M o r t e n s e n a n d C . A . P i s s a r i d e sS u p p o s e t h a t e a c h o f p o t e n t i a ll y m a n y m i d d l e m a n o f f e r s a p a r t i c u la r e l e m e n t ( / 3 ,0 ) i n

a n a v a i l a b l e s e t o f s h a r i n g r u l e a n d w a i t i n g t i m e p a i r s , d e n o t e d a s g ~ . G i v e n t h a t e a c he m p l o y e r a n d w o r k e r c a n f r e e l y c h o o s e to p a r t i c i p a t e in a n y o n e o f th e m a r k e t s o r g a n i z e db y t h e s e m i d d l e m e n , t he v a l u e s o f h o l d in g a v a c a n c y a n d o f s e a rc h in g f o r e m p l o y m e n ts o l v e

~(1 1 - RFV (/~,0)cs?(a n dr U = m a x _ b + / 3 h (O ) pb e c a u s e t h e s u r p l u s s h a r i n g r u l e r e q u i r e s

1 - R J ( 1 ) - V W ( 1 ) - Ur + a 1 - / 3 /3

F r e e e n t r y , V = O , i m p l i e s t h a t1 - R ) c( 1 - 1 3 ) ~ p - - rt(-O )

(46)

(47)

(48)

m u s t h o l d i n e v e r y s u b - m a r k e t . F i n a ll y , t h e r e s e r v a t i o n p r o d u c t R is t h e s u b m a r k e t c h a r -a c t e r i z e d b y a g i v e n p a i r ( / 3 , 0 ) m u s t s o l v e( )+ ( x - R ) d F ( x ) p = r U = b + - - . (49)r + 6 R 1 - f lI n c o m p e t i t i v e e q u i l ib r i u m i n t h is e n v i r o n m e n t , n o m a r k e t m a k e r e a r n s p u r e p ro f it . O u t o fe q u i l i b ri u m , h o w e v e r , a m a r k e t m a k e r c a n p r o f i t b y c h a r g i n g a n a r b i t r a ri l y s m a l l f e ep r o v i d e d t h a t t h e s u b - m a r k e t c h a r a c t e r i s t i c s o f f e r e d ( / J , 0 ~) (E g 2 a t t r a c t b o t h e m p l o y e r sa n d w o r k e r s . 12 T o d o s o , b o t h o f t h e f o l l o w i n g i n e q u a l it i e s m u s t h o l d a n d o n e h o l d ss t r ic t ly :

1 - R( / - f l : )~ l ( O ' ) ~ 7 - g ) p - c >-- r V ,

i [ 1 - R \b + / 3 A ( o ) t 7 V - ) P >- r UT o e l i m i n a t e s u c h p u r e p r o f i t p o t e n t i a l , t h e n , e v e r y e l e m e n t ( /3 , 0 ) o f t h e e q u i l i b r i u m s e t g~m u s t s a t i s f y t h e t a n g e n c y c o n d i t i o n

~2 For si mplicity, we assum e that mid dle men incur no costs.

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Ch 39: New Developments in Models of Search in the Labor Market 2591d~o z _ A ( O ) - OA t(O)., A~(O) d~o U0 a ( 0 ) t i - ~ ) - - A ( 0 ~ / 3 = ( 5 0 )

g i v e n t h e d e fi n it i o n 0 7 ( 0 ) = A (0 ). I n o t h e r w o r d s , f o r a n y m e m b e r o f t h e e q u i li b r iu m s e t o fs u b - m a r k e t s , t h e r a t e a t w h i c h a n u n e m p l o y e d w o r k e r w h o p a r t i c i p a t e s i s w i l l i n g t oe x c h a n g e m a r k e t t i g h t n e s s f o r a s h a r e o f s u r p l u s o n c e m a t c h e d m u s t e q u a l t h e r a t e a tw h i c h a n y p a r t i c i p a t in g e m p l o y e r s w i t h a v a c a n t j o b i s w i ll i n g t o t ra d e t h e t w o .

S i n c e t h e z e r o p r o f i t c o n d i t i o n f o r m a t c h m a k i n g , E q . ( 5 0) , i s e q u i v a l e n t t o t he H o s i o scond i t i on (40 ) fo r a soc i a l l y e f f ic i en t s ea rch equ i l i b r ium, M oen (1997) ca l l s any pa i r (g2 ,R)that sa t i s fy Eqs . (49) , (50) , and V = 0 a competitive search equilibrium. W h e n w o r k e r s a n de m p l o y e r s a r e r e s p e c t i v e l y i d e n t i c a l a s a s s u m e d h e r e , X2 i s a s i n g l et o n p a i r d e t e r m i n e d b yt h e u n i q u e s o l u ti o n t o E q s . ( 4 8 ) a n d ( 4 9 ) a n d th e a s s o c i a t e d e q u i l i b r i u m w o r k e r ' s s h a r e o fma tch su rp lus i s t ha t wh ich so lves Eq . (50 ) .

On e in t e rp re t a t i on o f t he t a l e t o ld in t h i s s ec t ion i s t ha t the gen e ra l i ne f f i c i ency o f s ea rche q u i l i b r iu m i s d u e t o i n c o m p l e t e m a r k e t s . I n p a r ti c u l a r, i n a c o m p l e t e m a r k e t m o d e l s u c has t ha t j u s t de sc r ibed , w a i t i ng t ime s a r e app ro pr i a t e ly p r i ced , s ea rch ex t e rna l i t i e s a r ein t e rna l i zed , and the ove ra l l marke t equ i l i b r ium i s Pa re to op t ima l . As i t t u rns ou t , Pa re too p t i m a l i t y i m p l i e s s o c i a l e f f i c ie n c y g i v e n t h e li n e a r p r e f e r e n c e s a s s u m e d . O b v i o u s l y ,w a g e s d e t e r m i n e d b y b a r g a i n i n g o r m a n y o t h er m e c h a n i s m s a r e n o t l ik e l y to y i e ld t h iso u t c o m e .

4.2. Monopoly unionI n m a n y l a b o r m a r k e t s , t e r m s o f e m p l o y m e n t a r e d e t e r m i n e d b y c o l l e c t i v e b a r g a i n i n ga g r e e m e n t s. T h e m o n o p o l y u n i o n f o r m u l a t io n r e p r es e n t s t h e s t an d a r d a p p r o a c h t o m o d e l -i n g w a g e f o r m a t i o n in t h is c o n t e x t. I n th i s S t a c k e l b e r g g a m e o f w a g e a n d e m p l o y m e n td e t e r m i n a t i o n , t h e u n i o n f i rs t s e ts t h e w a g e a n d t h e n e m p l o y e r s r e s p o n d b y d e t e r m i n i n ge m p l o y m e n t . T h e w a g e d e t e r m i n a t io n m e c h a n i s m c a n b e p l a c e d i n a s e ar c h e q u il ib r iu mf r a m e w o r k b y s u p p o s i n g t h a t e m p l o y e r s c r e a te a n d d e s t r o y j o b s g i v e n a w a g e c o n t r a c t t h a ts p e c if i e s t h e s h a r e / 3 o f m a t c h s u r p lu s o b t a i n e d b y w o r k e r s . 13

P i s sa r ides (1990) sho ws tha t a un ion wo u ld se t t he wo rke r ' s sha re /3 a t i t s e f f i c ien t va lueg i v e n b y th e H o s i o s c o n d i t io n ( 4 0 ) w e r e a l l o f i ts m e m b e r s u n e m p l o y e d . H o w e v e r , i f t h eu n i o n a c t s i n t he i n t e re s t o f e m p l o y e d w o r k e r s i n s te a d , t h e s h a r e e x c e e d s t h e s o c i a lo p t i m u m , b u t s t i l l t h e u n i o n d o e s n o t f u l l y e x p l o i t m a r k e t p o w e r b y a p p r o p r i a t i n g t h ee n t ir e m a t c h s u r p lu s . T h u s , a n " i n s i d e r - o u t s i d e r " c o n f l i ct b e t w e e n m e m b e r s o f t h e u n i o ne x i s t s . E m p l o y e d w o r k e r s , t h e " i n s i d e r s " , w a n t t h e u n i o n t o c h o o s e h i g h e r w a g e s t h a nu n e m p l o y e d w o r k e r s , t he " o u t s i d e r s " .

T h e p r o o f i s s im p l e . I f a u n i o n w e r e t o re p r e s e n t u n e m p l o y e d w o r k e r s , it w o u l d c h o o s e/3 to m a x i m i z e t h e e q u i l i b r i u m r e t u r n t o s e a r c h , w h i c h c a n b e w r i tt e n a s

13 The wo rker's share of m atch rents is set as the p arame terof a contingent wage contract rather than a fixedwage lev el to quarantee ob destruction that is individually rational from the view o f bo th worker and em ployer.

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2592 D. T. Mortensen and C. A. P issarides( )U ~-- R + ( x - R ) d F ( x ) p (51)r + 6 Rby Eqs . ( 37 ) and (39) . A s t he r i gh t hand s i de i s i nc r eas i ng i n R , t he op t i ma l cho i cemax i mi zes t he r e se rva t i on p roduc t . A s no t ed above , a compl e t e d i f f e r en t i a t i on o f t hee q u a t i o n s y s t e m c o m p o s e d o f E q s. ( 3 7 ) - ( 3 9 ) i m p l i e s t h a t t h e e q u i l ib r i u m v a l u e o f R isa c o n c a v e f u n c t i o n o f / 3 w h i c h i s m a x i m i z e d w h e n t h e w o r k e r ' s s h a r e o f m a t c h s u rp l ussa t i s f ies the Hosios condi t ion , Eq. (40) .

I n p r a c ti c e , u n i o n s r e p re s e n t e m p l o y e d w o r k e r . S u p p o s e t h a t t h e u n i o n i s a d e m o c r a c ya n d t h e m e d i a n v o t e r i s e m p l o y e d i n s o m e j o b w i t h m a t c h p r o d u c t 2 R . As Eqs . (23) , (30) ,and (36) i mpl yw ( ~ ) = u + / 3 p ( r ~ ) '- Rt h e s h a r e / 3 t h a t m a x i m i z e s W ( 2 ) i s d e f in e d b y

2 - R/ 3 = a r g m a x ~ U + / 3 P ( r ~ ) } ./3E{0,11 l (52)Eq . ( 51 ) i mpl i e s

Ou ( r + 6 F ( R ) "~ ORr ~ = p r + 6 ] O f f 'so t he f i rs t o rde r cond i t i on fo r a i n t e r i o r so l u t ion t o t he op t i mi za t i on p rob l em i s

- P ~ + -r 7~--~ o~ =0 . (53)H e n c e , i f m o r e t h a n h a l f o f a ll m e m b e r s a r e e m p l o y e d , 2 > R f o r th e m e d i a n v o t e r , t h e nOR/O~3 < 0 a t / 3 = / 3 . B u t , w e h a v e a l re a d y n o t e d t h a t R d e c r e a s e s w i t h / 3 o n l y f o r v a l u e sabov e t he e f f i c ien t shar e .

B ec ause t he a s soc i a t ed r e se rva t i on p roduc t i s le s s t han t he soc i a l op t i ma l , t he i nc i dence o fu n e m p l o y m e n t , 6 F ( R ) , i s a lso l e s s than t ha t f o r a com pe t i t i ve s ea r ch equ i l ib r i um. H ow ever ,t he exp ec t ed dura t i on o f an une m pl oy m ent spe ll , l / A (0 ) is l onge r than i n t he com pe t i t i vecase . A s t hese t w o e f f ec t s o f a h i ghe r w o rke r ' s sha re on t he s t eady s t a te une m pl oy m ent ra t ea r e o f f se t t i ng , t he qua l i t a t i ve r e l a t i onsh i p be t w een t he compe t i t i ve and monopo l y unem-p l oy m ent r a te s i s unc l ea r . In t e r e s t ing l y , unem pl o ym ent du ra t i on is l onge r and i nc i dence i sl o w e r i n E u r o p e a n c o u n t r i e s t h a n i n t h e U S . A s E u r o p e a n l a b o r m a r k e t s a r e a l s o m o r eu n i o n i z e d , t h e m o d e l m a y h a v e e x p l a n a t o r y p o w e r f o r t h e s e d i f fe r e n ce s .4 . 3 . S t r a t e g i c b i l a t e r a l b a r g a i n i n gT h e m a t c h s u rp l u s s h a r in g r u l e c h a r a c t e r iz e d b y t h e g e n e r a l iz e d N a s h b a r g a i n i n g o u t c o m ehas a l so been i n t e rp r e t ed a s a so l u t i on t o a s t r a t eg i c ba rga i n i ng game . For exampl e ,

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25944 .4 . R e n t s h a r i n g w i t h t u r n o v e r c o s t s

D. 7". Mo rtensen and C. A. P issaride s

T h e d i f fe r e n c e s in p r e f e re n c e o v e r w a g e s o f th e u n e m p l o y e d a n d th o s e o f th e e m p l o y e d int h e m o n o p o l y u n i o n c a s e is a n e x a m p l e o f w h a t L i n b e c k a n d S n o w e r ( 1 9 8 8) h a v e t e r m e d" i n s i d e r - o u t s i d e r " c o n f l ic t . T h e c o n f l i c t a r i s es i n th a t c a s e b e c a u s e t h e c o s t s o f f i n d i n g a

j o b a r e s u n k f o r a n e m p l o y e d w o r k e r b u t n o t f o r a n u n e m p l o y e d w o r k e r . I n s i d e r - o u t s i d e rc o n f l i c t a l s o a r i s e s i n h o l d - u p p r o b l e m s . L4 H i r i n g a n d f i r i n g c o s t s , w h i c h w e i n t e r p r e t a s t h ef i x e d c o s t s o f j o b c r e a t i o n a n d j o b d e s t r u c t i o n r e s p e c t i v e l y , m o t i v a t e h o l d - u p s .

A s s u m e t h a t t h e l eg a l a n d e c o n o m i c e n v i r o n m e n t s a r e s u c h t h a t t h e f i rm i s l i ab l e f o ri n i ti a l h i r i n g a n d t r a i n i n g c o s t s a n d f o r s u b s e q u e n t f i ri n g c o s t s i n th e e v e n t o f j o b d e s t r u c -t i o n b u t w o r k e r a n d e m p l o y e r a re a b le t o p r e c o m m i t t o a n e n fo r c e a b l e w a g e c o n t r a c tw h i c h d e t e r m i n e s t e r m s o f e m p l o y m e n t c o n t i n g e n t o n f u t u r e e v e n t s w h e n t h e y f o r m am a t c h . I n g e n e r a l , t h e w a g e s t r u c t u r e t h a t ar i s es a s a N a s h b a r g a i n i n g s o l u t i o n h a s t w o -t i e r s u n d e r t h e s e c o n d i t i o n s w i t h t h e p r o p e r t y t h a t t h e w o r k e r s h a r e s t h e i n i t i a l h i r i n g c o s ta n d p r e p a y s e x p e c t e d f i r in g c o s t b y a c c e p t i n g a l o w e r in i t ia l w a g e b u t l a te r e n j o y s a h i g h e rw a g e . 15 T h e l o w e r f ir st t ie r w a g e r e f le c t s t h e f a c t t h a t h i r i n g c o s t s a r e d i r e c t l y r e l e v a n t t ot h e d e c i s i o n t o a c c e p t a m a t c h a n d t h a t t h e p o s s i b i l i t y o f i n c u r r i n g f i r i n g c o s t s i n t h e f u t u r ea f fe c ts t h e v a l u e t h e e m p l o y e r p l a c e s o n t h e m a t c h . T h e h i g h e r s e c o n d t ie r w a g e a p p l ie s a ts o m e l a te r t e n u r e w h e n f i r in g c o s t s a r e d i r e c t l y r e l e v a n t t o c o n t i n u a t i o n d e c i s i o n a n d w h e ns e p a r a t i o n w i t h o u t r e n e g o t i a t i o n w o u l d o t h e r w i s e v i o l a t e t h e i n t e re s t s o f b o t h p a r ti e s .

B e c a u s e t h e s e c o n d t i e r w a g e i s g e n e r a l l y h i g h e r t h a n t h e f ir s t t ie r , a w o r k e r , o n c e" i n s i d e " , h a s a n i n c e n t iv e t o d e f a ul t o n th e o r i g in a l t w o - t i e r a g r e e m e n t b y d e m a n d i n g t or e n e g o t i a t e i m m e d i a t e l y a f te r b e in g h i r ed . I n d e e d , a s L i n b e c k a n d S n o w e r ( 1 9 8 8 ) a r g u e s,e v i d e n c e s u g g e s t s t h at w o r k e r s o n c e e m p l o y e d d o t a k e a c t io n s d e s i g n e d t o e x t r a c t t h e q u a s i-r e n t s c r e a t e d b y r e c r u i t in g , h i r i n g , a n d f i ri n g c o st s , i. e. , a " h o l d - u p " p r o b l e m e x i st s . T h ee m p l o y m e n t e f f e c ts o f s u c h b e h a v i o r c a n b e s t u d i ed i n t h e s e a l' ch e q u il i b r iu m f r a m e w o r k b yc o m p a r i n g e q u i l ib r i u m c o n d i t i o n s w i t h a n d w i t h o u t t h e tw o - t i e r e d w a g e s tr u c tu r e.

W e f i r s t i n t r o d u c e f i x e d h i r in g a n d f i r i ng c o s t s a n d d e r i v e t h e t w o - t i e r w a g e s t r u c t u r et h a t th e y i n d u c e . S u p p o s e t h e e m p l o y e r i s o b l i g e d t o p a y h i r i n g c o s t C i n o r d e r t o b e g i np r o d u c t i o n a n d f i ri n g c o s ts T a t ti m e o f j o b d e s t ru c t io n . T h e f o r m e r c a n b e v i e w e d a s th es u m o f a p p l i c a t i o n , p r o c e s s i n g , a n d t r a in i n g c o s t s , f o r m s o f m a t c h s p e c i f ic i n v e s t m e n t .E x a m p l e s o f t h e l at te r in c l u d e t h e c o s ts i m p l i c it i n m a n d a t e d e m p l o y m e n t p r o t e c t io nl e g i s l a t i o n a n d i n e x p e r i e n c e r a t e d u n e m p l o y m e n t i n s u r a n c e t a x e s . H o w e v e r , a p u r es e v e r a n c e t r a n s fe r , a p a y m e n t t o t h e w o r k e r b y t h e e m p l o y e r , i s n o t i n c lu d e d i n T . F o rr e a s o n s p o i n t e d o u t b y L a z e a r ( 1 9 9 0 ) a n d B u r d a ( 1 9 9 2) , t h e e q u i l ib r i u m v a l u e s o f t h e

14 Caballero and H amm our (1994, 1999) and A cemo glu (1996) discuss hold-up problems in search and m atch-ing problems.i5 Alternatively, he worker could "buy" rights to the job by mak ing an initial transfer to the employer.This kindof side payme nt is ruled out here on empirical grounds. One theoretical explanation for w hy side payments of thisform are not observed s that no employercan precomm it o a future employm entduration under the "employmentat will" doctrine.

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Ch 39: New Developments in Models of Search in the Labor Market 2595r e l e v a n t d e c i s i o n v a r i a b l e p a i r ( R , O ) i s u n a f f e c t e d b y a s e v e r a n c e p a y m e n t a l t h o u g h t h em a g n i t u d e o f th e p a y m e n t w i l l e f fe c t t h e w a g e s o v e r t h e d u r a t i o n o f th e m a t c h .

L e t t h e s u b s c r i p t i = 0 i n d e x t h e i n it i a l w a g e a n d v a l u e s o f a j o b a n d e m p l o y m e n t u n d e rt h e t e r m s o f t h e t w o - t i e r c o n tr a c t . F o r s i m p l i c i t y , w e a s s u m e t h a t t h e s e c o n d t i e r o f t h ew a g e c o n t r a c t a p p l i e s t o a l l m a t c h e s o n c e a n i d i o s y n c r a t i c s h o c k t o m a t c h p r o d u c t i v i t yo c c u r s. 16 T h e v a l u e o f a n e w j o b m a t c h t o t h e e m p l o y e r u n d e r t h e s e a s s u m p t i o n s a n dn o t a t i o n a l c o n v e n t i o n s i s

rJo = p - w 0 + 6 [ f ' R J ( 2 ) d F ( 2 ) - F ( R ) T - J 0 ] , (56)w h i l e t h e i n i t i a l v a l u e o f t h e m a t c h t o t h e w o r k e r i sr W o = w o + W ( s ) d F ( s ) - F ( R ) U - W o , (57)Rw h e r e J ( x ) a n d W ( x ) a r e th e v a l u e s o f c o n ti n u i n g t h e m a t c h t o w o r k e r a n d e m p l o y e r u n d e rt h e s e c o n d t i e r c o n tr a c t. T h e s e v a l u e f u n c t i o n s s o l v e t h e a n a l o g o u s f n n c t i o n a l e q u a t io n s

an d(59)

w h e r e w ( x ) r e p r e s e n t s t h e p r o d u c t i v i t y c o n t i n g e n t s e c o n d t i e r w a g e c o n t r ac t .T h e v a l u e o f u n e m p l o y m e n t s a ti s fi e s

rU = b + h(0)[W0 - U] (60)a n d , b e c a u s e t h e c r e a t i o n c o s t C i s i n c u r r e d w h e n t h e m a t c h f o r m s , t h e v a l u e o f a v a c a n tj o b s o l v e sr V = - c + ~/(0)[J0 - V - C] = 0 (61)g i v e n f r e e e n t ry . F i n a l l y , t h e j o b d e s t r u c t i o n c o n d i t i o n r e q u i r e s a f u t u r e e x p e c t e d l o s sg r e a t e r in e x p e c t e d p r e s e n t v a l u e t h a n t h e c o s t o f t e r m i n a t i o n , i .e . ,J ( R ) = - T . (62)The in i t i a l f i r s t t i e r wage r a t e s a t i s f i e s t he Nash cond i t i on

~6 MacLeod and Malcom son (1993) mgue that the initial wage agreement wi ll be renegotiated only if not doingso would resu lt in an inefficient separation, i.e., the destructio n of the job when the sum of both partners futureincome given continuation exceeds the firing cost. The contract we study has the same two-tier feature, yields theidentical ob creation and job destruction decision, and is much easier to characterize. None the less, our contractgenerates a much more "flexible" wag e than theirs.

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2596 D . T